UF'iSS UK AB.,1870; A(HEROFI PLANE TRIGONOMETRY WITH TABLES F -dmmmmmrmmmmr McGraw-Hill DookCompany 6P~6is2des of Ioosfobr Electrical World The Engineering andMining Journal Lngineering Record Engineering News RailwayAge Gazette American Machinist Signal Enginoer American Engineer Electric Railway Journal Coal Age Metallurgical and Chemical Engineering Power i IrlWllillm[millimlllillalillllliilmllll ill liiillllll[l[lIiiiilIIIH[iiili;illlil IliillllllllllliililllllllIillliilllIlli illilllmiilililli1lllllllillllIlllli BlmrmlVBIP PLANE TRIGONOMETRY WITH TABLES BY CLAUDE IRWIN PALMER Associate Professor of Mathematics, Armour Institute of Technology, Author of a series of Practical Mlathematics AND CHARLES WILBUR LEIGH Associate Professor of Mathematics, Armour Institute of Technology FIRST EDITION McGRAW-HILL BOOK COMPANY, INC. 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET,"LONDON, E. C. 1914 COPYRIGHT, 1914, BY THE McGRAW-HILL BOOK COMPANY, INC. Ztanbope flrezz F. H. GILSON COMPANY BOSTON, U.S.A. PREFACE THIS text has been written, because the authors felt the need of a treatment of trigonometry that duly emphasized those parts necessary to a proper understanding of the courses taken in schools of technology. Yet it is hoped that teachers of mathematics in classical colleges and universities as well will find it suited to their needs. It is useless to claim any great originality in treatment or in the selection of subject matter. No attempt has been made to be novel only; but the best ideas and treatment have been used, no matter how often they have appeared in other works on trigonometry. The following points are to be especially noted: (1) The measurement of angles is considered at the beginning. (2) The trigonometric functions are defined at once for any angle, then specialized for the acute angle; not first defined for acute angles, then for obtuse angles, and then for general angles. To do this, use is made of Cartesian coordinates, which are now almost universally taught in elementary algebra. (3) The treatment of triangles comes in its natural and logical order and is not forced to the first pages of the book. (4) Considerable use is made of the line representation of the trigonometric functions. This makes the proof of certain theorems easier of comprehension and lends itself to many useful applications. (5) Trigonometric equations are introduced early and used often. (6) Anti-trigonometric functions are used throughout the work, not placed in a short chapter at the close. They are used in the solutions of equations and triangles. Much stress is laid upon the principal values of anti-trigonometric functions as used later in the more advanced subjects of mathematics. (7) A limited use is made of the so-called "laboratory method" to impress upon the student certain fundamental ideas. (8) Numerous carefully graded practical problems are given and an abundance of drill exercises. v vi PREFACE (9) There is a chapter on complex numbers, series, and hyperbolic functions. (10) A very complete treatment is given on the use of logarithmic and trigonometric tables. This is printed in connection with the tables, and so does not break up the continuity of the trigonometry proper. (11) The tables are carefully compiled and are based upon those of Gauss. Particular attention has been given to the determination of angles near 0~ and 90~, and to the functions of such angles. The tables are printed in an unshaded type, and the arrangement on the pages has received careful study. The authors take this opportunity to express their indebtedness to Professor D. F. Campbell of the Armour Institute of Technology, Professor N. C. Riggs of the Carnegie Institute of Technology, and Professor W. B. Carver of Cornell University who have read the work in manuscript and proof and have made many valuable suggestions and criticisms. THE AUTHORS. CHICAGO, September, 1914. CONTENTS CHAPTER I INTRODUCTION ART. PAGE 1. Introductory remarks....................................... 1 2. Angles, definitions.......................................... 1 3. Quadrants................................................. 2 4. Graphical addition and subtraction of angles................... 3 5. Angle m easurem ent......................................... 3 6. General angles............................................. 7 7. Directed lines and segments................................. 9 8. Rectangular coordinates.................................... 10 9. Polar coordinates........................................... 11 CHAPTER II TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 10. Functions of an angle....................................... 13 11. Trigonometric ratios........................................ 13 12. To each and every angle there corresponds but one value of each trigonom etric ratio...................................... 14 13. Signs of the trigonometric functions.......................... 15 14. Exponents of trigonometric functions........................ 15 15. Calculation of trigonometric functions by measurement......... 15 16. Trigonometric functions by computation...................... 16 17. Given the function of an angle, to construct the angle........... 20 18. Trigonometric functions applied to right triangles.............. 22 19. Relations between the functions of complementary angles........ 23 20. Given the function of an angle in any quadrant, to construct the angle................................................... 24 21. Fundamental relations between the functions of an angle......... 27 22. To express one function in terms of each of the other functions.... 29 23. To express all the functions of an angle in terms of one function of the angle, by means of a triangle........................... 29 24. Transformation of trigonometric expressions so as to contain but one function............................................. 31 25. Identities.................................. 32 26. Inverse trigonometric functions.............................. 34 27. Trigonometric equations................................... 34 vii Viii CONTENTS CHAPTER III RIGHT TRIANGLES ART. PAGE 28. General statement....................................... 37 29. Solution of a triangle....................................... 37 30. The graphical solution...................................... 37 31. The solution of right triangles by computation.............. 38 32. Steps in the solution........................................ 39 33. Solution of right triangles by natural functions................. 40 34. Remark on logarithms................................... 43 35. Solution of right triangles by logarithmic functions............. 43 36. D efinitions................................................ 45 37. Accuracy............................................... 51 38. Tests of accuracy.......................................... 52 39. Orthogonal projection....................................... 53 40. Vectors................................................... 53 41. Distance and dip of the horizon........................ 56 42. Areas of sector and segment........................... 56 CHAPTER IV GRAPHICAL REPRESENTATION OF TRIGONOMETRIC FUNCTIONS 43. Line representation of the trigonometric functions.............. 61 44. Functions of 1 7r + 0 in terms of functions of 0................. 63 45. Functions of 7r - 0 in terms of functions of 0.................. 63 46. Functions of wr + 0 in terms of functions of 0................... 64 47. Functions of 3 7r - 0 in terms of functions of 0................. 64 48. Functions of 3 7r + 0 in terms of functions of 0................. 65 49. Functions of 2 -r - 0 and -'0 in terms of functions of 0......... 65 50. Functions of an angle greater than 2........................ 65 51. Summary of the reduction formulas......................... 66 52. Proof of the reduction formulas for any value of 0........... 68 53. Given the function to find the angle....................... 68 54. Values for all angles that have a given sine or cosecant.......... 69 55. Values for all angles having the same cosine or secant........... 70 56. Values for all angles that have the same tangent or cotangent.... 71 57. Method by corresponding angles....................... 71 58. Changes in the value of the sine and cosine as the angle increases from 0~ to 360......................................... 74 59. Graph of y = sin 0......................................... 75 60. Mechanical construction of graph of sin 0...................... 76 61. Inverse functions....................................... 777 62. Graph of y = sin- x, or y = arc sin x......................... 78 63. Relation between sin 0, 0, and tan 0........................... 79 CONTENTS ix CHAPTER V FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES PAGE ART. 65. Addition and subtraction formulas........................ 66. Derivation of formulas for sine and cosine of the sum of two angles 67. Derivation of the formulas for sine and cosine of the difference of two angles.......................................... 68. Proof of the addition formulas for other values of the angles...... 69. Proof of the subtraction formulas for other values of the angles.... 70. Formulas for the tangents of the sum and the difference of two angles.................................................. 71. Functions of an angle in terms of functions of half the angle...... 72. Functions of an angle in terms of functions of twice the angle..... 73. Sum and difference of two like trigonometric functions as a product................................................. 74. To change the product of functions of angles to a sum.......... 83 83 84 85 86 89 90 92 94 97 CHAPTER VI OBLIQUE TRIANGLES 75. General statement........................................ 76. Sine theorem............................................. 77. Cosine theorem............................................ 78. Case I. The solution of a triangle when one side and two angles are given........................................ 79. Case II. The solution of a triangle when two sides and an angle opposite one of them are given............................. 80. Case III. The solution of a triangle when two sides and the included angle are given. First method..................... 81. Case III. Second method.................................. 82. Case III. Third method.................................... 83. Case IV. The solution of a triangle when the three sides are given 84. Case IV. Formulas adapted to the use of logarithms........... 101 101 103 109 110 112 113 114 CHAPTER VII MISCELLANEOUS TRIGONOMETRIC EQUATIONS 85. Types of equations....................................... 86. To solve r sin 0 + s cos 0 = t for 0 when r, s, and t are known.... 87. Equations in the form p sin a cos 3 = a, p sin a sin = b, p cos a = c, where p, a, and 3 are variables........................... 88. Equations in the form sin(a + A/) = c sin a, where 3 and c are know n.................................................. 89. Equations in the form tan(a + A3) = c tan a, where / and c are known................................................. 90. Equations of the form t = 0 + 4 sin t, where 0 and 0 are given angles.................................................. 127 129 129 x CONTENTS CHAPTER VIII COMPLEX NUMBERS, DEMOIVRE'S THEOREM, TRIGONOMETRIC SERIES ART. 91. rThe imaginary unit j....................................... 92. Graphical representation of a complex number................ 93. Complex number in terms of amplitude and modulus........... 94. Multiplication of complex numbers........................... 95. Division of two complex numbers............................ 96. Square roots of a number................................... 97. Cube roots of a number.................................... 98. To find the nth roots of a number............................ 99. Expansion of sin nO and cos n............................... 100. Computation of trigonometric functions...................... 101. Exponential values of sin 0, cos 0, and tan 0................... 102. Series for sinn0 and cos 0 in terms of sines or cosines of multiples of..................................................... 103. Hyperbolic functions....................................... 104. Relations between the hyperbolic functions................... 105. Relations between the trigonometric and the hyperbolic functions. 106. Expression of sinh x and cosh x in a series. Computation....... Summary of formulas....................................... Useful constants......................................... The contents for the Logarithmic and Trigonometric Tables and Explanatory Chapter is printed with the tables. PAGE 134 134 135 136 138 138 139 141 143 145 145 146 148 149 150 150 152 154 GREEK ALPHABET A, a B, / r, 7 A, S E, E Z, g H, v 0, 0 I, t K, K A, X M, /u.......Alpha.......Beta...... aGamma........ Delta...... Epsilon...... Zeta...... J Eta...... r.Theta.......... Iota...... Kappa...... Lambda.......Mu N, v 0, o 11, 7r P P T, T Y, v X, X x,Q I,, qi 0 (). *.. Nu.... Xi.... Omicron.....Pi.... Rho.... Sigma.... Tau.... Upsilon.... Phi.... Chi.... Psi.... Omega PLANE TRIGONOMETRY CHAPTER I INTRODUCTION 1. Introductory remarks. - The word trigonometry is derived from two Greek words, TrpyWovov (trigonon) meaning triangle, and wETrpta (metria) meaning measurement. While the derivation of the word would seem to confine the subject to triangles, the measurement of triangles is merely a part of the general subject which includes many other investigations involving angles. To pursue the subject of trigonometry successfully, the student should know the subjects usually treated in algebra up to and including quadratic equations, and be familiar with plane geometry, especially the theorems on triangles and circles. Frequent use is made of the protractor, compasses, and the straightedge. While parts of trigonometry can be applied at once to the solution of various interesting and practical problems, much of it is studied because it is very frequently used in more advanced subjects in mathematics. 2. Angles, definitions. - The definition of an angle as given in geometry admits of a clear conception of small angles only. In trigcnometry we wish to consider positive and negative angles and these of any size whatever, hence we need a more comprehensive definition of an angle. If a line, starting from the position OX, Fig. 1, is revolved about the point 0 and always kept in the yI G "C A \B FIG. 1. same plane, we say the line generates an angle. If it revolves from the position OX to the 1 Copyright, 1914, by the MCGRAW-HILL BOOK COMPANY, INC. 2 PLANE TRIGONOMETRY position OA, in the direction indicated by the arrow, the angle XOA is generated. The original position OX of the generating line is called the initial side, and the final position OA, the terminal side of the angle. If the rotation of the generating line is counter-clockwise, as already taken, the angle is said to be positive. If OX revolves in a clockwise direction to a position, as OB, the angle generated is said to be negative. In reading an angle, the letter on the initial side is read first to give the proper sense of direction. If the angle is read in the opposite sense the negative of the angle is meant. Thus, Z AOX= - ZLXOA. It is easily seen that this conception of an angle makes it possible to think of an angle as being of any size whatever. Thus, the generating line, when it has reached the position OY, having made a quarter of a revolution in a counter-clockwise, direction, has generated a right angle; when it has reached the position OX', it has generated two right angles. A complete revolution generates an angle containing four right angles; two revolutions, eight right angles; and so on for any amount of turning. The right angle is divided into 90 equal parts called degrees (~), each degree is divided into 60 equal parts called minutes ('), and each minute into 60 equal parts called seconds ("). Starting from any position as initial side, it is evident that, II I p2 for each position of the terminal side, there are two angles _X' _ (0 4;__yless than 360~, one positive and one negative. Thus, in Fig. 1, OC is the terminal side for the III / (IV positive angle XOC or for the ~Pi ~ / ~ negative angle XOC. l' y/ 3. Quadrants. -It is conFIG. 2. venient to divide the plane formed by a complete revolution of the generating line into four parts by the two perpendicular lines X'X and Y'Y. These parts are called first, second, third, and fourth quadrants, respectively. They are placed as shown by the Roman numerals in Fig. 2. INTRODUCTION 3 If OX is taken as the initial side of an angle, the angle is said to lie in the quadrant in which its terminal side lies. Thus, XOP1, Fig. 2, lies in the third quadrant, and XOP2, formed by more than one revolution, lies in the first quadrant. An angle lies between two quadrants if its terminal side lies on the line between two quadrants. 4. Graphical addition and subtraction of angles. - Two angles are added by placing them in the same plane with their vertices together and the initial side of the second on the terminal side of the first. The sum is the angle from the initial side of the first to the terminal side of the second. Subtraction is performed by adding the nega- / tive of the subtrahend to the minuend. B Thus, in Fig. 3, ZAOB + ZBOC = ZAOC. ZAOC - ZBOC = ZAOC + ZCOB = ZAOB..A ZBOC - ZAOC = ZBOC + ZCOA = ZBOA. FIG. 3. EXERCISES 1. Choose an initial side and lay off the following angles. Indicate each angle by a circular arrow. 75~; 145~; 243~; 729~; 456~; 976~. State the quadrant in which each angle lies. 2. Lay off the following angles and state the quadrant that each is in: -40~; -147~; -295~; -456~; -1048~. 3. Lay off the following pairs of angles, using the same initial side for each pair: 170~ and -190~; -40~ and 320~; 150~ and -210~. Unite graphically, using the protractor: 4. 40~ + 70~; 25~ + 36~; 95~ + 125~; 243~ + 725~. 5. 75~ - 43~; 125~ - 59~; 23~ - 49~; 743~ - 542~; 90~ - 270~. 6. 450 + 30~ + 25~; 125~ + 46~ + 950; 327~ + 25~ + 400~. 7. 45~ - 56~ + 850; 325~ - 256~ + 400~. 5. -Angle measurement. - Two systems for measuring * angles are in ordinary use. They are the sexagesimal system and the circular system. The sexagesimal system has for its fundamental unit the degree, which is defined to be the angle formed by 30 part of a revolution of the generating line. This is the system used by engineers and others in making practical numerical computations. The * The protractor is the instrument generally used for measuring angles. It is graduated into degrees and half degrees. 4 PLANE TRIGONOMETRY subdivisions of the degree are the minute and the second as defined in Art. 2. In the circular system for measuring angles, sometimes called radian measure or ir-measure, the C B fundamental unit is the radian. The \ / radian is defined as the angle at the D/_~ \ /C\ u center of a circle measured by an arc [ ^,S;D \ 9equal in length to the radius of the 0 /' adianA circle. Or it is defined as the positive G angle generated when a point on the E8< \ v generating line has passed through an arc equal in length to the radius of the circle being formed by that point. The circular system is used almost exclusively in the higher branches of mathematics for measuring angles. Another system for measuring angles was proposed in France somewhat over a century ago. This is the centesimal system. In it the right angle is divided into 100 equal parts called grades, the grade into 100 equal parts called minutes, and the minute into 100 equal parts called seconds. While this system has many admirable features, its use could not become general without recomputing with a great expenditure of labor many of the existing tables. The definition of the radian is based on geometrical facts: (1) Given several concentric circles and an angle AOB at the cenfter as in Fig. 5, then arc PiQi = arc P2Q2, OPi OP2 = arc P3Q3 OP3 That is, the ratio of the intercepted arc to the radius of that arc is a constant for all circles FIG. 5. when the angle is the same. The angle at the center which makes this ratio unity is then a convenient unit for measuring angles. (2) In the same or equal circles, two angles at the center INTRODUCTION 5 are in the same ratio as their intercepted arcs. That is, in Fig. 6, Z AOB arc AB ZAOC are AC arc AB Here if ZAOC is unity when arc AC = r, ZAOB =, or, in general, 0 = s, where 0 is the angle at the center measured in radians, s the arc length, and r the radius of the circle. From this we have s = rO. Which states that the arc length equals the B product of the radius and the angle at the center measured in radians. Since the circumference equals 2 7r / A times the radius of the circle, 2 r radians = 360~. 3600 180~ 1 radian= 36- 18~ 57.29578 - = 206264.8"+ = 57~ 17' 44.8"+. FIG. 6. For less accurate work 1 radian is taken as 57.3~. Conversely, 180~ = ir radians..'. 10 = _r = 0.0174533- radians. 180 To convert radians to degrees, multiply the number of radians by 180 -, or 57.29578-. 7T To convert degrees to radians, multiply the number of degrees by 80 or 0.017453+. In writing an angle in degrees, minutes, and seconds, the signs ~, ', ", are always expressed. In writing an angle in circular measure, usually no abbreviation is used. Thus, the a'ngle 2 means an angle of 2 radians, the angle ~ r means an angle of - 7r radians. Sometimes radian is abbreviated as follows: 3, 3(r), 3 p, or 3 rad. When the word "radians" is omitted, the student should be careful to supply it mentally. The most frequently used angles are conveniently expressed in 6 PLANE TRIGONOMETRY radian measure by using 7r. Thus, 180~ = 7r radians, 90 = 7r 7v radians, 60~ = ~ radians, 30~ = 6 radians, etc. 3 6 EXERCISES 1. Separate an angle formed by one complete revolution of the generating line into radians. Use the protractor. 2. Compute all of the results given in Art. 5. 3. Construct the following angles and state what quadrant each is in: 2 radians; ~ T radians; 3 7r radians; 3~ radians; 6.2832 radians; 5 ir radians; - T radians; -}- radians. 4. What is the measure of 135~ when the radian is taken as the unit? of 150~? of 225~? of 250~? of 260~? What is the measure of each when the right angle is taken as the unit? When 7 radians is taken as the unit? 5. What is the measure of each of the following angles when the right angle is taken as the unit: 1 rad., 4 -r, -r, 750~, 6.2832 rad.? 6. Add the following angles graphically: (a) 7r and 23 r; (b) r and 6 r; (c) 1 T and T r; (d) (2 n + 1) and 7. 7. Subtract the following angles graphically taking the second from the first: (a) 270~ and T7r; (b) 3r and r; (c) 2 r and 5r; (d) t r and 270~; (e) 0 and 2 ir. 8. How many radians at the center of a circle of radius 5 in. if the sides of the angle intercept an arc 19 in. in length? Of radius 7 in. and arc 11| in.? Ans. 3.8; 1.67857. 9. Change the following angles to sexagesimal measure, expressing the results in degrees to four decimals: 2.5 rad.; - r; t 7r; 1; 0.025. Ans. 143.2394~; 150~; 210~; 66.8451~; 1.4324~. 10. Change to sexagesimal measure giving the results in degrees, minutes, and seconds: 2.35 radians; 7r; ~; 1.75. Ans. 134~ 38' 42.3"; 128~ 34' 17.1"; 19~ 5' 54.9"; 100~ 16' 3.4". 11. Express the following angles in radians: 25~ 16'; 46~ 32' 14"; 227~ 43' 20". Ans. 0.44099; 0.81223; 3.9745. 12. Express the interior angles of the following regular polygons in radians: triangle, square, pentagon, hexagon, octagon, 59-gon. Ans. I 7; I 7r; 7r; 2T; 3T r; ' 7. 13. How many radians does the minute hand of a watch turn through in 10 minutes? In 35 minutes? Ans. ~ T; rTT. 14. In a circle 5 in. in radius, how long an arc will subtend an angle at the center of 12 radians? An angle of 75~ 16'? Ans. 8~ in.; 6.5683 in. 15. Find correct to three decimals the diameter of a circle in which an arc of 27~ 15' 16" is 7.23 in. long. Ans. 30.399 in. 16. In what quadrant is each of the following angles: I Tr, 21 r, 17 r, 7.1886 rad., 41 rad.? Draw each, using circular arrows to show size. 17. Find the distance measured on the surface of the earth from the equator to a city in latitude 42~ 30' north. Use 3963 miles as the radius of the earth. Ans. 2939.6 mi. INTRODUCTION 7 18. The diameter of a graduated circle is 5 ft. and the graduations on the circumference are 5' apart, find the length of arc between the graduations in inches to four decimal places. Ans. 0.0436 in. 19. Find the radius of a circle in which an arc of 20 ft. measures an angle of 2.3 radians at the center. In this circle, find the angle at the center measured by an arc of 3 ft. 8 in. Ans. 8.6957 ft.; 0.4217 rad. 20. Find the angular velocity per minute of the minute hand of a watch. Express in degrees and in radians. Ans. 6~ or 0.10472 rad. 21. A train of cars is going at the rate of 15 miles per hour on a curve of 600 ft. radius. Find its angular velocity in radians per minute. Ans. 23 rad. per min. 22. A flywheel 22 ft. in diameter is revolving with an angular velocity of 9 radians per second. Find the rate per minute a point on the circumference is traveling. Ans. 5940 ft. per min. 23. How large a target at a distance of 30 ft. will subtend the same angle at the eye as a target 4 ft. in diameter at a distance of 1000 yd.? What angle do they subtend? Ans. 0.48 in.; 4' 35". 24. Find the radius of a globe such that the distance between two places on the same meridian, whose latitudes differ by 1~ 20', shall be 1.5 in. Ans. 64.46 in. 25. The perimeter of the sector of a circle is equal to the arc of a semicircle having the same radius. Find the angle of the sector in radian measure. In degrees, minutes, and seconds. Ans. 1.14159 rad.; 65~ 24' 30". 26. Find the length (arc) which at a distance of 1 mile will subtend an angle of 10' at the eye. An angle of 1". Ans. 15.36 ft.; 0.3072 in. 27. The radius of the earth's orbit which is about 92,700,000 miles subtends at the star Sirius an angle of about 0.4". Find approximately the distance of Sirius from the earth. Ans. 48 x 1012 miles. 28. It has been observed that the earth's radius subtends an angle of 8.82" at the sun. Taking the radius of the earth as 3960 miles, find the distance of the sun from the earth. Ans. 92,700,000 mi. 29. Express an angular velocity of 3.4 revolutions per second in degrees per second; in radians per second; and in 7r radians per second. Ans. 1224~ per sec.; 21.363 rad. per sec.; 6.8 t rad. per sec. 30. A wheel is revolving at an angular velocity of 6 7 radians per second. Find the number of revolutions per minute and per hour. Ans. 180; 10,800. 31. A carriage wheel 2 ft. in radius rolls along a level road, the axle moving at the rate of 8 miles per hour; find the angular velocity in radians per second. Ans. 5.867 rad. per sec. 32. Find the angular velocity of a 34-inch wheel on an automobile that is moving 5 miles per hour. Express in revolutions per second and in radians per second. Ans. 0.824 rev. per sec.; 5T7 rad. per sec. 6. General angles.- In Fig. 7, the angle XOP1 is 30~; or if the angle is thought of as formed by one complete revolution and 30~, it is 390~; if by two complete revolutions and 30~, it is 750~. So an angle having OX for initial side and OPI for terminal side 8 PLANE TRIGONOMETRY may be 30~, 360~ + 30~, 2 X 360~ + 30~, or, in general, n X 360~ + 30~, where n takes the values 0, 1, 2, 3,..., that is, n is any integer, zero included. In radian measure this is 2 nr- + 6 7r. The expression n X 360~, + 30~ or 2 nr + - Xr, is called the general measure of an angle having OX as initial side and OP1 as terminal side. If the angle XOP2 is 30~ less than 180~, then the general measure of an angle having OX as initial side and OP2 as terminal side yr ~ is an odd number times 180~ less 30~; and may be written A P, (2 n + 1) 180~ - 30~, or (2n+ 1)r - -Tr. X' A-X Similarly n7r ~t Tr means an integral number of times 7r is taken and P3 p P4 then - ir is added or subtracted. This gives the terminal side in one of the four positions shown in Fig. 7 FIG. 7. by OP1, OP2, OP3, and OP4. It is evident that throughout this article n may have negative as well as positive values, and that any angle 0 might be used instead of 30~, or - 71. EXERCISES 1. Using the same initial side, draw angles of 40~; 360~ + 40~; n 360~ + 40~. 2. Using the same initial side, draw angles of 50~; 180~ + 50~; 2 X 180~ + 50~; n X 1800 + 50~. 3. Using the same initial side, locate the terminal sides for 180~ + 60~; 3 X 180~ + 60~; 5 x 180~ + 60~; (2n + 1) 180~ + 60~. 4. Draw the following angles: 90~ + 200; 90~ - 20~; 3 X 90~ L 20~; (2 n + 1) 900 - 20~. 5. Draw the following angles: 2 n X 180~ t 60~; (2 n + 1) 180~ ~ 60~; 1 1 r 1 r r (2 n +1) 7r 7r; 2 nr + r; (2 n + 1)-; n i4-r; (4 n + 1); 7'3 2 3' 4 2 (4n - l) i. 6. Give the general measure of all the angles having the lines that bisect the four quadrants as terminal sides. Those that have the lines that trisect the four quadrants as terminal sides. INTRODUCTION 9 7. Directed lines and segments. -For certain purposes in trigonometry it is convenient to give a line a property not often used in plane geometry. This is the property of having direction. In Fig. 8, RQ is a directed straight line if it is thought of as traced by a point moving without change of direction from R toward Q or from Q toward R. The direction is often shown by an arrow. Let a fixed point 0 on RQ be taken as a point from which to measure distances. Choose a fixed length as a unit and lay it R P P, 0 P1 Ps Q _ I I t r I 'I - - I I I >-5 -4 -3 -2 -1 0 1 2 4 5 FIG. 8. off on the line RQ beginning at 0. The successive points located in this manner will be 1, 2, 3, 4,... times the unit distance from 0. These points may be thought of as representing the numbers, or the numbers may be thought of as representing the points. Since there are two directions from 0 in which the measurements may be made, it is evident that there are two points equally distant from 0. Since there are both positive and negative numbers, we shall agree to represent the points to the right of 0 by positive numbers and those to the left by negative numbers. Thus, a point 2 units to the right of 0 represents the number 2; and conversely the number 2 represents a point 2 units to the right of 0. A point 4 units to the left of 0 represents the number -4; and conversely the number -4 represents a point 4 units to the left of 0. The point 0 from which the measurements are made is called the origin. It represents the number zero. A segment of a line is a definite part of a directed line. The segment of a line is read by giving its initial point and its terminal point. Thus, in Fig. 8, OP1, OP2, and PiP3 are segments. In the last, Pi is the initial point and P3 the terminal point. The value of a segment is determined by its length and direction, and it is defined to be the number which would represent the terminal point of the segment if the initial point were taken as origin. It follows from this definition that the value of a segment read 10 PLANE TRIGONOMETRY in one direction is the negative of the value if read in the opposite direction. In Fig. 9, taking 0 as origin, the values of the segments are as follows: P1 = 3, OP3 = 8, OP5 = -5, P2P3 = 3, P3P1 = - 5. P4P6 = -6, P6P5 = 3, P1P2 = -P2P1 = 2. Two segments are equal if they have the same direction and the same length, that is, the same value. If two segments are so placed that the initial point of the second is on the terminal point of the first, the sum of the two segments is the segment having as initial point the initial point of the first, and as terminal point the terminal point of the second. The segments are subtracted by reversing the direction of the subtrahend and adding. Thus, in Fig. 9, PG P5 P4 0 P1 P2 P3. I I I I I I I I ] I I I I I I I -10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10 FIG. 9. P5P4 + P4P1 = P5P1 = 8. P2P4 + P4P6 = P2P6 = -13. P1P3 - P2P3 = P1P3 + P3P2 P1P2 = 2. P2P3 - P1P3 = P2P3 + P3P1 = P2P1 = -2. 8. Rectangular co6rdinates. - Let X'X and Y'Y, Fig. 10, be two fixed directed straight lines, perpendicular to each other and intersecting at the point 0. Choose the positive direction towards the right, when parallel to X'X; and upwards, when parallel to Y'Y. Hence the negative directions are towards the left, and downwards. The two lines X'X and Y'Y divide the plane into four quadrants, numbered as in Art. 3. Any point P1 in the plane is located by the segments NP1 and MP1 drawn parallel to X'X and Y'Y respectively, for the values of these segments tell how far and in what direction Pi is from the two lines X'X and Y'Y. It is evident that for any point in the plane there is one pair of values and only one; and conversely, for every pair of values there is one point and only one. INTRODUCTION 11 The value of the segment NP1 or OM is called the abscissa of the point P1, and is usually represented by x. The value of the segment MP1 or ON is called Y the ordinate of the point P1, and is usually represented P2 by y. Taken together the N abscissa x and the ordinate y are called the coordinates of the point P1. They are I written, for brevity, within parentheses and separated P3 by a comma, the abscissa always being first, as (x, y).Y' The line X'X is called the FIG. 10. axis of abscissas or the xaxis. The line Y'Y is called the axis of ordinates or the y-axis. Together these lines are called the coordinate axes. It is evident that, in the first quadrant, both coordinates are positive; in the second quadrant, the abscissa is negative and the ordinate is positive; in the third quadrant, both coordinates are negative; and, in the fourth quadrant, the abscissa is positive and the ordinate is negative. This is shown in the following table: Quadrant I II III IV Abscissa Ordinate Y ' 0 q - - _ + + _ + Thus, in Fig. 10, Pi, P2, P3, and P4 are respectively the points (4, 3), (-2, 4), (-4, -3), and y (3, -4). The points M, 0, N, X and Q are respectively (4, 0), (0, 0), x M (0, 3), and (-4, 0). 9. Polar coordinates. - The point P1, Fig. 11, can also be located if the angle 0 and the length of the line OP1 are known. The line OP1 is called the radius X 0 I y' FIG. 11 vector and is usually represented by r. Since r denotes the distance of the point P1 from 0 it is always considered positive. 12 PLANE TRIGONOMETRY It is seen that r is the hypotenuse of a right triangle of which x and y are the legs; hence r2 = x2 + y2, no matter in what quadrant the point is located. EXERCISES 1. Plot the points (4, 5), (2, 7), (0, 4), (5, 5), (7, 0), (-2, 4), (-4, 5), (-6, -2), (0, -7), (-6, 0), (3, -4), (7, -6). 2. Find the radius vector for each of the points (7, 6),-(-5, 6), (-4, -6), (9, -8), (9, -10). Plot in each case. Ans. 9.2195; 7.8102; 7.2111; 12.0416; 13.4536. 3. Where are all the points whose abscissas are 2? Whose ordinates are 3? Whose abscissas are -5? Whose radii vectores are 6? 4. The positive direction of the x-axis is taken as the initial side of an angle of 45~. A point is taken on the terminal side with a radius vector equal to 10. Find the ordinate and abscissa of the point. Ans. Each is 7.071. 5. In Exercise 4 what is the ratio of the ordinate to the abscissa? The ratio of the radius vector to the ordinate? Show that you get the same ratios if any other point on the terminal side is taken. 6. The radius vector of a point is 8 and makes an angle of 60~ with the positive x-axis. Find the coordinates of the point. Find the coordinates of the points if the radius is the same length and the angles are respectively 120~, 150~, 240~, and 330~. Ans. (4, 4 /3); (-4, 4 /3); (-4 /3,4); (-4, -4 3); (4 3, -4). 7. With the positive x-axis as initial side construct angles of 30~, 150~, 210~, and 330~. Take a point on the terminal side so that the radius vector is 2 a in each case, and find the length of the ordinate and the abscissa of the point. 8. The hour hand of a clock is 4 ft. long. Find the coordinates of its outer end when it is twelve o'clock; when one o'clock; when two; five; eight; ten; half-past four. Use perpendicular and horizontal axes intersecting where the hands are fastened. Ans. (0,4); (2, 2/3); (2 3,2); (2,-2 /3); (-2 /3,-2); (-2 /3,2); (2 /2, -2 V2). CHAPTER II TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 10. Functions of an angle. - Connected with any angle there are six ratios that are of fundamental importance, as upon them is founded the whole subject of trigonometry. They are called trigonometric ratios or trigonometric functions of the angle. One of the first things to be done in trigonometry is to investigate the properties of these ratios, and to establish relations between them. y P P Al 0 I -0 (a) (b) Y M. //, O 0 -X P (c) FIG. 12. 11. Trigonometric ratios. - Draw an angle 0 in each of the four quadrants as shown in Fig. 12, each angle having its vertex at the origin and its initial side coinciding with the positive part of the x-axis. Choose any point P (x, y) in the terminal side of such angle at the distance r from the origin. Draw MP lOX, 13 14 PLANE TRIGONOMETRY forming the coordinates OM = x and MP = y, and the radius vector, or distance, OP = r. Then in whatever quadrant 0 is found, the functions are defined as follows: sine 0 (written sin 0) cosine 0 (written cos 0) tangent 0 (written tan 0) cotangent 0 (written cot 0) secant 0 (written sec 0) cosecant 8 (written csc 0) ordinate MP y distance oP r abscissa OM x distance oP r ordinate _ I y abscissa 01M x abscissa OM x ordinate ALP y distance oP r abscissa OM1~ x distance O_ P r ordinate M1P y Two other functions frequently used are: versed sine 0 (written vers 0) = 1 - cos 0, coversed sine 0 (written covers 0) = 1 - sin 0. The trigonometric functions are abstract numbers, and therefore subject to the ordinary rules of p Y algebra, such as addition, subtracps2 tion, multiplication, and division. \P1 0o-C 12. To each and every angle X there corresponds but one value M3 M2 Ml 0 of each trigonometric ratio.Draw any angle 0 as in Fig. 13. FIG. 13. Choose points Pi, P2, P3, etc., on the terminal side OP. Draw MP, M2P, 2, M3P3, etc., perpendicular to OX. From the geometry of the figure, M1P1 M2P2 _ M3P3 -P1 -P2 - M3P= etc. = sin 0, OP1 OP2 OPs OM1 OM2 3OM OP1 OP2 O- _P- = etc. = cos 6, MP1_ M2P2 MPs3 -MIP M22.= M33 = etc. = tan 0, OM1 OM2 OM3 and similarly for the other trigonometric ratios. Hence the six ratios remain unchanged as long as the value of the angle is unchanged. TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 15 Definitions. When one quantity so depends on another that for every value of the first there are one or more values of the second, the second is said to be a function of the first. Since to every value of the angle there corresponds a value for each of the trigonometric ratios, the ratios are called trigonometric functions. They are also called natural trigonometric functions in order to distinguish them from logarithmic trigonometric functions. A table of natural trigonometric functions for angles from 0~ to 90~ for each minute is given on pages 108 to 130 of Tables.* An explanation of the table is given on page 27 of Tables. 13. Signs of the trigonometric functions. - The sine of an angle 0 has been defined as the ratio of the ordinate to the distance of any point in the terminal side of the angle. Since the distance r is always positive (Art. 9), sin 0 will have the same algebraic sign as the ordinate of the point. Therefore sin 0 is positive when the angle is in the first or second quadrant, and negative when the angle is in the third or fourth quadrant. In a similar manner the algebraic signs of the remaining functions of 0 are determined. The student should verify the following table: Quadrant sin 0 cos 0 tan 0 cot 0 sec 0 csc 0 I............ + + + - II........... - - - - III.......... - - + + — IV........... - + - - + 14. Exponents of trigonometric functions. - When the trigonometric functions are to be raised to powers they are written sin2 0, cos3 6, tan4 0, etc., instead of (sin 0)2, (cos 0)3, (tan 0)4, etc., except when P the exponent is - 1. Then the,' t function is enclosed in parentheses.! 250 1 _ Thus, (sin 0)-1= (see Art. 26). o0 -— 2-M sin 6 15. Calculation of trigonometric functions by measurement.- IG. 14. Functions of 25~. By means of the protractor draw angle XOP = 25~, Fig. 14. Choose P in the terminal side, say, 23 * The reference is to Logarithmic and Trigonometric Tables by the Authors. 16 PLANE TRIGONOMETRY in. distant from the origin. Draw PM L OX. By measurement, OM = 2 in. and MP = 5 in. From the definitions, MP 1 5 OM 2 sin 250 = OP = 2c25 = 0.91, OP 23 ~OP 2,3 MP 1 5 OM 2 tan 25~ - - = 0.47, cot 25 = — = 2.13, OM 2 MP.1 OP OP 2 3, see 25~ - -, Cse 25'~ 2.33, serOM 2- ' es MP = 5 vers 25 = 1 - cos 25 = 1 - 0.91 = 0.09, covers 25~ = 1 - sin 25~ = 1 - 0.43 = 0.57. Exercise. Construct several acute angles at random, with their vertices at the origin and their initial sides on the positive part of the x-axis. Choose a point in the terminal side of each angle, draw and measure its coordinates, and calculate the trigonometric functions for each angle. Measure each Y angle with the protractor and determine its trigonometric functions from the P (a,) Table IV. Compare these functions O ' -* X with those computed. 16. Trigonometric functions by computation. - Trigonometric functions of 0~. The initial and terminal sides of 0~ FIG. 15. are both on OX. Choose the point P. on OX as in Fig. 15, at the distance of a from O. Then the coordinates of P are (a, 0). By definition then we have: Y 0 x a sin0~ = -~ 0. *cot0~= x a = o r a y 0 x a r a cos0~ = = 1. sec = 1. r a x a tan0~ = = ~ 0. *csc 0 - r a = x a y 0 a a * By the expression a = oo is understood the value of - as x approaches zero 0 x a a a a as a limit. For example, a = a; = 10 a; = 100 = 1000 a; I 0.1 0.01 0.001 a a.0000001 = 10,000,000 a; etc. That is, as x gets nearer and nearer to zero - 0.0000001 x gets larger and larger, and can be made to become larger than any number N. The value of - is then said to become infinite as x approaches zero. The symbol is usually read infinity. symbol oc is usually read infinity. TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 17 Trigonometricfunctions of 30~. Draw angle XOP = 30~ as in Fig. 16. Choose P in the terminal side and draw PM _OX. By geometry, MP, the side opposite the 30~ angle, is one half the hypotenuse OP. Take y = MP = 1 unit and r = OP = 2 units, then x = OM = V/3. By definition sin 30 = - =. cot 3( r 2 x V\~ 1 cos 300 = = sec 3( tr 2 2/ y 1 /3 1 tan 300= -V3. csc 31 x \/3 3 3 FIG. 16. then we have: 0~= 2 = a/3. or 2 2 0~" = - /3. 2 r 2 0 = - =2. y 1 Trigonometric functions of 45~. Draw angle XOP = 45~ as in Fig. 17. Choose the point P in the terminal side and draw its co6rdinates OM and MP, which are necessarily equal. Then the coordinates of P may be taken as (1, 1) and r = /2. By definition then we have: FIG. 17. sin 45 = -/1 1 2 r V2 2 cos 45 0- x 1 Y 1 tan 45~ = y = 1. x 1 x 1 cot 45 = - = 1. y 1 r V2/ sec 450-= - = 2. x 1 r V2 csc 450 = =- = -. y I Trigonometric functions of 90~. Draw angle XOY = 90~ as in Fig. 18. Choose any point P in the terminal side at the distance a from the origin. Then the coordinates of P are (0, a) and r = a. By definition then we have: YP(o,a) 0 0 FIG. 18. 18 PLANE TRIGONOMETRY sin 90~ = y a 1. cot 90~ = ~ = 0. r a y a x 0 r a Cos 90~ = -= 0. sec 90~ =-=- o. r a x 0 tan 90~ = y =-=. csc 90~ = a 1. x 0 y a Trigonometric functions of 120~. Draw angle XOP = 120~ as in Fig. 19. Choose any point P in the terminal side and draw its coordinates OM and MP. Triangle p Y MOP is a right triangle with ZMOP = 60~. Then, as in computing the functions 120~ of 30~, we may take OP = 2, MO = 1, and MP = V3. But the abscissa of P is OM M-1 =- 1. Then the coordinates of P are (-1, /3), and r = 2. By definition then FIG. 19. we have: sin 120~ y= y / 1 r 2 2 x -1 1 cos 120~ - -. r 2 2 tan 120~ = = y -/ 3. x -1 cot 120~ ==-I 3. y v=3 3 r 2 sec 120~ = = -2. x -1 r 2 2 csc120~ -= - -=V3. y v'3 3 Exercise. Verify the following table by actually constructing the angles and computing the functions. TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 19 00 00 300 450 600 900 1200 1350 1500 1800 2100 2250 2400 2700 3000 3150 3300 3600 o in radians 0 W 6 3w 27w 4w 5wr 7w Tr 11 2w sin 0 0 1 2 1 -\/3 1 1 2 - 0 1 0 Cos 6 1 2 -1 2 <2 2-Y 1 1 tan 0 0 1 cot 0 sec 0 1 2<-/3 3 2 -1 3 0 +3 1 1 0 3 - 1 00 1 0 -1I - 2 3 - 1 3 - 2 eSO 0 00 2 1 3 2 00 -2 2< 3 - 1 2<3\/ 3 - 2 00 -1 -3 0 2 23 1 EXERCISES Find the numerical values of the following expressions correct to three decimal places: 1. sin 300 + 3 cos 450. Ans. 2.621. 2. sin 2600 + sin' 900. Ans. 1.75. 3. 10 cos' 450 X sec 300. Ans. 2.886. 4. sec 0' * cos 450 + csc 900.- sec2 300. Ans. 2.040. 5. cos 1500. sin2 22700 - sin 900- tan', 1350. Ans. 0.134. In the following expressions, show that the left hand member is equal to the right, by using the preceding table or the table of natural functions when necessary. 6. sin 900 cos 00 ~ cos 900 sin 00 = sin 900. 20 PLANE TRIGONOMETRY 7. cos 30~ cos 60~ - sin 30~ sin 60~ = cos 90~. 8. sin 60~ cos 30~ - cos 60~ sin 30~ = sin 30~. 9. cos 120~ cos 90~ - sin 120~ sin 90~ = cos 210~. tan 1200 + tan 150 _ 10. tan 120~ tan 150~ tan 270~. i - tan 120~ tan 150~ tan 240~ - tan 30~ 11. -a 4~0Ot3o= tan 210~. i + tan 240n tan 300 17. Given the function of an acute angle, to construct the angle. - Example 1. Given sin 0 = 4. Construct the angle 0 and find the other functions. y 4 Solution. By definition sin 0 - = - Since we are conr 5 cerned only with the ratios of the lines, we may take y = 4 and r = 5 units of any size. Draw A N p/ B AB parallel to OX and 4 units above, Fig. 20, intersecting OY at 5/ N. With the origin as a center and a radius of 5 units, draw an X arc intersecting AB in the point 0 3 M P. Draw OP forming ZXOP, and draw MP _OX. Then OP = 5, FIG. 20. MP = 4, and OM = /OP2 - MP2 = V25- 16 3. MP 4.. ZXOP = 0 is the required angle since sin 0 = = 4 OP 5 The remaining functions may be written as follows: OM 3 MP 4 OM 3 cos0= O = tan0= OM= cot M OP 5' OM 3' MP 4' OP 5 OP 5 sec 0 M csc 0 MP OM 3' MP 4 Example 2. Given cos 0 = 2. Construct angle 0 and find the other functions. Solution. By definition cos 0 =. 3/ 2 Choose x = 2 and r = 3. Draw AB I OY and 2 units to the right, Fig. 21, O MX intersecting OX at Mi. With the origin as a center and a radius of 3 units, draw an arc cutting AB at P. Join 0 and P FIG. 21. TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 21 forming Z XOP. Then OP = 3, OM = 2, and MP = \/OP2 - OM2 = /5... Z XOP = 0 is the required angle since cos 0 = M = OP 3 The remaining functions are as follows: MP V/5 MP V/5 OM 2 sin 0 p=, tan 0- = - cot 0 - OP 3' OM 2MP 5 V ' OP 3 OP 3 OM- 2' M sec 0 OM = 2 csco 0 = MP = V/. Example 3. Given tan 0 = 2. Construct angle 0 and find the other functions. Solution. By definition tan 0 = y 2 A = Choose y =2 and x =5. x 5' C P D Draw AB 11 OY and 5 units to the 2 right, Fig. 22, intersecting OX at M; O MX also draw CD 11 OX and 2 units above intersecting AB at P. Then OM = 5, MP = 2, and OP = F/29. MP 2.. Z XOP = 0 is the required angle since tan 0 = OM = The other functions are as follows: 2 5 5/29 sin = _, cos0=, cot0=, sec 0 =_29 V29 V2\9 2 5 csc 0 = 2 2 EXERCISES In Exercises 1 to 8, construct 0 from the given function, and find the other functions of 0 when 0 is in the first quadrant. _/3 5 1. sin = - 5. sec0 = - 2 3 2. cos 0 =- 6. csc 0 = 2. 1 a 3. tan 0 =- 7. cos = 3 a 4. cot0 =- 8. tan = 2 b sin 0sec 9 o 1 9. Find the value of - 0 c - when cos 0 = and 0 is an acute angle. COAns.0csCO' 41 Ans. 15. 22 PLANE TRIGONOMETRY 10. Find the value of ee tan when tan 0 = 2 and 0 is an acute cos 0 + vers 0' angle. Ans. 4.236. csc ~ + sec 0 1 11. Find the value of C0 + see0 when cos 0 = and 0 is an acute sin 0 + cos 0' angle. Ans. 1-A. 18. Trigonometric functions applied to right triangles. — When the angle 0 is acute, the abscissa, ordinate, and distance for any point in the terminal side, form a right triangle, in which the Y A C given angle 0 is one of the acute angles. B On account of the many applications of the right triangle in trigonometry, the definitions of the trigonometric functions will be stated with special reference to Cb` X the right triangle. Draw the right triangle ABC, Fig. 23, 23. with the vertex A at the origin, and AC on the initial line. Then AC and CB rdinates of B in the terminal side AB. Let AC = b, 1 AB = c. ion: FIG. are the cooI CB = a, anc By definit] ordinate sin A = - =dis distance a side opposite c hypotenuse abscissa b side adjacent cos A = = - = distance c hypotenuse ordinate a side opposite tan A = a i- - adj abscissa b side adjacent abscissa b side adjacent cotA =. = - = - ordinate a side opposite distance c hypotenuse see A =. - abscissa b side adjacent distance c hypotenuse cscrA -a side o - p ordinate a side opposite A Again, suppose the triangle ABC placed so that Z B has its vertex at the origin, BC for the initial side, and BA for the terminal side, as in Fig. 24. The coordinates of A are BC = a and CA = b. b FIG. 24. TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 23 By definition: sin B b side opposite c hypotenuse a side adjacent cos = - c hypotenuse b side opposite tanB = - side adj a side adjacent a side adjacent cot B - = b side opposite c hypotenuse secB - = a side adjacent c hypotenuse cse B = - = b side opposite Then, no matter where the right triangle is found, the-functions of the acute angles may be written in terms of the legs and the hypotenuse of the right triangle. 19. Relations between the functions of complementary angles. - From the formulas of Art. 18, the following relations are evident: sinA = cosB = - c cosA = sinB =c tan A = cotB = - b cot A = tanB =a sec A = cscB = - b cscA = seeB = - a But angles A and B are complementary; therefore, the sine, cosine, tangent, cotangent, secant, and cosecant of an angle are respectively the cosine, sine, cotangent, tangent, cosecant, and secant of the complement of the angle. They are also called co-functions. For example, cos 75~ = sin (90~ - 75~) = sin 15~; tan 80~ = cot (90~ - 80~) = cot 10~. Note. The term cosine was not used until the beginning of the 17th century. Before that time the expression, sine of the complement (complementi sinus) was used instead. Cosine is a contraction of this. Similarly, cotangent and cosecant are contractions of complementi tangens and complementi secans respectively. The abbreviations, sin, cos, tan, cot, sec, and csc did not come into general use until the middle of the 18th century. EXERCISES 1. In a right triangle, given sin A = % and c = 100; find a and b. a a 3 Solution. By definition, sin A = - = - c 100 5.. 5a = 300 and a = 60. b2 = c- a2 = 10,000 - 3600 = 6400..'. b = 80. 24 PLANE TRIGONOMETRY 2. Express each of the following functions in terms of angles less than 45~: sin 68~, cot 88~, sec 75~, csc 47~ 58' 12", cos 71~ 12' 56". In the following right triangles, calculate the required parts from the given parts: 3. cosB = -, c = 200.5; find a and b. Ans. b = 160.4. 4. cosA = 0.55, b = 25; find a and c. Ans. c = 45.45. 6. cotA = -, b = 41; find a and c. 6. a = 8, b = 15, c = 17; write all the functions of A and B. /r2 s 7. a = Vr2 + s2, b = 2rs; find cosA. Ans. cos A = - r -t s 8. a = Vr2 + 2, c = r + s; find cot A. Ans. cotA = 22 s2 r2 - s2 9. a = 2rs, b = r2 - S2; find sin B. Ans. sinB = -2 r2 t- s2 10. Given sin A = cos 2 A; find A. Solution. cos 2 A = sin (90~ - 2 A) by Art. 19... sin A = sin (90~ - 2 A). Since the functions are equal the angles are equal..-. A = 90~ - 2 A, from which A = 30~. 11. Given tan A = cot 3 A; find A. Ans. A = 22~~. 12. Given cos fi = sin (45~ - ~ P); find F. Ans. f = 90~. 13. Given tan 0 = cot (45~ + 0); find 0. Ans. 0 = 220~. 20. Given the function of an angle in any quadrant, to construct the angle. - Example 1. Given sin 0 =. Construct the angle 0 and find all the other functions. Solution. By definition, sin 0 = - Take y = 3 units and r = 5 units. Draw AB 11 OX and 3 units above it as in Fig. 25. y Construct the arc of a circle with center at 0 and radius 5 units, A P2 H P1 B intersecting AB at P1 and P2. 3 5 ^^ Then for Pi, x = 4, y = 3, and 03\ \ 2z4 3 r = 5; for P2, x - 4, y= 3, -4 1, r and r = 5. Now OP1 and OP2 are the terminal sides respectively of ZXOP1 = 01 and ZXOP2 = FIG. 25. 02, each of which has its sine equal to 5. Then from the definitions of the trigonometric functions we have the following: Quadrant Angle sin 0 cos 0 tan 0 cot 0 sec 0 csc 0 I........., - 4 t t 5 II........ 52 3 4 3 -t - 5 5 TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 25 Example 2. Given cos 0 = —. Construct the angle 0 and find all the other functions. Solution. By definition, cos 0 x 2.. r- —. Since r is always positive r 3 we take x = - 2 units and r = 3 units. Draw AB II OY and 2 units to the left as in Fig. 26. Construct a circle of radius 3, with its center at 0, and intersecting AB at P1 and P2. Draw OP1 and OP2. As in Example 1, it may be shown that Z XOP1 = 0 and Z XOP2 = 02 are the required angles. tions are as follows: BI Y, P1 -X M 9 PI A FIG. 26. The remaining func Quadrant Angle sin 0 cos 0 tan 0 cot 0 sec 0 csc 0 <s5 2 v' 2 3 3 II........ 01 2 III.6.. 2 3 3 2 2 I5 V 2 2 3 3...3 3 2 - 2 -5 Example 3. Given tan 0 = -. all the other functions. Construct the angle 0 and find Solution. By definition, tan 0 =. Hence = -= and x 4 -4'and FIG. 27. we may takey = ~3 and x = ~4. Then r = '/(V-4)2+(43)2 = 5. With 0 as a center and 5 as a radius construct a circle as in Fig. 27. Draw AB and CD 11 OY and 4 units to the right and left 26 PLANE TRIGONOMETRY respectively of OY. Also draw EF and GH 11 OX and 3 units above and below OX respectively. These lines and the circle intersect at the points P1, P2, P3, and P4. Since x and y must both be positive or both negative, the required points must be P1 and P3 located in the first and third quadrants. Draw OP1 and OP3 forming the angles XOP1 = 01 and XOP3 = 03. The remaining functions are as follows: Quadrant Angle sin 0 cos 0 tan 0 cot 0 sec 0 csc 0 I......... 01 i 4 3 III. 03 — 3 4 3 4 III....... -03 -i -4 -3 EXERCISES In Exercises 1 to 5 tabulate the functions of 0 in each of the two quadrants in which the angle is found. 1. sin 0 = -1. 4. cot 0 = 3. 2. tan 0 =-1. 6. cos = -- 2 3. sec 0 = -2. 6. What is the greatest value that the sine of an angle may have? That the cosine of an angle may have? What is the least value for each? 7. Between what two numbers will sec 6 and csc 0 have no values? In Exercises 8 to 10 show by substitution that the right-hand member is equal to the left. 8. cos 0 tan 0 + sin 0 cot 0 = sin 0 + cos 0, when tan 0 = 2 and 0 is in the third quadrant. 9. (1 + tan2 0) (1 - cot2 6) = sec2 0 - csc2 0, when cos 0 = ~ and 6 is in the fourth quadrant. sin 6 1 10. cot 0 + 1 - = csc 0, when sin 0 = 3 and 0 is in the second 1 + cos60 2 quadrant. 11. Find the value of sin tan cos 0 when sin and i in the cot 0 sec 0 - cos 0 5 fourth quadrant. Ans. 3-7. sin 6 + tan 6 4 12. Find the value of 0 + when cos 6 = -and is in the third cos 0 +- vers ' 5 quadrant. Ans. -!-. sec 0 - csc 0 13. Find the value of when cot 0 = -2 and 0 is in the sec 0 + csc 0' second quadrant. Ans. -3. TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 27 21. Fundamental relations between the functions of an angle.- From the figures of Art. 11, it is evident that for an angle in any quadrant (1) x2 + y2 = r2 2 y2 r2 Dividing (1) by r2, - + =2 1. But - = cos 0 and = sin 0. r r [1].-. sin2 0 + cos2 = 1. 2 r2 Dividing (1) by x2, 1 + Y2 = X2 X2 But tan 0= and sec 0 = x x [2].~. 1 + tan2 0 = sec2 0. x2 [r2 Dividing (1) by y2, + 1= x r But cot 0 = - and csc 0 = -* Y Y [3].~* 1 + cot2 0 = csc2 0. Also, from the definitions of the trigonometric functions, the following reciprocal relations are evident: [4] sin 0 = - and csc 0= sin. csc 9 sin 9 _ _ _ _ _ 1 [5] cos = 0 and sec0 = sec 0 cos 0 [6] tan 9 = o and cot = t cot tan The following are formulas that are easily derived: [7] tan O = sin 0 cos 0. cos 0 [8] cotO = csi The eight formulas of this article are identities for they are true for any angle whatever. They are often spoken of as fundamental identities. 28 PLANE TRIGONOMETRY EXERCISES In the following exercises determine the remaining functions from the given functions by means of the fundamental identities. 1. Given tan 0 =, and 0 in the first quadrant. Solution. By [2], sec 0 = /1 + tan2 9 = V1 + 1-6- =. 1 1 3 By [6], cot = t L- c tan 0 4- 4 By [3], csc 0 = V/1 + cot2 0 = V1 + -t =. 1 1 4 By [4], sin 0 = = 2. Given sin 0 = b, and 6 in the second quadrant. Solution. By [1], cos 0 = - V1 - sin2 0 = -V1 - = - V/3. By [7], tan 0 3. i 13 By [6], cot 0 = = = - tan 0 1 /. 1 1 2 By [5], see 0 = = - = - cos _ 3 1 1 By [4], csc = = 2. tasin 0 12 Note. The proper algebraic sign is determined by referring to the table of signs of Art. 13. 3. Given cos 6 = -4, and 0 in the third quadrant. 4. Given cot 0 = - 5, and 0 in the fourth quadrant. 5. Given sec 0 = V/2, and 0 in the first quadrant. 6. Given cos 0 = - 1, and in the third quadrant. 7. Given tan 0 = - 23, and 0 in the fourth quadrant. 8. Given cot 0 = 5, and 0 in the third quadrant. 9. Given cos ac = -a, and a in the second quadrant. 10. Given cot a = - -, and 0 in the fourth quadrant. 11. Given sec 0 = -—, and 0 in the third quadrant. 12. Given cs c = -—, a nd a in the fourth quadrant. c. 1en t/s (s - a) a + b + c 13. If cos = -, where s = 2 2 show that sin 0 = \(-b s2 be 14. If cos 0 = s (Sa, where s - a +b + show that tan 1 0 = a) ( 2 s (s - c) __ TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 29 15. If tan 3 = c, show that cse S is real for all values of c. 2 mn 2 mn 16. Given sin =2 2; show that tan y = -4 — n2 + n2 I m2_ n2 22. To express one function in terms of each of the other functions. - Suppose that we wish to express sin 0 in terms of each of the other functions. By [1], sin 0 = /1 - cos2 0. By [5], cos == * sin 0 = /1 - cos2, == '/1 _ 1 ~~~Vsec 0 sin0= \/i —cos20 / 16 _ /sec2 6 - 1 sec 0 By [2], sec2 0 = 1 + tan2 0. si V/sec2 - 1 Vtan2 0. sin 0 = -. sec /1 V +- tan2 0 tan 0 Also sin 0 = - = - c /1 + tan2 0 1 V1 + cot2 0 V/I + cot2 0 By [4], sin = csc 0 The algebraic sign of sin 0 is determined from the quadrant in which 0 is P found. 23. To express all the functions of an angle in terms of one function of the 0 X angle, by means of a triangle.- 0 1\-sin2 M Example 1. Express all the functions FIG. 28. of 0 in terms of sin 0. Solution. Construct angle 0 in the first quadrant, Fig. 28, and choose the point P in the terminal side with coordinates OM and 30 PLANE TRIGONOMETRY MP MP. Then by definition sin 0 =, and if OP is taken equal to 1, MP = sin 0, and OM = VOP2 - MP2 = V/ - sin2 0. The remaining functions may then be written as follows: OM cos = = VI - sin2 6. OP MP sin 0 tan 0 - = - OM /1 - sin2 0 OP 1 sec 0 = = - OM -V- sin2 0 OP 1 C 0 = - MP sin 0 OM /1 - sin2 0 cot 0 = -MP MP sin 0 Example 2. Express all the functions in terms of cos 0. Solution. Construct angle 0 in the first quadrant, Fig. 29, and choose the point P in the terminal side with coordinates OM Y P 0o c0M x cos 0 M 0 FIG. 29. OM and MP. Then by definition cos 0 =-p and if OP is taken equal to 1, OM = cos 0, and MP = VOP2 - OM2 = VI - cos2 0. The remaining functions may then be written as follows: Psin = - os2. sin 0 n/1 cos2 0. OP OM cos 0 cot 0 = _-_ = MP /1 - cos2 0 OP 1 sec 0= OM =c OM - cos 0 MP /1 - cos2 0 tan 0 = - = OM cos 0 OP 1 csc 0 = - os2 MP -/1 - coss 0 In the following table, the student is asked to show that each function in the first column is equal to every expression found in the same row with the function. TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 31 tanG 1 V5\/seC 1 1 sinG sin B 1 -cos'G 0 V+cot2G secS cscG CosG0 V-\j-0' CosG 1 cotG0 1 V\/csC2 8 V1 +taoan2 i-\j1 + Cot2 sec c csCG sin 8 tanG 01 - cosG n 1 1 cos t cot 0 5-c'C V/CSC2G 1 cotG 'V -sisn2 cos 1 1 _I sin V1' j - Cos2 s tan co VseC2 \ S- 1 1 1 N/I _ Ct2 _ csc 5 sec Cs1 1+ tan2 V1cot 0 sec V5C8S-i V1/~s in2G8 co cot 5 ccz1 1 1-\1 +tan2 I sec 5 sin i V/1 - cosS tanG V1 Cot2G VseC6 - The table has been prepared under the assumption that 6 is an acute angle. Should 0 be in any other quadrant, the proper sign for each function may then be determined. 24. Transformation of trigonometric expressions so as to contain but one function. - In all transformations, avoid radicals if possible. If the expression is in a factored form, it is often desirable to reduce each factor separately and multiply the results. EXERCISES cos 6 1. Express in terms of tan 6. sin 6 Cot2 6 cos6 cos _ cos _ sinO0 Solution. cot 6 = then tan 0. sin 0 sin 0 * Cot2 6 cos'6 - Cosc2 0 Cos t sin 6 tan4 6 2. Express 1 - 2 (1 - covers 6)2 (1 tan' in terms of cos 6. (I + tan 2 0)2 sin 6 Solution. By definition, covers 6 = 1 - sin 6, and tan 6 = cos 6 1 + tan'2 = sec2 6, and cos 6 Substituting these values, we have see 0 sin4 6 sin4 6 cos4 6 cos4 6 1 - 2 [I - (1- sin 6)12 + C 14 - 2 sin2 ~ 1S sec" 4 f cos4 6 1 - 2 sin'2 + fin4 6 = (1 - sin2 6)2 = (COs2 )2 COS4 6. 3. Express (2 - vers 6) vers 6 in terms of sin 6. Ans. sin' 6. sin 6 4. Express cot 6 + in terms of cse 6. Ans. csc 6. 1 + cos 6 1 - cos 6 5. Express (csC 6 - cot 6)' in terms of cos 6. Ans. I + C os 0 32 PLANE TRIGONOMETRY 6. Express cot 0 - sec 0 csc 0 (1 - 2 sin2 0) in terms of tan 0. Ans. tan 0. (I - covers 0)2 CsCe 0 7. Express (1- covers 0) csc4 0 in terms of tan 0. Ans. tan2 0 + tan4 0. (csc2 0 - 1) Cot2 0 sec 0 csc 0 - 4 sin 0 cos 0. 8. Express in terms of sin 0. sin 0 sec 0. (1-2 sin2 0)2 sin2 0 sec2 0 csc2 + sec2 0- csc2 0 - 1 9. Express sem 2 0 in terms of cot 0. tan2 0 - csc2 0 + 1 cot2 0 + 2 Ans. 1 - cot4 0 25. Identities. - When two expressions in some letter x are equal for all values of that letter they are said to be identically equal. The equation formed by equating the two expressions is called an identity. The symbol denoting identity is -. When there can be no misunderstanding as to the meaning, the sign of equality is often used to denote identity. The symbol-= is read "identically equals," or "is identically equal to." Thus, x2 - 1 = (x - 1) (x + 1) because the equation is true for all values of x. Since the fundamental formulas are true for all values of 6, they are identities. In showing that one trigonometric expression is identically equal to another, we either transform both expressions to the same form, or transform one expression into the other, by means of the fundamental formulas of Art. 21. It is usually best, especially for the beginner, to express all the functions of the expression which is to be transformed, in terms of sine and cosine before attempting to simplify. Avoid radicals whenever possible. When the expression that is to be transformed is given in a factored form, it is usually best to simplify each factor separately before multiplying them together. EXERCISES Prove the following identities by transforming one member of the identity into the other. 1. tan 0 sin 0 + cos 0 = sec 0. sin 0 Proof. Substituting os0 for tan 0, we have sin 0 sin20 + cos2 0 1 - sin 0 + cos 0 = = - = se 0. cos 0 cos 0 cos 0 TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 33 cot ca COS c COt a - COS a cot a + Cosa cot a Cos a COS a Proof. Substituting CoS for cot a, we have COS a COS2 a * cos oa cot a Cos a sin a CO sin a Cos2 a cot a + cos a cos a cos a (1 + sin a) cos a (1 + sin a) sin aC sin a 1 -sin2a 1 - sin a cos a (1 + sin a) cos a Now multiply the numerator and denominator by cot a, and we have 1 - sin a cot a cot a - sin a cot a cot a - cos a COS a cot a Cos a cot a Cos a cot a 3. cot j cos $- + sin 3 = csc i. 4. (tan a - sin a)2 + (1- cos a)2 = (1-sec a)2. 5. (cosy + sin ). (sec y + cscy) = cos rsiny. 6. (1 + cot 0)2 + (1 + tan 0)2 = (csc 0 + sec 0)2. 7. = 1 + sin,3. 1 - sin B CSC2 a q- sec2 a 8. cot a + tan a =c a seca csc a ~ sec a 9. (r * cos 0)2 + (r * sin 0 sin 3)2 + (r * sin 0 cos /)2 = r2. sin a +- tan a 10. in a = sin a tan a. cot a + csc a 1 - 2 cos2 g 11. i-2 cos2 =tan -ecot,. sin, cos, 12. (sin y + cos 7)2 + (sin - - cos 7)2 = 2. 13. sin4 - cos4 0 = sin2 - cos2 0. 14. sin2 0 + vers2 = 2 (1 - cos 0). 15. (tan 0 - cot 0) sin 0 cos 0 = sin2 0- cos2. 16. cot2 0 - cos2 0 = cot2 0 * cos2 0. cos 0 Proof. Substituting s 0for cot 0, c-s2 - cos2 0 = cos2 0 cos2 0 -- in = co2 * c s 2 1 -sin2 = sos2..2 sin 2O sin26 - sin2 6 sin2 9 = cos2 cot2 0. tan a - cot a 2 17. 1. tan a + cot a csc2 a 18. cot, - sec / csc f (1 - 2 sin2 f) = tan 3. 19. vers 0 (sec 0 + 1) - covers 0 (csc 0 + 1) = sin 0 tan 0 + cos 0 cot 0. 20. sin2 y (tan2 - 1) + cos2 (cot2 y - 1) = (1-2 cos2 )2 sec4 tan2 y 34 PLANE TRIGONOMETRY 21. cos4 0 - sin4 0 = cos 0 (1 - tan 0) (sin 0 + cos 0). Suggestion. Factor the first member. sin4 0 + cos4 0 22. 2 + -sin 0 + 0 = se2 0 + csc2 0. sin2 0 cos2 0 sin a - tan2 a covers a sin4 a csc a cot2 a (1 + sin a) cos2 a 24. 1 (1- tan2 )2 4 sin2 C cos2 4 tan2, 26. Inverse trigonometric functions. - The equation sin 0 = a means that 0 is an angle whose sine is a. The expression sin-la is an abbreviation for the expression "an angle whose sine is a." Then we may write 0 = sin-la. The form sin-la is also read "anti-sine a," "inverse-sine a," "arc sine a." It is also written invsin a and arc sin a. Analogous forms with analogous meanings are given for the other functions. Illustrations. sin- = 30~ or 150~. cos-'1 = 0~. tan-'1 = 45~ or 225~. EXERCISES Prove the following relations for angles not greater than 90~. 1. cos~-1 }5 = sin-l -'. Proof. Let 0 = cos-1 1,. Then from the definition of the inverse functions, we have cos 0 = 'I.__.-. sin 0 = /1 )2 =.. = sin-' i. cos-1 1 = sin-1 8a. 2. sin-1 -A = cos-1 1. 3. sec-i 1 = tan-1 A. 65 16 c 4. sec-1 6 = sin-1 6. 5. tan-1 x = cot- 1 b b65 x 6. sin (sin-l a) = a. 7. sec- a = tan-li/ 8. cos-1 (cos 0~) = 0~. + a2. 27. Trigonometric equations.- A trigonometric equation is an equation in which the unknown is involved in a trigonometric function. The solution of a trigonometric equation is a value of the angle which satisfies the equation. Example 1. Solve the equation cos2 a + 2 cos a - 3 = 0 for values of a not greater than 90~. TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 35 Solution. Factoring the equation, (cos a + 3) (cos a - 1) = 0. Equating each factor to 0 and solving for cos a, cosa = 1 or -3... c = cos-1 or a = cos-1(- 3). Since there is no angle with a cosine equal to -3, the only solution admissible is a = cos- 1 = 0~. Example 2. Solve sin 0 = 4 for values of 0 not greater than 90~. Solution. 0 = sin-1 0.25000. From the table of natural functions (see Table IV), sin-1 0.25000 = 14~ 28.6'. EXERCISES 1. Find the values of the angles not greater than 90~ which satisfy the following anti-trigonometric equations: _V2 -\/3 \/3 (a) a = sin-1 — (e) 3 = cot-1 - (i) y = sin-1 - 2 3 2 (b) 0 = tan-l 1. (f) 3 = sin- 0. (j) 0 = cos-1' (C) 0 = COS-1 2. (g) ' = cos-1 0. (k) 0 = sin-l -. (d) a = sin-l 4. (h) 0 = tan-l oo. 2. Solve the following equations for angles not greater than 90~. (a) sin0 = 4. (d) cot a=. (b) cos = ~. (e) sina = 1. (c) tan 0 = —. (f) cos a = V2. 3. Solve 6 sin2 0- 5 sin 0 + 1 = 0, for the angle 0 < 90~. Solution. Factoring the left-hand member, (2 sin -1) (3 sin -1) = 0. Equating each factor to zero, 2 sin 0 - 1 = 0, and 3 sin 0 - 1 = 0, whence 2sin 0 = 1, and 3 sin0 = 1, from which sin 0 =, and sin 0 =. From these values we find 0 = 30~ or 19~ 28' 16". Solve the six following equations for values of 0 not greater than 90~. 4. /2 cos2 0 = cos 0. Ans. 0 = 45~, 90~. 5. 2 tan2 0 - 5 tan0 + 3 = 0. Ans. 0 = 45~, 56~ 18.6'. 6. 3 tan 0 = 4 sin 0. Suggestion. Apply [7], and factor. Ans. 0 = 0~, 41~ 24' 35". 7. 8 cos 0 - 10 cos 0 + 3 = 0. Ans. 0 = 41~ 24' 35", or 60~. 8. sin4 0-1 = 0. Factor. Ans. 0 = 90~. 9. 2 sin 0 + 3 sin 0-2 = 0. Ans. 0 = 30~. 36 PLANE TRIGONOMETRY In the following equations, express all the functions in terms of. one function and then solve. 10. 3sina = 2COS2 a. Ans. a = 300. 11. sin,6= 3 cos~3 Ans. 03 = 600. 12. sec atan a= 2V\3. Ans. a = 600. 13. Sec2 y + cot2 -y = 13 Ans. -y = 300, 600. 14. secO0 + tanO0 = </3. Ans. 0 = 300. 15. csc 0 = 3 sin 0. Ans. 0 = 350 15' 52". 16. 4 cos 0 = secc0. Ans. 0 = 600. 17. 4 sin a = 3cesc a. Ans. a = 600. 18. tan,3 = 3cotf3. Ans. 03= 600. 19. 1 + tan2 y 4 cos2 y. Ans. 'y = 450. 20. tan 0 v'N2 sin 0. Ans. 0 = 0', 450. CHAPTER III RIGHT TRIANGLES 28. General statement. - One of the direct applications of trigonometry is the solution of triangles both right and oblique. It is well to note that the solution of triangles is not the phase of trigonometry that is of most importance to the student who is to pursue more advanced subjects in mathematics. He will more often find use for the relations existing between the different functions, and in transforming one form of an expression involving trigonometric functions into an equivalent one. In attacking the triangle, trigonometry, in many ways, is a more powerful tool than geometry, which makes little use of the angles while trigonometry makes use of the angles, as well as of the sides, of a triangle. 29. Solution of a triangle. - Every triangle, whether right or oblique, has six parts, viz., three sides and three angles. When certain ones of these are given the others can be found. The process of finding the parts not given is called the solution of the triangle. By means of trigonometry a triangle can be solved when the parts given are sufficient to make a definite geometrical construction of the triangle. By geometry, a triangle can be constructed when three parts are given at least one of which is a side. The remaining parts can then be measured and so a solution of the triangle obtained. There are two ways of solving a triangle: (1) the graphical solution, (2) the solution by computation. 30. The graphical solution. - This consists in drawing a triangle such that its angles are equal to the given angles, and its sides equal to or proportional to the given sides. Of course it is necessary that the given parts be consistent and sufficient to determine a definite triangle. For instance, two angles must 37 38 PLANE TRIGONOMETRY not be given such that their sum is greater than 180~; nor can a construction be made if three sides are given such that one of them is as great as or greater than the sum of the other two. EXERCISES 1. Construct triangles by means of the straightedge and compasses having given: (a) Two sides and the included angle. (b) Two angles and the included side. (c) Three sides. (d) Two sides and an angle opposite one of them. Discuss and give drawings for all the possibilities. (e) Three angles. Is the construction definite? 2. Construct right triangles by means of the straightedge and compasses having given: (a) Two legs. (b) An acute angle and the hypotenuse. (c) An acute angle and one leg. (d) The hypotenuse and a leg. 3. Use the protractor in measuring the angles and construct the following: (a) A right triangle with an acute angle equal to 32~ and adjacent side 2.25 in. (b) An oblique triangle with an angle equal to 43~ 25' and the including sides 14 in. and 21 in. respectively. (c) A triangle with two angles 43~ and 57~ respectively, and the side opposite the first angle 7.5 in. (d) A triangle with sides 5.2 in., 3.1 in., and 2.4 in. respectively. (e) A triangle with sides 11.5 ft. and 4.7 ft. and the angle opposite the second side 115~. 31. The solution of right triangles by computation. - In the two previous articles, what was said referred to the oblique triangle as well as to the right triangle; here reference is to the right triangle only. Since in a right triangle the right angle is always a given part, it is necessary to have given only two other parts at least one of which is a side. In what follows a, b, and c represent the altitude, base, and hypotenuse respectively, and A, B, and C, the angles opposite the respective sides. The solutions depend upon the following relations, the first two of which are from geometry and the last eight from the definitions of trigonometric functions: RIGHT TRIANGLES 39 (1) c2 = a2 + b2, (2) A+B = 90~, a a (3) sin A = - (4) cos B = -, C C b b (5) cos A=-, (6) sin B = - C C a a (7) tanA =, (8) cot B = b b (9) cot A =-, (10) tanB = - a a Number (2) shows that no other part can be derived from the two acute angles alone. In each of the other formulas, three parts are involved. If any two of these parts are given the third can be a found. Thus, in (3) if a and A are given, c -= sin if c and A are sin A' given, a = c sin A; and if a and c are given, A = sin1-a Exercise. Solve each of the above formulas for each letter in terms of the others. 32. Steps in the solution. - In solving a triangle, it is of the greatest importance to follow some regular order. The following is suggested: (1) Construct the triangle carefully to scale, using compasses, protractor, and ruler. The required parts can then be measured and a check obtained on the computed values. (2) State the given and the required parts, and write down the formulas which are needed in the solution, solving them for the part required. In choosing these formulas, select for each part required a formula that shall contain two known parts and one required part. Thus, if A and a are the given parts and c the required part, then sin A =- contains the given parts and the C a required part c. This solved for c gives c = inA. In general avoid the use of c2 = a2 + b2 unless a table of squares and square roots is at hand. (3) Compute by substituting the given values in the formulas and evaluating. (4) Arrange the work neatly and systematically, as this conduces to accuracy and therefore speed. 40 PLANE TRIGONOMETRY (5) Always check. This can be done by making a careful construction, and also by using other formulas than those used in the solution. 33. Solution of right triangles by natural functions. Example 1. Given a = 3.25 and A = 47~ 25.6', find b, c, and B. Solution. Construction B Given { a = 3.25 Gven A = 470 25.6' b = 2.986 C a To find* c = 4.413 B = 42~ 34.4' A C b Formulas FIG. 30. (1) tan A = b = bv / b tan A a a (2) sin A = c = c sin A (3) A+B = 90.. B = 90 - A. Computation 3.25 b - = 2.986. 1.0885 3.25 c = 2 = 4.413. 0.7364 B = 90~ - 47~ 25.6' = 42~ 34.4'. Check a2 = c-b2 = (c + b) (c - b). 3.252 = (4.413 + 2.986) (4.413 - 2.986). 10.5625 = 10.5584. These agree to four significant figures. The formula sin B = - could also be used in checking. Results to be inserted when work is completed. Results to be inserted when work is completed. RIGHT TRIANGLES 41 Example 2. Given a = 6.72 and b = 3.27, find c, A, and B. Solution. Construction B Given a = 6.72 Given 3.27 c = 7.473 C a To find A = 640 3.11 B = 250 56.9' A b C Formulas FiG. 31. a a (1) tan A = A = tan a a (2) cot B = B = cot-6 b a a (3) sin A c= sin i c sin A Computation 6 72 A = tan-' = tan-' 2.0550 = 64' 3.1' -. 3.27 6.72 B = cot' = cot-' 2.0550 = 250 56.9' A-. 3.27 6.72 c 7.473 - 0.89919 Check a2 = c2 - b2 = (c -1 b) (c - b). 6.722 = (7.473 + 3.27) (7.473 - 3.27). 45.158 = 10.743 X 4.203 = 45.153. EXERCISES In Exercises 1 to 12, the first two parts are given and the last three are to be found. Use natural functions. 1. a = 3, b = 2, A = 560 18.6', B = 330 41.4', c = 3.606. 2. a = 92, b = 47, A = 620 56.3', B = 270 3.7', c = 103.31. 3. a = 46.8, c = 63.1, A = 470 52.5', B = 420 7.5', b = 42.324. 4. a = 0.236, c = 1.273, A = 100 41' 2", B = 79018' 58", b = 1.2509. 5. b = 4.92, c = 5.31, A = 220 5.8', B = 67' 54.2', a = 1.9974. 42 PLANE TRIGONOMETRY 6. a = 72.6, A = 33~, B = 57~, b = 111.8, c = 133.3. 7. b = 2.93, B = 17~ 19', A = 72~ 41', a = 9.397, c = 9.844. 8. a = 2.367, A = 17~ 16.4', B = 72~ 43.6', b = 7.612, c = 7.972. 9. b = 89.17, A = 4~ 46.3', B = 85~ 13.7', a = 7.444, c = 89.48. 10. a = 2.46, B = 87~ 6.4', A = 2~ 53.6', b = 48.67, c = 48.74. 11. c = 0.232, B = 49~ 13.1', A = 40~ 46.9', a = 0.1515, b = 0.1757. 12. c = 476.3, A = 23~ 47.3', B = 66~ 12.7', a = 192.12, b = 435.83. 13. What angles do the diagonals of a rectangle 7 in. by 10 in. make with each other? Ans. 69~ 59' 2" and 110~ 0' 58". 14. What is the angle of inclination of a stairway with the floor if the steps have a tread of 8 in. and a rise of 6~ in.? Ans. 39~ 5' 38". B C FIG. 32. 15. What angle does a rafter make with the horizontal if it has a rise of 6 ft. in a run of 12 ft.? Ans. 26~ 33' 54". 16. Certain lots in a city are laid out by lines perpendicular to B St. and C/2 OQ B St. FIG. 33. running through to A St. as shown in Fig. 33. Required the width of the lots on A St. if the angle between the streets is 28~ 40'. Ans. 113.97 ft. 17. Find the angle between the rafters and the horizontal in roofs of the following pitches: two-thirds, half, third, fourth. Ans. 53~ 7.8'; 45~; 33~ 41.4'; 26~ 33.9'. Note. By the pitch of a roof is meant the ratio of the rise of the rafters to twice the run, or, in a V-shape roof, it is the ratio of the distance from the plate to the ridge to the width of the building. RIGHT TRIANGLES 43 18. A regular hexagon of side 6 ft. is placed with its center at the origin and a diagonal on the x-axis. Find the coordinates of its vertices. Ans. (6, 0); (3, 3 /3); (-3, 3 3); (-6, 0); (-3, -3 3); (3, -3 3). 19. A man whose eyes are 5 ft. 6 in. above the ground is on a level with, and 150 ft. distance from, the foot of a flag staff 72 ft. high. What angle does his line of sight when looking at the top of the staff make with the horizontal line from his eye to the staff? Ans. 23~ 54' 33". 20. The sides of a triangle are 30, 42, and 42 respectively. Find the angles. Ans. 69~ 4' 31"; 69~ 4' 31"; and 41~ 50' 58". 21. In a circle a chord 48 in. long subtends an angle of 46~ 45' at the center. Find the radius of the circle. Ans. 60.491 in. 22. Find the radius of a circle inscribed in an equilateral triangle whose perimeter is 66 in. Ans. 6.3509 in. 34. Remark on logarithms. - By the use of logarithms, the processes of multiplication, division, raising to powers, and extracting roots may be shortened. In the solution of triangles, logarithms are very advantageous in saving time and labor, and thus conduce to accuracy. However, the student should bear in mind that logarithms are not necessary for this work. The computer must decide for himself whether or not it will be of advantage to use logarithms in any given problem. Formulas which have been so arranged that they involve only operations of multiplication, division, raising to powers, and extracting roots are said to be adapted to computation by logarithms. 35. Solution of right triangles by logarithmic functions. - The solution of a right triangle is the same by logarithms as by natural functions except that logarithms are used to avoid the long multiplications 'and divisions. The tables of logarithmic functions are used instead of the tables of natural functions. Example 1. Given a = 33.75 and c = 45.72, find A, B, and b. Solution. Construction B Given I a = 33.75 c = 45.72 A =-470 34.6' / a To find* B = 42~ 25.4' b = 30.843 A = C FIG. 34. * Results to be inserted when work is completed. 44 PLANE TRIGONOMETRY (1) (2) (3) Formulas sinA = a C a cos B= - b cos A = -, or b = c cos A. Logarithmic formulas log sin A = log a - log c. log cos B = log a - log c. log b = log c + log cos A. (1) (2) (3) Computation log a = 1.52827 a2 log c = 1.66011 log sin A = 9.86816 - 10 A = 470 34.6' log cos B = 9.86816 - 10 B = 420 25.4' log c = 1.66011 log cos A = 9.82905 - 10 logb = 1.48916 b = 30.843 Example 2. Given b = 8.724 and a. Solution. {ie b = 8.724 Given 1 A = 290 52.3' J = 600 7.7k To find c = 10.061 a= 5.011 Check = c2 - b2 = (c + b) (c - b) = 76.563 X 14.877 log (c + b) = 1.88402 log (c - b) = 1.17251 log a2 = 3.05653 log a = 1.52827 a = 33.75 and A = 290 52.3', find B, c, Construction B A/ 'C~ a A b C FIG. 35. RIGHT TRIANGLES 45 (1) (2) (3) Formulas A +B = 90~ or B = 90~ -A. a tanA=, or a= btanA. b b cosA = -, or c = c cos A Logarithmic formulas log a = log b + log tan A. log c = log b - log cos A. (1) (2) Computation B = 90~ - 29~ 52.3' = 600 7.7' log b = 0.94072 log tan A = 9.75919 - 10 log a = 0.69991 a = 5.0109 log b = 0.94072 log cos A = 9.93809 - 10 logc = 1.00263 c= 10.061 Check b2 = c2- a2 = (c + a)(c - a) = 15.072 X 5.050 log (c + a) = 1.17817 log (c - a) = 0.70329 log b2 = 1.88146 log b = 0.94073 b = 8.7242 Note. It is best to make a full skeleton solution before proceeding to the use of the tables. The skeleton solution can be seen in this example by erasing the numerical quantities. In using the tables, plan so as to save time as much as possible. For instance, if both log sine and log cosine of some angle are required, look up both of them while the tables are open at that page. 36. Definitions. - The angle of elevation is the angle between the line of sight and the horizontal plane through the eye when the object observed is above that horizontal plane. When the object observed is below this horizontal plane, the angle is called the angle of depression. Thus, in Fig. 36 (a), an object 0 is seen from the point P. The angle 0 between the line PO and the horizontal PX is the angle of elevation. In (b) an object 0 is seen from the point of obser 46 PLANE TRIGONOMETRY vation P. The angle 0 between the line PO and the horizontal AP is called the angle of depression. 0 I; 2z - -'- ( / P A. 6 (6) FIG. 36. Directions on the surface of the earth are often given by directions as located on the mariner's compass. As seen from Fig. 37, these directions are located with reference to the four cardinal -W^bY -8, -\ j,:s: s 1 8 a-' ~ ' ^h S - FIG. 37.- Mariner's Compass. points, north, south, east, and west. The directions are often spoken of as bearings. When greater exactness is required, the direction may be given as a certain number of degrees from a cardinal point. Thus, a direction given N. 10~ E., means a direction 10~ east of north; S. 40~ W. means 40~ west of south. EXERCISES Solve the following right triangles for the parts not given. Consider the first two parts as the given parts. Use logarithms. 1. a = 476.5, A = 17~ 44.8', B = 72~ 15.2', b = 1488.9, c = 1563.3. 2. a = 27.435, B = 57~ 23.7'. Check results. RIGHT TRIANGLES 47 3. a = 32.754, c = 45.237, A = 46~ 23.4', B = 43~ 36.6', b = 31.202. 4. b = 9.276, c = 19.243, A = 61~ 10.9', B = 28 49.1', a = 16.860. 5. b = 0.0246, A = 43~ 17.4'. Check results. 6. b = 86.435, B = 27~ 29.3', A = 62~ 30.7', a = 166.12, c = 187.26. 7. c = 42.426, A = 31~ 44' 25". Check results. 8. c = 1.1467, B = 41027'30". Check results. 9. c = 812.36, a = 476.32, A = 350 53.9', B = 54~ 6.1', b = 658.05. 10. c = 2291.2, b = 1426, A = 51 30' 38", B = 38 29' 22", a = 1793.4. 11. a = 122.43, B = 18~ 45.8', A = 71~ 14.2', b = 41.59, c = 129.3. 12. a = 184.29, A = 28~ 47' 45", B = 61~ 12' 15", b = 335.29, c = 382.6. 13. b = 54.318, B = 850 26.8'. Check results. 14. b = 49.409, A = 59~ 43.6', B = 30~ 16.4', a = 84.642, c = 98.008. 15. b = 51.85, c = 761.05, A = 86~ 5.6/, B = 3~ 54.4', a = 759.28. 16. a = 34.357, c = 36.72, A = 69~ 19.5', B = 20~ 40.5', b = 12.96. 17. The shadow cast on a horizontal plane by a vertical pole 10 ft. long is 15.5 ft. Find the angle of elevation of the sun. Ans. 32~ 49.7'. 18. The angle of elevation of the top of a tree at a point 145.5 ft. from the tree and 5.5 ft. above its base is 51~ 32'. Find the height of the tree. Ans. 188.63 ft. 19. A ladder 48 ft. long rests against a building and makes an angle of 75~ 36' with the ground. Find the distance it reaches up the building. Ans. 46~ ft. 20. From the top of a tower 305 ft. high, the angle of depression of a man on the horizontal plane through the foot of the tower is 33~ 14.8'. Find the distance the man is from the foot of the tower. Ans. 465.27 ft. 21. At 60 ft. from the base of a fir tree the angle of elevation of the top is 75~. Find the height of the tree. Ans. 224 ft. nearly. 22. What is the inclination from the vertical of the face of a wall having a batter of 8, that is, slants 1 ft. in a height of 8 ft.? Ans. 7~ 7' 30". 23. What is the angle of slope of a road bed having a grade of 5 per cent? One with a grade of 0.25 per cent? (A road with a rise of 5 ft. in 100 ft. has a grade of 5 per cent.) Ans. 2~ 51' 45"; 0~ 8' 36". Suggestion. Use S. and T. scheme. C B 35 26/ 35 26 A 114.78'- * < E -— B - D Coal Pile \ A FIG. 38. 24. In surveying along the Lake Front in Chicago, a pile of coal was encountered. Measurements were taken as shown in Fig. 38. Find the distance on a straight line from A to E. Ans. 338.41 ft. 48 PLANE TRIGONOMETRY 25. Locate the centers of the holes B and C, Fig. 39, by finding the distance each is to the right and above the center 0. The radius of the circle is 1.5 in. Compute correct to four decimals. Ans. B 1.2135 in., 0.8817 in.; C 0.4635 in., 1.4266 in. D C 36Eo \ / B60 A 0 1.5 In. FIG. 39..26. In the parallelogram of Fig. 40, b = 17.6 in., c = 9.3 in., and 0 = 127~ 25'. Find the value of the altitude h of the parallelogram. State its value as a formula in terms of c and 0. (See Art. 51.) Ans. 7.3863 in., h = c sin 0. I b FIG. 40. 27. A ladder 32 ft. long is resting against a wall at an angle of 72~ 30'. If the foot of the ladder is drawn away 2 ft., how far down the wall will the top of the ladder fall? Ans. 0.704 ft. 28. A man surveying a mine, measures a length AB = 220 ft. due east with a dip of 6~ 15'; then a length BC = 325 ft. due south with a dip of 10~ 45'. How much deeper is C than A? Ans. 84.57 ft. 29. Find the number of square yards of cloth in a conical tent with a circular base, and the vertical angle 72~, the center pole being 12 ft. high. Ans. 45.14. 30. A wheel 4 ft. in diameter and an axle 6 in. in diameter and 5 ft. long are fastened rigidly together. If one end of the axle rests on the ground, find the inclination of the plane of the wheel to the ground. If the wheel and the axle are rolled along, find the radii of the circles formed by the end of the axle and the wheel. Ans. 9.08 in. and 72.65 in. RIGHT TRIANGLES 49 31. In the side of a hill which slopes upward at an angle of 32~, a tunnel is bored sloping downwards at an angle of 12~ 15' with the horizontal. How far below the surface of the hill is a point 115 ft. down the tunnel? Ans. 94.63 ft. 32. Find the areas of the following isosceles triangles: (a) Altitude = 17 ft. and base angles each 49~ 19'. Ans. 248.43 sq. ft. (b) Base = 23 ft. and vertical angle 47~ 16'. Ans. 302.23 sq. ft. (c) Each leg = 18 ft. and base angles each 57~ 17.5'. Ans. 147.32 sq. ft. 33. Find the radius of a circle circumscribed about a regular polygon of 128 sides if one side is 2 in. What is the difference between the circumference of the circle and the perimeter of the polygon? Ans. 40.747 in.; 0.02 in. 34. Find the areas of the following regular polygons with sides 6 ft.: (a) pentagon, (b) hexagon, (c) octagon, (d) decagon, (e) 12-gon, (f) 20-gon. Ans. 61.937 sq. ft.; 93.530 sq. ft.; 173.82 sq. ft.; 276.99 sq. ft.; 403.06 sq. ft.; 1136.5 sq. ft. 35. Find the difference in the areas of a regular hexagon and a regular octagon, each of perimeter 72 ft. Ans. 16.99 sq. ft. 36. If R is the radius of a circle, show that the area of a regular circumscribed polygon of n sides is given by the formula 1800 A = nR2 tan n 37. Show that the area of a regular inscribed polygon of n sides is given by the formula 1800 1800 1 2. 360~ A = nR2 sin - cos = nR sin n n 2 n 38. The radius of a circle is 62 ft. Find the perimeter of a regular inscribed pentagon. Ans. 364.43 ft. 39. The radius of a circle is 75 ft. Find the area of the regular inscribed octagon. Ans. 15,910 sq. ft. 40. Find the area of a regular dodecagon inscribed in a circle of radius 24 in. Ans. 12 sq. ft. 41. A circle 10 in. in diameter is suspended from a point and held in a horizontal position by 10 strings each 6 in. long and equally spaced around the circumference. Find the angle between two consecutive strings. Ans. 29~ 50' 42". 42. A girder to carry a bridge is in the form of a circular arc. The length of the span is 120 ft. and the height of the arch is 25 ft. Find the angle at the center of the circle such that its sides intercept the arc of the girder; and find the radius of the circle. Ans. 90~ 28.6', 84.5 ft. 43. In a circle of 60 in. radius, find the area of a segment having an angle of 63~ 15'. Find the length of the chord and the height of the segment; take W of their product and compare with the area. Ans. 379.69 sq. in. 44. Find the area of the following segments by the approximate rule h3 2 A = 2- + _- hw, and by the exact method of trigonometry. Find the per 2i W 50 PLANE TRIGONOMETRY cent of error by the approximate rule in each case. Here w stands for the width and h for the height of the segment. (a) w =10 ft., h = 2 ft. (b) w = 16 ft., h = 1.5 ft. (c) w = 23.34 in., radius = 15.5 in. 45. A tree stands upon the same plane as a house whose height is 60 ft. The angle of elevation and depression of the top and base of the tree from the top of the house are 41~ and 35~ respectively. Find the height of the tree. Ans. 134.49 ft. 46. From a point 20 ft. above the surface of the water, the angle of elevation of the top of a tree standing at the edge of the water is 41~ 15', while the angle of depression of its image in the water is 58~ 45'. Find the height of the tree, and its horizontal distance from the point of observation. Ans. 65.50 ft.; 51.886 ft. 47. At a certain point the angle of elevation of a mountain peak is 44~ 30'; at a distance of 3 miles further away in the same horizontal plane, its angle of elevation is 29~ 45'. Find the distance of the top of the mountain above the horizontal plane, and the horizontal distance from the first point of observation to the peak. Ans. 4.0984 mi.; 4.1705 mi. B FIG. 41. Suggestion. Find two simultaneous equations involving the unknowns h and x representing the distances as shown in Fig. 41. These are tan 440 30' = h and tan 29~ 45' =. Solve these algebraically for h and x. 48. At a certain point A the angle of elevation of a mountain peak is a; at a point B that is a miles further away in the same horizontal plane its angle of elevation is 3. If h represents the distance the peak is above the plane and x the horizontal distance the peak is from A, derive the formulas: a tan a tan 3 h tan a - tan (' a tan 3 tan a - tan 3 Note. In using these formulas, it is convenient to use natural functions. On page 120 is given a solution of the same problem, obtaining formulas adapted to logarithms. RIGHT TRIANGLES 51 49. Find the height of a tree if the angle of elevation of its top changes from 37~ to 44~ 30' on walking toward it 80 ft. in a horizontal line through its base. Ans. 258.52 ft. 50. From a point A on the level with the base of a steeple, the angle of elevation of the top of the steeple is 42~ 30'; from a point B 22 ft. directly over A, the angle of elevation of the top is 36~ 45'. Find the height of the steeple and the distance of its base from A. Ans. 118.87 ft.; 129.72 ft. 51. A ship sailing east at 24 miles per hour observes a lighthouse at noon to be E. 26~ N. At 12:30 the lighthouse is E. 39~ N. Determine when the ship is nearest the lighthouse and what the distance is. Ans. 15 min. 26.1 sec. past 1 o'clock; 14.717 mi. 52. A ship sailing due north observes two lighthouses in a line due west; after an hour's sailing the bearings of the lighthouses are observed to be southwest and south-south-west. If the distance between the lighthouses is 8 miles, at what rate is the ship sailing? Ans. 13.66 mi. per hour. 53. The description in a deed runs as follows: "Beginning at a stone (A), at the N.W. corner of lot 401; thence east 112 ft. to a stone (B); thence S. 361~ W. 100 ft.; thence west parallel with AB to the west line of said lot 401; thence north on west line of said lot to the place of beginning." Find the area of the land described. Ans. 6612.88 sq. ft. 37. Accuracy. - It is of very great importance that one should bear in mind as far as possible the limitations as regards accuracy. The degree of accuracy that can be depended upon in a computation is limited by the accuracy of the tables of trigonometric functions and logarithms used, and by the data involved in the computation. The greater the number of decimal places in the table, the more accurately, in general, can the angles be determined from the natural or logarithmic functions; but, in a given table, the accuracy is greater the more rapidly the function is changing. Since the cosine of the angle changes slowly when the angle is near 0~, small angles should not be determined from the cosine. For a like reason, the sines should not be used when the angle is near 90~. The tangent and cotangent change more rapidly throughout the quadrant and so can be used for any angle. Most of the data used in problems are obtained from measurements made with instruments devised to determine those data more or less accurately. The inability to be precise in the data depends not only upon the instruments used, but upon the person making the measurements and upon the thing measured. A man in practical work uses instruments which are of such accuracy as to secure results suitable for his purpose. The data 52 PLANE TRIGONOMETRY given in problems for practice are supposed to be of such accuracy as the instruments that are used in such measurements would warrant. In the solution of a problem it is useless to carry out the computations with a greater degree of accuracy than that of the data. That is, if the data are accurate only to, say, four significant figures, there is no necessity to compute accurately to more figures than this. If the measuring instrument can be read only to minutes of angle, in the computation, there is no object in carrying the work to seconds of angle. 38. Tests of accuracy. - The practical man endeavors in one way or another to check both his measurements and his computations. In our work here we are interested in checks on the computation. (a) Often a graphical construction to scale will give results that will check the numerical work. If the construction is made free hand, only the gross mistakes in computation will be discovered; but if the construction is made carefully with accurate instruments, results may be obtained as accurate as the data will warrant. This then may be considered a graphical solution of the problem. (b) Mistakes in the computations may be found by making another computation using a different set of data; or by recomputing, using the same data but using a different set of formulas. Many ways will present themselves to the thoughtful student. EXERCISES 1. In determining an angle by means of a table of natural functions that is correct to five places, if the angle is near 1~ can seconds be determined from the cosine of the angle? Can tenths of minutes? Can minutes? 2. Answer the same questions as in Exercise 1 if the sine is used instead of the cosine. If the tangent is used. If the cotangent. 3. Answer similar questions if the angle is near 89~, 80~, 10~, 20~, 70~, 45~. 4. From the results obtained in the first three exercises, state conclusions as to what sized angles can be determined most accurately from sine, cosine, tangent, and cotangent of the angle. 5. Compare the logarithms of 92.8766 and 92.876; 99.8375, 99.837, and 99.838; 121.575, 121.57, and 121.6. 6. Can a number be determined correct to six figures by using a five-place logarithm table? When? When is it not possible to determine five figures of a number by means of a five-place table of logarithms? RIGHT TRIANGLES 53 39. Orthogonal projection. - If from a point P, Fig. 42 (a), a perpendicular PQ be drawn to any straight line RS, then the foot of the perpendicular Q is said to be the orthogonal projection, or simply the projection, of P upon RS. P ", B p n B S B.C ID x R Q (a) 0 C () D A ( E FIG. 42. The projection of a line segment upon a given straight line is the portion of the given line lying between the projections of the ends of the segment. In Fig. 42 (b) and (c), CD is the projection of AB on OX. In each case AE = CD and AE = AB cos 0. The projections usually made are upon a horizontal line OX and a vertical line OY, as in Fig. 43. Hence, if 1 is the length of the segment of line projected, x the projection on OX, y the projection on OY, Y and 0 the angle of inclination, that is, __ B the angle that the line segment makes S with the x-axis, then X R iA [9] x = cose, R and 0 M x N [10] y = i sin O. This may be stated in the following: FIG. 43. THEOREM. The projection of any line segment upon a horizontal line equals the length of the segment multiplied by the cosine of the angle of inclination; the projection upon a vertical line equals the length of the segment multiplied by the sine of the angle of inclination. 40. Vectors. - In physics and engineering, line segments are often used to represent quantities that have direction as well as magnitude. Velocities, accelerations, and forces are such quantities. For instance, a force of 100 pounds acting in a northeasterly direction may be represented by a line, say 10 inches long, drawn 54 PLANE TRIGONOMETRY in a northeasterly direction. The line is drawn so as to represent the force to some scale; here it is 10 pounds to the inch. An arrow head is put on one end of the line to show its direction. Y In Fig. 44, OP = v is a line representing a directed quantity. Such a line is R ---- P called a vector. 0 is the beginning of vy V, the vector and P is the terminal. OQ = x ~o~ 0 -- -X is the projection of the vector on the x4Q horizontal OX, OR = y is the projection FIG. 44. on the vertical OY, and 0 is the inclination of the vector. The vectors x and y are called components of the vector v. As before, x = v COS o, and y = v sin 0. Suppose the vectors OQ and OP, Fig. 45, represent the magnitude and direction of two forces acting at the point 0, and having any angle ~ between their lines of action. If the parallelogram OQRP is completed, then the diag- / onal OR represents in magnitude / and direction a force that will 0 Q produce the same effect as the FIG. 45. two given forces. The vector OR is called the resultant of the vectors OQ and OP. The process of finding the resultant of two or more given forces is called composition of forces. Conversely, the vectors OQ and OP are components of OR. Since QR is equal and parallel Y to OP, it follows that the two R_ — _-_-_ p components and their resultant _A_ i~ p form a closed triangle OQR. W20~ 2 X The relations between forces //>Q and their resultant may then be found by solving a triangle FIG. 46. which is, in general, an oblique triangle. Example 1. Suppose that a weight W is resting on a rough horizontal table as shown in Fig. 46. Suppose that a force of RIGHT TRIANGLES 55 40 lb. is acting on the weight in the direction OP, making an angle of 20~ with the horizontal; then the horizontal pull on the weight is OQ = 40 cos 20~ = 37.588 lb., and the vertical lift on the weight is OR = 40 sin 20~ = 13.68 lb. Example 2. A car is moving up an incline, making an angle of 35~ with the horizontal, at the rate of 26 ft. per second. What is its horizontal velocity? Horizontal velocity = 26 cos 35~ = 21.3 ft. per second. Vertical velocity = 26 sin 35~ = 14.9 ft. per second. EXERCISES 1. Find the projection of a segment 27 ft. long upon a straight line making an angle of 47~ 16.4' with the segment. Ans. 18.319 ft. 2. The line segment AB 21.75 in. long makes an angle of 33~ 37.7' with the line OX. Find the projection on OX. Find its projection on the line OY perpendicular to OX and in the same plane as OX and AB. Ans. 18.11 in.; 12.045 in. 3. A steamer is moving in a southeasterly direction at the rate of 24 miles per hour. How fast is it moving in an easterly direction? In a southerly direction? * Ans. 16.971 mi. per hr. in each direction. 4. The eastward and northward components of the velocity of a ship are respectively 5.5 miles and 10.6 miles. Find the direction and the rate at which the ship is sailing, that is, the resultant. Ans. 11.94 mi. per hr., 27~ 25.4' east of north. 5. A roof is inclined at an angle of 33~ 30'. The wind strikes this horizontally with a force of 1800 pounds. Find the pressure perpendicular to the roof. Ans. 993.48 lb. 6. A roof 20 ft. by 25 ft. and inclined at an angle of 27~ 25' with the horizontal will shelter how large an area? Ans. 443.84 sq. ft. 7. A hillside is on a slope of 16~ and contains 5.2 acres. How much more is this than the projection of the hillside on a horizontal plane? Ans. 0.2015 acre. 8. Show in general that the projection of a plane area upon a fixed plane is equal to the given area times the cosine of the angle between the planes. 9. Two men are lifting a stone by means of ropes that are in the same vertical plane. One man pulls 85 lb. in a direction 23~ from the vertical and the other 125 lb. in a direction 42~ from the vertical. Determine the weight of the stone. Ans. 171.136 lb. 10. Two forces of 240 lb. and 180 lb. act upon a heavy body, the first at an angle of 40~ with the horizontal and the second at an angle of 65~. Find the total force tending to move the body horizontally, to lift it vertically. Ans. 259.92 lb.; 317.41 lb. 56 PLANE TRIGONOMETRY 41. Distance and dip of the horizon. - In Fig. 47, let 0 be the center of the earth, r the radius of the earth, and h the height of a point P above its surface; to find P ____ C the distance from the point P to the h 0- horizon at A. / B~ X A By geometry, PA2 = P02 - OA2 = (r + h)2 - 2 = 2 rh + h2. J/ < r \. P. PA =V/2rh+ h. C6 / |For points above the surface that are reached by man, h2 is very small compared with 2 rh,.. PA = V/2hr, approximately. FIG. 47. In the above PA, r, and h are in the same units. However, a very simple formula can be derived if h be taken in feet, r and PA in miles, and r = 3960 miles. Then PA = 2 X 3960 5280 = miles. The following approximate rules may then be stated: The distance of the horizon in miles is approximately equal to the square root of 3i times the height of the point of observation in feet. The height of the point of observation in feet is 2 times the square of the distance of the horizon in miles. Definition. The angle APC = 0 A in Fig. 47 is called the dip of the horizon. PA I Evidently tan 0 =- ( r BL X 42. Areas of sector and segment. - Formulas for solving for the areas of the sector and segment of a circle are derived here so that F 48 they may be used for reference. From geometry, the area of the sector of a circle as XOA, Fig. 48, equals the arc XnA times one half the radius OX. RIGHT TRIANGLES 57 By Art. 5, arc XnA = OX X 0, where 0 is expressed in radians. Hence, using r for the radius and S for the area of sector, S = 1r20. Evidently the area of the segment XAn = S - area of triangle XOA. But area of triangle XOA = ~ OX. BA = ~ OX. OA sin 0 =- r2 sin 0. Hence, using G for area of segment, G = r2 0 - r2 sin. 1 [11] (0 -sin 0). As an exercise the student may later show that this formula holds when 0 is an obtuse angle. Also when 7r < 0 < 2,r. (See Art. 51.) EXERCISES (More difficult exercises for advanced work and review.) 1. A building 80 ft. long by 60 ft. wide has a roof inclined at 36~ 45' to the horizontal. Find the area of the roof, and show that the result is the same whether the roof does or does not have a ridge. Ans. 5990.6 sq. ft. S E___ F A _hQ E F FIG. 49. 2. Find the width of the shadow of the wall shown in Fig. 49. If the height of the wall is h ft., the angle of elevation of the sun a, and the angle between the vertical plane through the sun and the plane of the wall 0, show that width of shadow = h cot a sin 0. 3. A wall extending east and west is 8 ft. high. The sun has an inclination of 49~ 30' and is 47~ 15' 30" west of south. Find the width of the shadow of the wall. Ans. 4.637 ft. 4. A cliff 2000 ft. high is on the seashore; how far away is the horizon? What is the dip of the horizon? Ans. 54.77 mi.; 0~ 47' 33". 5. Find the greatest distance at which the lamp of a lighthouse can be 58 PLANE TRIGONOMETRY seen from the deck of a ship. The lamp is 85 ft. above the surface of the water and the deck of the ship 30 ft. Ans. 18 mi. approx. 6. Find the radius of one's horizon if located 1250 ft. above the earth. How large when located 3 miles above the earth? Ans. 43.3 mi.; 154.17 mi. 7. How high above the earth must one be to see a point on the surface 50 miles away? Ans. 1666.7 ft. 8. If R and r are the radii of two pulleys, D the distance between the centers, and L the length of the belt, show that when the belt is not crossed, Fig. 50, the length is given by the following formula where the angle is taken in radians: L = 2 /D2 (R -r)+ -(R+r)2(R-rsin-l R r D P KILN N T FIG. 50. FIG. 51. 9. Using the same notation as in Exercise 8, show that when the belt is crossed, Fig. 51, the length is given by the following formula: L = 2 /D2 -(R +r)2 +(R + r) (- + 2 sin-l R+ ) Note. These formulas would seldom be used in practice. An approximate formula would be more convenient, or the length would be measured with a tape line. A rule often given for finding the length of an uncrossed belt is: Add twice the distance between the centers of the shafts to half the sum of the circumferences of the two pulleys. 10. Using the formula of Exercise 8, and given R = 18 in., r = 8 in., and D = 12 ft., find the length of the belt. Find the length by the approximate rule. Ans. 30.87 ft.; 30.81 ft. 11. Use the same values as in Exercise 10, and find by the formula of Exercise 9 the length of the belt when crossed. Ans. 31.20 ft. 12. An open belt connects two pulleys of diameters 6 ft. and 2 ft. respectively. If the distance between their centers is 15 ft., find the length of the belt. Ans. 42.83 ft. 13. Two pulleys of diameters 7 ft. and 2 ft. respectively are connected by a crossed belt. If the centers of the pulleys are 16 ft. apart, find the length of the belt. Ans. 47.41 ft. 14. A ray of light after reflection at a plane mirror makes with the perpendicular to the mirror at the point of incidence an angle equal to the angle RIGHT TRIANGLES 59 it makes with this perpendicular at incidence. Prove that if the mirror is turned through an angle a, the reflected ray is turned through an angle 2 a. 15. Compute the volume for each foot in the depth of a horizontal cylindrical oil tank of length 30 ft. and diameter 4 ft. 16. A cylindrical tank in a horizontal position is filled with water to within 10 in. of the top. Find the volume of the water if the tank is 10 ft. long and 4 ft. in diameter. Ans. 106.7 cu. ft. 17. Find the angle between the diagonal of a cube and one of the diagonals of a face which meets it. Ans. 35~ 15.8'. A FIG. 52. 18. The slope of the roof in Fig. 52 is 33~ 40'. Find the angle 0 which is the inclination to the horizontal of the line AB, drawn in the roof and making an angle of 35~ 20' with the line of greatest slope. Ans. 26~ 53' 14". 19. A hill slopes at an angle of 32~ with the horizontal. A path leads up it making P an angle of 47~ 30' with the line of steepest slope, find the inclination of the path with the horizontal. Ans. 20 58' 40". 20. Two roofs have their ridges at right angles, and each is inclined to the horizontal at an angle of 30~. Find the inclination of B their line of intersection to the horizontal. Ans. 220 12' 28". 21. Two set squares whose sides are 3, 4, \ and 5 in. are placed as in Fig. 53, so that ----- their 4-in. sides and right angles coincide, and the angle between the 3-in. sides is 50~ 46' 20". Find the angle 0 between the\ longest sides. Ans. 29~ 48' 40". A 22. Show that placing the carpenter's F square as shown in Fig. 54 (b) will determine the miter for making a regular pentagonal frame as shown in (a). What is the angle 0 of the miter? Ans. 0 = 54~. 60 PLANE TRIGONOMETRY 23. If 12 in. is taken on the tongue of the square, how many inches must be taken on the blade to cut miters for making regular polygons of the followC D d b A (b) B (a)A FIG. 54. ing numbers of sides: 3, 4, 6, 8, and 10? Express results to the nearest 16th of an inch. Ans. 201 3; 12; 61; 5; 37. 24. In the frame of a tower shown in Fig. 55, determine the distances from A and B, C and D, etc., to make the holes in the braces so that they may be bolted at points a, b, c, etc. These distances should be accurate to tenths of an inch. Can these distances be determined by means of geometry? Ans. Aa = 10 ft. 5.3 in., etc.; Yes.,- N - - EC> D B A L- --- 8 -- - FIG. 55. FIG. 56. 25. What diameter of stock must be chosen so that a hexagonal end 2| in. across the flats may be milled upon it? Answer the same question for an octagon. The meaning of "across the flats" is shown in Fig. 56. Ans. 2.74 in.; 2.57 in. 26. A tripod is made of three sticks each 4 ft. long, by tying together the ends of the sticks, the other ends resting on the ground 2~ ft. apart. Find the height of the tripod. Ans. 3 ft. 83 in. CHAPTER IV GRAPHICAL REPRESENTATION OF TRIGONOMETRIC FUNCTIONS 43. Line representation of the trigonometric functions.Construct a circle of radius OH, with its center at the origin of co6rdinates, Fig. 57. Since, in finding the trigonometric functions of an angle with its vertex at the origin of coordinates and its y (c) FIG. 57. initial side on the positive part of the axis of abscissas, any point may be chosen in the terminal side of the angle, we may take the point where the terminal side cuts the circumference of the circle. Draw angle 0 = angle XOP in each of the four quadrants, and draw MP.L OX in each case. Now choose OH as the unit of measure, that is, OH = 1. Then in each of the four quadrants, MP MP MP sinS - =o = OH __ = MP OP OH I OM OM OM cos0 = - = - OM. OP OH 16 61 62 PLANE TRIGONOMETRY Stated in words these are as follows: The sine of an angle 0 is represented by the ordinate of the point where the terminal side cuts the circumference of the unit circle. The cosine of an angle 0 is represented by the abscissa of the point where the terminal side cuts the unit circle. Draw tangents to the circle at H and E, Fig. 57, to meet the terminal side OP extended or produced back through the origin as the position of the angle requires. In each of the four figures triangles OMP, OHD, and OEF are similar. Assume that HD is positive when measured upward, and negative when measured downward; also that EF is positive when measured to the right, and negative when measured to the left. MP HD OM EF From the similar triangles, and OM OH 3MP - OE Then in each of the four quadrants, P HD HD = tan= OM OH = D OM EF EF cot 0 = OM = E EF. MP OE I Or, in words, these are: The tangent of an angle 0 is represented by the ordinate of the point where the terminal side of 0 is cut by a tangent line drawn to the unit circle where the circle cuts the positive part of the axis of abscissas. The cotangent of an angle 0 is represented by the abscissa of the point where the terminal side of 0 is cut by a tangent line drawn to the unit circle where the circle cuts the positive part of the axis of ordinates. Let it be assumed that OD and OF are positive when measured on the terminal side OP of the angle, and that they are negative when measured on OP produced back through the origin. Then in each of the four quadrants, OP OD OD sec 0 =o= OD= OD' 1OM OH I^0 OP OF OF csc 0 =.-E 1 = OF. MP OE 1 Or, in words, these are: The secant of an angle 0 is represented by the segment of the termi TRIGONOMETRIC FUNCTIONS 63 nal side of 0 from the origin to the point where the line representing the tangent of 0 cuts the terminal side. The cosecant of an angle 0 is represented by the segment of the terminal side of 0 from the origin to the point where the line representing the cotangent of 0 cuts the terminal side. It is not to be understood that the functions are lines; but that when the radius is taken as the unit of measure, and the lines are expressed in terms of this unit, the numbers which then represent the lines are the functions. Thus, if MP, Fig. 57, is 4 in. and the radius is 7 in., MP in terms of OH is 4, which is then the sine of 0. Exercise. Prove the six fundamental relations of Art. 21 by means of the line values of the trigonometric functions. Note. The trigonometric functions are sometimes called circular functions. The origin of the term is apparent from the line representation of the functions. The origin of the terms tangent and secant are also apparent from the figures. 44. Functions of inT + 0 in terms of functions of 0. - Construct Z HOP1 = 0, an acute angle, Fig. 58. center at 0 and radius OH = 1. Draw P1IM1 OX, and OP2 L OP1. Then ZHOP2 = 7r + 0. Draw P2M21 OX. Triangles OM2P2 and OMiP1 are equal having ZM2P20 = Z M1OP1 and OP1 = OP2. Then M2P2 = OM1 and 0112 = - MP1, being equal in numerical values, but OM2 read negative and M1P1 positive. Then Draw a circle with its By Ir G 0. 5.M FIG. 58. sin (1 r + 6) = M2P2 = OM1 = cos, cos (2- T + 0) = OM2 = -M1P1 = -sin 0, tan (-r + 0) -M2P_ m = -cot, (2 )l OM2 - MlP1, 0OM2 _ -MYIP1 cot^ (7 + 6) M2P2 =-Ml1 - -tan 0. 45. Functions of rr- 0 in terms of functions of 0. - Construct a circle with radius OH = 1, Fig. 59, and construct Z HOP1 = ZP20B = 0, an acute angle. Draw P1M1 and P2M2 0OX. 64 PLANE TRIGONOMETRY Then ZHOP2 = T - 0. And since triangles OM2P2 and OMNPi are equal, M2P2 = MiP1 and OM2 = -OM1. Then sin (r - ) = M2P2 = M1P1 = sin 0, cos (r - ) = OM2 = -OM1 = -cos 0, M2P2 M1P1 tan ( - ) = OM - -M -tan 0, OMp2 - OM,1 cot (7r - 6) = 2 Pcot P2 P1 -cot FIG. 59. FIG. 60. 46. Functions of Tr + 0 in terms of functions of 0.- Construct a circle with radius OH = 1, Fig. 60, and draw OP3 forming L HOP3 = T + 6. Draw P3M3 1OX. Triangles OM1P1 and OM3P3 are equal, M3P3 = -MiP3, and OM3 = - OM1. Then sin (wr + 0) cos (7r + 0) tan (7r + 0) cot (7r + 0) Y M, = M3P3 = - M1P1 = -sin 0, = OM3 = -OM1 = — cos, M3P3 -M 1P1 OMN3 - -OM1 = tan0, OMN3 -OM1!M3P3 - MlPl= cot 0. 47. Functions of -'r - 0 in terms of functions of 0. -Construct a circle with radius OH = 1, Fig. 61, and draw OP3 so that ZP3OY' = 0. Then /HOP3 = 2 IT -. Draw P3M3 10X. Triangles OM3P3 and OM1P1 are equal, M3P3 = -OM1, and OM3 = - M1P1. Then P1 L H r P 1 -Y FIG. 61. TRIGONOMETRIC FUNCTIONS 65 sin (3r - 0) = M3P3 = cos (37 - 6) = OM3 = 2 M3P3 = tan (g7r- ) = P3-= 2 t3,_ OM3 cotM3 cot (3 7r - O) - MP =.T 3P 3 -OM1 = - cos 0, -MiPi = - sin 0, -OMi -OM d1 = cot 0, - M1lP1 -M,1P1 = tan 6. — OM1 48. Functions of -ir + 0 in terms of functions of 8. - Construct a circle with radius OH = 1, Fig. 62, and draw OP4 so that Y'OP4 =.. Then ZHOP4 = 3 T + 0. Draw P4M41OX. Triangles OM4P4 and OMiP1 are equal, M4P4 = -OM1, and OM4= MIPi. Then sin (Q r + 0) = M4P4 = -OM1 = -cos 0, cos (3 T + 0) = OM4 = MPI = sin 6, tan (_- +0) M4P4 -OM cot 0, 2( + 6) -O0l4 M1P1 t( +a) OM4 M1P1 O -j 1 M4P4 -OM - tan. tant.+ =^ =-^ -— ot FIG. 62. FIG. 63. 49. Functions of 2 r - 0 and - 0 in terms of functions of 0. - Using Fig. 63, the student is asked to prove the following relations: sin (27r - 0) = -sin 0. cos (2 r - 0) = cos 0. tan(27r - 0) = -tan0. cot (27r - 0) = -cot 0. sin (-0) = -sin 0. cos (- ) = cos O. tan (- 0) = -tan 0. cot (- ) = -cot 0. 50. Functions of an angle greater than 2 Tr. - Any angle a greater than 2 w has the same trigonometric functions as a minus an integral multiple of 2 r, because a and a - 2 nr have the 66 PLANE TRIGONOMETRY same initial and terminal sides. That is, the functions of a equal the same functions of a - 2 nfr, where n is an integer. 51. Summary of the reduction formulas.- The formulas of the previous articles and Art. convenient for reference. 19 are here collected so as to be sin (2 7 - 0) cos ( T 6 - 0) tan (1 wT - 0) cot ( 7r - 0) sin ( w - 0) cos ( 7r - ) tan ( r - 6) cot ( r - 0) = cos 0. = sin 0. = cot 0. = tan 6. = sin 0. = -cos 6. = -tan 0. = -cot 0. sin (2w r + 0) = cos 0. cos (~ r + 0) = -sin 0. tan (1 r + 0) = -cot O. cot (I 7 + 0) = -tan 0. sin ( r + 0) =-sin 0. cos ( r'+ 0) = -cos. tan( + + 0) =tan 0. co ) cot( + =cot. sin (3- r + 0) = -cos. cos (2-7r + 0) =sin 0. tan ( 7r + 0) = -cot O. cot ( 7r + 0) = -tan 0. sin (-0) = -sin 0. cos (-0) = cos 6. tan (-) = -tan O. cot (-0) = -cot 0. sin (3 2 - 0) = -cos 0. cos (3 T - 0) = -sin 0. tan (31 r- 0) = cot O. cot (3- - 0) = tan 0. sin (2 r - 0) = -sin 0. cos (2 7 - 6) = cos 0. tan (2 r — 0) = -tan 0. cot (2 7r - ) = -cot 0. While the proof of the reduction formulas have all been based upon the assumption that 0 is an acute angle, they are true for all values of 0. Tables of trigonometric functions, in general, do not contain angles greater than 90~. Since the principal application of the reduction formulas is made in determining the numerical values of functions of angles greater than 90~, it will be found convenient to have a rule for the application of the formulas for 0 an acute angle. The rule is as follows: RULE. Express the given angle as a multiple of 90~ plus or minus an acute angle O. If the multiple of 90~ is even, take the same function of 0 as of the original angle; if the multiple of 90~ is odd, take the co-function of 0. In either case prefix the algebraic sign of the original function to the function of 6. If the given angle is negative, first express its function as the function of the given angle with its sign changed, and then proceed by the rule. TRIGONOMETRIC FUNCTIONS 67 EXERCISES In the following exercises, express the given function as a function of an acute angle. 1. sin 290~ = sin (270~ + 20~) = -cos 20~. Since 270~ = 3 X 90~ is an odd multiple of 90~, take the cosine of the acute angle 20~. Since sin 290~ is negative, the minus sign precedes cos 20~. Again, sin 290~ = sin (360~ - 70~) = -sin 70~. Here the multiple of 90~ is even, therefore the same function, that is, the sine of the acute angle 70~, is taken. The sign is determined as before. These results do not differ, for -cos 20 = -sin 70~, the angles being complementary and the functions co-functions. It is to be noted that any angle that is not a multiple of ~ ir lies between two consecutive multiples of 21r, and either multiple may be used in the reduction. 2. tan 165~ = tan (90~ + 75~) = -cot 75~. Since 90~ is an odd multiple of 90~ we take cot 75~, and since tan 165~ is negative a minus sign is prefixed to cot 75~. 3. cos 210~ = cos (180~ + 30~) = -cos 30~ = - V/3. 4. cos 825~ = cos (9 x 90~ + 15~) = -sin 15~. Another solution of this is as follows: cos 825~ = cos (2 X 360~ + 105~) = cos 105~ = cos (90~ + 15~) = -sin 15~. 5. cot (-1115~) = -cot 1115 = -cot (12 X 90~ + 35~) = -cot 35~. 6. By the use of the table of natural functions, find correct to four decimal places the sine, cosine, tangent, and cotangent of the following angles: -115~, 153~, 212~, 265~, 295~. 7. Find sine, cosine, tangent, and cotangent of 120~, 135~, 150~, 210~, 225~, 300~, and 330~ by expressing them in terms of functions of 30~, 45~, or 60~. Compare the results with the table of values given in Art. 16.. sin ( 7 - 0) cos (I r +- 0) sin ( 7r - 0) 8. Simplify s (2- + 0 ) sn (2 -) Ans. -1. tan (17r + 0) sec (7r + 0) Verify Exercises 9 to 15. tan 1800 + tan 0 1 - tan 180 tan 0 tan(8 10. cos 3 ir cos 0 + sin 7rsin = cos ( - 0). 11. sin 7r cos 0 - cos sin 0 = sin (or - 0). 12. cos ( Xr + aC) cos (r - a) + sin (I T + a) sin (- + a) = 0. sin (-0) - cos (-6) _ sin (90~ + 0) + cos (270~ - 0) tan (-0) - cot (-0) cot (180~ + 0) + tan (360~ - 0) 5 (sin 2 ar - tan 2 7r + cos 3 7r) 14. - tan2 -os3) 5. 2 csc 3 7r * sec 5 r 2 sec 3r - 2sin 7r + 2cos 7 r 5 15. __2 3 csc 5 Tr +- 7 cos 2 - sec 7 r 2 16. If sin a = -, prove that tan (a - 12 -) = -, when a is in the third quadrant. 1 1 17. If cos 340~ =, prove that csc 110 = b, and tan 110 = - ---- 1 tan 205~ sin 335~ /i - c2 18. If csc 1150 =, prove that = -- c' cos 2450 c 68 PLANE TRIGONOMETRY 52. Proof of the reduction formulas of Art. 51 for any value of 0.- Although the reduction formulas of Art. 51 were proved ~Y ~ for acute values only of 0, they are true in general. The proof can readily be carried out for any desired value of 0. Proof for Tr + 0 when 0 is in the third N H x t 1o \H X quadrant. Construct Z XOP = 0 in a unit circle, Pf \ / Fig. 64. Draw ORLOP, forming Z XOR R = 2r + 0. From the construction, it is FIG. 64. easily seen that A OPM = A ORN, PM = ON, and OM = NR. Because the radius of the circle is unity sin ( + 0) = NR, sin0=MP, cos (2 - + 0) = ON, cos 0 = OM. Then sin (~ ~r + 0) = NR = OM os 0, cos ( I + 0) = ON = PM = -MP = -sin 0, tan (i- 7r+ 0) = in(1 ( 0) =_ 0 =-cot 0. cos (~ q- + 0) - sin 0 These results are seen to agree with the results previously determined for the functions of ~ wr + 0 where 0 is acute. In a similar manner, the formulas for functions of wr t- 0, X ~-t 0, etc., for any value of 0 may be proved. 53. Given the function to find the angle. - In Art. 12 it was proved that for any angle there is but one value for each of the trigonometric functions. The converse of this proposition is not true, however. It will now be shown that for every function of an angle 0, there are any number of angles, both positive and negative, which satisfy the relation expressed. If 0 is restricted to positive values not greater than 360~, it will be shown that there are, in general, two values of 0. Since angles with the same terminal side have the same values as functions, it will then follow that an integral multiple of 360~, or 2 r radians, added to or subtracted from these two values will give other angles having the same values as functions. In the following articles only positive values of 0 less than 360~ will be considered. The general expressions for such angles may be obtained by adding multiples of 2 r to each angle. TRIGONOMETRIC FUNCTIONS 69 54. Values for all angles that have a given sine or cosecant. - Given sin 0 = c, where c is a positive or negative number not greater than 1 nor less than -1. (1) Suppose that c is positive. Draw a unit circle with its center at the origin, Fig. 65. Draw RN X1 X'X at a distance c above it. RN intersects the circle at P1 and P2. Draw OPi and OP2 forming the angles 01 and 02, and draw M1P1 and M2P2-lX'X. Now sin 01 = M1P1 and sin 02 = M2P2, but MlP1 = M2P2. sin 01 = sin 02. From the geometry of the figure 02 = 7 - 01, that is, two angles less than 360~ having the same positive sine are supplementary. y FFFP P4 FIG. 65. FIG. 66. Also ZP1OY = Z YOP2 = ), that is, 01 = T- and 02 r + I. It is to be noted that, if c =, 0 = 02 1, 1 and f =0. (2) Suppose that c is negative. A similar discussion for the case when c is a negative number, as in Fig. 66, shows that sin 03 = sin 04, and that 03 = 3 - ) and 04 = t +. In general, then, the terminal sides of angles that have the same sine differ by the same angular magnitude from the axis of ordinates. That is, from 1 T, 3- 7, 5 T, etc. Thus, sin 30~ = sin 150~, because 90 - 30~ = 150~ - 90~, that is, both 30~ and 150~ differ from 90~ by the angle 60~. sin 20~ = sin 160~ because 90~ - 20~ = 160~ - 90~. sin 220~ = sin 320~ because 270~ - 220~ = 320~ - 270~. Example. Given sin 0 = -0.25882, find all values of 0 < 360~. 70 PLANE TRIGONOMETRY Solution. From the tables we find that when sin. 0 = +0.25882, 0 = 15~. But when sin 0 = -0.25882, 0 is in the third and fourth quadrants. Now all angles having equal sines differ by the same angle 0 from 90~, 270~, etc., 0 = 90~ - 15~ = 75~..'. = 270~ t 75~ = 345~ or 195~. A similar discussion shows that the rule for determining all angles that have a given cosecant is the same as that for all angles which have a given sine. 55. Values for all angles having the same cosine or secant. - Given cos 0 = c, where c is a positive or negative number not greater than 1 nor less than -1. (1) Suppose that c is positive. Draw a unit circle as in Fig. 67. Draw RN 1\ Y'Y, at the distance c from it and cutting the circle at P1 and P4. Draw OPi and OP4. Then cos 09 = OM1 = cos 04. (2) Suppose that c is negative. Y R Y P,~P Pi 2 That is, from 0, ~, 2 ~, etc. Y' FIG. 67. FIG. 68. A similar discussion applied to Fig. 68 shows that cos 02 = OM2 = cos 03. But Z P4OH = Z HOP1 and Z P2OM2 = Z M20P3, therefore, in general, the terminal sides of angles that have the same cosine differ by the same angular magnitude from the axis of abscissas. That is, from 0, wr, 2 r, etc. Thus, cos 50~ = cos (3600 - 50~) = cos 310~. cos 160~ = cos (180~ + 20~) = cos 200~. Example. Given CQS 0 = 0.57358, find all values of 0 < 360~. Solution. From the tables we find that 0 = 55~ when cos 0 = 0.57358. Since the cosine is positive, 0 is in the first and fourth quadrants. Therefore, by the rule, 0 = 0~ + 55~ = 55~ or 360~ - 55~ = 305~. TRIGONOMETRIC FUNCTIONS 71 A similar discussion shows that the rule of this article applies when the angle is determined from the secant. 56. Values for all angles that have the same tangent or cotangent. - Given tan 0 = c, where c is any number positive or negative. Suppose c is positive. Draw the unit circle, Fig. 69. On the tan- R gent RN to the circle at H lay off HP1 = c. Draw OP1 and extend it Pi backwards through the origin to P3. Then from the line values of the functions tan 01 = HP1 = tan (7r + 01). \H \ H If c is negative, similarly tan 02 = HP2 = tan (r + 02). " — P2 Therefore, in general, all angles that N have the same tangent differ by integral FIG. 69. multiples of 7. Thus, tan 30~ = tan (180~ + 30~) = tan 210~, tan 160~ = tan (180~ + 160~) = tan 340~. Example. Given tan 0 = -0.70021, find all values of 0 < 360~. Solution. From the tables we find when tan 4 = 0.70021 that ) = 35~. But when tan 0 = -0.70021 the angle is in the second and fourth quadrants. Now since tan 4 and tan 0 are equal in absolute value but opposite in sign, angle 0 in the second quadrant is the supplement of 4... 0 = 180~ - 35~ = 145~. Also by the rule, 0 = 145~ + 180~ = 325~. A similar discussion shows that the rule of this article applies when the angle is determined from the cotangent. 57. Method by corresponding angles. - The rules of Arts. 54, 55, and 56 for determining angles from their trigonometric functions may be stated in one general rule as follows: RULE. First find the acute angle 4 which corresponds to the absolute value of the given function. The remaining, or corresponding, angles which have the same trigonometric function in absolute value are -r 4 0 and 2 7r - 4. From these the angles can be chosen which satisfy the given function. Example 1. Given sin 0 =-; find 0 < 360~. Solution. First find 4 = sin-1 = 30~. By the rule, the remaining angles which have their sine equal to ~ in absolute value are 180~ - 30 = 150~, 180~ + 30~ = 210~, and 360~ - 30~ = 330~. 72 PLANE TRIGONOMETRY Since the sine is negative, 0 must be in the third and fourth quadrants. 0 = 210~ and 330~. Example 2. Given cos 0 = - V/2; find 0 < 360~. Solution. Find 6 = cos-1 -/2 = 45~. The corresponding angles are 135~, 225~, and 315~. But the cosine is negative in the second and third quadrants,. = 135~ and 225~. EXERCISES In the following exercises find all positive values of the angles less than 360~ which satisfy the given equations. 1. sin 0 = ~. Solution. The acute angle whose sine is I is 30~. Then by Art. 54, 0 = 30~ or 150~. 2. cos 0 = - - V2. Solution. The acute angle whose cosine is 2 is 45~. By Art. 55, since the cosine is negative and therefore the angle in the second and third quadrants, 0 = 180~ - 45~ and 180~ + 45~, or 135~ and 225~. 3. tan 0 = - /3. Solution. The acute angle whose tangent is + V/3 is 60~. Then 0, which must be in the second and fourth quadrants, is 180~ - 60~ = 120~ and 180~ + 120~ = 300~. Therefore 0 = 120~ and 300~. 4. (a) cos 0 = -1. (f) 0 = tan-'1 3. (b) sin 0 = -1. (g) sin 0 = 0.3469. (c) 0 = sin-. (h) cos 0 = -0.7863. 2 (d) 0 = cos-l 0. (i) tan 0 = -0.7833. (e) cot 0 = -V3. (j) cot 0 = 0.5469. 5. Given sin 0 = -cos 220~; solve for 0 < 360~. Solution. In solving problems of this type, express the function of the given angle in terms of the same function that appears in the other member. cos 220~ = cos (270~ - 50~) = -sin 50~. Then sin 0 = -(-sin 50~) = sin 50~. Therefore 0 = 50~ and 130~. 6. Given cos 0 = sin 250~. Solution. sin 250~ = sin (270~ - 20~) = -cos 20~. Then cos 0 = -cos 20~. Since cos 20~ is positive, cos 0 is negative. The values of 0 are then 180~ - 20~ and 180~ + 20~, or 0 = 160~ and 200~. 7. (a) sin 0 = -sin 325~. (g) cot 0 = tan 345~. (b) sin2 0 = cos2 200~. (h) cot2 0 = tan2 140~. (c) cos 0 = sin 170~. (i) sec 0 = see 245~. (d) cos 0 = -cos 115~. (j) sec 0 = -csc 195~. (e) cos2 0 = sin2 100~.! (k) csc 0 = -csc 305~. (f) tan 0 = -cot 195~. (1) csc 0 = see 250~. TRIGONOMETRIC FUNCTIONS 73 8. Given 2 sin 6 ~ cos 6 = 2; solve for 6 < 3600. Solution. First express all the functions in terms of one function as in Art. 24. Then since cos 6 = VI - sin2 6, we have 2 sin + vI - sia26 2. Transposing and squaring, 4 sin2 6 - 8 sin 6 + 4 = 1 - sin2 6. Transposing, 5 sin2 6 - 8 sin 6 + 3 = 0, a quadratic equation in sin 6. Solving for sin 6 by the formula, 8 A VN64 - 60 3 sin6= 1 or 10 5 6 = sin-' 1 = 900, and 6 = sin-' 3 = 360 52.2' or 1430 7.8'. 9. Given tan 6 sec 6 = - \/2; solve for 6 < 3600. Solution. Substituting sec 6 = N/I + tan2 0, tanO0 N/ + tan2 6 = - V2. Squaring, tan2 6 (1 + tan2 6) = 2. tan4 6 + tan2 6 - 2 = 0, a quadratic equation in tan2 6. tan2 6 = 1 or -2, and tan6 = 4+1 or 4\ -2. tan 4 4 4 ar,~ r, i r Since \/-2 is imaginary, no such angle as tan-l(J~ < —2) exists. From the original equation the product of tan 6 and sec 6 is negative. Therefore these functions must be opposite in sign, and the angle 6 must be in the third or fourth quadrants. It is necessary then to reject 1 ir and ' ir. 6 = 5 or 7 x, Ans. 10. Given tan 6 + cot 6 = 2; solve for 6 < 3600. Solution. Expressing in terms of cot 6, t6 + cot 6 = 2. cot 0 Solving for cot 6, cot 6 = 1. 0 = cot-' 1 = 7~ r or5 rr. Solve the following equations: 11. 2 sin 6 + 2 cos 6 <2. Ans. 1050, 3450 12. \<3 sin 6 sec 6 - 1 = /2 sec 6. Ans. 750, 1650. 13. sec2 6 + tan 6 = 3. Ans. 450, 2250, 1160 33' 56", 2960 33' 56". 14. sin 6 tan 6 = 1. Ans. 510 49.6', 3080 10.4'. 15. tan 6 = 3.26 tan 1980 13' cos 130 17'. Ans. 250 7' 56", 2050 7' 56". 4.7 sin 280 16' 16. cos, 6 Cos 6. Ans. 1 4I, 3w, 4w, 4w, 4w. 17. cot = ese - 1. Ans. 4,7-. 18. sin 6 cos 6 (1 + 2 cos 6) = 0. Ans. 0, 2 7r, 7r, 2 4 19. cot 6 = 2 cos 6. Ans. 300, 1500, 900, 2700. 20. cos 4 = -. Ans. 1200. 21. sin 2 6 = '4 2 Ans. 2240, 6740, 20210, 24710. 2 2 2 2 2Llfl 22. tan 3 6 = -1. Ans. 450, 1050, 1650, 2250, 2850, 3450. 23. sin 2 6 (3- 4 cos2) =0. Ans. 0, r, 3w, -wr 5w rw T. 74 PLANE TRIGONOMETRY 58. Changes in the value of the sine and cosine as the angle increases from 0~ to 360~. - Draw a circle with unit radius, Fig. 70, and construct an angle 0 in each of the four quadrants. Since in a unit circle y. 70. FIG. 70. the sine of an angle 0 is represented by the ordinate of the point where the terminal side of the angle intersects the circle, the variation in the ordinate will represent the variation in the sin 0. At 0~ the ordinate is 0. As the angle increases from 0~ to 90~, the ordinate increases from 0 to 1. As 0 increases from 90~ to 180~, the ordinate decreases from 1 to 0. From 180~ to 270~, the ordinate becomes negative and decreases from 0 to -1. From 270~ to 360~, the ordinate increases from -1 to 0. Therefore as the angle varies from 0~ to 360~, the sine varies from 0 at 0~ to 1 at 90~, to 0 at 180~, to -1 at 270~, and back to 0 at 360~. The cosine, being represented by the abscissa of the point where the terminal side of the angle intersects the unit circle, will then decrease from 1 to 0 as the angle increases from 0~ to 90~. From 90~ to 180~, the cosine is negative and decreases from 0 to -1. From 180~ to 360~, the cosine increases from -1 through 0 at 270~ to 1 at 360~. It has been shown in Art. 50 that all angles which differ by multiples of 2 r have the same trigonometric functions. Then the values of the sine and cosine as the angle changes from 0 to 2 r will be repeated as the angle changes from 2 7r to 4 r, from 4 wr to 6 7r, etc., or from 0 to -2 7r, from -2 r to -4 r, etc. A function that repeats its values in this manner is called a periodic function. Definition. A periodic function of a variable 0 is a function whose value is not changed when the variable is increased by a constant quantity. The least positive value of this constant quantity is called the period. Since sin 0 = sin (0 + 2 nwr) and cos 0 = cos (0 + 2 nfr), where n is any integer, positive or negative, sine and cosine are periodic functions, and the period is 2 7r. TRIGONOMETRIC FUNCTIONS 75 EXERCISES 1. Trace the changes in tangent and cotangent as the angle varies from 0 to 2 r. Are they periodic functions? If so, what is the period? 2. Trace the changes in sin2 a. Is this a periodic function? If so, what is the period? 3. Trace the changes in sin a + cos a. What is the maximum value? Find the minimum value. Find the value of a for these values of sin a + cos a. For what values of a is sin a + cos a = 0? 59. Graph of y = sin 0. - The changes which take place in sin 0, as indicated in the preceding article, are best shown by a graph. Referring again to Fig. 70, Art. 58, let OA be the unit of measure. Then the complete circumference is the measure of 360~, that is, 360~ may be represented by a line 2 7 units in length. Lay off OB = 6.2832 on OX, Fig. 71. OB is then the radian measure of 27r, or OB = 2 7. Then lay off the proportional Y only. (Other 7angles could be used as well as or in a57ddition to3 these, making the curve more nearly accurate; but for our pur6 2 r3 3 6y o,^-~ ~~~ Pg y= sin 9 FIc. 71. parts as indicated in the figure, using multiples of 6 7r and 7r only. (Other angles could be used as well as or in addition to these, making the curve more nearly accurate; but for our purpose the easy proportional parts of 2 7r are used.) Lay off OA- o the y-axis. This will represent the unit for plotting the sines of the angles. Select various values of 0 from 0 to 2 r, determine the corresponding values of y, and plot the points of which these values are the coordinates. Values of 0: 0 rT r _ ~ r _ T T _ ] _ 2r T 7r T _ etc. Values of y: 0 ~ /V2 ~/ 3 1 V/3 1 2 ~ 0 -~ -4\/2 etc. Points: 0 Pi P2 P3 P4 P5 P6 P7 Ps P91 P10 etc. Draw a curve through these points. The curve is the graph of y = sin 0. It shows the change in sin 0 as the angle changes from 0 to 2 r. 76 PLANE TRIGONOMETRY It is evident that the curve will repeat its form if 0 were given values from 2 7 to 4 7, from 4 7 to 6 r, etc., or from 0 to - 2 7, etc. The curve is then periodic. Here the angle and the function are both plotted to the same unit or scale. Often, however, for convenience when plotting on coordinate paper, the angles are plotted according to the divisions on the paper. For example, 1 space = 6~ or 10~, or some other convenient angle, depending on the size of the plot. 60. Mechanical construction of graph of sin 0. - On one of the heavy horizontal lines of a sheet of coordinate paper, choose an origin and lay off angles every 15~ from 0~ to 360~, using each lllllllltlllli ill-l- tN X11 -FIG. 72. small space to represent 30, as in Fig. 72. With any convenient point on this horizontal axis as a center, describe a circle with a radius equal to 30 spaces. Choose the initial side of all the angles on the axis of the angles. By means of the protractor lay off the central angles every 15~ from 0~ to 360~, such as ZAOB, ZAOC, etc. Let the radius of the circle be the unit of measure. Then the sines of the angles are the ordinates of the points A, B, C, etc. Through B draw a horizontal line to intersect the vertical line through 15~ as plotted TRIGONOMETRIC FUNCTIONS 77 on the horizontal axis. The point b, thus determined, has as co6rdinates (15~, sin 15~). In the same way locate c, (30~, sin 30~); d, (45~, sin 45~); e, (60~, sin 60~); etc. Through these points draw a curve. With the sine curve thus constructed, one can determine the value of the sine of an angle approximately by measurement. For example, find the sin 51~. By measurement the ordinate for sin 51~ is 23.3 spaces. Since the unit is 30 spaces, sin 51~ 23.3 -30 0.7766. From the table of natural functions sin 51~ 30 0.77715. A comparison of the results for a number of angles will give an idea of the accuracy of the graph. Exercise. Measure the ordinates for the angles given in the following table, compute the sines, and tabulate the results. Find the sines of the same angles from the Tables and tabulate. Compare the results. 0 Sine from curve Sine from table Difference 18~ 57~ 780 99~ 123~ 1380 171~ EXERCISES 1. Plot y = cos 0 by both of the methods employed in plotting sin 0. 2. Plot y = tan 0 and y = cot 0 on the same set of axes. 3. Plot y = sin 2 t, y = sin t, and y = sin 2 t on the same set of axes. Compare the curves as to the maximum and minimum values, and as to their periods. 4. Plot y = ~ sin t, y = sin t, and y = 2 sin t on the same set of axes. Compare the curves as directed in Exercise 3. 5. Plot y = sin x + cos x. Suggestion. Plot yj = sin x, and y2 = cos x, using the same units and axes. Then y = y1 + qY2 may be computed by means of the dividers. 61. Inverse functions. - We have seen that sin-' t means the angle whose sine is t. In Art. 58, it was shown that the sine function varied from -1 to +-1. Then the equation 0 = sin- t has real solutions when and only when t is not less than -1 nor 78 PLANE TRIGONOMETRY greater than +1. In the same way it can be shown that 0 = cos-1 t has a solution when and only when t is not less than -1, nor greater than +1. Since tan 0 and cot 0 can have any value from -oo to + oo the equation 0 = tan-' t and 0 = cot-' t have solutions for all values of t. The two expressions sin 0 = t and 0 = sin-' t both express the same thing, namely, that 0 is an angle whose sine is equal to t. In the first expression t is a function of 0 and in the second 0 is a function of t. Y h In sin 0 = t, there is but one value of t for every value of 0. sin 0 is then said to h~[ \ ~ be a single valued function of 0..: \, ~ In 0 = sin-l t for every value of t, there are an indefinite number of values of 0, as was seen in Art. 50. sin-' t is then said to be a.jad f multiple valued function of t..t4L 62. Graph of y = sin-' x, or y = arc sin x. - Stating y = sin-' x in the form sin y = 9-1' 1 AAX x, it is readily seen by comparison with y ^/, *,1 sin x, that sin y = x is obtained from y = sin I/ ',J x by interchanging x and y. Then the graph of y = sin-' x is obtained by plotting the A ail sine curve on the y-axis instead of the xaxis as in Art. 59. The curve is shown in Fig. 73. In many mathematical operations where sin-lx enters, it is often desirable and, inY deed, necessary to consider a portion of the 7 i= si-l x curve, Fig. 73, for which there will be but one value of y for every value of x. A glance at the figure will show that for the portion AOC of the curve, the function is single valued. That is, for every value of x between and including -1 and +1, y takes values between and including - 7r and - 7r. Definition. The values of sin-' x between and including -~ T and ~ Tr for each value of x, are called the principal values of sin-' x. To represent the principal value of the function the s is often written a capital, thus, Sin-1 x. The other functions are denoted in a similar manner. TRIGONOMETRIC FUNCTIONS 79 EXERCISES In the following exercises, find the numerical values of the given expressions, using the principal values of the angles.* 1. -25 [sin-l 1 - sin-1 (-1)]. Ans. 225-r. 2. 8 [sin-1 (0.4) - sin-1 (0.2)]. Ans. 1.681. 3. tan-1 ' + tan-1 ~. Ans. 7r. 4. sin-1 - sin- (-). Ans. ' 3r. 5. sin- 1 - sin-1 (-1\V3). Ans. - r. 6. Given x V225-9 x2 + 15 sin-1 * In this expression substitute 5 15 x = 5, then substitute x = 2, and subtract the second result from the first. Determine the result correct to the third decimal. Ans. 11.891. 7. Given x </12-x2 +6 sin-'-. Substitute x = 2 /3, and then x = 2 <2 2, and subtract the second result from the first. Ans. 2.9035. 63. Relation between sin 0, 0, and tan 0, for small angles. - Draw angle BOE = 0, Fig. 74. With 0 as a center and OB = 1 as radius, describe the arc BD. Draw DA 1. to OB and BE tangent to the arc at B. Then sin 0 =AD, 0 = are DB and tan 0 = BE. 0 A B A OBD < sector OBD < A OBE. FIG. 74. But A OBD = 1 OB X AD, sector OBD = ~ O2. 0, where 0 is in radians, see Art. 5, and A OBE = ~ OB * BE. Then ~ OB. AD < ~ OB2. 0 < ~ OB * BE. Dividing by ~ and substituting OB = 1, AD = sin 0, and BE = tan 0, [12] sin0 < 0 < tan0. sin 0 Dividing [12] by sin 0 and remembering that tan 0 = s 0 COS 0' 1 < < sec 0. sin 0 Now as 0 approaches 0 as a limit sec 0 approaches 1 as a limit, written lim sec 0 = 1. o-0 * In many applications of anti-functions, as in the calculus, they enter into the expressions for areas, volumes, etc., and the angles must be expressed in radians. 80 PLANE TRIGONOMETRY Then since. — is always less than a quantity which approaches sin u 1 as a limit, and at the same time is greater than 1, we have lim 1 0 a- sin Again dividing [12] by tan 0 and simplifying, cos0 <t <1. But limcos 0 = 1, therefore lim 0 tan 8 8o0 o-O tan 0 By computing the following table the student will find the theorem verified for several angles. Angle in sin 0 0 in radians tan sin degrees sin 0 200 10~ 5~ 4~ 3~ 2~ 1~ 10 These results show that for small angles, sin 0 and tan 0 may be replaced by 0 in radians and the results will be approximately correct. For example, sin 50 9.4' = 0.0899, 5~ 9.4' = 0.0900 radian, and tan 5~ 9.4' = 0.0902. For a smaller angle the agreement would be still closer. EXERCISES 1. A tower is 125 ft. high. The angle of elevation of the top of the tower, from a point in the same horizontal plane as the base, is 1~. Find the distance from the point of observation to the tower. BC 125 Solution. Let BC = 125 ft., Fig. 75, and AB = x. Then tan 1~ = B = 1 AB x Replacing tan 1~ by 1~ = - radians, by Art. 5, 125 125 x 180. x = = = 7162 ft. zr/180 3.1416 2. A railway track has a 2% grade for a certain distance. Find the inclination of the track to the horizontal. TRIGONOMETRIC FUNCTIONS 81 Solution. The per cent of a grade is the ratio of the number of feet rise per mile to the number of feet in a mile. Then for a 2% grade the rise per mile would be 2% of 5280 = 105.6 ft. C A x B FIG. 75. In Fig. 76, let CB = 105.6 ft. and AC = 5280 ft. 105.6 Then tan 0 = 0 in radians = 15. 5280 105.6 180 18 18 7 63 in degrees = 5 X =X 1==10 8.7'. 5280 w7 5 5X 22 55 3. A railway track rises 80 feet to the mile. Find the angle of inclination of the track. Ans. 52' 5". B A --- 5280 ft. C FIG. 76. 4. A certain plane is inclined to the horizontal at an angle of 45'. Find the per cent of the grade of a railway track constructed on this plane. Ans. 1.309%. 64. Many problems arise which involve triangles, two sides of which are practically equal, and each very great compared with the third side, as shown in Fig. 77. The long sides may be taken C A- B B FIG. 77. as the radii of an arc, of which the third side BC is the chord. Since the limit of the ratio of the arc to its chord is unity as the angle approaches zero, chord BC may be replaced by the arc BC. arc BC Then since Z BAC (in radians) = AB if the chord BC is substituted for the arc, the results will be accurate enough for many problems.* * In solving exercises in which radian measure is changed to degrees or vice versa, it is convenient to use Table V. 82 PLANE TRIGONOMETRY EXERCISES 1. Telescopes at the end of a base line, 250 feet long, on the deck of a ship are turned upon a distant fort. The lines of sight of the telescopes are found to make angles of 89~ 12' and 89~ 40' with the base line. Find the distance from the ship to the fort. Ans. 2.39 miles. Suggestion. In Fig. 77, let B and C be the positions of the telescopes and A the position of the fort. 2. The diameter of the moon subtends an angle of 31' 5" at the earth. The moon is approximately 240,000 miles from the earth. Find the diameter of the moon in miles. Ans. 2170 miles. CHAPTER V FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 65. In the previous chapters, we have worked with, and established the relations between, the functions of a single angle. But, in solving oblique triangles and in many of the applications of trigonometry to other subjects, formulas are used which are derived from the functions of the sums or differences of angles. These functions are expressed in terms of the functions of the angles and are as follows for the sine and cosine: [13] sin (a -+ P) = sin a cos P + cos a sin P, [14] cos (a + P) = cos a cos - sin a sin P, [15] sin (a - P) = sin a cos - cos a sin P, [16] cos (a - p) = cos a cos P + sin a sin P. Formulas [13] and [14] are often called addition formulas, and [15] and [16] subtraction formulas. 66. Derivation of the formu- C las for the sine and cosine of p the sum of two angles.-Let ZAOB = a and ZBOC = 3, Fig. 78, each of which is acute, B and so chosen that a+ t = Z AOC is less than 90~. In order that the functions of a, 3, and a + / may be involved in the same for- A mula, we may form right triangles A which have a, 3, and a + a as H K F.IG 78. acute angles. Choose any point P in the terminal side OC. Draw PHIOA, PD I OB, DK OA, and DL I PH. A KOD is similar to A LPD, since their sides are perpendicular each to each. Then Z LPD = a. oHP KD + LP KD LP By definition, sin (a + -F) -= = +P - + LP OP= OP OP OP KD Now multiply numerator and denominator of Ko by OD, the 83 84 PLANE TRIGONOMETRY common side of the two triangles of which KD and OP are sides LP respectively. Also multiply -D in the same way by PD, the OD common side of triangles DOP and LPD. Then KD OD LP PD sin (aI) = OD OP PDOP KD OD LP PD But OD = sin a, = cos 3, = cos a, and - = sin (. [13].'. sin (a + 3) = sin a cos l + cos a sin 3. By definition, OH OK -HK OK LD OK OD LD PD os (O-+o) = p - OP O OP PD OP OK LD But D = cos a and = sin a. OD PD [14].. cos (a -+ o) = cos a oss - sin a sin S. 67. Derivation of the formulas for the sine and cosine of the difference of two angles. —Let Z AOB = a and Z COB = B R p s___ A D H FIG. 79. be the two acute angles, Fig. 79. Then angle AOC = a - P. For reasons similar to those given in the preceding article, choose any point P in the terminal side OC of (a - 3). Draw PH IOA, PR IOB, RDIOA, and PE IDR. A DOR is similar to A PER and ERP = a. FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 85 By definition, HP DR DRER DR ER DR OR ER RP Dsn(ad - OP OP OOP O P OR'OP RP OP DR OR ER RP But R= sin a Cos, O = cos a, and - = sin. [15].. sin (a - s) = sin a cos - cos a sin 3. By definition, OH OD + EP OD EP OD OR EP RP cos(a-o)- OP OP OP OP OR OP RP OP [16].. cos (a - 3) = cos a cos 3 + sin a sin 3. In the proof of [15] and [16], it was assumed that a > 3. Now suppose f > a. Then a - = ( - - a) sin (a - 3) = sin [ -( - a)] = -sin ( - a) by Art. 49. -sin (3 - a) = - (sin / cos a - cos / sin a) by [15] = sin a cos / - cos a sin p, which is the same result as was obtained before. Exercise. Show that cos (a - f) gives the same result whether a > P or a < 3. 68. Proof of the addition formulas for other values of the angles.- In Art. 66 formulas [13] and [14] were proved when a, 3, and a + 3 are each less than 90~. They are, however, true for all values of the angles. (1) Suppose that a and 3 are acute and such that a = 90~ - and f = 90~ - y, where f and y are each less than 45~. On this assumption, (a + 3) > 90~, (0 + r) < 90~, sin a = cos 4, cos = sin 0, sin 3 = cos y, and cos f = sin y. sin (a + 3) = sin [(90~ — ) + (90~ - )]=sin [180 - (4+y)] = sin (~ + y) = sin 0 cos 7 + cos 0 sin y. Substituting for the functions of f and y their values in terms of the functions of a and 3, sin (a + f) = cos a sin + sin a cos / = sin a cos / + cos a sin /. That is, the formula for sin (a + 3) is true when (a + /) is an angle in the second quadrant and a and f as stated. 86 PLANE TRIGONOMETRY In the same way we may show that the formula for cos (a + 3) is true for values of the angles as given above. (2) Suppose that a is in the second quadrant and 3 in the third, such that a = 90~ + ~ and 3 = 180~ + -. On this assumption, sin a = sin (90~ + )) = cos ), cos a = cos (90~ + -) = -sin e, sin 3= sin (180~ + y) = -sin y, cos 3 = cos (180~+ ) = -cos y, sin (a + 3) = sin [(90~ + - ) + (180~ + y)] = sin [270~ + ( + y )] = -cos (4 + y) = - s c cos y + sin 0 sin y. Substituting for the functions of f and 7 their values in terms of the functions of a and 3, sin (a -+ ) = -(sin a) (- cos /) + (-cos a) (-sin /) = sin a cos /3 + cos a sin 3. In the same manner it may be shown that the addition formulas are true for any angles. 69. Proof of the subtraction formulas for all values of the angles. - Since the addition formulas are true for all values of a and 3, they are true when -3 is put for 3. Then sin (a - 3) = sin [a + (-3)] = sin a cos (-3) + cos a sin (-/), and cos (a - 3) = cos [a + (-/3)] = cos a cos (- )-sin a sin (-3). But sin (-a)= -sin a, and cos (-a) = cos a,.'. sin (a - 3) = sin a cos - cos a sin /, and cos (a - 3) = cos a cos A + sin a sin A. That is, the subtraction formulas are true in general. EXERCISES Prove that formulas [13] and [14] are true in the following cases: 1. a in the fourth quadrant and i3 in the first. 2. a in the third quadrant and f in the third. 3. a in the first quadrant and 3 in the third. Solve the following exercises by means of the addition and subtraction formulas, assuming a and 3 less than 90~. 4. Find the sine and cosine of 90~ by assuming that 90~ = 60~ + 30~. Solution. sin 90~ = sin (60~ + 30~) = sin 60~ cos 30~ + cos 60~ sin 30~. Substituting the values of the functions of 30~ and 60~, sin 90~ = /3. /3- + ~. = + = 1. cos 90~ = cos (60~ + 30~) = cos 60~ cos 30~ - sin 60~ sin 30~ = I * / - /3. = I 3 - 3 = 0. -- j 0. FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 87 5. Given sin a - - and cos 3 = -1-; find sin (a +i3) and cos (a +13). Solution. Construct the right triangles ABC and DEF, Fig,. 80, with a an acute angle of A ABC, and 1 an acute angle of A DEF. By [13], sin (a + 1) = sin a cos 1 + cos a sin 1. Substituting the values for sin a, cos a, sin 1, and cos 1 from the triangles, 3 _ 1= L+A sin (ar + P> - I-~ -t ~ 1 3 1 + 8 - 63 cos (a +13) = cosa cosf - sin a sin13 4.5 _ 3 12 20.36 16 5 3 63 -65 65. E B 3 A C DF (a) (b) FiG. 80. 6. Given sin a ='2, and cos13 '\2; find sin (a ~13) and cos (a +13). 7. Find the sine and cosine of 750, having given the functions of 450 and 300. Ans. '3(v'6 + x2), '3(V6 - 42). 8. Given cos a =2, and cos13 ='3; find sin (a +13), cos (a+13), sin (a-13), and cos (a - 1). Ans. (V3+2V\/3 2\2), '1 -2V6), 6 ' /3V - 2 V2), '(1 +2V \6). 9. Show that sin (450 + 300) s sin 450 + sin 30'. Proof. sin (450 + 300) = sin 750 = 0.9659. sin 45' = 0.70711, sin 30' = 0.5, sin 450 ~ sin 300 = 0.70711 + 0.5 = 1.20711. But 0.9659 r 1.20711. sin (450 + 300) # sin 450 + sin 30'. Note. The symbol s4 means is not equal to. 10. Show that sin (250 + 370) sX sin 25' + sin 37'. 11. Show that cos (350 + 280) # cos 350 + cos 28'. 12. Prove from Fig. 78 that sin (a +13) s sin a + sin 1. HP. KD. DP Proof. sin (a+3) + sin a na, sinfl = OP) OD' OP Now OD < OP and DP > LP. (Why?) DK DK dP L ( 6D — and 0 — O.(hy?) 88 PLANE TRIGONOMETRY Adding, + D)> (D + L. (Why?) DK-DP- HO P (Why?) OD+op)>O* (Why?) Therefore sin a - sin /3 4 sin (a + /3). 13. Prove from Fig. 78 that cos (a + 13) $ cos a + cos,3. Note. A very common mistake made by beginners in trigonometry is to assume that sin (a + 3) = sin a + sin 3, etc. Exercises 10, 11, 12, and 13 are given for the purpose of impressing the student with the fact that such relations are not true. In the following exercises, the angles may have any values. 14. Find the sin 120~ by using 120~ = 90~ + 30~. 15. Find cos 150~ by using (a) 150~ = 120~ + 30~, (b) 1500 = 2100 - 60~, (c) 150~ = 750 + 75~. 16. Find sin 240~ and cos 240~ by using (a) 240~= 210~+ 30~, (b) 240~ = 300~ - 60~, etc. 17. Given sin a = -5, cos =3 -2, a in the third quadrant, and 3 in the second. Find sin (a + 1 ), cos (a -+ 1), cos (a - 1), sin (a - 3). Ans. sin (a +- ) = I, cos (a +3- ) 6= 6. 18. Given sin a = 5, a in the second quadrant, and tan f = -1, 1 in the third quadrant. Find sin (a + -), cos (a -+ 3), sin (a - 3), and cos (a - 3). Ans. sin (a + A) = -16, cos (a + 3) = 60 Find the value of 0 in the following exercises. 19. cos (20~ + a) cos (20~ - a) + sin (20~ + a) sin (20~ - a) = cos 0. Ans. 0 = 2 a. 20. cos 50~ cos (85~ - a) - sin 50~ sin (85~ - a) = cos 0. Ans. 0 = 135 - a. 21. sin (90~ + - 3) cos (90~ - ~) + cos (90~ + /3) sin (90~ - 1 3) = sin 0. Ans. 0 = 1800. 22. cos (45~ - x) cos (45 + x) - sin (45~ - x) sin (45~ + x) = cos 0. Ans. 0 = 90~. 23. If sin-l x + sin-l y = a, show that y = V1 - x2 sin a - xcos a. Suggestion. Let a = sin-' x and 3 = sin-l y. Then3 = a - a. Take the sine of each side of the equation. 24. By means of [13], [14], [15], and [16] prove the following relations. (a) sin-(90~ + 0) = cos0. (f) cos (90~ + 0) = -sin 0. (b) sin (180~ - 0) = sin 0. (g) cos (180~ + 0) = -cos 0. (c) sin (180~ + 0) = -sin 0. (h) cos (270 ~- 0) = -sin 0. (d) sin (270~ - 0) = -cos 0. (i) cos (270~ + 0) = sin 0. (e) sin (270~ + 0) = -cos 0. (j) cos (360~ - 0) = cos 0. 25. If sin a = h and sin = a,-, a and 3t acute angles, prove that a + 3 = 45. 26. Show that sin-l' + sin-1' = i X, using only the principal values of the anti-functions. FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 89 Proof. Let a - sin-'l- and 3 = sin'A. sin a = ' and sin 4 Then sin (a + 3) = sin a cosf3 + cos a sin 3 Find the value of the following expressions, using only the principal values of the angles: 27. sin (sin-' ' + sin — ` '/3). Ans. 1. 28. cos (sin-' 3 - tan-' 3). Ans. 1. 29. sin (sin-' m - cos-' n). 30. sin (sin-' 4 -I- eos'1 1). Ans. 1. 31. cos (tan-' i - eos' 1). Ans. 0.617. 70. Formulas for the tangents of the sum and the difference of two angles. - By [7], [13], and [14], sin (a + 3) sin a Cos/3 + eos a sin/ tan (a + 0) = Cotan( + + ) cos a Cos/3 - sin a sin/3 Dividing. both numerator and denominator by cos a cos /, and applying [71, sin a eos 3 eos a sin/3 tan +a = eos a Cos Cos a cos _ tan a + tan/3 eos a Cos _ sin a sin 1 - tan a tan/3 Cos a Cos Cos a Cos tan c -- tan P [1171 tan (cL + P) = t 1 - tan cL tan P tan cL - tan [.81 Similarly tan (a - )= tancL tan [18] ~~~~~~1 +~ tan a tan P Since formulas [13], [14], [15], and [16] are true for all values of a and /, the formulas [17] and [18] are true in general. EXERCISES 1. Find tan 750, and tan 150 by means of 450 and 300. Ans. tan 750 = 2 + V3; tan 150 = 2 - V3. 2. If sin a = 2, and sin / = 3, a and 3 acute angles, find tan (at + /), and tan (a - /). Ans. tan (a + /3) = -3%, tan (ao - /) = 3. If cos a =5, and sin 3 = -, a in the fourth and 3 in the third quadrant, find tan (a + /) and tan (a - /). Ans. tan (a+/) 7-,ZT tan (a -p)= o* In the following problems use only the values of the angles < 900. 90 PLANE TRIGONOMETRY 4. If a = tan-1 " and j3 = tan-l 3, find (a + i/). Solution. Take the function of (a + /) which involves the given functions if possible. tan (a + ) tan a + tan 3 tan (a + -) = 1 —tan-an-* 1 - tan a tan /3 But tan a = and tan / =. 1 3.~. tan( + -) = D.- 1-6. (a+ 3) = 45~. 4 1 r 5. Prove that tan-' - - tan-l = 3 7 4 6. Prove tan-l +- sin-l = tan-1 2. 5 1 7. Prove that tan- - + tan- = - 7 6 4 8. Find the value of tan [sin-1 2 - cos-1 f]. cot a cot /3 - 1 9. Prove that cot (a + 3) = cot a cot 3 -cot a +- cot o 10. Prove that cot-' 3 + csc-1 V/5 = i r. 11. Prove that tan-l( /3-+ /) + tan-' = 3 xtana3 — 2 /14 12. If cos-1 x + tan-l y = a, show that y =x taax + 1 - x2 tan a Suggestion. Let a = cos-l x and A = tan-1 y. Then 3 = a - a. Take the tangent of each side of the equation and substitute for tan a and tan, their values in terms of x and y. a b 13. Prove that tan-' a i- tan-l b = tan- a --- —b I ab 3 3 8 7r 14. Prove that tan-l' + tan-l -tan- = ' 15. Prove that tan-' n + cot-l (n + 1) = tan-l (n2 + n + 1). 71. Functions of an angle in terms of functions of half the angle. - Since the formulas for the sum of two angles are true for all values of a and 3 they will be true when 3 = a. Then sin (a -+ /) = sin (a + a) = sin a cos a + cos a sin a. That is, [19] sin 2 a = 2 sin a cos a. This formula may be stated as follows: The sine of any angle is equal to twice the product of the sine and cosine of the half angle. Thus, sin 40~ = 2 sin 20~ cos 20~. sin a = 2 sin ~ a cos I a. FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 91 And conversely, 2 sin 3 a cos 3 a = sin 2 (3 a) = sin 6 a. 2 sin 250 cos 250 = sin 2 (250) = sin 500. Also cos (a ~ 3) = cos (a + a) = cosaoz cos a - sin a sin a cos2 a - sin2 a - 1 - sin2 a - sin2 a = 1 - 2 sin2 a, or = COS2 a -(1- COs2 a) = 2cos2a -1. That is, [20] cos 2 cL= cos2 CL sin2 a = I - 2 sin2cL = 2 cos2a.- 1. Thus, cos 30' = cos2 15' - sin2 15', or 1 - 2 sin2 15', etc. 36 36 cos3 = cos2 — sin2 2 2 tan a + tan a Also tan (a +-P tan (a + a)= I - tan a tan a. That is, [21] tan 2ci 2 tan cL 1-tan2 cL Thus, tan 600 = 2 tan 300 1000 2 tan 500 1 - tan2 300' 1 - tan250"' 2 tan' 6 tan 8 = 1 - tan2 2 6 EXERCISES 1. Given the functions of 300, to find the functions of 600. Solution. sin 600 = 2 sin 300 cos 300 = 2 (1) (1 \'3) = I V'3. cos600' = 2Cos2300 1 = 2 3)2 1- -— 1 =, or = 1 - 2sin230 = I - 2 ()2 = tan6O0 2 tan 30' 3 1- tan2300 = 1 -2. Find the functions of 1200, 1800, 2400, 2700 and 3000 by means of the preceding formulas. 3. Given tan 0, 0 < 900; find the sine, cosine, and tangent of 2 0. Ans. sin 2 8 =2 ', Cos 2 0= I, tan 2 = 24. Suggestion. Construct a right triangle with 0 as one of the acute angles. 4. Prove that 2tana sin 2 a. 1 + tan2 a sin a Proof. tan a and 1+ tan2 a = see2 a. COS a 2sin a sin a 2 2, 2 tan a Cos a Cos 0z Then 2taa= 2osina ccos = sinoa. 1 + tan2 a sec2a - 1 COS2 aZ 92 PLANE TRIGONOMETRY 5. Given cos 600; find tan 30. 1 - tan2 p3 6. Prove that 1 -tan' /3 = cos 2 3. 1 +- tan1 p 7. Prove that tan 0 + cot 0 = 2 cse 2 0. 8. If tan0 -, show that sinO = and sin 2 = 4xy (X' - yl) 2 x x2 _ y _2 (X2+ y2)2 Y x - y 2 +y ~ / — _ cos 0 9. If tan - showthat x + ~ x x n - y x +~ Y \/COS 2 O' 10. Prove that sin 3 0 = 3 sin 0 - 4 sin3 0. Suggestion. In formula [131 let a = 2 0 and / = 0. 11. Prove that cos 3 0 = 4 cos3 0 - 3 cos 0. 12. Find the value of tan (2 tan-1 2). Solution. Let 0 = tan-1 2, i.e., tan 0 = 2. 2 tan _ 4 tan (2 tan-12) = tan (2 0) = ta 1 = 4 - tan 2 0 1 - 4 W. Find the values of the following, using the prineipal values of the angles. 13. sin (2 sin-' D). 16. sin (2 sin-' V \3). 14. cos (2 cos-1 - )/2 17. sin (2 sin-1 2 X2 X 15. tan (2 are tan V3). 18. cos (2 are tan 2 V3). Derive the formulas in 19 to 25. 19. cos 4 = eos4 0 - 6 Cos2 0 sin20 + sin' 0. 20. sin 4 = 4 cos' sin 0 - 4 cos 0 sin3'0. 3 tan 0 - tan3 0 1 - 3 tan2O0 4 tan 0 (1 - tan2 0) 22. tan40 6a28a~ I - 6 tan 2 8 + tan 4 0 23. sin (a +/3 + -) = sin a cos/3 eos y + cos a sin/3 cosy+- cos a cos/3 siny - sin a sin /3 sin y. 24. eos (a +/3 + -y) = cos a cos/3 cos y - sin a sin/3 cos-y - sin a cos/3 sin y - cos a sin /3 sin y. tan a ~ tan / + tan y - tan a tan 3 tan y 21 -tan a tan/3 - tan/3 tan y - tan -ytan a 72. Functions of an angle in terms of functions of twice the angle. - By [20], cos 2 a = 1 - 2 sin2 a. Solving this for sin a, I/1- cos2 a we have sin a == A a 2 Let a = 0 0 and we have [22] sin + /1Cos 2 2 That is, the sine of an angle is equal to the square root of one half of the quantity, one minus the cosine of twice the angle. FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 93 Thus, sin 50~ = V1 - cos 100~ sin 10~ -V - 2 Also by [20], cos 2 a = c2 cos2 a - 1. Solving this for cos a, /I + cos 2 a we have cos a = 1 + cos2 /. 2 Let a = ~ 0 and we have [23] cos = + cos That is, the cosine of an angle is equal to the square root of one half of the quantity, one plus the cosine of twice the angle. Thus, cos 30O = 1 + cos 60, cos 500 = + cos 100 By dividing [22] by [23], we can derive [2] 1 = /1 -cosO _ -cos _ sin 9 [24] tanO + cos = sin O 1 + cos ' The last two forms given in [24] may be obtained as follows: Multiplying numerator and denominator of / + cos0 by V1 - cos 0, a _ /1- COS 0 1- cos0 _V( - cos 0)2 1 - cos 0 tan2 V 1 + os-cos 0 s 0 / - cos2 0 sin 0 Again, multiplying numerator and denominator by VI + cos 0, 1 /1 - cos 6 1 + coso _ / - cos2 0 sin0 tn2 VI + cos 0 + cos 0 - V(i +cGos0)2 1 +Gos, I - cos 80~ 1 - cos 80~ sin 800 Thus, tan 400 = V 1 + cos 800 = sin 800 1 + cos 80' EXERCISES 1. Given the functions of 45~; find the functions of 22-~. Ans. sin 22~~ = /2- / 2, etc. 3 0 0 2. Given tan 0 = -; find sin - and cos 0* 4' 2 2.0 1 Ans. sinm = - etc. 2 94 PLANE TRIGONOMETRY 1 0 0 0 3. Given sin = - X 0 in the third quadrant; find sin 2, cos, tan. 2t 2iL 2' i Ans. tan =-2- 3, etc. 4. Having obtained the functions of 20~, 36~, and 72~ from the tables, find by computation, sine, cosine, and tangent of 10~, 18~, and-'360 respectively. 5. Find the value of sin (~ cos-1 D). Solution. Let 0 = cos-1 i. Then cos 0 =.in [1,\.1 I ~ - 1- cos 9 /1 - _ A I /2 sin cos-' = sin = \ -2o \/ \/= = Find the values of the three following expressions, using only the principal values of the inverse functions. 6. cos [~ tan-' 1]. Ans. 3 / l-. 7. tan [I sin-1 3]. Ans. 3. 8. tan sin-' (2 x )]. Ans. Vx. b2 - C' - a2 9. Given cos 0 = c, and 2 s =a b + c; prove that 1 (s - b) (s - c) (a) sin2 0 = b s 2 bc 1 s (s - a) (b) cos2 0 = -, 2 bc 1 (s - b) (s -c) (c) tan' - o= 2 s (s - a) 10. Prove that tan 2 and cot 2 are the roots of x2 - 2 x csc 0 + 1 = 0. 2 2 1 - cos 0 11. In [24], show why the sign ~ is not necessary before sin 0 and sin 0 1 + cos 0 73. To express the sum and difference of two like trigonometric functions as a product. - In this article the following formulas are proved. [25] sin a + sin P = 2 sin - (a + P) cos (a - A). [26] sin a- sin P = 2 cos 1 (a + P) sin 2 (a - ). [27] cos a + cos = 2 cos (a + ) cos (a - ). [28] cos a - cos = - 2 sin 2 (a + p) sin (a - ). The object of these four relations is to express sums and differences of functions as products. In this manner formulas can be made suitable for logarithmic computations. Proof of [25] and [26]. Let a = x + y and = x - y. FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 95 Solving simultaneously for x and y, x = 2 (a + /) and y = ~ (a - ). Now sin a = sin (x + y) = sin x cos y + cos x sin y, (a) and sin / = sin (x - y) = sin x cos y - cos x sin y. (b) By adding (a) and (b), sin a + sin: = 2 sin x cos y. Substituting the values of x and y, we have sin a + sin / = 2 sin I (a + /) cos ~ (a - ). Subtracting (b) from (a) and substituting for x and y, sin a - sin 3 = 2 cos x sin y = 2 cos ~ (a + /) sin ~ (a - j). Proof of [27] and [28]. cos a = cos (x + y) = cos x cos y - sin x sin y. (c) cos 3 = cos (x - y) = cos x cos y + sin x sin y. (d) Adding (c) and (d), cos a + cos 2 = 2 cos x cos y = 2 cos ~ (a + -) cos 2 (a - /). Subtracting (d) from (c), cos a - cos = -2 sin x sin y = -2 sin (a + /) sin ~ (a -/ ). EXERCISES In Exercises 1 to 4, find the values of the functions from the tables, carry out the indicated operations on each side of the equality sign, and compare results. 1. sin 60~ + sin 40~ = 2 sin 50~ cos 10~. Solution. The right-hand member is best computed by logarithms. sin 60~ = 0.8660 Let x = 2 sin 50~ cos 10~ sin 40~ = 0.6428 log 2 = 0.30103 sin 60~ + sin 40~ = 1.5088 log sin 50~ = 9.88425 log cos 10~ = 9.99335 log x = 0.17863 x = 1.5088 The two results are found to, agree. 2. sin 70~ - sin 40~ = 2 cos 55~ sin 15~. 3. cos 110~ + cos 80~ = 2 cos 95~ cos 15~. 4. cos 75~ - cos 25~ = - 2 sin 50~ sin 25~. E cos 70~ - cos 30~ 5. Express cos 70~ cos30 as a product. cos 70' + cos 30' Solution. By [28], cos 70~ - cos 30~ = -2 sin ~ (70~ + 30~)sin ~(700-30~) -2 sin 50~ sin 20~. By [27], cos 70~ + cos 30~ = 2 cos 1 (70~ + 30~) cos ~ (70~ - 30~) = 2 cos 50~ cos 20~. 96 PLANE TRIGONOMETRY cos 700 - cos 300 -2 sin 500 sin 20 tan 50 tan 20. Then 70 30 = 2 -tan 50~ tan 20~. cos 70~ + cos 300 2 cos 50~ cos 200 Express the following as products: sin 100~ - sin 40~ cos 100~ + cos 40~ sin 120~ + sin 80~ cos 120~ - cos 80~ cos 150~ + cos 70~ sin 150~ + sin 70~ Prove the following identities: sin x + sin y tan ~ (x + y) sin x - sin y tan ~ (x - y) Suggestion. Apply [25] and [26]. 10. cos30 - cos70 = 2sin 5 0sin 2 0. 11. sin 7, - sin 5 3 = 2 cos 6 3 sin f. 2. sin 2 a + sina tan3 cos 2 a + cos a 2 13. cos (60~ + 0) + cos (60~ - 0) = cos 0. cos (a - 3) 14. cota + tan3 = sacos ( sin a cos 3 Proof. Express cot a and tan 3 in terms of sine and cosine; then n cos a sin cos cos sin ao sin a cos (a - o ) cot a$ + tan =.- + = -- = sin a cos sin a cos 3 sin a cos cos (0 + a) 15. cot - tan a = cos (0 sin 0 cos a 16. tan ) ( )0. If a +,3 + y = 180~, prove the identities in the following problems. 17. sin a + sin 3 + sin - = 4 sin ~ (a + 8) cos i a cos /3. Proof. sin a + sin f = 2 sin ~ (a. 3) cos I (a -f ), by [25]. Now - = 180~ - (a + 3).. sin. = sin [180~ - (a +,)] = sin (a + 3), Art. 45. sin (a -+ 3) =2 sin + (a + 3) cos ~ (a + ), by [19]. sin a + sin 3 + sin = 2 sin (a +) os i (a - 3 ) co 2 sin (a ) cos (a + /3) = 2 sin 4 (a + 1) [cos 4 (a - ) + cos 4 (a + 1)]. But cos i (a - /) + cos 4 (a + /) = 2 cos 2 [2 (a - 13) + 2 (a + 1)] cos [2 (a - 3) - (a +)] = 2 cos I a cos 13. sin sin a + sin = 2 sin ( = 2 sin (o + ) 2 c os CS = 4 sin 4 (a + ) cos 2 a Cos 4 1. 18. cos a + cos, + cos y = 4 cos 4(a +- ) sin ^ a sin 3 + 1. 19. cos 2 a + cos 2 3 + cos 2 y = -4 cos a cos, cos - 1. 20. sin 2 a + sin 2 3 + sin 2 = 4 sin a sin 3 sin y. FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 97 74. To change the product of functions of angles into the sum of functions. - From Art. 65, sin (a + /) = sin a cos 3 + cos a sin 3. (a) sin (a - 3) = sin a cos - cos a sin: (b) cos (a + 3) = cos a cos 3 - sin a sin 2. (c) cos (a - ) = cos a cos, + sin a sin 3. (d) Adding (a) and (b), sin (a + 3) + sin (a - 3) = 2 sin a cos 3. [29]. sin a cos P = sin (a + ) + sin (a - ). Subtracting (b) from (a), sin (a+3) - sin (a-3) = 2 cos a sin 3. [30].. cos a sin P =1 sin (a +P)-,- sin (a - ). Adding (c) and (d), cos (a + 3) + cos (a - 3) = 2 cos a cos 3. [31].. csa cos = Cos (a + ) + os (a - ). Subtracting (d) from (c), cos (a +- 3) - cos (a - 3) = -2 sin a sin 3. [32].'. sin a sin p = - cos (a + P) + - cos (a -, ). EXERCISES 1. Show that sin 4 0 cos 2 = 0 sin 6 0 + ' sin 2 0. Proof. Applying [29], where a( = 4 0 and / = 2 0, sin 4 0 cos 2 0 = sin (4 + 2 ) + sin (4 - 2 0) = sin 6 0+ sin 2 0. In the following exercises verify the indicated relations. 2. cos8 0 sin 20 = sin 10 0 -sin6 0. 3. cos2 sin4 0 = sin6 0 + sin2. 4. cos80cos4 = cos12 + cos 4 0. 5. sin 10 sin 6 = - cos 16 0 + 2 cos4. 6. sin2 cos 0 = - cos 3 0 + cos 0. Proof. By [19], sin 2 0 = 2 sin 0 cos 0. Then sin 0 cos 0 = ~ sin 2 0, therefore sin2 0 cos 0 = sin 0 (sin 0 cos 0) = ~ sin 2 0 sin 0 = ~ [ - cos (2 0 + 0) + ~ cos (20 - 0)], by [32], -cos 3 0 + cos0. 7. sin2 0cos2 = I (1 - os 4 ). Proof. sin2 0 cos2 0 = (sin 0 cos 0)2 = (~ sin 2 0)2 = sin22 20. I - cos 460 But sin 2 2 0 \/by [22]. sin20 cos2 0 = /1 (/ 04 (1 - cos 4 0). 8. sin2 0 cos3 0 = 1 (cos - I Cos 3 0 - cos 5 0). 9. sin3 0 cos3 0 = 13 (3 sin 2 0 - sin 6 0). 10. cos3 0 = - (cos 3 0 + 3 cos 0). 98 PLANE TRIGONOMETRY EXERCISES In the following, when an identity is to be proved, transform one member into the other by the application of methods previously suggested. 1. (sin 0 - cos 0)2 = 1 - sin 2 0. 2. V2 sin (0 + 45~) = sin 0 + cos 0. 1 - cos 0 t 0 3. T c = tan2 -I d- cos 0 2 4. sin (a -+ 3) sin (a - 3) = sin2 a - sin2 = cos2 - cos2 a. 5. cos (a + F) cos (a -3) = cos2 a - sin2 3 = cos2 - sin2 a. 6. If tan a = (x - 1) and tan, = (x + 1), prove that 2 cot (3 - a) =2. cot2 2 + 1 7. --- = sec 0. cot2 -1 1 + tan - cos O 2 1 1-sin 0 0 1 - tan - 2 1 + sin 2 0 + cos 2 0 ot 1 - sin2 0 - cos22 0 10 os3 0 + sin3 10 2 - sin 0 cos I 0 + sin ~ 0 2 sin 3 x cos 3 x 11. - =2. sin x cos x 12 cosx /csc + 1. 2 V/sin x / - sin x 2 sin 4 x 13. sin x cos3 x - cos x sin3 x = -. 4 sin 7r co s 7r 14. si6 COS 6 = 2. sin - r cos 6 T tan2 (45~ + ) - 1. 15. tan 2 sin 0. 1 tan2 (450 + I 0) + 1 2tan 0 1 1 + tan2 0 csc 2 0 17. (1 cot2 )3 = 8 CS3 2 x. cot3 x 18. sin6 0 + cos6 0 = sin4 0 + cos4 0 - sin2 0 cos2 0. 19. sin2 a tan a + cos2 a cot a + 2 sin a cos a = sec a csc a. 20. se2 a - sos2 a =sin2 a (1 + cos2 a) 20. sec2a-cos2a = c2 COS" ao 1 + sin a sin a 21. sin + sn = sec2 a (csc a +1). sin a 1 - sin a 22. sec4 a - sec2 a = tan2 a + tan4 a. 23. Given al2 cos 0 + bI cos 0 = cO, and an22 - bnl = c 0 prove that anI2 = c cos 0 FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 99 Suggestion. Multiply the first by and add the second. cos 0 24. Eliminate 0 in the following pair of equations: x sin 0 - y cos 0 = VX2 + y2. sin2 0 Cos2 0 1 X2 Y2 + b Ans. - + =1 a b2 +Y2 *a2 b' Suggestion. Square the first equation and collect the terms in X2 and y'. y This gives the square of x sin 0 + y cos 6=0. Then tan 0= -. From this find sin 0 and cos 0 and substitute in the second equation. 25. Given P = W sin 0, P cos 0 = x sino0; show that P2 = W - X2. 26. Given a sin 0 + b cos 0 = c, and a cosO0 - b sin 0 = d; eliminate 0. Ans. a2 +- b' = C2 + d2. 1 2 sin a sini3 27. Show that if tan (0 —a) tan (0-fl) = tan2 0, 0 = -tan-' 2 sin (a +) 28. Show that sin-' 4 + sin-' J- + sin-' 14 3 4. Suggestion. Find the sine of the first two angles and then combine with the third. 29. Given sin —' 2 x - sin-' x V3 = sin-' x;_find x. Ans. x = 0 or L4. 30. Prove that sin x = 41~2cosx)1 CosX 2 2~~~~~~ 31. Given sin - r - 2 tan-' $ + = a; find x. Ans. x = a. 32. Prove that sin-' T5 + sin-' 3 = cot-' -.a 1 3 5 5 6, /z 33. Given that sin-1 x + sin-' 1 x = find x. Ans. x = 0.505. 34. Given tan a = 4 and tan f = 4; show that tan (, - 2 a) = -. 1 + tan' (450 - a) 35. C 2 a. I - tan' (450 - a) tan (454+ 5 0+l) -tan (450 -4 36. J sin P tan (45~ + ')+ tan (450-4) - nI 37. cot - tan= 2. 8 8 - cos ac + os 2 a 38. sin 2 a - sin a cota. 1a 1 2 a 39. Show that tan-' + tan-' + tan —4 = n~r. i t- a a a a2 cos (a-f) 1~+tan a tanfl cos (a3) 1 -tan a tanj3 4 sin a sin (600 + a) 41. csc(600-a) sin 3 a. cot'08 + tan'08 1 42. = sec 20 —tan 2 0 sin 2 0. cot2 0 - tan2 0 2 2 x - y + 2 - 43. Show that tan-' - tan ' ' = tan-' V3. ySw i 3 =cV3 44. Show that if tan q = A sin 0+B cos 0= A'~T~~ + 'in0ta') 100 PLANE TRIGONOMETRY 4 (b_ a0) 4sin3 a0O 45. V2 -cOs 4 sin b see2 (7 - i 1X) 46. 2t ( -r ) - sec x. 2tan ( r + X) 47. If 2 tan 2 a = tan 2 f sin 2 /, find the relation between the tangents of a and _. sin a - /1 + sin 2 a 48. Prove that (a) = cot a. cos a - V1 + sin 2 a (b) 2cos2 +) = 1-sinx. 49. In any triangle with angles a, 3, y, and opposite sides a, b, c, respectively, show that (1) b = a cos + ccos a. cosa cos/i cos-y a2 + b2 + c2 (2) a + b + -c 2 abc sin a + sinf + sin y a + b + c (3) -(3 -- -sin y c V/1 - x2 1 50. Show that tan-1 + sin-' x = r. x 2 51. Show that vers1 a- - 2 sin-l \/ is equal to a fixed number, no matter what value is assigned to a. 52. Given I = W sin 0, and P cos 0 = W sin 0; show that 1 1 1 Y2 P- + W2 CHAPTER VI OBLIQUE TRIANGLES 75. General statement. -In the present chapter will be developed methods for solving any triangle. As pointed out in Art. 29, it is possible to solve a triangle whenever there are enough parts given so that the triangle can be constructed. The constructions and, likewise, the solutions fall under four cases depending upon the parts given and required. CASE I. Given one side and two angles. CASE II. Given two sides and an angle opposite one of them. CASE III. Given two sides and the included angle. CASE IV. Given the three sides. Since there are six parts to a triangle, and, in each of the four cases, three parts are given, then, in general, there are three unknown parts to be found in solving a triangle. Also, since three independent equations are necessary and sufficient to determine three unknowns, it is necessary to have three independent formulas or relations connecting the parts of a triangle. These three relations are: (1) The sum of the angles of a triangle is equal to 180~. (2) The sine theorem, or the "sine law." (3) The cosine theorem, or the "cosine law." For greater convenience in carrying out the numerical work of the solutions, various other relations are derived from the formulas growing out of the sine theorem and cosine theorem. 76. Sine theorem. - In any triangle the sides are proportional to the sines of the opposite angles. First proof. In Fig. 81, let ABC be any triangle, and let h be the perpendicular from B to AC. The following applies to each of the triangles (a), (b), and (c); but note that in (b) sin ' = sin (180~ - y) = sin -y, and in (c) h = a. h (1) sin a = -. (2) sin a=10 101 102 PLANE TRIGONOMETRY Dividing (1) by (2), there results sin a a a c (3) - = -, or. = - sin y c' sin a sin 7 Similarly, drawing perpendiculars from A to CB, (4) sin 3 b - - = or sin c' b c sin p sin y Hence, uniting (3) and (4), there results a b c [33] - c sin a sin p sin y B /A cc A I D b (a) B?y Y I b CG (b) FIG. 81. B Second proof. In Fig. 82, let ABC be any triangle. About the triangle circumscribe a circle. Let 0 be the center. Draw the radii OA, OB, and OC. Draw OD B B^-^ ~ perpendicular to AC. Then Z AOD = f or is the supple'/ / \ \ ment of f. \ a \ In triangle AOD, C 7 ~ \ AD=AO sinZAOD. \.. / \.'.- ~/\ /*.2b = r sin f. In a similar manner, ~ c = r sin y, and a = r sin a. FIG. 82. These give a b c sin a sin 3 sin y COROLLARY. The constant ratio of a side of the triangle to the sine of the opposite angle is equal to the diameter of the circumscribed circle. OBLIQUE TRIANGLES 103 EXERCISES b c 1. Derive the proportion = sinm3 sin Y a ~ 2. Derive 2 r = also 2r = sin sin 7 a b 3. Solve = for each part involved. sin a sin 3 77. Cosine theorem. - In any triangle the square of a side equals the sum of the squares of the other sides minus twice the product of these sides by the cosine of their included angle. Proof. In each triangle of Fig. 81, a2 = h2 + DC2. But h2 = c2- AD2 and DC2 = (b - AD)2. (Notice that in (a) DC is positive, in (b) negative, and in (c) it is zero because D falls on C.) a2 =c2- AD2 + (b -AD)2 = c2- AD2 + b2 - 2b.AD + AD2 = c2 + b2 - 2 b.AD. But AD = c cos a, [34]. a2 = b2 + c2 - 2 be cos a. By cyclic change, [342] b2 = a2 + 2 - 2 ac cos P, [34] C2 = a2 + b2 - 2 ab cos y. EXERCISES 1. Are the formulas [341], [342], and [343] adapted to solving by logarithms? 2. Derive [342] and [343] independently. 3. Solve each of the three formulas for the angles in terms of the sides. 4. Solve a2 = b2 + c2-2 bc cos a for b. Ans. b = c cosa \V /a2-c2 sin2 a. 78. Case I. The solution of a triangle when one side and two angles are given. - In this case, it is evident that the third angle can always be found from.the equation a+ +y = 180~. 104 PLANE TRIGONOMETRY The sides can then be found by using the relations stated in the sine theorem, namely, a b a c b c =- = and = sin a sin f' sin a sin y' sin f sin y Since in each of these there are four parts of the triangle involved, therefore, if any three of these parts are known, the fourth can be found. Also since, in any example, only two sides are to be found, two of the above relations may be used for solving and the third for checking. The same suggestions as were given in Art. 32 for the solution of right triangles should be carried out here. Draw the triangle, state the formulas, make out a careful scheme for all the work, and, lastly, fill in the numerical part by the use of the tables. Remember that in computations time and accuracy are of very great importance. Time will be saved by carefully planning the arrangement of the work. Accuracy can be secured by checking the work at every step. Verify at every step the additions, subtractions, multiplications, and divisions. Check interpolations when using tables, by repeating the work at each step. From geometry, the area of a triangle equals one half the product of the base and altitude. Using b for base, h for altitude, b sin and K for area, K = bh. But h = c sin a, and c = i. n sin p ~~~~[35] K = ~b2 sin a sin y [2 sin P Example. Given a = 53~ 23.7', y = 75~ 46.3', and a = 27.64; find f, b, and c. Solution. Construction B a = 530 23.7' Given -y = 75~ 46.3' a = 27.64/ / = 50~ 50' To find* b = 26.695 c = 33.375 / A, C * Values to be put in after solving. FIG. 83. OBLIQUE TRIANGLES 105 Formulas a + + 7 t = 180~ a b sin a sin 3 a c sin a sin -y.. = 180~ - (a + - ). a sin.'. b= sin a a sin y sin ao Logarithmic formulas log b = log a + log sin F + colog sin a. log c = log a + log sin y + colog sin a. Computation = 180~ - (53~ 23.7' + 75~ 46.3') = 50~ 50'. log a = 1.44154 log a = 1.44154 log sin 1 = 9.88948 log sin y = 9.98647 colog sin a = 0.09541 colog sin a = 0.09541 log b = 1.42643 log c = 1.52342 b = 26.695 c = 33.375 EXERCISES 1. Given a, i, and c; to find 7, a, and b. Give formulas and scheme for solution. 2. Give the formula for area when a is the given side. When c is the given side. 3. Given a = 60~ 25' 31", = 69~ 26', and c = 173; find a = 196, b = 211, and y = 50~ 8' 29". 4. Given / = 43~ 44' 18", y = 75 2' 42", and b = 81.5; find a = 61~ 13', a = 103.32, and c = 113.89. 5. Given a = 11~ 11' 18", y = 57~ 37' 24", and c = 444.79; find a = 102.19, b = 491.06, and 3 = 111~ 11' 18". 6. Given f = 20~ 20.2', y = 12~ 28.5', and a = 673.75; find b = 432.13, c = 268.58, and a = 25~ 19.9'. 7. Given a = 28~ 14' 44", 7 = 109~ 32' 30", and b = 730.8; find 3 = 42~ 12' 46", a = 514.74, and c = 1025.0. 8. Given f = 102~ 38' 16", -r = 20~ 3' 8", and b = 479.36; find a = 57~ 18' 36", a = 413.45, and c = 168.44. 9. Given 3 = 30~ 36' 48", 7r = 107~ 15' 30", and b = 14.397; find a = 42~ 7' 42", a = 18.964, and c = 26.998. 79. Case II. The solution of a triangle when two sides and an angle opposite one of them are given. - It is known from 106 PLANE TRIGONOMETRY geometry that when two sides and an angle opposite one of them are given the triangle may not be uniquely determined. With these parts given: (1) it may not be possible to construct any triangles; (2) it may be possible to construct just one triangle; (3) it may be possible to construct two triangles - the ambiguous case. EXERCISES Construct carefully the following triangles: 1. (a) a = lin.,c=3in., and a =40~. (b) a = 2 in., c = 3 in., and a = 140~. 2. (a) a = 1 in., c = 2 in., and a =30~. (b) a = 3 in., c = 2 in., and a = 35~. (c) a = 3 in., c = 2 in., and a = 120~. 3. a = 2 in., c = 3 in., and a = 30~. a c c \a (1)(a) b ac- Va (2)( c (2)(a) I' cc cc (1) (b) a (2) (b) (2)(b) b a (3) CP7Y a b' 8 FIG. 84. Corresponding to Exercises 1, 2, and 3 above, we have the following which should be compared with the corresponding constructions in Fig. 84. (1) No solution when: (a) Angle is acute and opposite side less than adjacent side times the sine of the angle. (b) Angle is obtuse and opposite side not greater than adjacent side. OBLIQUE TRIANGLES 107 (2) One solution when: (a) Angle is acute and opposite side is equal to adjacent side times the sine of the angle. This gives a right triangle. (b) Angle of any size and opposite side greater than adjacent side. (3) Two solutions when angle is acute and the opposite side greater than the adjacent side times the sine of the angle, and less than the adjacent side. The ambiguity of (3) is also apparent from the solution of y found from the relation sin = sin. This equation has two values of y less than 180~ a each of which may enter into the triangle when a is acute. With each of these values of 7 there may be found values of 3 and b, thus making two triangles. When logarithms are used, proper conclusions can be drawn from the following, where a, b, and a are given. For other given parts, the proper change can easily be made. If log sin 3 = 0, sin 3 = 1, / = 90~; hence a right triangle. If log sin 3 > 0, sin 3 > 1 which is impossible; hence no solution. If log sin 3 < 0 and b < a, and therefore / < a, only the acute value of /3 can be used; hence there is one solution. If log sin / < 0 and b > a, both acute value of P and its supplement may be used; hence there are two solutions. If the given parts are a, c, and a, with a acute and a < c, the formulas for the solution are: a c si n = i-, which gives two values for -y, say, y and -y'; sin a sin 7 J = 180 - (a + y); '= 180~ - (a + y'); b a b. a b - a gives b;. _ - gives b'; sing sin a sin' sin a b c. b c or. = gives b; gives b. sinm sin 7 sin/3 sin a The area K can be determined as follows: suppose b, c, and 7 are given. Then K = bh = ~ bcsin a, anda = 180~- + 7) where g can be determined from sin f = 7 c Example. Solve the triangle when a = 11.75, c = 15.61, and a = 34~ 15.3'. 108 PLANE TRIGONOMETRY Solution. Here a is acute, a < c, and are two solutions. a > c sin a, hence there Construction Given To find a = 11.75 c = 15.61 a = 340 15.3' y = 48' 23.9' 1 = 97' 20.8' b = 20.704 yy = 1310 36.1' 3' = 14' 8.6' b' = 5.1008 FiG. 85. Formulas a c c sin a sin y=y in siny'. sina sin - a 13=1800-(ay); 3/ = 1800 - (a+.y'). b a a sin sin sina sin a bl a a sin 3' b' s sin p'sin a sin. a Logarithmic formulas log sin y = log c + log sin a ~ colog a = log sin y'. log b = log a + log sin 3 + colog sin a. log b' = log a + log sin 1' + colog sin a. Computation log c = 1.19340 log sin a = 9.75041 colog a = 8.92996 log sin'y = 9.87377 7 = 48' 23.9' 7/ = 131' 36.1' 1 = 970 20.8' 13 = 14' 8.6' log a = 1.07004 log sin 1 = 9.99642 colog sin a = 0.24959 logb = 1.31605 b = 20.704 log a = 1.07004 log sin 1' = 9.38801 colog sin a = 0.24959 log b' = 0.70764 b' = 5.1008 OBLIQUE TRIANGLES 109 EXERCISES 1. Given a = 86.425, c = 73.463, and y = 49~ 18.7'; find a = 63~ 8', / = 67~ 33.3', b = 89.544, and K = 2934.1. 2. Given b = 223.46, c = 327.92, and - = 116~ 19.6'; find a = 26~ 1.7', /3 = 37~ 38.7', and a = 160.55. 3. Given b = 32.492, c = 52.392, and 3 = 27~ 49.1'; find a = 67.736, a = 103~ 22.5', y = 48~ 48.4', a' = 24.938, a' = 20~ 59.3', and -' = 131~ 11.6'. 4. Given a = 67.291, c = 110.97, and a = 37~ 19.8'; find 7 = 90~, / = 52~ 40.2', and b = 88.236. 5. Given b = 45.872, c = 56.321, and 3 = 20~ 14' 18"; find a = 94.374, a = 134~ 37' 44", - = 25~ 7' 58". a' = 11.314, a' = 4~ 53' 40", and 7' = 154~ 52'2". 6. Find the areas of the triangles given in Exercises 3 and 4. 7. Given a = 57.147, b = 46.703, and 3 = 19~ 17' 42"; find a = 23~ 50' 54", y = 136~ 51' 24", c = 96.652. a' = 156~ 9' 6", 7' = 4~ 33' 12", and c' = 11.221. 8. Given a = 15.8, b = 17.9, and a = 21 17' 22"; find B = 24~ 17' 18", y = 134~ 25' 20", c = 31.08. -' = 155~ 42' 42", r' = 2~ 59' 56", and c' = 2.277. 9. Given a = 36~ 42.7', c = 45.63, and a = 23.62; find b, 3, and 7. 80. Case III. The solution of a triangle when two sides and the included angle are given. First method. - Let the given parts be a, b, and y. Then from the cosine law, c = /a2 + b2 - 2 ab cos 7, and a and: may be found from a sin y b sin r sin = a- and sin p =,respectively. C c As a check, a + A + y = 180~ may be used. The area K = ~ hb = ~ ab sin y; or, in words, the area equals one half the product of the two sides and the sine of the included angle. [36] K = ab sin y. It is evident that the formula for finding c is not adapted to the use of logarithms. However, this method is often convenient when the numbers expressing the sides contain few figures or when only the third side is to be found. 110 PLANE TRIGONOMETRY EXERCISES Solve the following triangles by the formulas of this article without logarithms. 1. a = 3, b = 4, and - = 42~ 37.6'. 2. a = 5, c = 9, and 3 = 106~ 31.4'. 3. b = 27, c = 81, and a = 63~ 42' 32". 81. Case III. Second method. - For a solution by logarithms the following theorem is needed: TANGENT THEOREM. In any triangle the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. a sin a Proof. a = sin a from sine theorem. b sin f3' T a+b sin a + sin 8 Thensin a -sin ' by a proportion theorem, a - b sin a - sin p' 2 sin (a + ) cos (a- by [25] and [26 2 cos (a + ) sin (a - tan1 (a +) cot (a - ) tan ( 3) tan2 2 = tan~ (a- f)'.a + b _ tan ( (a + P) [37] a - b tan ( - ) This can be put in another form for a + A = 180~- y and (a + ) = 90~- 17 tan ( ) ta n an (90~ - n 7) = cot ~ 7. Substituting this in [37] gives 1 a-b 1 [38] tan (a_- p) a b cot ly 2 a + b 2 Formula [37] or [38] makes it possible to find I (a - 3) when a, b, and y are given, while ( (a + f) can readily be found because 2 (a + /) = 90~ - 7, therefore a and 3 can be found from the relations: a= (a + ) + 2 (- ), and P = (a + )- (a- A). It is evident now that the other side can be found by the sine theorem, which may also be used as a check. OBLIQUE TRIANGLES 111 A discussion similar to the above can be given when any two sides and the included angle are given. A convenient set of formulas for solving the triangle when a, b, and y are given is (a- + B) = 900- -,. 1 / ^ ba- b tan (-a -) cot 2, 2 a +b = 1 (a + ) + ~ (a - ), = (a+ )- (a - ), a sin - b sin 7 sin a sin 3 It should be noted that negatives are avoided if the larger angle and side come first in [38]. Thus, if A > a and hence b > a, b-a 1 write [38] in form tan ~ (3 - a) = - cot y. 2 6b+a ' Exercise. Derive formulas like [38] for finding tan (a,- 7); for tan ~ (7 - /). Example. Solve the triangle when a = 42.367, c = 58.964, and 3 = 79~ 31' 44". Solution. Construction a = 42.367 B Given c = 58.964 / = 79~ 31'44" c/ \ b = 66.057 To find a = 39~6' 1" A Y = 61~ 22' 15" b FIG. 86. Formulas 1 1 1 c-a 1 ( + a) = 90~ —, tan (7-a)c a cot 2, a sin p b sin y b = -, and for a check c = - sin a sin 6 112 PLANE TRIGONOMETRY Computation c = 58.964 a = 42.367 c - a = 16.597 c + a = 101.331 1( col log tan 1 = 39~ 45' 52" og (c - a) = 1.22003 og (c + a) = 7.99426 log cot /3 = 0.07981 2 (y - a) = 9.29410 (7 - a) = 11~ 8' 7" (Qy + a) = 50~ 14' 8" y = 61~ 22' 15" a = 39~ 6' 1" log a = 1.62703 log sin f = 9.99270 colog sin a = 0.20019 logb = 1.81992 b = 66.057 logb = 1.81992 log sin y = 9.94337 colog sin 3 = 0.00730 logc = 1.77059 c = 58.964 82. Case III. Third method. - Among several other methods for solving Case III, the following is selected because it offers some advantages over the second method. Take as the given parts a, b, and 7. Choose the auxiliary angle x so that a tanx = - b mtanx - 1 a-b Then t —an -a b by a proportion theorem. tan x +- 1 a + b ' But tan x - 1 tanx +1 tan (x - 450), by [18]. tan x q- 1 a-b + b =tan( x -450). Substitute this in [37] and tan (a - /3) = tan (x - 45~) tan 2 (a + /). In choosing tan x, put the larger of the given sides in the numerator, and thus make x > 45~. This avoids the negative angle in tan (x - 450). Example. Solve by this method the example given under the second method. OBLIQUE TRIANGLES 113 Formulas tan 1 ( - a) = tan (x - 45~) tan (y + a) and tan x = - 2 2 a log c = 1.77059 log tan (x - 45~) = 9.21429 log a = 1.62703 log tan ~ (y + a) = 0.07981 log tan x = 0.14356 log tan ~ (7 - a) = 9.29410 x = 54~ 18' 7" (7- a) = 11~ 8' 7" x - 45~ = 9~ 18' 7" ( + a) = 50~ 14' 8" 7 = 61~ 22' 15" a = 39~ 6' 1" This method can be used to good advantage when the triangle is in a series of triangles where the logarithms of c and a have already been found. This saves finding c + a and c - a and the logarithms of them. EXERCISES 1. Given a-= 49.366, b = 26.437, and y = 47~ 18.6'; find c = 36.962, a = 100~ 58.3', 3 = 31~ 43.1', and K = 479.65. 2. Given a = 283.4, b = 268.5, and y = 630 38' 11"; find c = 291.25, a =60~ 40' 26", and P = 55~ 41' 23". 3. Given b = 247.81, c = 513.48, and a = 137~ 8' 49"; find a = 715.25, = 13~ 37' 43", and y = 29~ 13' 28". 4. Given a = 36.518, b = 8.8196, and - = 132~ 18'; find c = 42.954, a = 38~ 57' 52", and / = 8~ 44' 8". 5. Given a = 67.375, c = 26.858, and / = 20~ 20.2'. find b = 43.213, a = 147~ 11.3', and y = 12~ 28.5'. 6. Find the areas of the triangles in Exercises 3 and 5. 7. As in Case III, third method, is it always possible to choose an angle x such that tan x = b? Show why. 83. Case IV. The solution of a triangle when the three sides are given. - In this case the angles can be found by means of the cosine theorem, from which the following formulas are derived: b2 + c2 - a2 COS ao = — 2 bc a2 + c2 - b2 cos P = 2-ac -' 2 ac a2 + b2 - c2 cos 2ab ' 2 ab 114 PLANE TRIGONOMETRY These formulas give the cosines of the angles and therefore the angles; but they are not adapted to logarithms and so are convenient only when the sides are expressed in numbers of few figures, or when a table of squares is at hand. EXERCISES Find the angles when the sides are given as follows: 1. a = 2, b = 3, and c = 4. 2. a = 10, b = 8, and c = 3. 3. a=5, b=6, andc =7. 4. a = 12, b =21, and c = 14. 84. Case IV. Formulas adapted to the use of logarithms. — b2 + c2 - a2 (a) Start with the equation cos a = 2b and subtract each 2 be member of it from 1. This gives b2 + C2 - a2 1 - cos a = 1- - 2 bc 2 1 a2 - (b2 - 2bc + c2)_ (a - b + c) (a + b- c).. 2sin2 - = 2sn2 22bc 2bec Let a+b+c==2s. Thena-b+ c = 2 (s-b),and a + b-c =2 (s - c). Substituting these values in the above, 1 2 2 (s - b)2(s- c) 2sin2,- 2bc= ~2 ___ _2 bc' (s — b) (s — c) [391]. sin 2 a = In like manner are obtained [3921 sin=1 /(s - a) (s - c) [393] sin - P = ' -b 2 ac 9 I(s - a)(s - b) [3931 sin b2 + c2 —a2 (b) By adding each member of the equation cos a = 2 bc to 1, and carrying out the work in a manner similar to the above, there are obtained the following: [401 Cos CL _]a 2 be [402] cosa = s (\ - b) [403] yosy= (8 -) = OBLIQUE TRIANGLES 115 (c) By dividing each formula of the set under (a) by the corresponding formula of the set under (b) there results 1. CC /(s -b) (s -c) [411 tan -a ) 2 s (S - a) (s - a) (s - c) [4121 tan - P 2 s (s - b) [413] tan 1-y = a) = - b) 2 S31 ~tan 2Y8 (S - C)) These last three can be put in a form slightly more convenient. Since (S - b) (s - c) (s - a) (s - b) (s - c) s (s - a) s (s - a)2 = si V(S - a) (s - b) (s - c) s - a s by writing r=V a) (s b)(s c) there results r [421] tan cL = 2 s - ac Similarly there result r. [422] tan r P= - 2 s- b r [42] tan1 S - C In using any of these sets of formulas, the work may be checked by 2~ - a + +Z=y = 900. The area can be found from K = b bh = bc sin a = bc sin - a cos a -=bcV( - b)-(s - c) a) - Vs(s-a) (s - b) (s - c). = be~~b [43].'. K s ~ S- ~ s-c. * Formula [43] was discovered by Hero (or Heron) of Alexandria about the beginning of the Christian era. 116 PLANE TRIGONOMETRY Since the sine varies most rapidly for small angles, and the cosine most rapidly for angles near 90~, formulas [39] should be used when the angles are small, and [40] when the angles are near 90~. In all cases the tangent varies more rapidly than either sine or cosine. Hence formulas [41] or [42] are always more nearly accurate than [39] or [40]. Again formulas [41] or [42] are more convenient since, for a complete solution of the triangle, they require only four logarithms to be taken from the table; while [39] and [40] require, respectively, six and seven. B b F FIG. 87. Formulas [42] may be derived by taking from geometry the fact that the area of a triangle, when the three sides are given, is K = Vs (s - a) (s - b) (s - c); and, from Fig. 87, K = sr, where r is the radius of the inscribed circle.. sr = s (s - a) (s - b) (s - c), a(- a) (s - b) (s - c) and r V S Also AF+ EC + EB = s. AF = s - (EC + EB) = s - a. But tana = AF1 r tan a = - 2 s - a OBLIQUE TRIANGLES 117 EXERCISES 1. Derive sin = V a) (s b) from the cosine theorem. 2 ab 2. Derive cos 2 = / - b8 ) from the cosine theorem.,~2. Drveos =ac 3. Derive K = /s (s - a) (s - b) (s - c) by geometry. 4. What is the tabular difference for each of log sine, log cosine, and log tangent when the angle is near 11~? How accurately can the angle be found from each? 5. Answer the same questions for 82~ and 46~. 6. In Fig. 87, show that BE = s - b. 7. Can s - a be less than O? Show why. 1 (s - b) (s - C)? 8. How many values of a will satisfy sin = /( — )? 9. In solving the triangle when two sides and an angle opposite one of them are given, an ambiguity was introduced because from the sine of the angle two values of the angle were found. Why is there not an ambiguity when formulas [39] are used? 10. Solve for the angles when a = 23.764, b = 42.376, and c = 31.166. Solution. Construction Use formulas [42] with that for r. a = 23.764 b = 42.376 c = 31.166 2 s = 97.306 s = 48.653 s - a = 24.889 s -b = 6.277 s- c = 17.487 2 s = 97.306 A check. B c \ A /M \C b FIG. 88. log (s - a) = 1.39600 log (s - b) = 0.79775 log (s - c) = 1.24272 colog s = 8.31289 log r2 = 1.74936 log r = 0.87468 log tan ~ a = 9.47868 a = 16~ 45' 21" log tan 1 0 = 0.07693. 1 = 50~ 2' 53" log tan ~ 7 = 9.63196. Y. 7 = 23~ 11' 45" Check. 2 a + I / + - 7 = 89~ 59' 59". Remark. The sum of s, (s - a), (s - b), and (s - c) is 2 s, and hence is a check on the additions and subtractions. 118 PLANE TRIGONOMETRY To facilitate the subtractions, write the values of s on the margin of a slip of paper, when it can be placed above the values of a, b, and c, successively. In like manner log r can be written on a margin and placed above logs of (s - a), (s - b), and (s - c). 11. Solve 10 by using formulas [39]. By using formulas [40]. Compare the work with that in 10. 12. Given a = 125.26, b = 176.23, and c = 91.23; find 2a = 21~ 16' 21", 3i = 53~ 58' 33", 7 = 14~ 45'6', and K = 5435.6. 13. Given a = 24.568, b = 24.743, and c = 10.047; find a a = 38~ 38', - /i = 39~ 36' 42", and ~ 7 = 11~ 45' 18". 14. Given a = 10.037, b = 9.4367, and c = 15.067; find a = 40~ 46' 54", 3 = 37~ 53' 18", and 7y = 101~ 19' 48". 15. Given a = 22.438, b = 24.692, and c = 31.256; find a = 45~ 26' 4", 3 = 51~ 37' 42", and 7 = 82~ 56' 18". 16. Given a = 7252, b = 5684, and c = 10,012; find a = 45~ 22' 20", 3 = 33~ 54' 14", and y = 100~ 43' 28". 17. Given a = 0.011, b = 0.015, and c = 0.024; find a = 19~ 11' 18", 3 =.26~ 37' 38", and 7 = 134~ 11' 4". EXERCISES a2 sin /3 sin y 1. Prove that in any triangle K = 2 sin (3 + ) 2. Solve the following triangle for the parts not given: K = 7298.1, a = 370, and 3 - 7 = 13~. Ans. 3 = 78~; 7 = 65~; a = 99.543; b = 161.79; c = 149.91. 3. Use the corollary of Art. 76 and the formula K = ~ ab sin 7, and abe show that the radius of the circumscribed circle is given by R = - Also 4 K abc show that K =a. 4R 4. Find the area of a triangle with sides 12.5 ft. and 17.65 ft. and included angle 101~ 45' 16". Ans. K = 108 sq. ft. 5. Find the area of a triangle with the three sides respectively 46.45 ft., 27.3 ft., and 32.75 ft. Ans. 438.89 sq. ft. 6. Two sides of a parallelogram are 46.3 rd. and 76.45 rd. respectively, and the included angle is 75~ 46'. Find the area. Ans. 3431 sq. rd. 7. The angles of a triangle are in the ratio of 2: 3: 5; and the longest side is 105 ft. Solve the triangle. Ans. Angles, 36~, 54~, 90~; sides, 61.717, 84.948. 8. The sides of a triangle are in the ratio of 7: 6: 9; find the sine of the smallest angle. The cosine of the largest angle. Ans. 0.66592; 0.04762. 9. The base of a triangle is 825 ft. and the two angles at the base are, respectively, 102~ 46' and 43~ 21'; find the other two sides and the area of the triangle. Ans. 1015.8 ft.; 1443.2 ft.; 408,660 sq. ft. 10. Two angles of a triangle are, respectively, 55~ 49' 13" and 75~ 13' 41". If the included side is 14.31 in., find the area. Ans. 108.6 sq. in. OBLIQUE TRIANGLES 119 11. The sides of a triangular field of which the area is 10 acres are in the ratio of 3: 5: 7. Find the length of the shortest side. Ans. 47.086 rd. Solve the following three exercises without the use of logarithms: b 12. Given a = 21, b=26, and c =21; J '( find a, 3, and -. C 13. Given a = 73, b =82, and c =91; find a,, and y. 14. Given a = /0, b = /14, and c= B i-7; find a, 3, and y. FIG. 89. 15. To find the distance AB through the swamp in Fig. 89, the following measurements were made: a = 102 rd., b = 145 rd., and angle C = 41~ 25'. Compute the distance AB. Ans. 96.87 rd. 16. At a certain point the length of a lake subtends an angle of 47~ 43.6', and the distance from this point to the two extremities of the lake are 144 rd. and 87.5 rd. respectively. Find the length of the lake. Ans. 106.96 rd. FIG. 90. 17. Compute the inaccessible distance AB, Fig. 90, from the measurements, b = 450 ft., angle A = 82~ 30', and angle C = 57~ 42'. Ans. 594.23 ft. 18. In a parallelogram, given a diagonal d and the angles a and (3 which this diagonal makes with the sides; find the sides. Given d = 14.67, a = 23~ 24.3', and, = 36~ 47.4'; find the sides. Ans. 10.125; 6.7157. 19. In a parallelogram are given a side a, a diagonal d, and the angle 0 between the diagonals; find the other diagonal and side. Find the other diagonal when a = 15.42 in., d = 19.23 in., and 0 = 430 16' 14". Ans. 41.923 in. 20. In a trapezoid, given the parallel sides a and b, and the angles a and 3 at the ends of one of the parallel sides; find the non-parallel sides. Find these sides if a = 16.8 ft., b = 9.4 ft., a = 71~ 25', and,3 = 42~ 46'. Ans. (a - b) sin (a - b) sin; 76888 ft.; 5.5081 ft. sin (a o- +) ' sin (a + ) 120 PLANE TRIGONOMETRY 21. Show that the area of any quadrilateral is equal to one half the product of its diagonals and the sine of the included angle. 22. One side of a parallelogram is 46.4 rd., and the angles which the diagonals make with that side are 36~ 30' and 54~ 25'. Find the length of the other side. Ans. 47.112 rd. 23. Two circles whose radii are 14 in. and 18 in. intersect. The angle between the tangents at the point of intersection is 29~ 25'. Find the distance between their centers. Ans. 8.8995 in. or 30.997 in. 24. When light passes from a rarer to a denser medium the index of resin i fraction y is determined by the relation, = si- where i and i' are the angles between the ray of light, in the rarer and denser mediums respectively, and the line vertical to the separating surface. When /, = 1.167, and i = 18~ 30', find i'. When, = 1.2 and the deflection is 12~, find i and i'. Ans. 15~ 46' 39"; i = 55~ 8' 33". 25. B is 42 miles from A in a direction W. 22~ N., and C is 58 miles from A in the direction E. 73~ N. What is the position of C relative to B? Ans. 68.581 mi. E. 35~ 24.1' N. 26. In Fig. 91, find the height CD = x, and the distance AC = y of an inaccessible object, having measured on a horizontal plane the distance a in the line CAB and the angles a and 3. Suggestion. AD = sn si sin (a -/) CD = AD sin a sin a3 sin (a - p) a cos a sin 3 AC = AD cos a = in 3 -sin (a + (3) P D,J / / A o / - C aX /, 2 B C/3J a A Y B FIG. 91. FIG. 92. 27. In Fig. 92, the point P is an inaccessible object above the horizontal plane ABC. The straight line AB = a is measured, also the angles a, /, 0, and 0. Find the height x of the point P above the plane, giving two solutions which will check each other. State the result in the form a sin / tan _ a sin a tan 0 sin (a + ) sin (a + 3) OBLIQUE TRIANGLES 121 28. In Exercise 27, given a = 465 ft., a = 49~ 51' 47", / = 52~ 46' 30", q = 39~ 16' 14", and 0 = 40~ 25' 5"; find x. Could this exercise be solved if 0 were not given? Ans. 310.26. 29. Two observers at A and B, 100 rods apart on a horizontal plane, observe at the same instant an aviator. His angle of elevation at A is 68~ 25', and at B 55~ 58.2'. The angles made by the projections of the lines of sight with the line AB are 43~ 27' at A and 23~ 45' at B. Find the height of the aviator. Ans. 1822 ft. 30. Two points P and Q, Fig. 93, are on opposite sides of a stream and invisible from each other on account of an island in the stream. A straight line AB is run through Q and the following measurements taken: AQ = 756 ft., QB = 562 ft., angle QAP = 47~ 28.6', and angle QBP = 57~ 45'. Compute QP. Ans. 851.77 ft. A B / -. B I C FIG. 93. FIG. 94. 31. To measure the height x of an inaccessible tower, Fig. 94, the line a in the horizontal plane through the foot of the tower and the angles a, 3, and y are measured. Show that a sin a sin y sin (3 + y) 32. Two railroad tracks intersect at an angle of 65~ 47'. At a certain time a train going 30 miles an hour passes the point of intersection; two minutes later a train going 60 miles an hour on the other track passes this point. Write a formula showing their distance apart t minutes after the first train passes the intersecting point. How far will they be apart in 22 minutes? Ans. 18.453 mi. 33. P and Q, Fig. 95, are two inaccessible A \\ \ points both of which are visible from one point A only. Show what measurements may be made so that the distance PQ can be computed. 34. A statue, of height 2 h, standing on the 9 top of a pillar, subtends an angle a at a point FIG. 95. on the ground distant d from the foot of the pillar. Prove that the height of the tower is V/h2 + 2 hd cot a - d2 - h. 122 PLANE TRIGONOMETRY 35. A flagstaff 40 feet tall stands on the top of the end of a building 90 feet high. At what distance from the base of the building will the flagstaff subtend an angle of 10~? P Ans. 79.294 ft. or 147.56 ft.,< 36. From a point h feet above the surface of a lake the angle of elevation of the top of o' / a tree, standing at the edge of the water, is ca;, / x and the angle of depression of its reflection in // / / the water is A. Prove that the height of the tree /X. / o ^. hsin( +a) A 2 / < sin ( -a) "d_' 7 ' C_ \ 37. In taking measurements for finding e7 a i/ e/',/ the height of P, Fig. 96, above the horizontal, 7'/ ~line AC, a line AB = a was measured in a \B,7/ plane making an angle DAB = - with the \Bw/ horizontal. Other angles measured were D ZDAC = a, zADC = f, zCAP = ~, and FIG. 96. ZEBP = 0. Find the height x that P is above C, and put in the form a cos sin 3 tan 0 a cos sin a tan 0 sin (a +/) sin ( /a +, s 38. In Exercise 37, given a = 127 ft., a = 47~ 16' 33", fi = 64~ 43' 42", ~ = 59~ 47' 15", y = 4~ 16' 30", and 0 = 63~ 37' 32"; find x. Ans. 211.84 ft. 39. Compute the inaccessible distance PQ, Fig. 97, when given the line AB = a and the angles a, 3, y, and 6. Are the data sufficient for a check? 40. In Exercise 39, given a = 250 ft., a = 41~ 36.5', 3 = 62~ 43.7', y = 39~ 47.6', and a = 37~ 53.8'; find PQ. Ans. 266.39 ft. P AA ~ ~ ~ A P y x C i Q A E- D FIG. 97. FIG. 98. 41. From the data given in Fig. 98, find x and y in the forms: x = a (cos 8 tan a - tan 5) cos a cos f csc (3 - a), y = a (cos 6 tan a - tan 5) cos a sin f csc (3 - a). OBLIQUE TRIANGLES 42. Given the data as shown in Fig. 99, find the distance x in form: (a + x) (b + x) sin a sin 3 = ab sin (a + a) sin (3 + -y). 123 After numerical values are substituted, this can be solved as a quadratic equation in x. D, a D -F -,%~3' B B b m 4 /~A r C FIG. 99. FIG. 100. 43. Given data as shown in Fig. 100, solve for x and state the result in a formula. Ans. x = m [tan (a + /) - tan 3]. 44. Given data as shown in Fig. 101, solve for x and state the result in a formula. Ans. x = (b sec a - m) sin 3 ~~~~~~~~~~~~cosformula. Ans. x -) cos (a + A) A b x FIG. 102. FIG. 101. 45. Solve for x, using the data given in Fig. 102 and state the result as a formula. Ans. x = tan 2 a - tan c' 46. An automobile is traveling W. 36~ N. at 27 miles per hour, and the wind is blowing from the N. E. at 30 miles per hour. What velocity and direction does the wind appear to have to the chauffeur? Ans. 37.09 mi. per hour from N. 58.3' W. 47. A train is running at the rate of 30 miles per hour in the direction W. 35~ S., and the engine leaves a steam track in the direction E. 10~ N. The wind is known to be blowing from the N. E.; find its velocity. Ans. 22.1 mi. per hr. 124 PLANE TRIGONOMETRY 48. In a river flowing due south at 4 miles per hour a boat is drifted by a wind blowing from the southwest at the rate of 15 miles per hour. Determine the position of the boat after 40 minutes if resistance reduces the effect of the wind 70 per cent. Ans. 2.19 mi. E. 14~ 25' S. 49. An aeroplane rises from a point P and proceeds with uniform velocity in a straight line in a northeasterly direction. From a point Q, 400 ft. east of P, the aeroplane is observed 30 seconds from the start, and its angle of elevation is 15~ 30'. At that time its direction is 26~ 30' east of north. Determine the height of the aeroplane at that instant, and its speed in miles per hour. Ans. 247.2 ft.; 26.25 mi. per hr. 60. A ship S is 10 miles to the north of a ship Q. S sails 12 miles per hour and Q 15 miles per hour. Find the distance and direction Q should sail in order to intercept S which is sailing in a northeasterly direction. Ans. 38.62 mi. N. 34~ 27' E. 51. A tug that can steam 11 miles per hour is at a point P. It wishes to intercept a steamer as soon as possible that is due east at a point Q and making 18 miles per hour in a direction W. 32~ S. Find the direction the tug must steam and the time it will take if Q is 2 miles from P. Ans. S. 29~ 52.3' E.; 5 min. 47 sec. P ~ a |< / / / A B FIG. 103. 52. In Fig. 103, ABCD is the ground plan of a rectangular barn of known dimensions a and b. A surveying party wishing to locate a point P measure the angles a and 3. Determine the lengths of the lines PB = x, PC = y, and PD = z. Ans. b cos 4'. a sin (4 + oa) b cos (0 -/) a sin 4, sin ' sin a sin 3 sin a a + b cot /3 and tan = - + cot, where q = angle DCP. b +- a cot ca' 53. The eye is 25 in. in front of a mirror; and an object appears to be 20 in. back of the mirror while the line of sight makes an angle of 32~ 30' with the mirror. Find the distance and direction of the object from the eye. Ans. 70.8 in. in a direction making an angle of 4~ 3' with plane of mirror. OBLIQUE TRIANGLES 125 54. Two lighthouses, one 95 ft. high and the other 80 ft., are just visible from each other over the water. Find how far they are apart. Ans. 22.9 mi. 55. At each end of a horizontal base line of length 2 a, the angle of elevation of a mountain peak is 3, and at the middle of the base line it is a. Show that the height of the peak above the plane of the base line is a sin a sin 3 Vsin (a + 3) sin (a - 3) 56. The jib of a crane makes an angle of 35~ with the vertical. If the crane swings through a right angle about its vertical axis, find the angle between the first and last positions of the jib. Ans. 47~ 51' 18". 57. If the jib of a crane makes an angle f with the vertical and swings about the vertical axis through an angle 0, show that the angle a between the first and last positions of the jib is given by the equation sin 1 a = sin 4 sin ~ 0. 58. An umbrella is partly open and has n ribs each inclined at an angle 0 with the center stick of the umbrella. Show that the angle 0 between con1 secutive ribs is given by the equation sin 6 0 = sin - sin 4. u2 n 59. If the angle of slope of a plane is 0, find the angle of slope x of the line of intersection of this plane with a vertical plane making an angle a with the vertical plane containing the line of greatest slope. (Note the difference between this exercise and Exercise 18, page 59.) E F //D >^G /C A B / FIG. 104. Suggestion. In Fig. 104, AD = a cot 0, AG = a cot x. a cot 0 cos a = and tan x =tan 0 cos a. a cot x 60. Two vertical faces of rock at right angles to each other show sections of a geological stratum which have dips (angles with the horizontal) of a and 13 respectively. If 8 is the true dip (angle between the stratum and a horizontal plane), show that tan2 8 = tan2 a + tan2 3. 61. To determine the dip of a stratum that is under ground, three holes are bored at three angular points of a horizontal square of side a. The depths at which the stratum is struck are, respectively, p, q, and r feet. Show that the dip 8 of the stratum is given by the equation tan = (p- q)2 + (q-r)2 a 126 PLANE TRIGONOMETRY 62. To lay out a pentagon in a circle, draw two perpendicular diameters AB and CD, Fig. 105, and bisect AO at E. With E as a center and ED as a radius, draw the arc DF. The length of the chord DF is the side of the inscribed pentagon. Prove this. A A C FIG. 105. FIG. 106. 63. To lay out a regular heptagon in a circle, make a construction as shown in Fig. 106. AB is very nearly the side of the inscribed regular heptagon. Determine the error in one side for a circle with a radius of 10 in. and determine the per cent of error. Determine the angle at the center intercepting the chord found by this process. Ans. 0.2% too small. CHAPTER VII MISCELLANEOUS TRIGONOMETRIC EQUATIONS 85. Types of Equations. -In this chapter equations of the following types will be considered: (1) Where there is one unknown angle involved in trigonometric functions. (2) Where the unknown is not an angle but is involved in inverse trigonometric functions. (3) Where there are other unknowns, as well as unknown angles, involved in simultaneous equations; but only the angle involved trigonometrically. (4) Where the unknown angle is involved both algebraically and trigonometrically. It is not possible to give general solutions of equations of all these types. They offer algebraic as well as trigonometric difficulties. Methods of solution are best shown by examples. EXERCISES 1. Given tan 2 0 = 2 —; find sin 0 and cos 0 without finding 0, for values in the first and second quadrants. 2 tan 0 Solution. By [21], tan 2 0 = - tan2 1L - tan2 0 2 tan 0 24 -1 —tan20 7-~, or 12 tan2 0 +7 tan0 -12 =0. 1 - tan o 7 — 7 = V49 + 576 4 3 Solving for tan 0, tan 0 = or24 3 4 When tan 0 = 4, sin 0 = ] and cos 0 =. When tan 0 = -4, sin 0 =4 5 and cos 0 = -. The student can easily verify these by triangles or formulas. 2. Given tan-' (a + 1) + tan-l (a - 1) = tan- 2; find a. Solution. Let 0 = tan-l (a + 1); then tan 0 = a + 1. Let 3 = tan-l (a - 1); then tan = a - 1. ta ( + tan 0 + tan,3 a 1 + a - 1 2a tan(O ) 1- tan tanj3 1 - (a + 1) (a-1) 2-a2 2a 0- +3 = tan-12 _ = tan-12. 2 - a2 127 128 PLANE TRIGONOMETRY That is, 2 = whence a2a a - 2 0 2 - a 2 or (a + 2) (a - 1) = 0, whence a = -2 or 1. I -1 -3 Check. When a = -2, tan-' (-1) + tan-' (-3) = tan —' (-1) (-3) -4 = tan-' = tan-' 2. -2 3. Given sin 2 0 = 2 sin 6; find 6 < 3600. Ans. 0, ir. Suggestion. Use sin 2 6 = 2 sin 6 cos 6 and factor. 4. Given tan 2 6 = -i,-2; find sin 6 and cos 6 for 6 in quadrants I and II. 2 3 Ans., - 5' 2cos2 2 6 + cos2 6 -1 = 0; find 6. Ans. (n~- ) ir,(2n + 1) 7r. 6. sin 2 o + sin 4 o+sin 6=0; find 6. Ans. j, (2n + 1 ~Dw 4 3 l Suggestion. Apply [25] to sin 2 0 + sin 6 6. Factor the resulting equation and equate each factor to zero. 7. cos 2 6 = sin 0; find 6. Ans. (2 n~+ 7r, (2 n + r. 8. Given tan-' (a + 1) + tan-' (a - 1) = tan-' (-sz~); solve for a. Ans. 7.137 or - 0.280. 9. Given r sin 6 = 2 and r cos 6 = 4; solve for r and 0. Ans. r = 2 V\/-5; 6 = 260 33' 53", 2060 33' 53." Suggestion. Square both equations and add, to obtain r. Divide the first by the second to obtain 6. 10. Given tan-' (x + 1) + tan-' (x - 1) - tan-' " find x. Ans. 0.610 or - 3.277. 11. Given cos- (1 - a) + cos'1 a = cos-' (-a); solve for a. Ans. 0 or 1 12. Given r sin (6 - tan-' ) = 3, and r cos (6 - tan-'') - 6; find r and 6. Ans. r = 3 \5; 6 = 400 36' 5", 2200 36' 5". Suggestion. Obtain r as in 9. To obtain 6 divide one equation by the other; expand the functions and solve for tan 6. 13. sin 4 o = 2 cos 2 0; find 6. Ans. (n + 1) 22 14. tan-a + tan-' = tan-' x. Find a when (a) x=1, a -i a (b) x = 2, (c)x= -7. Ans. (a) a = 0, (b) a = or -1, (c) a =2. 15. tan 2 a tan a = 1; find a. Ans. (n+7)ir. 16. sin (1200 - x) - sin (1200 + x) = -\/3; find x. Ans. 60,0 1200. 17. cos (300 + 6) - sin (600 + 6) = - / V3; find 6. Ans. 600, 1200. 18. V/1 - Vsin x + sin2X = sin x - 1; solve for x. 19. \7 sin a - 5 + V/4 sin a -1 = \7sin a - 4 + V4 sinc, -2; solve for a. Ans. (2n+ )-ir. 20. Given tan (80' - 0 6) = cot 2 0; find 6. Ans. 600. MISCELLANEOUS TRIGONOMETRIC EQUATIONS 129 21. Given 3 sin-l x + 2 cos-1 x = 240~; solve for x. Ans. 1 v3. 22. Given tan- x + 2 cot-' x =135~; solve for x. Ans. 1. 23. Given tan 2 x (tan2 x - 1) = 2 sec2 x - 6; solve for x. Ans. 45~, 225~, 116~ 33' 56", 296~ 33' 56". 24. Given tan- 12 x + tan-' 3 x = -17r; solve for x. Ans. x = 6 or -1. 25. Given 10 cos 0 - 5 sin 0 = 2; show that 0 = 2 tan-1 2. 86. To solve r sin 0 + s cos 0 = t, for 0, when r, s, and t are known. - Solution. Either sin 0 or cos 0 can be eliminated by means of the relation sin2 0 + cos2 0 = 1, but logarithms are not applicable to this solution. A solution will now be given in which the computations may be done by logarithms. (1) Let msiny = r, and mcos- = s, where m is a positive constant, and y an auxiliary angle. Such an assumption is always permissible, for squaring both equations of (1) and adding, m2 sin2 y + m2 cos2 y r2 + s2, or m2 = r2 + s2, or m Vr2 + s2. Then m is real if r and s are real quantities. Dividing the first equation of (1) by the second, (2) tan = Since the tangent may have any real value from -oo to + o, when r and s are real, the angle y will always exist. Substituting (1) in the original equation, m sin y sin 0 + m cos y cos 0 = t, which by [16] gives (3) m cos (0 - ) = t. Now m and y can be determined from (1) and (2), and then 0 - y from (3). From this 0 is determined. p sin a cos p = a, 87. Equations in the form p sin a sin P = b, p cos a = c, where p, a and p are variables. - Solution. Squaring all three equations and adding p2 sin2 a cos2 i + p2 sin2 a sin2 2 + p2 cos2 a = a2 + b2 + c2. p2 sin2 a (cos2 3 + sin2 %) + p2 cos2 a = a2 + b2 + 2. p2 (sin2 a + cos2 a) = p2 = a2 + b2 + c2. 130 PLANE TRIGONOMETRY From the third equation, c c c coS a =- a = -, or a = cos-1 - 2 P Va2 + b2 + c2 V/a2+b2+c2 p sin a sin 3 b Dividing the second equation by the first, p sin osin b p sin a cos p a Whence tan A = b; = tan-1 b a a 88. Equations in the form sin (ac+P) = c sin a, where 3 and c are known. - Solution. Dividing by sin a, sin (a + ) c sin a 1 Taking the proportion by composition and division, sin (a + f- ) + sin o c + 1 sin (a + ) - sin a c- 1 2 sin (a - )cos ~ _ c +- 1 By [25] and [26], 2 cs (a+ )osi c2cos (a + q-~) sin c - -1 tan (a 3- ) c + 1 Applying [7], tan tan C1 1 or tan (a +- ) = c tan2. From which, since 3 and c are known, a may be found. Example. Solve sin (a + 50~) = 2 sin a. Solution. Substituting 50~ for 3 and 2 for c in the above formula, we have tan (a + 25~) 2 1 tan 25~ = 3 tan 25~. log 3 = 0.47712 log tan 25~ = 9.66867 log tan (a + 250) = 0.14579 a + 25~ = 54~ 26' 29" or 234~ 26' 29". a = 29~ 26' 29" or 209~ 26' 29". MISCELLANEOUS TRIGONOMETRIC EQUATIONS 131 89. Equations in the form tan (a + P) = c tan a, where P and c are known. - Solution. Dividing by tan a and taking the resulting equation in proportion by composition and division, tan (a -+ ) tan a tan (a + ) + tan a c+ 1 tan (a- + f)-tan a c- 1 sin (a + ) sin a cos (a- +) cos a sin (2 a + ) c + 1 sin (a -+ t) sin a sin (a- + - a) c - cos (a + t) cos a cAsin (2a+A)= sin. c- L Since c and: are known, a may be found. Example. Given tan (a + 24~) = 4 tan a; find a. Solution. Substituting 24~ for f and 4 for c in the above formula, we have sin (2 a + 24~) = 5 sin 24~. log 5 = 0.69897 log sin 24~ = 9.60931 colog 3 = 9.52288 log sin (2 a + 24~) = 9.83116 2 a + 24~ = 42~ 40.7', 137~ 19.3', 402~ 40.7', 497~ 19.3'. 2 a = 18~ 40.7', 113~ 19.3', 378~ 40.7', 473~ 19.3'. a = 90 20.4', 56~ 39.6', 189~ 20.4', 236~ 39.6'. 90. Equations of the form t = 0 + q sin t, where 0 and 4 are given angles. - First express 0 and 4 in radians if not already so given. Then t must satisfy the relation t - 0 = 4 sin t. Let yj = t - and y2= sint. Plot the straight line with equation y = t - 0, and the sine curve y2 = 0 sin t. An approximate value of t can be determined from the value of t where the line and curve intersect. The more nearly accurate the sine curve is plotted, the more nearly will the value of t come to the solution of the equation. Example. Given t = 2 + r sin t. Let y = t - 2 and y2 = T sint. 132 PLANE TRIGONOMETRY Now plot yj = t - 2, giving the line AB, as in Fig. 107. Also plot the modified sine curve with equation Y2 = 7r sin t. By measurement, the abscissa t for the point P of intersection is found to be 2.86 radians, or 164~. This is therefore an approximate solution for the equation. Substituting for t in the original equation, 2.86 = 2 + -r sin 164~ = 2 + 0.8659 = 2.8659. This result shows the value of t to be too small. B A y FIG. 107. Substituting t ='165~, 2.88 = 2 + Xr sin 165~ = 2.813. This result shows that 165~ is too large, which the intersection of the curves also verifies. The correct value may now be approximated by assuming values of t between 164~ and 165~, say 164~ 10', etc. EXERCISES 1. Given 3 sin 0 + 4 cos 0 = 2; find 0. Solution. This is of the form given in Art. 86, and r = 3, s = 4, and t = 2..'. m = /r2 + s2 = /9 + 16 = 5. r 3 tan, - r = 3 = 0.75, and y = 36~ 52' 12" or 216~ 52' 12". s 4 MISCELLANEOUS TRIGONOMETRIC EQUATIONS 133 Since r and s are both positive, sin 7 and cos y are positive. Therefore y is in the first quadrant, and so must be 36~ 52' 12" only. t 2 cos (0 - ) = - = = 0.4, by Art. 86, equation (3).? 5 0 - - = 66~ 25' 18" or 293~ 34' 42"... = 66~ 25' 18" + 36~ 52' 12" = 103~ 17' 30", and 0 = 293~ 34' 42" + 36~ 52' 12" = 330~ 26' 54". 2. Given 5 sin 0 - 2 cos 0 = 3; find 0. Ans. 55~ 39' 20", 167~ 56' 50". Suggestion. sin - is + and cos y is -, therefore - will be in second quadrant. 3. Given 2 sin 0 + 5 cos = -3; find 0. Ans. 145 39' 20", 257~ 56' 50". 4. Given 1.31 sin 0 - 3.58 cos 0 = 1.885; find 0. Ans. 99~ 32' 10", 220~16'. Note. Logarithms are not used in solving Ex. 1 because natural functions were just as convenient. In solving Ex. 4, logarithms are preferable. 5. Given p sin a cos / = 3, p sin ao sin 3 = 2, p cos a = 1; find a, /, and p. Ans. p = V/14, a = 74~ 29' 56," / =33~ 41' 24." 6. Given p sin 0 cos = 6, p sin 0 sin 0 = 2, p cos 0 = 0; find 0, 4, and p. Ans. p = 2 /10, 0 = 90~, 4 = 18~ 26' 5". 7. Given sin (x + 32~ 16') = 4 sin x; find x. Ans. 9~ 36' 23", 189~ 36' 23". 8. Given sin (y - 75~) = 3 sin y; find y. Ans. 160~ 35.3', 340~ 35.3'. 9. Given tan (r + 40~) = 5 tan r; find r. Ans. 17~ 18.6', 32~ 41.4', 197~ 18.6', 212~ 41.4'. 10. Given tan (s - 60~ 20') = 2 tan s; find s. 11. Given x = 1 + 30~ sin x; find x approximately. 12. Given S = 60~ + - sin S; find S approximately. 3 CHAPTER VIII COMPLEX NUMBERS, DEMOIVRE'S THEOREM, TRIGONOMETRIC SERIES 91. The imaginary unit j. - The square root of -a2 is often written V/-a2 = Va2 /-1 = aj, where j * stands for the symbol V/-1, and is called the imaginary unit. The powers of j form a cycle of numbers which are repeated for every fourth of the integral powers of j as follows: jO = 1, jl = V/-i = j, j2 = (V/_1)2 = _ 1, j3 =j2.j = _j, j4 = (j2)2 = (-1)2 = 1, etc. Complex number. - Any number of the form a + bj, where a and b are real numbers, is called a complex number. In algebra the following theorems relating to complex numbers are proved: (a) A real number cannot equal a complex number. (b) If two complex numbers are equal, then the real parts are equal, and the coefficients of the imaginary unit are equal. Thus, if a + bj = c + dj, then a = c and b = d. 92. Graphical representation of a complex number. - Take a positive real number a, and lay off OA = a on the axis OX, Fig. 108. Since aj2 = -a and aj4 = a are p~B 7opposite in sign, make aj2 = OC and aj4 = OA. Then the multiplication of '5r ~ a by j4 may be thought of as rotating C o A X the real number a = OA through 360~. ( ---- a)- X ~> Also since aj2 = -a, multiplying by j2 may be thought of as rotating the D ' real number a = OA through 180~. Since aj and aj3 -aj are opposite in FIG. 108. sign, just as a and aj2 are, the multi* The imaginary unit V - i is usually represented by i in algebra. In many of the applications of complex numbers, it is represented by j, as in electrical computations, where i represents electrical current. 134 COMPLEX NUMBERS 135 plication of a by j may be thought of as rotating a = OA through 90~, and of a by aj3 as rotating a = OA through 270~. That is, we may take the y-axis as the axis on which the pure imaginaries such as aj or - aj are measured. We therefore interpret a + bj as the sum of two directed lines or vectors at right angles to each other. But the sum of two vectors at right angles is equal to the diagonal of the rectangle formed by the two vectors and drawn from the point of intersection of the vectors. Thus, in Fig. 109, OA = a, OB = bj (j indicating the axis on which b is measured), and OH = a + bj. Another interpretation of a + bj is that it represents the point H with coordinates (a, b). Definition. The angle AOH = 0, which a + bj makes with OX, is called the amplitude of a + bj; and the length of OH = V/a2 + b2 = r, is called the modulus of a + bj. By definition r is always positive, and 0 is measured positive unless otherwise specified. YI B(x.y) (a,b) y H 0 0a --- —-— A FIG. 109. FIG. 110. 93. The complex number z = x + yj in terms of the amplitude and modulus.-In Fig. 110, let B have the coordinates (x, y). Then OB represents the complex number z = x + yj. Now x = r cos 0 and y = r sin 0.. z = x + yj = r cos 0 + jr sin O = r (cos 0 + j sin 0). This form is called the polar form. Also r = Vx2 + y2 and tan 0 = while cos 0 = x r 136 PLANE TRIGONOMETRY EXERCISES 1. Plot the complex number z = 3 - 4j. Find its amplitude and modulus and express z in the polar form. Solution. Plot the point B with coordinates (3, -4) as in Fig. 111. Then OB = 3 - 4j and r = X2.+ y2 = V9 + 16=5. tan 6 -4, and 0 is in the fourth qua0 drant as in the figure, since the cosine is A positive. Ox Then 3 - 4j = 5[cos tan-' (-4 -j sin tstn-l( —tan-' 5 Q9-; In the following exercises, plot the complex number, find r and 0, and express the I numbers in terms of r and 0. B (3i-4) 2. z = 2 - V3j. 5. z = V -j. FIG. 111. ~ 3. z = 5j. 6. z = -2j. 4. z = 4. 7. z = 12 + 5j. 94. Multiplication of complex numbers when expressed in the polar form. - Let z, = ri [ cos 01 + j sin 0,], and z2= r, [cos 02 + j sin 621. Then zlz = rjr2 [eos 01 cos 02 + j2 sin 01 sin 02 -- j (cos 0, sin 02 + COS 0, sin 0,)]. But j' = -1, Cos0,cos02 - sin 01 sin 02 = cOS (0, + 02), and cos 0, sin 0, A- COS 0, sin 0, = sin (01 ~ 02). (1).~. Z~s, =rlri3 [cos (01 + 02) Aj sin (0, +02)1. But this is the polar form of a complex number in which the modulus is rir, and the amplitude 01 + 0,. By the same reasoning, the product of z1z2 = rjr2 [COS (0 A+ 0,) A-i sin (0,+ 02)], and = r3 (COS 03 A-i sin 03), gives z1z2Z3 = rlr2r3 [COS (0, + 02 A- 03) A-i sin (0 A+ 02 + 03)], and so on for any number of factors. In general, (2) zjz2.. -=rjr... r, [cos (0,A-A-.0.. 0n) A- jsin (01 A-02 +A-.. -. COMPLEX NUMBERS 137 Hence the product of any number of complex numbers is a complex number which has for its modulus the product of the moduli, and for its amplitude the sum of the amplitudes of the factors. * If 01 = 02 = 03 =.. 0= and ri = r2 = r3= r, = r then z = z2z3=... = z, =, and z z~...z. to n factors rn [cos nO + j sin nO]. That is, for n a positive integer, (3) [r (cose0 +j sine0)]- = rn [cosn6 +j sinnej]. This is known as DeMoivre's Theorem, for n a positive integer. Now suppose n a negative integer = -in, then 1 1 [Ir (c0s 0 + j sin 0) ] — =I I [r (cos 0 + j sino)Jm rm[cos m0 + j sin m0] by (3). Rationalizing the denominator, 1 cos m0 - j sin m0 cos mO - j sin mO rm [cos m0 + i sin m0] cos m0 - j sin m0 Cos2 mO - j2 sin 2mO But cos mO = cos (-in) 0 and sin M0 = -sin (-in) 0. Then [Cos (- M) 0 + j sin (- i) 0] [r (cos0 8+j sin 0)>m = r-m- csm +snm Cos2?n + Sin2 MO = r-m [cos (-in) 0 + j sin (- M) 0] = rn (cos n0 + j sin nO). This proves DeMoivre's theorem for n = -i. Exercise. Prove that the theorem is true when n - 1? p and q being integers. EXERCISES The following exercises are to be solved by means of DeMoivre's Theorem. 1. Show that sin 2 0 = 2 sin 0 cos 0 and cos 2 0= cos2 0 - sjn2 0. Solution. In (3) let n = 2 and r = 1. Then [cos 0 + jisin 012 = cos 2 ~ +j sin 2 0. Expanding the first member, cos2 0 + 2j sinO0 cos 01 j2 sin2 0 = cos 20~ +j sin 2 0. But j2 = -1, therefore (cos2 0 - sin2 0) 2 sin 0 cos 0. j = cos 2 0 + j sin 2 0. By Art. 91, the real parts 'are equal and the coefficients of j are equal. That is, cos 2 0 = cos2 0 - sin2 0 and sin 2 0 = 2 sin 0 cos 0. 138 PLANE TRIGONOMETRY 2. Prove that [cos 450 + j sin 45012 = j. Solution. With 0 = 450 and n = 2, by (3) (cos 450 + j sin 450)2 = cos 900 + j sin 900 = j, since cos 900 = 0 and sin 900 = 1 3. Prove [cos 300 + j sin 300]4 = -+ + 2 2 4. Prove [cos 450 + isin 45'14 = -1. 5. Prove [cos 600 +1i sin 60015 =- 2 6. Prove [cos 900 + j sin 90014 = 1. 7. Prove that cos 3 0 = 4 cos3O0 - 3 cos 0, and sin 3 G = 3 sin 0 - 4 sin3O0. 8. Prove that cos 4 0 = cos4 0 - 6 cos2 0 sin2 0 + sin4 0, and sin 4 0 = 4 cos3 0 sin 0 - 4 cos 0 sin3 0. 95. Division of two complex numbers. - Let zi = r, (coS 01 + j sin 01), and Z2 = r2 (COS 02 + j sin 02). z1 ri (cos 01 + j sin 01) T n r2 (cOS 02 + jisinO02) Rationalizing the denominator, z1 r, (COS o1 +i+jsin 1 cos 02- j sin02Z Z2 r2 cos 02+ j sin02 COS 02 - jisinO20 - cos 01 cos 02 + sin 01 sin 02 + j (sin 01 cos 02 - COS 01 sin 02) r2 LCOS22 - j2 Sjn2 02 - [Cos (0- 02) +jsin (Oi - 02)]. Hence the quotient of two complex numbers is a complex number having an amplitude equal to the amplitude of the dividend minus the amplitude of the divisor, and a modulus equal to the modulus of the dividend divided by the modulus of the divisor. For example, 8 (cos 60' + j sin 60') 0) 2 [cos (600 - 300) + j sin (600 - 300)] 4 (cos 300 + j sin 30() -2 (cos 300 + j sin 300). 96. Square roots of a number. - Let Z2 = r2 (cos 0 + j sin 0) be the number. Now the square root of a number is one of the two equal factors of the number. By Art. 94, when a number is squared the modulus is squared and the amplitude is doubled. COMPLEX N UMBERS 139 Therefore, in extracting the square root, we extract the positive square root of the modulus and divide the amplitude by 2. Now cos 0 = cos (6 + 2 kwr) and sin 0 = sin (0 + 2 k7r). Then the most general form in which the original number may be written is z2 = r2 [cos (0 + 2 kr) + j sin (0 + 2 kw)]. 2 2... z = r[cos (+22k) + j sin (~ 2 kj, where k is an integer 0, 1, 2, etc. Let k = 0, then z0 = r cos +jsin ] Let k = 1, then z1 = r [os( +r) +jsin( +r) These are the only square roots, for if k = 2, the amplitude becomes + 2 7, the functions of which are the same as those of. 2i 2 If k = 3, we have + 3 wr, the functions of which are the same as + r, etc. Any other values of k will give either zo or z1. Example. Extract the square root of the number 1. Solution. The complex number 1 is plotted on the x-axis, one unit to the right of the origin. The modulus is 1 and the amplitude is 0 + 2 kr. Let z2= 1. Then z2 = 1 [cos (0 +2 k) + j sin (0 +2 kw)]. z = 1 cos 2 + j sin ]= cos kr + j sin ktr. Let k = 0... o = cos 0+ jsin0 = 1. Let k = 1... z = cos r + j sin 7r = -1. 97. Cube roots of a number. - The cube root of a number is one of the three equal factors of the number. The number is the cube of this factor. Since in raising a number to the third power, we cube the modulus, and multiply the amplitude by 3, then in extracting the cube root, we extract the cube root of the modulus and divide the amplitude by 3. 140 PLANE TRIGONOMETRY Suppose z3 = r (cos 6 -- j sin 6) is the number. Writing the number in the most general form we have z' = r [cos (6 — 2 kw) + j sin (6 + 2 k)]. B~i/;-[OS 0 + 2 kn + 0, + 2 kir z = cosO+ k i+ sinO k} 3 3 When k = 0, the amplitude is 3 When k 1, the amplitude is 6 2w 3 3 6 4w When k = 2, the amplitude is + 3 3 When k = 3, the amplitude is 0+ 2w7. But the functions of - and 0- +27w are equal and therefore the 3 3 roots corresponding to k = 0 and k = 3 are the same. Any other values of k will give an amplitude with the functions equal to those for k = 0, 1, or 2. Therefore there are but three cube roots of r [cos 6 + j sin 6]. They are O V= jcos0-vi sin ]0 cos 34 +jsin6 Z, = 0 + $ 2 7r + il 0+ 7 Z2 = 0+ 7r S 1A0 4 r] 3 3 Example. Find the cube roots of 1. Solution. Let z3 = 1 = cos (O + 2 k) +jsin (O + 2kw). 2kw 2kw,z = Cos ~ +j sin 3 3 Let k = 0, then z = cosO + A sin 0 = 1. 2.2 1. V3 Let k = 1, then z1 = cos 27w +j sin 2w= + 2 3 3 2 2~ 4 4 1 \V3 Let k = 2, thenZ2 = COS 7+j sin w -7 J o ~3 3 2 2 COMPLEX NUMBERS 141 These three roots may be represented graphically as follows: Draw a circle of radius 1, Fig. 112. Since the moduli of z0, z1, and z2. are each 1, their terminal points will lie on the circumference of the circle. The amplitudes of zo, z1, and z2 are 0, 2 Tr, and 4 ir respectively. Therefore z0 = OA, zi = OB, and Z2 = OC. Example. Find the three cube roots of z3 = -8. y 7r XX ~ ~ ~ ' 90 Xo it -X A. 5 Z FIG. 112. FIG. 113. Solution. Plot -8 = OA, Fig. 113. The amplitude is r and the modulus is 8. Then z3 = 8 [cos (ir + 2 kwr) + j sin (ir + 2 kcr)]. -' 7r + 2 kr.. 7r + 2 k7r z = 2 cos 3- +jsin 3- ] Let k = 0, then zo = 2 cos +j sin 7 = 1+ j 3. Let k = 1, then zi = 2 [cos 7 + jsinr] = -2. Let k = 2, then z2 = 2 [cos T- + j sin -] = 1 - /3. Now the moduli of zo, z1, and z2 are each equal to 2. The points representing the cube roots are therefore on the circumference of a circle of radius 2, Fig. 113. Constructing the amplitudes 60~, 180~, and 300~, we have z0 = OB, z1 = OC, and z2 = OD. 98. To find the nth roots of a number. - An nth root of a number is one of its n equal factors. By DeMoivre's Theorem, when a number is raised to the nth power, the modulus is raised to the nth power, while the amplitude is multiplied by n. Then to extract the nth root, we extract the positive nth root of the modulus, and divide the amplitude by n. * * The student is already familiar with the fact that there is one positive square root of a positive number, and one positive cube root of a positive number. It is true, in general, that there is one and only one positive nth root of any positive number. 142 PLANE TRIGONOMETRY Let Zn = rn [COS (0 + 2 kwx) + j sin (0 + 2 kwx)I be the number. Then z = r + 2 k+ i]0 2 k i n n Let k = 0, thenzo = r[COS 0~+jsin j. Let k = 1, then z, = rcos 0 + 2. sn 2 \n n \n n Let k = 2, then z2 = r COS 0 + 4 7r + sin the + 4z \n n \n n) Let o= n- 1,then z,, = rcos (0 + 2(n - 1)w).. / 2 (n - 1) )1r +jsi an _+3 \n n Let k = n, then Zn= r C 0 2nn + isin 2 nnr Ln n 2 n = r cos(0 +2r) ~jsin(+ 27r)]. But zo and z,, are seen to be the same, having the same moduli, and amplitudes differing by 3600. Any other integral value of l will lead to one of the n values of z already found as may be easily shown. There are therefore but n nth roots of a number. The roots zo, z,,... Zn,, all have their terminal points on the circumference of a circle of radius r. Their amplitudes differ by 2 0 integral multiples of -w, beginning with -, where 0 is the amplitude n n' of the original number. EXERCISES 1. Find and plot the five fifth rcots of z' = 2 ~ 5j. Solution. From the table 6 = tan-' ~ = 680 11.9'. The modulus = \4 + 25 = V2. Then z5 = V29 [cos (680 11.9' + 2 lcw) ~j sin (680 11.9' + 2 k)], and', = V/29lcos (130 38.4/ + 2 kw) + j sin (130 38.4' +2 kir)]. Assigning to k the values 0, 1, 2, 3, 4, we have ZO =!"29 [cos 130 38.4' +jisin 130 38.4'], zi =!N29 [cos 850 38.4' + j sin 850 38.4'], 22 = "29 [cos 1570 38.4' + j sin 1570 38.4'], 23 = k/29 [cos 2290 38.4' + j sin 2290 38.4'], 24 = '29 [cos 3010 38.4' + j sin 3010 38.4']. COMPLEX NUMBERS 143 In Fig. 114, OA represents the original vector 2 + 5 j. Construct a circle with radius t/29 = 1.4 approximately. Lay off OB with the amplitude 13~ 38.4', and OC, OD, OE, and OF respectively every 72~ from OB. Then OB = Zo, OC = zi, OD = Z2, OE = Z3, and OF = Z4. FIG. 114. In the following exercises, find and plot all the values of z. 2. 4 = 1. 6. z3 = -1 + /3j. 3. z= -1. 7. z3= -2 + /2j. 4. 3 = 1 8. z4 = -1 - 3j. 5. z5 =-32. 99. Expansion of sin nO and cos n0.-ByDeMoivre's Theorem and the binomial theorem, cos nO + j sin nO = (cos 0 + j sin )n = Cosn 0 + nj cosn-1 0 ~ sin 0 n (n- 1) jn (n -1) (n - 2) -3 - [2 e COS'-2 0 sin2 0 -- cosn-3 sin3 0 n (n-1) (n- 2) (n- 3) + — cos"-4 0 sin4 0 jn (n- 1) (n - 2) (n - 3) (n - 4) -+ ------- L ----- COSn-5 0 sin5 -.*.. (1) Equating the real parts, n(n - 1) cos nO = cos~ 0 - cos'-2 0 sin2 6 n (n - 1) (n - 2) (n- 3) +- con-4 0 sin4 6 +.. + 144 PLANE TRIGONOMETRY a a Let a = nO, then 0 = - and n =, where a is to be held constant n 6' while n and 0 are to vary. Substituting these values, cos a = cosn 0 - cos"-2 0 sin2 0 + a - 1 _2) ( - 3) _+ A cos-46-sin4+ ~. a (a - 0) c 2 sin 0 ) = cos 0 - cos2 a (a - 0) (a - 2 ) (a- 3 ) s-4 0(sin Now as n becomes infinite - = 0 approaches zero, cos 0 1, sin 0 1, and a - 0-. Therefore COS i 2 + 4 C6 (2) cos a = 1- + - + Equating the coefficients of the imaginary parts of (1), sin nO = n cosn-1 0 sin 0 (n 1) (n - 2) sin sin n6 = n cos1-1 6 sin 8 - cos7-3 6 sin3 0 n (n-1) (n-2) (n-3) (n-4) cos-0 sin 0 Making the substitutions for 0 and n, sin sin\ a(a - ) (a-20) /sin 0\3 sin cos n)- - -(l0 + (a -0) (a-2) (a -3) (a -40) n- sin 05 + ----- ^ --- COSn 0.. Then when n becomes infinite a3 a5 C7 (3) sin a = a - - + -... In (2) and (3), a is in radians. COMPLEX NUMBERS 145 If we divide (3) by (2) we get a3 c5 a7 sin a a ~ I * * a3 2 a tan a a =+ cos a a 2 a4 a-6 + 3 + 5-. Co n of tc f- F s ( 100. Computation of trigonometric functions. - Formulas (2) and (3) may be used to compute the functions of angles. Thus, let a = 10~= -^ r. Then I 1 (0~ z.)a ( — 7.)5 sin1 r = - T...1 + 8a -b + - +c -., where a, b, c,... may be computed as follows: log7r = 0.49715 3 log r = 1.49145 log 18 = 1.25527 colog 183 = 6.23419 - 10 log a = 9.24188- 10 colog L3 = 9.22185- 10 a = 0.17453 log b = 6.94749 - 10 b = 0.000886 5 log rr = 2.48575 colog 185 = 3.72365 - 10 colog 15 = 7.92082 - 10 log c = 4.13022 - 10 c = 0.000001349 sin 10~ = a - b = 0.17453 - 0.000886 = 0.17364. From the table of natural functions sin 10~ = 0.17365. By means of (2), cos 10~ may be computed. EXERCISES Compute the following functions correct to the fourth decimal place and compare with the tables. (a) sin 20~. (c) tan 30~. (b) cos 25~. (d) sin 45~. 101. Exponential values of sin 0, cos 0, and tan 0. - In algebra it is proved that if e is the base of the natural system of logarithms, then X2 X3 X4 (1) +e=l+x+-+ +- +... 146 PLANE TRIGONOMETRY Now if i0 is substituted for x, where i = V- 1, i202 i303 i404 e i (= +i+- +-4+ +.. ~ 02 04 06 ( 1- E —+ E...) But, by Art. 99, the expressions in the first and second parentheses are equal to cos 0 and sin 0 respectively. (2).. ei = cos 0 + i sin 0. Substituting x = - i in (1) and reducing as before, we have (3) e-i~ = cos 0 - isin 0. Subtracting (3) from (2), eiO _ e-ie (4) sin 0 = - e -i 2i Adding (2) and (3), ei9 + -it0 (5) cos 0=2 Dividing (4) by (5), eiO _ e-ie (6) tan 0 = (eiO e-i) i (e99 + 0 tv) Note. - The expressions for sin 0, cos 0, and tan 0, given in (4), (5), and (6), are called exponential -values of these functions. They are also called Euler's Equations after Euler their discoverer. Euler (1707-1783) was one of the greatest Physicists, astronomers and mathematicians of the eighteenth century. EXERCISES By means of the exponential values prove the following identities: 1. sin2 0 + cos2 0 = 1. 2. 1 + tan2 0 = sec2 0. 3. 1 + cot2 0 = csc2 0. 4. sin 2 = 2sin 0cos0. 5. cos 2 0 = cos2' - sin2 = 2 cos2 0-1 = 1 -2 sin2 0. 6. cos 3 0 = 4 cos3 0 -3 cos 0. 102. Series for sin"0 and cos"0 in terms of sines or cosines of multiples of 0. From (4) of Art. 101, 2 i sin 0 = ei~ - e-i COMPLEX NUMBERS 147 Expanding by the binomial theorem, (2 i sin O)n = (eiO - e -i)n = (ei~)+n (ei>-l1 (- e-i)+ n (n (eio)n-2 (_ e-io)2 +. (-1) (i0)2 (_ e-i)n-2 + ne' (_ e-i0)n-i + (- e-i). When n is odd, the number of terms in the series is even, and when n is even, the number of terms in the series is odd. Therefore, when n is odd, the terms can be grouped in pairs, the first with the last, the second with the last but one, etc. But, when n is even, there will be a certain number of pairs and one extra term, which is the middle term of the series. From this series, general formulas can be derived for expressing sin nO as a series of sines or cosines of multiples of 0. By using (5) of Art. 101, cos s can be dealt with in a similar manner. Here special cases only will be given. However, from these and other special cases, laws can easily be discovered that will determine the coefficients, and multiples of the angles. Example 1. Express sin5 0 in sines of multiples of 0. ei0 _ e-i0 Since sin 0 = - i 2i 1 ei5o - 5 ei0 + 10 eio - 10e-io+ 5 e-i3Oe-iol 2ln L 4 2 i Grouping in pairs, the first with the last, the second with the last but one, etc., sin52 - -- 2i -10 24 (ii + 2si sin50 = ~ (sin 5 0 - 5 sin 3 0 +- 10 sin 0). Example 2. Express sin6 0 in cosines of multiples of 0. eiO _ e-io Since sin 0 = 2i sin60= ei6o 6 ei4eiq~ X15 ei2~-20-15 e-i2~ —6e-i4 o+ e-i 6o 148 1PLANE TRIGONOMETRY Grouping in pairs, sin6 0 -1[ei6o+e-i606ei40 + ei40+ ei20-FeOe 1]. sin 2 = 2 + 15 2 -10 sinO= - (cos 6 6 - 6 cos 4 O - 15 cos 2 % - 10). Example 3. Express cos3 0 in cosines of multiples of 0. eieo +e-i0 Since cos0 = 2 cos3 o = 1[ei 3 ~3 ei0 +3 e-i0 +e-i36 4, 2 = Hei3 e + e-i3 eO + e-i0]Cos' 0 I [cos3 - - 3cos0]. Example 4. Express cos4 0 in cosines of multiples of 0. e0 + eCO Since cos 0 = 2 4 = 1[ei46 ~ 4 ei20+ 6 + 4 e-i20 + e- i4j 8L 2 ei4l + e-i4+ ei2O + ei20 +3]. 8 2 2 cos40=I(cos 4 0 + 4 cos 2 O + 3). EXERCISES Prove the following identities: 1. sin4O -= (cos 4 - 4ecos 2O0 + 3). 2. Cos7 8= -L (cos 7 O + 7 cos 5 0 + 21 cos 3 0 + 35 cos 0). 3. 128 cos8 0 = cos 8 0 + 8 cos 6 + 28 cos 4 0 + 56 cos 2 6 + 35. 4. 64 sin7 == 35 sinO0 - 21lsin 3 O + 7 sin 5 O - sin 7 0. 5. sin6 O + eos6O = ( (eos 4 - 4eos2O -01- 3). 103. Hyperbolic functions. - In Art. 43, note, the trigonometric functions were called Circular functions because of their relation to the arc of a circle. There is another set of functions whose properties are very similar to the properties of the trigonometric functions. Because of their relation to the hyperbola, they are called hyperbolic functions. They are defined as follows: (s - e-s (1) Hyperbolic sine x (written sinh x) e+ 2 (2) Hyperbolic cosine x (written cosh x) =e+ex COMPLEX NUMBERS 149. (3) Hyperbolic tangent x (written tanh x) = - -x x + e-x eX 1 CX(4) Hyperbolic cotangent x (written coth x) = e + e(5) Hyperbolic secant x (written sech x) = Cx + e-x (6) Hyperbolic cosecant x (written csch x) = 2 ex - C-x In these formulas e is the base of the Napierian system of logarithms, and so stands for the number 2.7182818 *. From the definitions, the following relations are evident: sinh x 1 1 tanhx =,h x tnh x = sechx =, etc. =cosh' tnx coth x cosh x 104. Relations between the hyperbolic functions. - Squaring (1) and (2) and subtracting the second from the first, e2 x _ 2 C- e-2 x e2x -2 + e -2x cosh2 x - sinh2x = - = 1. 4 4. cosh2 x - sinh2 x = 1. By analogy, from (1) we may write eX+y - e-(x+) sinh (x + y) = eC - e-X ey + e-y Also sinh x cosh y = 2 2 2 = - [+ - e-X eY + e e-Y - e-(+)]. z + e-X y_ -e-y And cosh x sinh y- 2 2 2 = [ex+y + e- ey - ex e-y - e-(x+2)] Adding the last two, sinh x cosh y + cosh x sinh y = [ex+y - e-(x+y)]. Comparing this with the first, sinh (x + y) = sinh x cosh y + cosh x sinh y. EXERCISES Prove the following identities: 1. sech2 x + tanh2 x = 1. 2. coth2 - csch2 x = 1. 3. sinh ( -x) = -sinhx. 150 PLANE TRIGONOMETRY 4. cosh ( - x) = coshx. 5. sinh (x - y) = sinh x cosh y - cosh x sinh y. 6. cosh (x -+ y) = cosh x cosh y - sinh x sinh y. 7. cosh (x - y) = cosh x cosh y -- sinh x sinh y. tanh x + tanh y 8. tanh (x 1 + tanh x tanh y 9. sinh 2 x = 2 sinh x cosh x. 10. cosh 2 x = cosh2 x + sinh2 x. 105. Relations between the trigonometric and hyperbolic functions. - If in (4) of Art. 101 we substitute iO for 0, i sin iO = [(i) _ e-i (i0) = - [e - e-0 = - sinh0. (1).. sin i0 = i sinh 6. Substituting iO for 0 in (5) of Art. 101, cos i = 1 [ei(i) + e-i(i1)] = 1 [eo + e-0] = cosh O. (2). cos i = cosh 0. Dividing (1) by (2), (3) tan ie = i tanh 0. 106. Expression of sinh x and cosh x in a series. Computation. - By definition and by (1) of Art. 101, sinh x = 1 [ex - e-] [ X x2 + X3 + 2 xX X 2 x3 X. 2 X3 ( (1).-. sinhx +=x+ -L++++*. Also cosh x = 1 [-x + e-x] =1[(l++2 + X - +)+ (l- +1- + +.)]. 1 _2 2 X3 2 X5 x3 X5 = [2 + - + -2- + X* * ] = + + ~ + Also cosh x = ~ [ex + e-x] -- l-+ - q2 -+ 3 ~ + X-2 X3 ~ ' ' 2 x4 X6 (2).. cosh =l+ -+ -+ -.. L2 L1 L6 Series (1) and (2) for sinh x and cosh x are convergent for all real values of x. Therefore, for any real value of x the hyperbolic functions of x can be computed. COMPLEX NUMBERS 151 For example, when x = 0 sinh O = and coshO = 1. When x = 1, sinh i = 1 + q + q- + I 4u + '- -. = 1.175 ~ * *. sh= 1 + - + ~a + 7 2- + * = 1.543 ~ ~ *. 152 PLANE TRIGONOMETRY FORMULAS [1] [2] [3] [4] sin2 0 + cos2 0 = 1. 1 + tan2 0 = sec2. 1 + cot20 = csc20. sin = - and csc 0 1 csc 0 -- sin 0 [5] cos0= s - sec 0 1 and sec 6= co- cos 0 1 1 [6] tan0= c and cotOt = cot60 tan60 sin 0 [7] tan8 = s cos cos 0 [8] cot6 sn sin 0 [9] x = I cos. (Projection on x-axis.) [10] y = sin. (Projection on y-axis.) [11] G = r2 (0 - sin 0). (Area of segment.) [12] sin < 0 < tan. lim[i i = 1. -0-oLs 6] ' [13] sin (a + ) = sin a cos + cos a sin 3. [14] cos (a ) = cos cos - sin a sin /. [15] sin (a - /) = sin a cos - cos a sin /. [16] cos (a - o) = cos a cos + sin a sin 3. tan a + tan [17] tan (a +) = — tantan' [18] t=/ I - tan a tan. tan a - tan/3 [18] tan(a —) - = -tan tan I + tan a tan p [19] [20] [21] sin 2 0 = 2 sin 0 cos 6. cos 2 = cos20 - sin2 = 1 -2sin2 = 2cos20 - 1. 2 tan 0 tan 2 0 - 1 - tan2 0 s 1 I - cos 0 2 2 o 1 / + cos 0 cos2 = -\/ 2 [22] [23] 1 / - cos 1 - cos sin 0 [24] tan0 = - 1 + cos= sin 1 + =cos 2[25]si + s cos + )sn 1a - s 3) [25] sin a q- sin 0 = 2 sin ~ (a q- 0) cos ~ (a -- t5). FORMULAS 153 [26] sin a - sin 3 = 2 cos ~ (a + ) sin (a - ). [27] cos a + cos = 2 cos (a + 3) cos (a -). [28] cosa - cos3 = - 2 sin 2 (a + /) sin (a - /). [29] sin a cos / = 2 sin (a + 3) -+ 2 sin (a - ). [30] cos a sin (a + ) - sin (a - 3). [31] cos a cos = 2 cos (a + ) + 2 cos (a - r). [32] sin a sin / = - cos (a ) + cos (a -). a b c [33] -a- -. (Sine Law.) sin a sin /3 sin [34] a2 = b2 + c2 - 2 bc cos a. (Cosine Law.) K b2 sin a sin y [35] K= 2sin/3 [36] K = ab sin y. [37] a+-b = tan(a+-) [37] 2 (a a - b tan (a - 3) 1 a-b 1 [38] tan1 (a - ) a + cot [39] sina = bc ( 2 s (s - a) [42] tan r h (s -a) (s b) (s - c) [41] tanla= s-a' r - 2 s - as [43] K = s (s - a) (s - b) (s - c). [44] [r (cos 0 +- j sin 0)]n = rn [cos nO + j sin nO]. (DeMoivre's Theorem, Art. 94.) a3 a5 a7 [45] sina=a — +- --— + -*. [46] cosa= l —a- - -+ ~* * * a3 2 a5 [47] tana = a + - -- + *. [48] log, N = 2.3026- logo N. [49] loglo N = 0.43429 log, N. 154 PLANE TRIGONOMETRY USEFUL CONSTANTS 1 cu. ft. of water weighs 62.5 lb. = 1000 oz. (Approx.) 1 gal. of water weighs 8~ lb. (Approx.) 1 gal. = 231 cu. in. (by law of Congress). 1 bu. = 2150.42 cu. in. (by law of Congress). 1 bu. = 1.2446- cu. ft. = 5 cu. ft. (Approx.) 1 cu. ft. = 7~ gal. (Approx.) 1 bbl. = 4.211- cu. ft. 1 meter = 39.37 in. (by law of Congress). 1 in. = 25.4 mm. 1 ft. = 30.4801 cm. 1 Kg. =2.20462 lb. 1 gram = 15.432 gr. 1 lb. (avoirdupois) = 453.5924277 g. = 0.45359+ Kg. 1 lb. (avoirdupois) = 7000 gr. (by law of Congress). 1 lb. (apothecaries) = 5760 gr. (by law of Congress). 1 liter = 1.05668 qt. (liquid) = 0.90808 qt. (dry). 1 qt. (liquid) = 946.358 cc. = 0.946358 liters, or cu. dm. 1 qt. (dry) = 1101.228 cc. = 1.101228 liters, or cu. dm. 7r = 3.14159265358979 = 3.1416 = 355 = 3-. (All Approx.) 1 radian = 57~ 17' 44.8" = 57.29577950+. 1~ = 0.01745329+ radians. e = 2.718281828+, the base of the Napierian logarithms. INDEX (Numbers refer to pages.) Abscissa, 11. Angle, addition and subtraction, 3. construction, 20, 24. definition of, 1. depression, 45. elevation, 45. functions of, 13. general, 7. generation of, 1. measurement of, 3. negative, 2. positive, 2. to find from function, 68. Accuracy, 51. tests of, 52. Circular measure, 3. Complex number, 134. amplitude of, 135. division of, 138. graphical representation of, 134. modulus of, 135. multiplication of, 136. polar form of, 135. Co6rdinates, polar, 11. rectangular, 10. Cosine Theorem, 103. Functions, multiple-valued, 78. of angles greater than 90~, 66. of double angles, 90. of half angles, 92. period of, 74. periodic, 74. principle value of, 78. relations between, 27. single-valued, 78. trigonometric, 13, 14. Formulas, addition and subtraction, 83. derivation of, for differences, 84, 89. for sums, 84, 89. for expansion of sin nO and cos nO, 143. for exponential values of sin 0, cos 0, and tan 0, 145. for powers of trigonometric functions, 146. for products to sums, 97. for sums to products, 94. lists of, 152. Graph, 75. mechanical construction of, 76. DeMoivre's Theorem, 137. Equations, trigonometric, 34, 127. Exponential values of sin 0 and cos 0, 145. Forces, composition of, 54. resultant of, 54. Functions, changes in value of, 74. hyperbolic, definition, 148. inverse, 77. Horizon, dip of, 56. Hyperbolic functions, definition, 148. relations between, 149, 150. Identity, 32. Infinity, 16. Limit, 79. Lines, directed, 9. segments of, 9. Logarithms, 43. 155 156 INDEX Ordinate, 11. Projection, orthogonal, 53. Quadrants, 2. Radian, definition of, 4. Radius of circumscribed circle, 102. Radius of inscribed circle, 116. Radius vector, 11. Roots of numbers, 138, 139, 141. Segment, area of, 56. value of line, 9. Sector, area of, 56. Sine Theorem, 101. Solution of oblique triangles, 103, 105, 109, 113. of right triangles, 40, 43. Series for powers of trigonometric functions, 146. Trigonometric functions, applied to right triangles, 22. calculation by computation, 16. calculation by measurement, 15. Trigonometric functions, changes in value of, 74. '. computation of, 100.' ' definition, 14. exponents of, 15. graph of, 75. inverse, 34, 77. line representation of, 61. logarithmic, 15. natural, 15. of angles greater than 90~, 66. of complementary angles, 23. of double angles, 90. of half angles, 92. of sums and differences of angles, 83. principle values of, 78. relations between, 27, 29. signs of, 15. table of, 15. transformation of, 31. Trigonometric ratios, 13. Tangent Theorem, 110. Vectors, 53. composition of, 54. FIVE-PLACE LOGARITHMIC AND TRIGONOMETRIC TABLES WITH EXPLANATORY CHAPTER ARRANGED BY CLAUDE IRWIN PALMER AND CHARLES WILBER LEIGH ASSOCIATE PROFESSORS OF MATHEMATICS, ARMOUR INSTITUTE OF TECHNOLOGY FIRST EDITION McGRAW-HILL BOOK COMPANY, INC. 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, E. 0. 1914 COPYRIGHT, 1914, BY THE McGRAW-HILL BOOK COMPANY, INC. Stanbope lprezs F. H. GILSON COMPANY BOSTON, U.S.A. CONTENTS LOGARITHMS AND EXPLANATIONS OF TABLES ART. PAGE 1. Use of logarithms........................................... 1 2. Exponents................................................ 1 3. D efinitions................................................. 2 4. Notation............................................. 2 5. Systems of logarithms.................................. 3 6. Properties of logarithms................................ 3 7. Logarithms to the base 10................................. 4 8. Rules for determining the characteristic........................ 6 9. T he m antissa............................................... 7 10. T ables....................................................... 7 11. To find the mantissa of the logarithm of a number............ 8 12. Rules for finding the mantissa............................... 9 13. Finding the logarithm of a number............................ 10 14. To find the number corresponding to a logarithm................ 10 15. Rules for finding the number corresponding to a logarithm....... 12 16. To multiply by means of logarithms........................... 13 17. To divide by means of logarithms............................ 13 18. C ologarithm s............................................ 14 19. To find the power of a number by means of logarithms.......... 14 20. To find the root of a number by means of logarithms........... 15 21. Proportional parts.......................................... 16 22. Suggestions............................................... 16 23. Changing systems of logarithms............................. 19 24. U se of Table II........................................ 20 26. Explanatory of Table III................................. 21 27. To find logarithmic function of an acute angle............... 22 28. To find the acute angle corresponding to a logarithmic function... 23 29. Angles near 0~ and 90~.................................... 24 30. Functions by means of S and T............................... 25 31. Examples in using S and T................................... 25 32. Functions of angles greater than 90~.......................... 26 33. Table IV. Explanatory..................................... 27 34. Table V. Explanatory..................................... 28 35. Errors of interpolation....................................... 28 Table I. Logarithms of Numbers............................. 31 Table II. Conversion of Logarithms.......................... 52 Table III. Logarithms of Trigonometric Functions........... 53 Table IV. Natural Trigonometric Functions.................... 107 Table V. Radian M easure.................................. 131 Table VI. Constants and Their Logarithms................... 132 LOGARITHMS AND EXPLANATION OF TABLES 1. Use of logarithms. - By the use of logarithms, the processes of multiplication, division, raising to a power, and, extracting a root, of arithmetical numbers are usually much simplified. The process of multiplication is replaced by one of addition, that of division, by one of subtraction, that of raising to a power by 'a simple multiplication, and that of extracting a root, by a division. Many calculations that are difficult or impossible by other mathematical methods are readily carried out by means of logarithms. It was said by the great French astronomer, Laplace, that the method of logarithms by reducing to a few days the labors of many months, doubled, as it were, the life of an astronomer, besides freeing him from the errors and disgust inseparable from long calculations. Of course these same advantages are shared by others who find it necessary to do numerical calculations.* 2. Exponents. -The student is already familiar with the following definitions and theorems from algebra, concerning exponents. For convenience they are restated here. Definitions. (1) a, = a *aa... to n factors. n an integer. (2) a-n = (3) a= 1. n (3) a~- 1. (4) am = 4/an Theorems. (1) an. am = anm. (2) an - a"m = an(3) (a b.c... )n= an b' cn ~.. (4) (b) =b(5) (an)m = anm * " The miraculous powers of modern calculations are due to three inventions: the Hindu Notation, Decimal Fractions, and Logarithms." Cajori, A History of Elementary Mathematics. 1 2 PLANE TRIGONOMETRY 3. Definitions. - If three numbers N, b, and x have such values that N = b, then x is called the logarithm* of N to the base b. In words this gives the following. DEFINITION. The logarithm of a number to a given base is the exponent by which the base must be affected to produce that number. If in the equation, N = bx, all possible positive values are given to N, while b is some positive number other than 1, the corresponding values of x form a system of logarithms. 4. Notation. - If 4 is taken as a base, then, in the language, or notation, of exponents, 43 = 64. In the language, or notation, of logarithms, the same idea is expressed by saying the logarithm of 64 to the base 4 is 3. This is abbreviated and written log4 4 = 3. Exponent Logarithmic notation notation 45 = 1024. log41024 = 5. 34 = 81. log 81 = 4. 53 = 125. log5 125 = 3. 405 = 2. log4 2 = 0.5. 16' = 64. loge6 64 = 3. 103 = 1000. logio 1000 = 3. EXERCISES Answer as many as possible orally. 1. Express in logarithmic notation: (1) 33 = 27. (4) 640.5 = 8. (7) 32~04 = 4. (2) 53 = 125. (5) 104 = 10,000. (8) 102.3979 = 250. (3) 73 = 343. (6) 125l = 5. (9) 101.5465 = 35.2. 2. Express in exponent notation: (1) log12256 = 8. (3) log162 = 0.25. (5) logio 429 = 2.6325. (2) log6216 = 3. (4) logio 643 = 2.8082. (6) logio 99.9 = 1.9996. * The word logarithm is derived from logos meaning ratio and arithmos meaning number. LOGARITHMS AND EXPLANATION OF TABLES 3 3. Find the logarithms of the following: (1) log636. (4) log9 729. (7) log82. (2) log3243. (5) log8512. (8) log16 128. (3) log53125. (6) log5i 100,000. (9) logo 0.001. 4. Find the value of x in the following: (1) logs x = 4. (5) logox = -3. (9) log25 X = 2. (2) logo0x = 4. (6) logs x =-2. (10) log16 x =. (3) log6 x = -. (7) log16x = 0.75. (11) log125 x =. (4) log1ox = 0. (8) logs x = 4. (12) log49 x =. 5. Find the value of x in the following: (1) logx 1000 = 3. (5) logx 8 = 0.5. (9) logx 27 = 0.75. (2) logs 81 = 4. (6) logs 4 = 0.25. (10) log, 2 = 0.125. (3) log, 256 = 4. (7) log, 16 = 4. (11) logx 36 = 2. (4) log, 1024 = 10. (8) log, 18 = 0.5. (12) logx 100 = 2. 5. Systems of logarithms.* - While theoretically any positive number other than 1 may be used as a base for a system of logarithms, in practice only two bases are used. (1) The common system, or Briggs' system, of which the base is 10. (2) The natural system, also called the hyperbolic, or Napierian system, of which the base is a number that to seven decimal places is 2.7182818. This base is usually represented by the letter e. The common system is the one commonly used in computing, and the natural system in more advanced and theoretical work. 6. Properties of logarithms. - The use of logarithms depends upon the following properties, which are true for any base greater than unity: (1) The logarithm of 1 is zero. Since b~ = 1, for any base, logb 1 = 0. (2) The logarithm of the base of any system is unity. Since bl = b, for any base, logb b = 1. *Logarithms were invented by John Napier, Baron of Merchiston, of Scotland, who lived from 1550 to 1617. They were described by him in 1614. A contemporary of Napier's, Henry Briggs (1556 to 1631) professor of Gresham College, London, modified the new invention by using the base 10, and so made it more convenient for practical purposes. See Cajori, A History of Elementary Mathematics, page 160, et seq. For a very complete account of logarithms, see History of the Logarithmic and Exponential Concepts, by Florion Cojori, appearing in the American Mathematical Monthly, January to June, 1913. 4 PLANE TRIGONOMETRY (3) The logarithm of zero in any system whose base is greater than 1 is negative infinity. *Since b-~ = = 0, logbO = -. If N = bx and M = by, then, by the definition of a logarithm, logbN = x and logb M = y. We have also by. the definitions and theorems of exponents and logarithms: (a) N X M = bx X by = bx+Y; logb (N X M) = x + y = logb N + logb M. (b) N- M = bx - by = bx-Y;.log (N - M) = - y = logb - log M. (c) Nn = (bx)n = bnx;.. logb (Nn) = nx = n logbN. 1 1 1 1 (d) -N = (bx)n = bnX;.. logb ='N = - x = - logb N n n The following theorems are therefore established: (4) The logarithm of a product equals the sum of the logarithms of the factors. By (a). (5) The logarithm of a quotient equals the logarithm of the dividend minus the logarithm of the divisor. By (b). (6) The logarithm of a power of a number equals the logarithm of the number multiplied by the exponent of the power. By (c). (7) The logarithm of a root of a number equals the logarithm of the number divided by the index of the root. By (d). The truth of the statements in Art. 1 follows from these theorems. That is, the process of multiplication is replaced by an addition; division by a subtraction; raising to a power, by a multiplication; and extracting a root, by a division. 7. Logarithms to the base 10.- In what follows, if no base is stated, it is understood that the base 10 is used. When the base is 10 we evidently have the following: 105 = 100,000.*. log 100,000 = 5. 104 = 10,000.'. log 10,000 = 4. 103 = 1000.'. log 1000 = 3. 102 = 100.'. log 100 = 2. 10' = 10.'. log 10 = 1. * By b — =0 is meantxl= [b] [ x] LOGARITHMS AND EXPLANATION OF TABLES 5 10~ =1.. log 1=0. 10- = 0.1.. log0.1 =-1 = 9 -10. 10-2 = 0.01.. log 0.01 =- 8 - 10. 10-3 = 0.001.'. log 0.001 = -3 = 7 - 10. 10-4 = 0.0001.. log 0.0001 = -4 = 6 - 10. 10-5 = 0.00001.-. log 0.00001 = -5 = 5 - 10. It is evident that these are the only numbers between 0.00001 and 100,000 which have integers for logarithms. Every other number in this range has then for a logarithm an integer plus or minus a fraction. This fraction is put in the form of a decimal. For instance, the logarithm of any number between 1000 and 10,000 is between 3 and 4, or it is 3 + a decimal. For a number between 100 and 1000 the logarithm is 2 + a decimal. Between 0.01 and 0.1, the logarithm may be -2 + a decimal or -1- a decimal; but, in order that the fractional part of the logarithm may always be positive, it is agreed to take the logarithm so that the integral part only is negative. Usually, then, the logarithm of a number consists of two parts, an integer and a fraction, the fraction being the approximate value of an irrational number. The integral part is called the characteristic. The fractional part is called the mantissa. The logarithm is the characteristic plus the mantissa. The mantissas of the positive numbers arranged in order are called a table of logarithms. The logarithm of 3467 consists of the characteristic 3 plus some mantissa because 3467 lies between 1000 and 10,000. The logarithm of 59,436 is 4 + a decimal because 59,436 lies between 10,000 and 100,000. The log 0.0236 = -2 + a decimal because 0.0236 lies between 0.01 and 0.1. It is readily seen that multiplying a number by 10n increases the characteristic by n, where n is an integer; and dividing a number by 10" decreases the characteristic by n. For, log (NXlOn) =log N+log lOn=log N+n log 10=logN+n, and log (N - 10n) = log N-log 10=log N-nlog 1O=log N-n. This establishes the following: THEOREM. The position of the decimal point in the number affects the characteristic of the logarithm only, the mantissa remaining unchanged for the same sequence of figures. 6 PLANE TRIGONOMETRY The advantages in using the base 10 are that the characteristic can be determined by inspection, and that the mantissa remains unchanged for the same sequence of figures. Thus, log 934,700 = 5.97067, log 9347 = 3.97067, log 9.347 = 0.97067, log 0.009347 = 3.97067 = 7.97067 - 10. 8. Rules for determining the characteristic. - From the foregoing considerations the following rules for determining the characteristic are evident: (1) When the number is greater than 1, the characteristic is positive, and is one less than the number of digits to the left of the decimal point. (2) When the number is less than 1 and expressed decimally the characteristic is negative, and is one more than the number of zeros immediately at the right of the decimal point. When the characteristic is negative the minus sign is placed above the characteristic to show that it alone is negative. Thus, in log 0.009347 = 3.97067, the 3.97067 means -3 + 0.97067. It should not be written -3.97067 for then the minus sign would indicate that both characteristic and mantissa were negative, while we have agreed that the mantissa shall always be considered positive. In computations involving negative characteristics, to avoid the use of the negative, 10 is usually added to the characteristic and subtracted at the right of the mantissa. In writing logarithms in this form, the characteristic, when 10 is added, is 9 minus the number of zeros immediately at the right of the decimal point. Thus, in the above, log 0.009347 = 7.97067 - 10. The characteristics of the following are as given: of 3426 is 3 by rule (1), of 3.2364 is 0 by rule (1), of 0.00639 is -3, or 7 - 10, by rule (2), of 2.04 is 0 by rule (1), of 0.000067 is -5, or 5 - 10, by rule (2). In each of the following state the characteristic to the base 10: 923 42,376 32.54 89,236 425.03 3.2067 0.0123 0.2146 1.111 0.00046 3.004 333.33 LOGARITHMS AND EXPLANATION OF TABLES 7 9. The mantissa. - The determination of the decimal part of a logarithm, the mantissa, is more difficult than the determination of the characteristic. Because of this difficulty the mantissas have been carefully determined and arranged in tables of logarithms.* They are given to three, four, five, or more places of decimals. The degree of accuracy in computations made by logarithms depends upon the number of places in the table used; the more places in the table the greater the degree of accuracy. The tables generally used are those having from four to seven places. 10. Tables. - Upon examining a five place table of logarithms (see Table I), it is noticed that the first column has the letter N at the top and the bottom. This is an abbreviation for number. The other columns have at top and bottom the numbers 0, 1, 2, 3.. 9. Table I contains the integers from 1000 to 11,009. Pages 32 to 49 have the numbers from 1000 to 10,009. Here the first three figures are printed in the column marked N and the fourth figure at the top and bottom of another column. Thus, to locate 4756, 475 is found in the N-column on page 39 and 6 in the column headed 6. Pages 50 and 51 contain the numbers from 10,000 to 11,009, where the first four figures are printed in the N-column. The columns of numbers after the first consist of the mantissas of the numbers located in the N-column and at the top, or bottom, of another column. These mantissas are printed correct to five decimals except on pages 50 and 51, where they are given to seven places. To save space, the first two figures of the mantissas are printed in the 0-column only. Any such two figures go with the other figures to the right and below until another two figures is found in the 0-column. Except that when an asterisk (*) is found * Professor Briggs' tables were computed to fourteen places, but were not finished by him. They were completed by Adrian Vlacq (1628), who shortened them to ten places, and finished a table including the numbers from 1 to 100,000. Briggs' and Vlacq's tables are essentially the same as those in use now. They have been checked and re-computed in part many times. At present, errors found in tables are typographical. The most complete check was undertaken by the French authorities in 1784. It required the labors of nearly one hundred mathematicians and computers for over two years. They computed to fourteen places the logarithms of all integers from 1 to 200,000, besides natural and logarithmic trigonometric functions. These tables were never printed. Two manuscript copies are preserved. 8 PLANE TRIGONOMETRY before the three figures given in the other columns, the first two figures of the mantissa are taken from the next line below. When a mantissa ends in a figure 5 it is printed 5 when it is really less than printed; otherwise a mantissa when ending in 5 is larger than printed. Thus, if the mantissa is 0.0273496, in contracting it to five places, it is printed 0.02735. This is to guide one wishing to write the mantissas correct to four places. For the meaning of the Prop. Parts, see Art. 21. For the meaning of the numbers at the foot of the pages and connected with S and T, see Art. 30. Notice that when advancing in the table, the mantissas increase. The difference between two consecutive mantissas is called the tabular difference. 11. To find the mantissa of the logarithm of a number. - Use Table I, pages 32 to 49. (1) When the number consists of four significant figures. Example. Find the mantissa of log 4673. Find the first three figures, 467, of the number in the N-column and the 3 at the top of the page. The mantissa of log 4673 is found to the right of 467 and in the column headed 3.. Mantissa of log 4673 = 0.66960. In like manner find the following: Mant. cf log 4799 = 0.68115. Mant. of log 23.78 = 0.37621. Mant. of log 2.955 = 0.47056. Mant. of log 0.0003964 = 0.59813. Mant. of log 3560 = 0.55145. Mant. of log 4930 = 0.69285. Mant. of log 556,700 = 0.74562. Mant. of log 0.001001 = 0.00043. (2) When the number consists of one, two, or three significant figures. The number is found in the N-column and the mantissa to the right in the 0-column. Thus, Mant. of log 4.78 = Mant. of log 4780 = 0.67943. Mant. of log 39 = Mant. of log 3900 = 0.59106. Mant. of log 4 = Mant. of log 4000 = 0.60206. LOGARITHMS AND EXPLANATION OF TABLES 9 (3) When the number consists of five or more significant figures. Example 1. Find the mantissa of log 39,467. Since 39,467 lies between 39,460 and 39,470 its mantissa must lie between the mantissas of these numbers. Mant. of log 39,460 = 0.59616. Mant. of log 39,470 = 0.59627. The difference between these mantissas is 0.00011, which is the tabular difference. Since an increase of 10 in the number increases the mantissa 0.00011, an increase of 7 in the number will increase the mantissa 0.7 as much, or the increase is 0.00011 X 0.7 = 0.000077 or 0.00008.. Mant. of log 39,467 = 0.59616 + 0.00008 = 0.59624. The process of finding the mantissa as above is called interpolation. As carried out, it is assumed that the increase of the logarithm is proportional to the increase of the number. This assumption is not strictly true as will be seen in Art. 35. Example 2. Find the mantissa of log 792,836. Mant. of log 792,900 = 0.89922 Mant. of log 792,800 = 0.89916 Tabular difference = 0.00006 Since an increase of 100 in the number increases the mantissa 0.00006, an increase of 36 in the number increases the mantissa 0.00006 X 0.36 = 0.00002, correct to the nearest fifth decimal place..'. Mant. of log 792,836 = 0.89916 + 0.00002 = 0.89918. These processes should seem reasonable; but, since they are to be performed so frequently, it is best to work by rule. 12. Rules for finding the mantissa. - (1) For a number consisting of four figures, find the first three figures of the number in the N-column and the fourth figure at the head of a column; then read the mantissa in the column under the last figure and at the right of the first three figures. (2) For a number consisting of one, two, or three figures, find the number in the N-column and the mantissa to the right in the column headed 0. 10 PLANE TRIGONOMETRY (3) For a number consisting of more than four figures, find the mantissa for the first four figures by rule (1) and add to this the product of the tabular difference by the remaining figures of the number considered as a decimal number. 13. Finding the logarithm of a number.- In finding the logarithm of a number, it is best to determine the characteristic first and then look up the mantissa. Perform all the interpolations without the aid of a pencil if possible. The use of the proportional parts is explained in Art. 21; but the student is advised to become familiar with interpolating without their help. Example 1. Find the logarithm of 92.36. The characteristic is 1, by rule (1) for characteristics. The mantissa is 0.96548, by rule (1) for mantissas... log 92.36 = 1.96548. Example 2. Find the logarithm of 3.4676. Rule (1) for characteristic gives 0. Rule (3) for mantissa gives 0.54003. log 3.4676 = 0.54003. Example 3. Find the logarithm of 0.00039724. Rule (2) for characteristic gives 4. Rule (3) for mantissa gives 0.59905. log 0.00039724 = 4.59905 = 6.59905 - 10. EXERCISES Verify the following by the tables: 1. log 9376 = 3.97202. 2. log 4.236 = 0.62696. 3. log 220 = 2.34242. 4. log 1.11 = 0.04532. 5. log 20 = 1.30103. 6. log 0.02 = 2.30103. 7. log 0.00263 = 7.41996 - 10. 8. log 26.436 = 1.42220. 9. log3.1416 = 0.49715. 10. log 0.492357 = 1.69228. 11. log276.392 = 2.44153. 12. log 0.027646 = 8.44164 - 10. 13. log 0.0049643 = 3.69586. 14. log 0.029896 = 2.47561. 15. log2.71828 = 0.43429. 16. log99.999 = 2.00000. 17. log 1111.11 = 3.04575. 18. log 100.03 = 2.00013. 14. To find the number corresponding to a logarithm.- If log 31.416 = 1.49715, then 31.416 is the number corresponding to the logarithm 1.49715. It is sometimes called the antilogarithm and is written 31.416 = log-' 1.49715. LOGARITHMS AND EXPLANATION OF TABLES 11 In nearly every problem involving logarithms, it is not only necessary to find the logarithms of numbers; but the inverse process, that of finding the number corresponding to a logarithm, has to be performed. Since the position of the decimal point in no way affects the mantissa, we should expect to determine the sequence of figures in the number from the mantissa. And since a change in the position of the decimal point increases or decreases the characteristic, the decimal point may be located in the number when the characteristic of the logarithm of the number is known. (1) When the mantissa of the given logarithm is given exactly in the table. Example. Find the number having 2.58939 for its logarithm. Find in the table the mantissa 0.58939. To the left of this mantissa, in the N-column, find the first three figures, 388, of the number, and at the head of the page find the fourth figure, 5, of the number. The number then consists of the sequence of figures 3885, but we do not know where the decimal point is until we consider the characteristic, which is 2. Hence there must be three figures at the left of the decimal point.. 2.58939 = log388.5. A change in the characteristic changes the location of the decimal point. Thus, 4.58939 = log 38,850, 2.58939 = log 0.03885, 7.58939 - 10 = log 0.003885. (2) When the mantissa of the given logarithm is not given exactly in the table. In this case two other consecutive mantissas can always be found between which the mantissa of the given logarithm lies. The number of four figures corresponding to the smaller of these mantissas gives the first four figures of the number sought. The fifth, and often the sixth figure, can then be found by interpolating, assuming that for comparatively small differences in the numbers, the differences in the numbers are proportional to the differences in the logarithms of the numbers. For using the proportional parts in interpolating see Art. 21. Example. Find the number whose logarithm is 1.49863. In the table find the mantissas 0.49859 and 0.49872, between 12 PLANE TRIGONOMETRY which the given mantissa lies. Thinking only of the sequence of figures in the numbers, 0.49872 = Mant. of log 3153 0.49859 = Mant. of log 3152 0.00013 1 Hence a difference of 0.00013 in the logarithm makes a difference of 1 in the number. Now the given mantissa is 0.00004 larger than the smaller one. Then the number having 0.49863 as the mantissa of its logarithm is 0.00004 4 0.00013 13 larger than 3152. Hence the sequence of figures for the number having 1.49863 as a logarithm is 31,523. Since the characteristic is 1, 1.49863 = log 31.523. The interpolation should be carried out mentally, leaving out zeros and taking -4 X 1 = 0.3. This could also be stated as a proportion, 13:4 =: x,.'. x = 0.3. 15. Rules for finding the number corresponding to a given logarithm. - (1) When the mantissa of the given logarithm is found exactly in the table, the first three figures of the number are found to the left of the mantissa in the N-column, and the fourth figure is at the head of the column in which the mantissa is found. (2) When the mantissa of the given logarithm is not found exactly in the table, find the mantissa nearest the given mantissa but smaller. The first four figures of the number are those corresponding to this mantissa, and are found by rule (1). For another figure, divide the difference between the mantissa found and the given mantissa, by the tabular difference. In both (1) and (2), place the decimal point so that the rules for determining the characteristic may be applied and give the characteristic of the logarithm. Example. Find the number corresponding to 3.87626. Mantissa nearest 0.87626 is 0.87622 = Mant. of log 7520. Tabular difference = 6. Difference between mantissas = 4. 4 + 6 = 0.7 to nearest tenth.. 3.87626 = log 7520.7. LOGARITHMS AND EXPLANATION OF TABLES 13 EXERCISES Find the values of x or verify the following: 1. 3.70944 = log 5122. 2. 2.58377 = log 0.03835. 3. 1.74819 = logx. 4. 7.94236 - 10 = log 0.008757. 5. 0.47712 = log x. 6. 3.47954 = log 3016.7. 7. 2.57351 = log 374.55. 8. 0.92876 = log x. 9. 9.23465 - 10 = log x. 10. 4.92317 = log 0.00083786. 11. 8.12112 - 10 = log 0.013217. 12. 6.28697 = logx. 13. 6.89909 = log. 14. 11.46729 = logx. 15. 9.92867 - 20 = log x. 16. 3.88888 = log x. 17. 3.33333 = log x. 18. 4.0002565 = log 10005.9. 19. 2.0331894 = log 107.9417. 20. 3.0275278 = log 1065.437. For Exercises 18 to 20 use pages 50 and 51 of Table I. 16. To multiply by means of logarithms. - Property (4) of Art. 6 gives the following: RULE. To find the product of two or more factors, find the sum of the logarithms of the factors; the product is the number corresponding to this sum. Example. Find the product of 34.796 X 0.0294 X 3.1416. Process. log 34.796 = 1.54153 log 0.0294 = 8.46835 - 10 log 3.1416 = 0.49715 log of product = 0.50703. product = 3.2139 17. To divide by means of logarithms. - Property (5),of Art. 6 gives the following: RULE. To find the quotient of two numbers, subtract the logarithm of the divisor from the logarithm of the dividend; the quotient is the number corresponding to this difference. Example 1. Find the quotient of 27.634 - 5.427. Process. log 27.634 = 1.44144 log 5.427 = 0.73456 log of quotient = 0.70688 quotient = 5.0919 7.246 X 0.8964 X 5.463 Example 2 Evaluate 4.27 X 0.3987 X 27.89 Here find logarithm of the numerator then that of the denominator. 14 PLANE TRIGONOMETRY Process. log 7.246 = 0.86010 log 4.27 log 0.8964 = 9.95250 - 10 log 0.3987 log 5.463 = 0.73743 log 27.89 log of Num. = 1.55003 log of Den. log of Den. = 1.67653 log of quotient = 9.87350 - 10 quotient = 0.74732 18. Cologarithms. - The logarithm of the number is called the cologarithm of the number. the arithmetical complement. = 0.63043 = 9.60065 - 10 = 1.44545 = 1.67653 reciprocal of a It is also called 1 1 Since the reciprocal of N is, the colog N = log = log 1 -log N. M 1 Also log W = log M + log N = log M + colog N, that is: The logarithm of the quotient of two numbers is equal to the logarithm of the dividend plus the cologarithm of the divisor. The logarithm of the quotient of two numbers is equal to the logarithm of the dividend plus the cologarithm of the divisor. To find the cologarithm of a number, subtract the logarithm of the number from 10 - 10. Do the work mentally beginning at the left and subtracting each figure from 9, excepting the last significant figure at the right which is to be taken from 10. Thus, colog 9.423 = log 1 - log 9.423. log 1 = 10.00000 - 10 log 9.423 = 0.97419. colog 9.423 = 9.02581 - 10 The solution of Example 2, Art. 17, takes the following form: log 7.246 = 0.86010 log 0.8964 = 9.95250 - 10 log 5.463 = 0.73743 colog 4.27 = 9.36957 colog 0.3987 = 0.39935 colog 27.89 = 8.55455 - 10 log of quotient = 9.87350 - 10.quotient = 0.74732 19. To find the power of a number by means of logarithms. - Property (6) of Art. 6 gives the following: LOGARITHMS AND EXPLANATION OF TABLES 15 RULE. To find the power of a number, multiply the logarithm of the number by the exponent of the power; the number corresponding to this logarithm is the required power. Example 1. Find the value of (2.378)6. Process. log 2.378 = 0.37621 6 X log 2.378 = 2.25726 = log (2.378)6 (2.378)6 = 180.83 Example 2. Find the value of (237.45)7. Process. log 237.45 = 2.37557 7 X log 237.45 = 1.69684 = log (237.45)r (237.45)" = 49.756 20. To find the root of a number by means of logarithms.Property (7) of Art. 6 gives the following: RULE. To find the root of a number, divide the logarithm of the number by the index of the root; the number corresponding to this logarithm is the root required. Example 1. Find /27.658. Process. log 27.658 = 1.44182 - X log 27.658 = 0.28836 = log ~/27.658.. /27.658 = 1.9425 Example 2. Find "/0.008673. Process. log 0.008673 = 7.93817 - 10 log ~/0.008673 = 6 (7.93817 - 10) = 1 (57.93817 - 60) = 9.65636 - 10. /0.008673 = 0.45327 REMARK. When a logarithm with a negative characteristic is to be divided by a number not exactly contained in the characteristic, it is best to first add and subtract such a number of times 10 that after dividing there will be a minus 10 at the right. In the above, before dividing (7.93817 - 10) by 6, 50 was added and subtracted. However, if the divisor had been 3, the division could have been performed by writing the logarithm in the form 3.93817 and dividing at once by 3. 16 PLANE TRIGONOMETRY 21. Proportional parts. -In Table I, the marginal tables, marked Prop. Parts, contain the products of the tabular differences by 1, 2, 3,... 9 tenths. These products are arranged for convenience in interpolating. The work should be done mentally. Thus, in interpolating, if the tabular difference is 35 35, then the marginal table is as given. In finding the 2 7.0 logarithm, it is required to multiply, say 35 by some 14.0 number; and, in finding a number corresponding to a 6 21 0 7 24.5 logarithm, it is required to divide some number by 35. 8 20 31.5 (1) Multiply 35 by 0.68. Process. 35 X 0.6 = 21.0 35 X 0.08 = 2.8. 35 X 0.68 = 23.8 (2) Divide 29 by 35. Process. Dividend..... 29 Next less........ 28.0 giving 0.8 Remainder...... 10 Next less........ 7.0 giving 0.02 Remainder...... 30 Next less........ 28 giving 0.008 Etc. 0.828... = quotient. In interpolating, the division is usually only to determine the nearest first figure, and therefore can easily be done mentally. 22. Suggestions. - In interpolating, do not carry logarithms beyond the number of decimal places given in the table. In writing numbers correct to a certain number of figures, take in the last place the figure that is nearest the true result when this is possible. If the next figure after the last one to be taken is 5 followed only by zeros, most computers take the nearest even figure for the last one. Thus, if the number is 0.02467500... it would be taken 0.02468 to five places; and if 0.02468500... it would also be taken 0.02468. In working with tables, use the pencil as little as possible. Work for accuracy first and then for speed. Write out a scheme for all logarithmic work before referring to the table. Be sure that your work is arranged so that it could be followed at any time by yourself or another person. LOGARITHMS AND EXPLANATION OF TABLES 17 Example. Write out a scheme for finding the value of 9.46 X (41.6)2 X -\/9.462 X = 276.2 X 3.4675 Scheme. log 9.46 = 2 log 41.6 = log 9.462 = colog 276.2 colog 3.4675 log x X In computing with negative numbers, disregard the negative sign in using the logarithms. The signs can be taken into account afterwards. EXERCISES Solve by logarithms. 1. 23.764 x 5.4326. 2. 9.0467 x 0.067948. 3. (-0.009237) x 4256.72. 4. 24.789 x 4.3648 x 0.03679. 5. 3.1416 x 1000 x 0.439647. 6. 39.6742 + 4.37893. 7. 0.093476 + 0.0046934. 8. 10.0467 + 95.649. 9. 479.996 + (-27.9395). 10. 9.49923 + 249.693. 11. (1.07)19. 12. (4.2367)1 x 0.09476. 13. (1.4641)i. 14. (4.56)4.56. 15. (7.2367)7.23. 16. V476.243. 17. 54.693 x V92.3764. 18. V5106.5 x 0.00003109. 19. <'0.0009657 ~. <0.0044784. 20. <'0.052734 X <'-0.019642. 6000 x 5 x 29 0.7854 x 25,000 X 81.7 47.96 x 9.2876 x 47 22. 29.439 x 29.999 9.25 x V/72.56 x 2.3672 1.27893 x <'927.86 Ans. 129.10. Ans. 0.61471. Ans. - 39.319. Ans. 3.9806. Ans. 1381.2. Ans. 9.0604. Ans. 19.917. Ans. 0.10504. Ans. -17.180. Ans. 0.038044. Ans. 3.6159. Ans. 7.2065. Ans. 1.1. Ans. 1011.3. Ans. 1,638,600. Ans. 3.4323. Ans. 104.40. Ans. 0.79450. Ans. 1.0695. Ans. -0.14979. Ans. 0.54234. Ans. 23.705. Ans. 21.638. 18 PLANE TRIGONOMETR Y Scheme of work. log 9.25 = ~ log 72.56 = 2 log 2.367 = 3 colog 1.2789 = ~ colog 927.86 = log result = result = 24. Evaluate (-b)(s-c) where 2s = a + b + c and a = 47.236, s-a b = 82.798, and c = 75.643. Ans. 31.750. 25. Evaluate Vs (s -a) (s - b) (s - c), where 2s = a + b +c and a = 4.2763, b = 9.9264, and c = 8.4399. Ans. 17.904. PLAN 26. Using the formula for horse-power, H = A; find H when P = 76.5, 33,000' L = 2.25, A = 231.8, and N = 116. Ans. 140.25. 27. Given 1T = 0.0033 X 10-7 n, find TW when n = 75,000. Ans. 0.000024749. 28. In finding the diameter of a wrought-iron shaft that will transmit 90 horse-power when the number of revolutions is 100 per minute, using a factor of safety of 8, it is required to find the diameter d from the formula: 90 d = 68.5 Ans. 3.5904. 100 X 50,000 8 Wl3 = 980 29. Find the value of M from the formula M = 3 when g = 980, 4b d3B' W = 75, 1 = 50, b = 0.98178, d = 0.5680, and B = 0.01093. Ans. 11.681 X 101. 360 Lnmgl 30. Find the value of n from the formula n = -, when L = 69.6, m = 10, g = 980, 1 = 28, 0 = 1.1955, and r = 0.317. Ans. 0.57704 X 101. 31. If m = ar-1-16, find r when m = 2.263 and a = 0.4086. Ans. 0.22864. 32. Given p = p. ), find the value of p in terms of po if y = 1.41. Ans. 0.5266 po. (For the meaning of this formula see Perry's Calculus, page 55.) 33. If an indebtedness is paid in installments, the payments being equal and each including the interest to the date of the installment, then the number of installments necessary to pay the debt is given by the formula: log p - log (p - Pr) log (1 + r) where P equals the total indebtedness, p = the amount of one installment, and r = the rate per cent for the period between installments. Find the LOGARITHMS AND EXPLANATION OF TABLES 19 number of installments necessary to pay an indebtedness of $1500 if the interest is 8% per annum and the installments are $15 a month. Ans. 164.5 nearly. 34. Using the formula of Exercise 33, find the number of installments if the indebtedness is $1800, the interest 5% per annum, and the installments $5 per week. Ans. 450 nearly. TABLE II 23. Changing systems of logarithms. - In what precedes, the computations have been made with logarithms to the base 10. It is often necessary to make computations when the logarithms used are the natural, or Napierian logarithms, in which the base is e = 2.71828 *~ ~. It will now be shown how to find the logarithm of a number to the base e from the table of logarithms to the base 10, and vice versa, by the help of Table II, page 52. For the sake of generality, the relation between the logarithms will be shown for any two bases. THEOREM. Given the logarithm of a number N to the base a, then the logarithm of N to the base b is given by the relation: logb N = o log N. log, b) Proof. Let x = logbN; then bx = N. loga (bx) = log, N; or x log b = logaN.. = log bOga N. But x=logb N, (1).~* logbN = log, N. The constant multiplier is called the modulus of the loga b system of which the base is b with reference to the system of which the base is a. If a is put for N in formula (1), 0gb a = loga a. (2).. logba = ob or logbalogab = 1. (2)' 10ogba =o b I or logbalogb,= 1. log,, b' 20 PLANE TRIGONOMETRY It follows from (1) that the modulus of the natural system with reference to the common system is, and the modulus of the log1o e' common system with reference to the natural system is lo log, 10 That is, (3) log N loglo N, logi0 e (4) and logio N = l loge N. loge 10 The modulus = logioe is usually represented by M. loge 10 i 1 Hence by (2), log M logio e M But logio e = logo 2.71828... = 0.43429448.. M = 0.43429448 and = 2.30258509. 1 Using these values for - and M, (3) and (4) become the following: 31 (5) loge N = 2.3026 loglo N. (6) loglo N = 0.43429 loge N. 24. Use of table. -In Table II are arranged multiples of M and I to facilitate changing natural logarithms to common logarithms and vice versa. Example 1. Find the Napierian logarithm of 225, its common logarithm being 2.35218. By (3), Art. 23, loge 225 = logo 225 = X 2.35218. Table II gives the products: X 2.3 = 5.295945714 M X 0.052 = 0.1197344248 1 X 0.00018 = 0.0004144653 1 X 2.35218 = 5.4160946041 loge 225 = 5.41609. LOGARITHMS AND EXPLANATION OF TABLES 21 Example 2. Find the common logarithm of 762, its natural logarithm being 6.63595. By (4), Art. 23, logio 762 = M log, 762 = M X 6.63594. M X 6.6 = 2.866343581 M X 0.035 = 0.01520030687 M X 0.00094 = 0.0004082368.. M X 6.63594 = 2.88195212467. logio 762 = 2.88195 EXERCISES Find the following logarithms: 1. loge 426 = 6.05444. 2. loge 1076 = 6.98101. 3. loge 0.0763 = -2.57309. 4. loge 1.467 = 0.38322. 5. Find x if log, x = 6.96319. Ans. 1057. 6. Find x if log, x = -3.46954. Ans. 0.031131. t 1 7. Given R = 106., where t = 120, Vo = 123, V= 115.8, and C logeV0 C = 0.082; find R. Ans. 2.426 x 1010. 8. The work W done by a volume of gas, expanding at a constant temperature from volume Vo to volume V1, is given by the formula: W = poVologe,( V Find the value of W if po = 87.5, Vo = 246, and V1 = 472. Ans. 14,026. TABLE III 25. On pages 54 to 61 are arranged the logarithms of sines and tangents of angles from 3~ to 8~ for every 10 seconds; and the logarithms of cosines and cotangents of angles from 82~ to 87~ for every 10 seconds. The method of using these pages will follow from the explanation for the remaining pages of the table. See Art. 29. 26. Explanatory. - On pages 62 to 106 are arranged the logarithms, to five decimal places, of the trigonometric sines, cosines, tangents, and cotangents, of angles from 0~ to 90~, for each minute. The logarithms in the columns headed log sin, log cos, or log tan are increased by 10 so as to avoid writing negative characteristics. Those in the column headed log cot are printed without this in 22 PLANE TRIGONOMETRY crease. The minus sign is printed over the final 5 in the logarithms, as explained in Art. 10. The columns marked d give the tabular differences for the log sin and log cos columns. The column marked c.d. (common difference) gives the tabular differences for both log tan and log cot columns. The marginal tables, marked Prop. Parts, give 9, ~6,..., i, o0. 6 0 of the tabular differences, and are arranged for convenience in interpolating for seconds. The use of them is similar to that explained in Art. 21. Since sec 0 = 1 and csc 0 =, the logarithms of the cos 0 sin 0 secant and cosecant of an angle are the cologarithms (arithmetical complements) of those of the cosine and sine respectively. 27. To find the logarithmic function of an acute angle.(1) When the angle is given in degrees and minutes. If the angle is less than 45~, the degrees are found at the head of the page, the minutes at the left, and the functions are taken as named at the tops of the columns. If the angle is between 45~ and 90~, the degrees are found at the foot of the page, the minutes at the right, and the functions are taken as named at the bottoms of the columns. The functions are found in the same line with the minutes. Thus, log sin 17~ 27' = 9.47694 - 10, log sin 68~ 23' = 9.96833 - 10, log cos 29~ 36' = 9.93927 - 10, log cos 76~ 14' = 9.37652 - 10, log tan 10~ 16' = 9.25799 - 10, log tan 86~ 14' = 1.18154, log cot 9~ 46' = 0.76414, log cot 56~ 43' = 9.81721 - 10. (2) When the angle contains seconds. Here the function is found for the degrees and minutes and an interpolation made for the seconds similar to the interpolations in Table I. The tabular difference is multiplied by the number of seconds and divided by 60. This product may be taken from the Prop. Parts tables. Since the sine and the tangent increase as the angle increases from 0~ to 90~, the correction for the seconds is to be added; but, since the cosine and cotangent decrease as the angle increases from 0~ to 90~, the correction for the seconds is to be subtracted. LOGARITHMS AND EXPLANATION OF TABLES 23 Example 1. Find log sin 51~ 26' 23". log sin 51~ 26' = 9.89314 - 10 Correction for 23" = 10 X 3 = 4.log sin 51~ 26' 23" = 9.89318 - 10. Example 2. Find log cos 27~ 49' 37". log cos 27~ 49' = 9.94667 - 10 Correction for 37" = 7 X 3 = 4. log cos 27~ 49' 37" = 9.94663 - 10. RULE. Find the function corresponding to the given degrees and minutes. Multiply the tabular difference by the number of seconds considered as 60ths. When finding the sine, or tangent, add this product to the function corresponding to the degrees and minutes; but when finding the cosine or cotangent, subtract this product. (3) When the angle has decimals of minutes. Here the only difference in procedure from that given in (2), is that, in interpolating, the tabular difference is multiplied by the decimal of a minute given. It is evident that the Prop. Parts tables cannot be used for this. 28. To find the acute angle corresponding to a given logarithmic function. - (1) When the function can be found in the table, locate the function and read the angle in degrees and minutes at the head and left, or at the foot and right of the page, as the case may be. (2) When the function cannot be found in the table, the method of procedure can best be shown by examples. Example 1. Find the angle 0 if log sin 0 = 9.81659 - 10. Nearest log sin but less from table, 9.81651 - 10 = log sin 40~ 57'. Tabular difference for a difference of 1' in angle is 14. Difference between given function and function found is 8. Hence if the increase in the angle necessary to increase the function by 8 is x, then 14: 8 = 60": x"... x" = 34". 9.81659 - 10 = log sin 40~ 57' 34"...0 = 40~ 57' 34". 24 PLANE TRIGONOMETRY Example 2. Find 0 if log cos 0 = 9.23764. Since the cosine decreases as the angle increases, locate in the table the nearest log cos but larger than the one given. 9.23823 - 10= log cos 80~ 2'. Tabular difference is 71. Difference between the function given and the one found is 59. T5 X 60" = 50".. 0 = 80~ 2' 50". Example 3. Find 0 if log tan 0 = 9.98773. From table, 9.98762 = log tan 44~ 11'. Tabular difference, 25. Difference between the function given and the one found, 11. Using the Prop. Parts table headed 25, find the nearest number to 11 which is 8.3, the difference for 20". Then subtract 8.3 from 11 leaving 2.7, the difference for 6".. 9.98773 = log tan 44~ 11' 26". Of course the interpolation should be done mentally when possible. The method of procedure may be stated in the following sine or tangent RULE. For a logarithmic find the degrees and cosine or cotangent less minutes corresponding to the function next than the given greater function. Find the difference between the given function and the one less next e Find the fractional part of 60" that this difference is greater of the tabular difference. The required angle is the degrees and minutes corresponding to the function found in the table together with the seconds found. 29. Angles near 0~ and 90~. - In what precedes, it has been assumed that the variation in the angle is proportional to the function with intervals of 1'. In angles near 0~, this is not very accurate with the sine and tangent; and near 90~, it is not very accurate with the cosine and cotangent. Table III (pages 54-61) gives the functions for every 10" between 3~ and 8~ and 82~ and 87~. This makes the interpolations LOGARITHMS AND EXPLANATION OF TABLES 25 more nearly accurate for these angles. For the angles less than 30 and greater than 87~, the S and T scheme is convenient. 30. Functions by means of S and T. - The quantities S and T which are used are defined by the equations: sin a S = log, or S = log sin a - log a, a tan a and T = log, or T = log tan a - log a, a where a is the number of seconds in the angle. For convenience, the values of S, T, and a for angles from 0~ to 3~ 4' are arranged at the bottom of pages 32 to 51. On pages 62 to 64 are columns headed cpl S and cpl T. These give the arithmetical complements of the values of S and T. From the above are derived the following: FORMULAS FOR THE USE OF S AND T (1) For angles near zero deg? log sin a = log a" + S. log tan a = log a" + T. log cot a = cpl log a" + cpl T = cpl log tan a. 'ees log a" = log sin a + cpl S = log tan a + cpl T = cpl log cot a + cpl T. (2) For angles near ninety degrees. log cos a = log (90~ - a)" + S. log (90~ - a)" = log cos a + cpl S log cot a = log (90 - a)"+ T. = log cot a + cpl T log tan a = cpl log (90~ - a)" = cpl log tan a + + cpl T. cpl T. = cpl log cot a. 31. Examples in using S and T.1. Find log sin 0~ 47' 19". 3. Find log cot 0~ 57'49". 47' 19" = 2839" 57' 49" = 3469" log 2839 = 3.45317 cpl log 3469 = 6.45980 - 10 S = 4.68556 - 10 cpl T = 5.31438.. log sin 0~ 47' 19" = 8.13873 - 10.. log cot 0~ 57' 49" = 1.77418 - 10 2. Find log tan 1~ 27' 14". 1~ 27' 14" = 5234" log 5234 = 3.71883 T = 4.68567 - 10.'. log tan 1~ 27' 14" = 8.40450 - 10 4. Find a if log sin a = 7.85387 - 10 log sin a = 7.85387 - 10 cpl S = 5.31443.'. log a" = 3.16830 a" = 1473.3".'. a = 0~ 24' 33.3" 26 PLANE TRIGONOMETRY 6. Find log cos 89~ 27' 32". 90~ -89~ 27' 32" = 0~ 32' 28" = 1948" log 1948 = 3.28959 S = 4.68557 - 10.'. log cos 89~ 27'32" = 7.97516 - 10 6. Find log cot 88~ 49' 51". 90~ - 88~ 49' 51" = 1~ 10' 9" = 4209" log 4209 = 3.62418 T = 4.68563 - 10. logcot88~ 49'51" = 8.30981 -10 7. Find log tan 89~ 47' 33.82". 90~ - 89~ 47' 33.82" = 12' 26.18". = 746.18" colog 756.18 = 7.12136-10 cpl T = 5.31442. log tan 89047'33.82" = 2.43578. 8. Find a if log cot a = 7.86432 - 10 log cot a = 7.86432 - 10 cpl T = 5.31442. log (90~ - a)" = 3.17874 (90 - a)" = 1509.2" 90 - a = 0~ 25' 9.2".a = 89034'50.8" In Example 4, to find cpl S, locate log sin a on page 62, and read cpl S in the adjoining column. In Example 8, to find cpl T, locate log cot a on page 62. 32. Functions of angles greater than 90~. - In trigonometry there is a rule which says: To find the function of an angle greater than 90~, express the angle as a multiple of 90~ plus an acute angle. If this multiple is even, take the same function of the acute angle as the one required; and, if the multiple is odd, take the co-function of the acute angle. In either case prefix the sign determined by the quadrant the original angle is in. Thus, sin 562~ = sin (6 X 90~ + 22~) = -sin 22~. tan 1042~ = tan (11 X 90~ + 52~) = -cot 52~. As a further convenience in finding the logarithmic functions of angles greater than 90~, there are arranged at the top and bottom of each page of Table III other angles. If a is the acute angle of the page, then 180~ + a is printed in light type and has the same functions as a; while 90~ + a and 270~ + a are printed in black type, and for the functions of these angles one must take the cofunction of a. In either case proper regard must be paid to the algebraic sign. Thus, log cos 128~ = log sin 38~ n. log tan 218~ = log tan 38~. log sin 308~ = log cos 38~ n. The small letter n is placed after the function to indicate that the natural function is negative. Of course the logarithm cannot take account of this. LOGARITHMS AND EXPLANATION OF TABLES 27 TABLE IV 33. In this table (pages 108 to 130) are arranged the natural trigonometric sine, cosine, tangent, and cotangent of angles from 0~ to 90~ for each minute. The values are given correct to five figures. The arrangement of and the method of using the table are practically the same as for Table III. This can easily be traced by leaving out the word logarithm and reading Arts. 26, 27, 28, and 32. EXERCISES Verify the following: 1. log sin 61~ 41' 31" = 9.94469. 8. log tan 1~ 14' 27" = 8.33566. 2. log cos 31~ 47' 27" = 9.92940. 9. log cos 88~ 47' 13" = 8.32572. 3. log tan 15~ 14' 36" = 9.43538. 10. log cot 89~ 12' 18" = 8.14227. 4. log sin 45~ 43' 28" = 9.85491. 11. log cos 216~ 14' 33" = 9.90662 n. 5. log cot 5~ 50' 47" = 0.98972. 12. log sin 138~ 48' 6" = 9.81867. 6. log tan 80~ 58' 17" = 0.79889. 13. log tan 325~ 17'29" = 9.84052 n. 7. logsin 0~29'47" = 7.93765. 14. log cot 227~ 28' 3" = 9.96253. Find the values of 0 less than 360~ in the following: 15. log sin 0 = 9.28762. Ans. 11~ 10' 53" and 168~ 49' 7". 16. log cos 0 = 9.87642. Ans. 41~ 12' 22" and 318~ 47' 38". 17. log tan 0 = 9.47632. Ans. 16~ 40' 13" and 196~ 40' 13". 18. log cot 0 = 0.49632. Ans. 17~ 41' 18" and 197~ 41' 18". 19. log tan 0 = 0.49936. Ans. 72~ 2' 38" and 252~ 2' 38". 20. log cos 0 = 8.32967. Ans. 88~ 46' 33" and 271~ 13' 27". 21. log sin 0 = 7.99892. Ans. 0~ 34' 17.5" and 179~ 25' 42.5". 22. log sin] 0 = 9.98762 n. Ans. 256~ 23' and 283~ 37'. 23. log cos 0 = 9.89263 n. Ans. 141~ 20' 54" and 218~ 39' 6". 24. log tan 0 = 0.96236 n. Ans. 96~ 13' 25" and 276~ 13' 25". 25. sin 0 = 0.49367. Ans. 29~ 34' 55" and 150~ 25' 5". 26. cos 0 = 0.89672. Ans. 26~ 16' 10" and 333~ 43' 50". 27. tan 0 = 2.4379. Ans. 67~ 41' 49" and 247~ 41' 49". 28. cot 0 = 1.8923. Ans. 27~51' 17" and 207~ 51' 17". 29. sin 0 = -0.89723. Ans. 243~ 47' 46" and 296~ 12' 14". 30. cos 0 = -0.42936. Ans. 115~ 25' 37" and 244~ 34' 23". 3.26 tan 198~ 13' cos 13~ 17' 3. tan 0 = 4.76 sin 28~ 16' Ans. 24~ 51' 15" and 204~ 51' 15". 17 sin 283~ 19' tan 47~ 16' 32. sin 0 = 32. s 0 = 39.2 cos 183~ 6' Ans. 27~ 13' 26" and 152~ 46' 34". _ tan (-2712~ 15' 40") sec 30500 40' 33 cos2 tan 1522~ 46' 30" csc 1898~ 17' Ans. 45~ 47' 30", 134~ 12' 30", 225~ 47' 30", and 314~ 12' 30". 28 PLANE TRIGONOMETRY TABLE V 34. This table (page 131) can be used to change an angle expressed in degrees to radians, or vice versa. It may also be used for finding the arc length in a circle when the angle at the center is given, or vice versa. Example 1. Express 143~ 27' 36" in radians. 143~ = 2.4958208 27' = 0.0078540 36" = 0.0001745. 143~ 27' 36" = 2.5038493 radians. The accuracy of the last figure in the sum cannot be relied upon. Example 2. Express 3.6678437 radians in degrees, minutes, and seconds. Given, 3.6678437 Next less in table, 3.1415927 = 180~ Difference, 0.5262510 Next less, 0.5235988 = 30~ Difference, 0.0026522 Next less, 0.0026180 = 0~ 9' Difference, 0.0000342 Next less, 0.0000339 = 0~ 0' 7" Difference, 0.0000003.3.6678437 radians = 210~ 9' 7". EXERCISES Verify the following: 1. 216~ 44' 44" = 3.78292 radians. 2. 47~ 23' 58" = 0.82728 radians. 3. 725~ 19' 33" = 12.65932 radians. 4. 3.96423 radians = 227~ 8' 22". 5. 1.49367 radians = 85~ 34' 52". 6. 0.0236784 radians = 1~ 21' 24". 35. Errors of interpolation. - In the process of interpolation in logarithms, values are inserted as if the change in the logarithm between the two nearest tabular values was directly proportional to the change in the number. This would mean that the graph of the equation y = log x for this interval is a straight line. LOGARITHMS AND EXPLANATION OF TABLES 29 If values of x and y = log x are plotted in the usual manner in rectangular coordinates, the graph of y = log x is as shown in Fig. 1, where the unit on the y-axis is 10 times as large as the unit on the x-axis. 1.5 1 0.5 5 10 15 y = log X 20 25 FIG. 1. The values of the logarithms of numbers can be read from this curve, but not to a very high degree of accuracy. The values of x and y given in the table fall so close together on this curve that.25240.25215 T.25190.25165 1.785 1.786 1.787 y = log X FIG. 2. the interpolating cannot be shown. Suppose, for example, that log 1.7854 is required. Take the portion of the curve near x = 1.7854 and magnify it in the ratio of 1 to 20,000 on the x-axis and 1 to 1000 on the y-axis; the resulting curve is shown in Fig. 2. Re 30 PLANE TRIGONOMETRY ferring to this figure, when x = 1.785, y = 0.25164; and when x = 1.786, y = 0.25188. These give respectively the two points S and T on the curve. When x = 1.7854, y = log 1.7854 has the value shown by the point P on the curve; but, by interpolation, the value of log 1.7854 = 0.251736 and is shown by the point Q. Therefore the interpolation gives an error equal to QP. By using a higher place table of logarithms the value of log 1.7854= -0.2517355. This shows that the error is such that the logarithm is not affected in the fifth decimal place. A similar discussion could be given for interpolating in trigonometric functions. TABLE I COMMON LOGARITHMS OF NUMBERS From 1 to 10,000 to five places. From 10,000 to 11,000 to seven places. (For explanations, see pages 7 to 12.) Also values of S. and T. from 0~ to 3~ 4. (For explanations, see page 25.) TABLE I 100-150 N. I L. 0o1 1| I 2 | 3 | 4 1 5 6 | 7 | 8 9 1 Prop. Parts 1 100 00 000 043 087 130 173 2171 260 303 346 389 101 432 475 518 561 604 647 689 732 775 817 44 43 42 102 860 903 945 988 *030 *072 *115 *157 *199 *242 1 4.4 4.3 4.2 103 01 284 326 368 410 452 494 536 578 620 662 2 8.8 8.6 8.4 104 703 745 787 828 870 912 953 995 *036 *078 3 13.2 12.9 12.6 105 02 119 160 202 243 284 325 366 407 449 490 4 17.6 17.2 16.8 106 531 572 612 653 694 735 776 816 857 898 5 22.0 21.5 21.0 107 938 979 *019 *060 *100 *141 *181 *222 *262 *302 626.4 25.8 25.2 108 03 342 383 423 463 503 543 583 623 663 703 3.8 30.1.4 109 743 782 822 862 902 941 981 *021 *060 *100 8 39.2 38 7 37 8 110 04 139 179 218 258 297 336 376 415 454 493.. 111 532 571 610 650 689 727 766 805 844 883 41 40 39 112 922 961 999 *038 *077 *115 *154 *192 *231 *269 1 4.1 4.0 3.9 113 05 308 346 385 423 461 500 538 576 614 652 2 8.2 8.0 7.8 114 690 729 767 805 843 881 918 956 994 *032 3 12.3 12.0 11.7 115 06 070 108 145 183 221 258 296 333 371 408 4 164 16.0 15.6 116 446 483 521 558 595 633 670 707 744 781 2.5 2.0 1. 117 819 856 893 930 967 *004 *041 *078 *115 *151 6 24.6 24.0 23.4 118 07 188 225 262 298 335 372 408 445 482 518 7 28.7 28.0 27.3 119 555 59! 628 664 700 737 773 809 846 882 8 32.8 32.0 31.2 120 918 954 990 *027 *063 *099 *135 *171 *207 *243 3 3 3 121 08 279 314 350 386 422 458 493 529 565 600 38 37 36 122 636 672 707 743 778 814 849 884 920 955 1 3.8 3.7 3.6 123 991 *026 *061 *096 *132 *167 *202 *237 *272 *307 2 7.6 7.4 7.2 124 09 342 377 412 447 482 517 552 587 621 656 3 11.4.11.1 10.8 125 691 726 760 795 830 864 899 934 968 *003 4 15.2 14.8 14.4 126 10 037 072 106 140 175 209 243 278 312 346 5 19.0 18.5 18.0 127 380 415 449 483 517 551 585 619 653 687 6 22.8 22.2 21.6 128 721 755 789 823 857 890 924 958 992 *025 7 26.6 25.9 25.2 129 11 059 093 126 160 193 227 261 294 327 361 8 34.2 33.3 32 4 130 394 428 461 494 528 561 594 628 661 694. 3 3 131 727 760 793 826 860 893 926 959 992 *024 35 34 33 132 12 057 090 123 156 189 222 254 287 320 352 1 3.5 3.4 3.3 133 385 418 450 483 516 548 581 613 646 678 2 7.0 6.8 6.6 134 710 743 775 808 840 872 905 937 969 *001 3 10.5 10.2 9.9 135 13 033 066 098 130 162 194 226 258 290 322 4 14.0 13.6 13.2 136 354 386 418 450 481 513 545 577 609 640 5 17.5 17.0 16.5 137 672 704 735 767 799 830 862 893 925 956 6 21.0 20.4 19.8 138 988 *019 *051 *082 *114 *145 *176 *208 *239 *270 7 24.5 23.8 23.1 139 14 301 333 364 395 426 457 489 520 551 582 8 28.0 27.2 269 7.4 140 613 644 6753 706 737 768 799 829 860 89 31.5 30.6 29.7 141 922 953 983 *014 *045 *076 *106 *137 *168 *198 32 31 30 142 15 229 259 290 320 351 381 412 442 473 503 1 3.2 3.1 3.0 143 534 564 594 6253 6553 685 715 746 776 806 2 6.4 6.2 6.0 144 836 866 897 927 957 987 *017 *047 *077 *107 3 9.6 9.3 9.0 145 16 137 167 197 227 256 286 316 346 376 406 5 16.0 15.5 15 0 146 435 465 495 524 554 584 613 643 673 702 6 19.2 18.6 18.0 147 732 761 791 820 850 879 909 938 967 997 7 1224 21.7 21 0 148 17 026 056 0853 114 143 173 202 231 260 289 7 22.4 241.7 2.0 149 319 348 377 406 435 464 493 522 551 580 9 25.6 274.8 2.0 150 6091 638 667 696 7253 754 782 811 840 869 N. |L. o I | 2 1 3 4 | 5 6 71 8] 9 Prop. Parts 0 1'= 60" S. 4.68557 T. 4.68 557 0~ 19'= 1140" S. 4.68557 T. 4.68 558 0 2 - 120 557 557 0 20= 1200 557 558 0 3 180 557 557 0 21= 1260 557 558 0 22 = 1320 557 558 0 16 = 960 557 558 0 23 = 1380 557 558 0 17 = 1020 557 558 0 24 = 1440 557 558 0 18 = 1080 557 558 0 25 = 1500 557 558 32 TABLE I 150-200 N. L. o I 2 3 4 | 5 6 7 8 9 Prop. Parts 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200, 17 609 898 18 184 469 752 19 033 312 590 866 20 140 412 683 952 21 219 484 748 22 011 272 531 789 23 045 300 553 805 24 055 304 551 797 25 042 285 527 768 26 007 245 482 717 951 27 184 416 646 875 28 103 330 556 780 29 003 226 447 667 885 30 103 638 926 213 498 780 061 340 618 893 167 439 710 978 245 511 775 037 298 557 814 070 325 578 830 080 329 576 822 066 310 551 792 031 269 505 741 975 207 439 669 898 126 353 578 803 026 248 469 688 907 125 667 955 241 526 808 089 368 645 921 194 466 737 *005 272 537 801 063 324 583 840 096 350 603 855 105 353 601 846 091 334 575 816 055 293 529 764 998 231 462 692 921 149 375 601 825 048 270 491 710 929 146 696 984 270 554 837 117 396 673 948 222 493 763 *032 299 564 827 089 350 608 866 121 376 629 880 130 378 625 871 115 358 600 840 079 316 553 788 *021 254 485 715 944 171 398 623 847 070 292 513 732 951 168 = — ` - 725 *013 298 583 865 145 424 700 976 249 520 790 *059 325 590 854 115 376 634 891 147 401 654 905 155 403 650 895 139 382 624 864 102 340 576 811 *045 277 508 738 967 194 421 646 870 092 314 535 754 973 190 7541 782 *041 *070 327 355 611 639 893 921 173 201 451 479 728 756 *003 *030 276 303 548 575 817 844 *085 *112 352 378 617 643 880 906 141 167 401 427 660 686 917 943 172 198 426 452 679 704 930 955 180 204 428 452 674 699 920 944 164 188 406 431 648 672 888 912 126 150 364 387 600 623 834 858 *068 *091 300 323 531 554 761 784 989 *012 217 240 443 466 668 691 892 914 115 137 336 358 557 579 776 798 994 *016 211 233 811 840 *099 *127 384 412 667 696 949 977 229 257 507 535 783 811 *058 *085 330 358 602 629 871 898 *139 *165 405 431 669 696 932 958 194 220 453 479 712 737 968 994 223 249 477 502 729 754 980 *005 229 254 477 502 724 748 969 993 212 237 455 479 696 720 935 959 174 198 411 435 647 670 881 905 *114 *138 346 370 577 600 807 830 *035 *058 262 285 488 511 713 735 937 959 159 181 380 403 601 623 820 842 *038 *060 255 276 -- 869 *156 441 724 *005 285 562 838 *112 385 656 925 *192 458 722 985 246 505 763 *019 274 528 779 *030 279 527 773 *018 261 503 744 983 221 458 694 928 *161 393 623 852 *081 307 533 758 981 203 425 645 863 *081 298 29 1 2.9 2 5.8 3 8.7 4 11.6 1 5 14.5 1 6 17.4 1 7 20.3 1 8 23.2 2 9 26.1 27 1 2.7 2 5.4 3 8.1 4 10.8 1 5 13.5 1 6 16.2 7 18.9 8 21.6. 9 24.3 25 1 2.5 2 5.0 3 7.5 4 10.0 5 12.5 6 15.0 7 17.5 8 20.0 9 22.5 24 1 2.4 2 4.8 3 7.2 4 9.6 5 12.0 1 6 14.4 1 7 16.8 1 8 19.2 1 9 21.6 6 22 1 2.2 2 4.4 3 6.6 4 8.8 5 11.0 6 13.2 7 15.4 1 8 17.6 9 19.8 28 2.8 5.6 8.4 11.2 14.0 16.8 19.6 22.4 25.2 26 2.6 5.2 7.8 10.4 13.0 15.6 18.2 20.8 23.4 23 2.3 4.6 6.9 9.2 11.5 13.8 16.1 18.4 20.7 21 2.1 4.2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 N. L. o| I 1'2 1 3 1 4 1 5 1 6 1 7 8 I 9 1 Prop. Parts 0~ 2'= 120" S. 4.68 557 T. 4.68 557 0 3 = 180 557 557 0 4= 240 557 558 0 25 =1500 557 0 26 = 1560 557 0 27 = 1620 557 558 558 558 0~ 28'= 1680" S. 4.68 557 T. 4.68 558 0 29 = 1740 557 559 0 30 = 1800 557 559 0 31 = 1860 557 559 0 32 = 1920 557 559 0 33 = 1980 557 559 0 34 = 2040 557 559 33 TABLE I 200-250 N. I L. o I I 2 | 3 4 5 6 7 8 9 Prop. Parts 1 I 200 30 103 125 146 168 190 211 233 255 276 298 201 320 341 363 384 406 428 449 471 492 514 22 21 202 535 557 578 600 621 643 664 685 707 728 1 2.2 2.1 203 750 771 792 814 835 856 878 899 920 942 2 4.4 4.2 204 963 984 *006 *027 *048 *069 *091 *112 *133 *154 3 6.6 6.3 205 31 175 197 218 239 260 281 302 323 345 366 4 1. 10. 206 387 408 429 450 471 492 513 534 555 576 6 12 126 207 597 618 639 660 681 702 723 744 765 785 154 47 208 806 827 848 869 890 911 931 952 973 994 176 1 209 32 015 035 056 077 098 118 139 160 181 201 198 18 9 19.8 18.9 210 222 243 263 284 305 325 346 366 387 408 211 428 449 469 490 510 531 552 572 593 613 20 212 634 654 675 695 715 736 756 777 797 818 1 2.0 213 838 858 879 899 919 940 960 980 *001 *021 2 4.0 214 33 041 062 082 102 122 143 163 183 203 224 3 6.0 216 445 465 486 506 526 546 566 586 606 626 6 1.0 217 646 666 686 706 726 746 766 786 806 826 7 14.0 218 846 866 885 905 925 945 965 985 *005 *025 16.0 219 34 044 064 084 104 124 143 163 183 203 223 9 18 0 220 242 262 282 301 321 341 361 380 400 420 221 439 459 479 498 518 537 557 577 596 616 19 222 635 655 674 694 713 733 753 772 792 811 1 1.9 223 830 850 869 889 908 928 947 967 986 *005 2 3.8 224 35 025 044 064 083 102 122 141 160 180 199 3 5. 4 7.6 225 218 238 257 276 295 315 334 353 372 392 5 95 226 411 430 449 468 488 507 526 545 564 583 6 11.4 227 603 622 641 660 679 698 717 736 755 774 7 13.3 228 793 813 832 851 870 889 908 927 946 965 8 15.2 229 984 *003 *021 *040 *059 *078 *097 *116 *135 *154 9 17.1 230 36 173 192 211 229 248 267 286 305 324 342 231 361 380 399 418 436 455 474 493 511 530 18 232 549 568 586 605 624 642 661 680 698 717 1 1.8 233 736 754 773 791 810 829 847 866 884 903 2 3.6 234 922 940 959 977 996 *014 *033 *051 *070 *088 3 5.24 235 37 107 125 144 162 181 199 218 236 254 273 5 9.2 236 291 310 328 346 365 383 401 420 438 457 5 90 237 475 493 511 530 548 566 585 603 621 639 6 10.8 238 658 676 694 712 731 749 767 785 803 822 7 12.6 239 840 858 876 894 912 931 949 967 985 *003 8 16 2 240 38 021 039 057 075 093 112 130 148 166 184 241 202 220 238 256 274 292 310 328 346 364 17 242 382 399 417 435 453 471 489 507 525 543 1 1.7 243 561 578 596 614 632 650 668 686 703 721 2 3.4 244 739 757 775 792 810 828 846 863 881 899 3 5.1 245 917 934 952 970 987 *005 *023 *041 *058 *076 8.5 246 39 094 111 129 146 164 182 199 217 235 252 6 102 247 270 287 305 322 340 358 375 393 410 428 7 11.9 248 445 463 480 498 515 533 550 568 585 602 8 1136 249 620 637 655 672 690 707 724 742 75 777 9 15 3 250 794 811 829 846 863 881 898 915 933 950 N. L. oI 2 3 4 5 6 7 8 9 Prop. Parts 0~ 3'= 180" S. 4.68 557 T. 4.68 557 0~ 36'=2160" S. 4.68 557 T. 4.68 559 0 4 = 240 557 558 0 37 =2220 557 559 0 5 = 300 557 558 0 38 =2280 557 559 0 39 =2340 557 559 0 33 =1980 557 559 0 40 =2400 557 559 0 34 = 2040 557 559 0 41 =2460 556 560 0 35 = 2100 557 559 0 42 =2520 556 560 34 TABLE I 250-300 N. L. o2 | 3 | 4 6 7 8 9 Prop. Parts I I 2 13 8 46 1 86 89 915 93 95 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283;284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 39 794 967 40 140 312 483 654 824 993 41 162 330 497 664 830 996 42 160 325 488 651 813 975 43 136 297 457 616 775 933 44 091 248 404 560 716 871 45 025 179 332 484 637 788 939 46 090 240 389 538 687 835 982 47 129 276 422 567 712 811 985 157 329 500 671 841 *010 179 347 514 681 8.47 *012 177 341 504 667 830 991 152 313 473 632 791 949 107 264 420 576 731 886 040 194 347 500 652 803 954 105 255 404 553 702 850 997 144 290 436 582 727 829 *002 175 346 518 688 858 *027 196 363 531 697 863 *029 193 357 521 684 846 *008 169 329 489 648 807 965 122 279 436 592 747 902 056 209 362 515 667 818 969 120 270 419 568 716 864 *012 159 305 451 596 741 846 863 *019 *037 192 209 364 381 535 552 705 722 875 892 *044 *061 212 229 380 397 547 564 714 731 880 896 *045 *062 210 226 374 390 537 553 700 716 862 878 *024 *040 185 201 345 361 505 521 664 680 823 838 981 996 138 154 295 311 451 467 607 623 762 778 917 932 071 086 225 240 378 393 530 545 682 697 834 849 984 *000 135 150 285 300 434 449 583 598 731 746 879 894 *026 *041 173 188 319 334 465 480 611 625 756 770 881 *054 226 398 569 739 909 *078 246 414 581 747 913 *078 243 406 570 732 894 *056 217 377 537 696 854 *012 170 326 483 638 793 948 102 255 408 561 712 864 *015 165 315 464 613 761 909 *056 202 349 494 640 784 898 9151 933 *071 *088 *106 243 261 278 415 432 449 586 603 620 756 773 790 926 943 960 *095 *111 *128 263 280 296 430 447 464 597 614 631 764 780 797 929 946 963 *095 *1 *127 259 275 292 423 439 455 586 602 619 749 765 781 911 927 943 *072 *088 *104 233 249 265 393 409 425 553 569 584 712 727 743 870 886 902 *028 *044 *059 185 201 217 342 358 373 498 514 529 654 669 685 809 824 840 963 979 994 117 133 148 271 286 301 423 439 454 576 591 606 728 743 758 879 894 909 *030 *045 *060 180 195 210 330 345 359 479 494 509 627 642 657 776 790 805 923 938 953 *070 *085 *100 217 232 246 363 378 392 509 524 538 654 669 683 799 813 828 950 *123 295 466 637 807 976 *145 313 481 647 814 979 *144 308 472 635 797 959 *120 281 441 600 759 917 *075 232 389 545 700 855 *010 163 317 469 621 773 924 *075 225 374 523 672 820 967 *114 261 407 553 698 842 18 I 1.8 2 3.6 3 5.4 4 7.2 5 9.0 6 10.8 7 12.6 8 14.4 9 16.2 17 1 1.7 2 3.4 3 5.1 4 6.8 5 8.5 6 10.2 7 11.9 8 13.6 9 15.3 log e = 0.43429 16 1 1.6 2 3.2 3 4.8 4 6.4 5 8.0 6 9.6 7 11.2 8 12.8 9 14.4 15 1 1.5 2 3.0 3 4.5 4 6.0 5 7.5 6 9.0 7 10.5 8 12.0 9 13.5 14 1 1.4 2 2.8 3 4.2 4 5.6 5 7.0 6 8.4 7 9.8 8 11.2 9 12.6 N L. o I 2 | 3 4 6 | 7 8 9 Prop. Parts 0~ 4'= 240" S. 4.68 557 T. 4.68558 0~ 45'=2700" S. 4.68 556 T 4.68 560 0 5 = 300 557 558 0 46 =2760 556 560 0 47 =2820 556 560 0 41 =2460 556 560 0 48 =2880 556 560 0 42 =2520 556 560 0 49 =2940 556 560 0 43 =2580 556 560 0 50 =3000 556 561 0 44 =2640 556 560 35 TABLE I 300-350 N. I L. | I 2 j 3 I 5 1 6 7 8 9 Prop. Parts 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 47 712 857 48 001 144 287 430 572 714 855 996 49 136 276 415 554 693 831 969 50 106 243 379 515 651 786 920 51 055 188 322 455 587 720 851 983 52 114 244 375 504 634 763 892 53 020 148 275 403 529 656 782 908 54 033 158 283 407 727 741 871 885 015 029 159 173 302 316 444 458 586 601 728 742 869 883 *010 *024 150 164 290 304 429 443 568 582 707 721 845 859 982 996 120 133 256 270 393 406 529 542 664 678 799 813 934 947 068 081 202 215 335 348 468 481 601 614 733 746 865 878 996 *009 127 140 257 270 388 401 517 530 647 660 776 789 905 917 033 046 161 173 288 301 415 428 542 555 668 681 794 807 920 933 045 058 170 183 295 307 419 432 756 900 044 187 330 473 615 756 897 *038 178 318 457 596 734 872 *010 147 284 420 556 691 826 961 095 228 362 495 627 759 891 *022 153 284 414 543 673 802 930 058 186 314 441 567 694 820 945 070 195 320 444 770 914 058 202 344 487 629 770 911 *052 192 332 471 610 748 886 *024 161 297 433 569 705 840 974 108 242 375 508 640 772 904 *035 166 297 427 556 686 815 943 071 199 326 453 580 706 832 958 083 208 332 456 784 929 073 216 359 501 643 785 926 *066 206 346 485 624 762 900 *037 174 311 447 583 718 853 987 121 255 388 521 654 786 917 *048 179 310 440 569 699 827 956 084 212 339 466 593 719 845 970 095 220 345 469 799 813 943 958 087 101 230 244 373 387 515 530 657 671 799 813 940 954 *080 *094 220 234 360 374 499 513 638 651 776 790 914 927 *051 *065 188 202 325 338 461 474 596 610 732 745 866 880 *001 *014 135 148 268 282 402 415 534 548 667 680 799 812 930 943 *061 *075 192 205 323 336 453 466 582 595 711 724 840 853 969 982 097 110 224 237 352 364 479 491 605 618 732 744 857 870 983 995 108 120 233 245 357 370 481 494 828 972 116 259 401 544 686 827 968 *108 248 388 527 665 803 941 *079 215 352 488 623 759 893 *028 162 295 428 561 693 825 957 *088 218 349 479 608 737 866 994 122 250 377 504 631 757 882 *008 133 258 382 506 842 986 130 273 416 558 700 841 982 *122 262 402 541 679 817 955 *092 229 365 501 637 772 907 *041 175 308 441 574 706 838 970 *101 231 362 492 621 750 879 *007 135 263 390 517 643 769 895 *020 145 270 394 518 15 1 1.5 2 3.0 3 4.5 4 6.0 5 7.5 6 9.0 7 10.5 8 12.0 9 13.5 log r =0.49715 14 1 1.4 2 2.8 3 4.2 4 5.6 5 7.0 6 8.4 7 9.8 8 11.2 9 12.6 13 1 1.3 2 2.6 3 3.9 4 5.2 5 6.5 6 7.8 7 9.1 8 10.4 9 11.7 12 1 1.2 2 2.4 3 3.6 4 4.8 5 6.0 6 7.2 7 8.4 8 9.6 9 10.8 N.i L.o I 2 | 3 | 4 5 6 7. 8 9 Prop. Parts 0~ 5'= 300" S. 4.68 557 T. 4.68 558 0~ 54'=3240" S. 4.68 556 T. 4.68 561 0 6 = 360 557 558 0 55 =3300 556 561 0 56 =3360 556 561 0 50 =3000 556 561 0 57 =3420 555 561 0 51 =3060 556 -.-.61 0 58 =3480 555 562 0 52 =3120 556 5&t 0 59 =3540 555 562 0 53 =3180 556 561 l~~~~~~~~~~ 36 TABLE I 350-400 N. L. o i I 2 3 I 4 1 5 6 8 I 9 Prop. Parts I.., =I I I 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 -389 390 391 392 393 394 395 396 397 398 399 400 54 407 419 531 543 654 667 777 790 900 913 55 023 035 145 157 267 279 388 400 509 522 630 642 751 763 871 883 991 *003 56 110 122 229 241 348 360 467 478 585 597 703 714 820 832 937 949 57 054 066 171 183 287 299 403 415 519 530 634 646 749 761 864 875 978 990 58 092 104 206 218 320 331 433 444 546 557 659 670 771 782.883 894 995 *006 59 106 118 218 229 329 340 439 450 550 561 660 671 770 780 879 890 988 999 60 097 108 206 217 432 555 679 802 925 047 169 291 413 534 654 775 895 *015 134 253 372 490 608 726 844 961 078 194 310 426 542 657 772 887 *001 115 229 343 456 569 681 794 906 *017 129 240 351 461 572 682 791 901 *010 119 228 444 568 691 814 937 060 182 303 425 546 666 787 907 *027 146 265 384 502 620 738 855 972 089 206 322 438 553 669 784 898 *013 127 240 354 467 580 692 805 917 *028 140 251 362 472 583 693 802 912 *021 130 239 456 580 704 827 949 072 194 315 437 558 678 799 919 *038 158 277 396 514 632 750 867 984 101 217 334 449 565 680 795 910 *024 138 252 365 478 591 704 816 928 *040 151 262 373 483 594 704 813 923 *032 141 249 469 593 716 839 962 084 206 328 449 570 691 811 931 *050 170 289 407 526 644 761 879 996 113 229 345 461 576 692 807 921 *035 149 263 377 490 602 715 827 939 *051 162 273 384 494 605 715 824 934 *043 152 260 481 605 728 851 974 096 218 340 461 582 703 823 943 *062 182 301 419 538 656 773 891 *008 124 241 357 473 588 703 818 933 *047 161 274 388 501 614 726 838 950 *062 173 284 395 506 616 494 617 741 864 986 108 230 352 473 594 715 835 955 *074 194 -312 431 549 667 785 902 *019 136 252 368 484 600 715 830 944 *058 172 286 399 512 625 737 850 961 *073 184 295 406 517 627 506 630 753 876 998 121 242 364 485 606 727 847 967 *086 205 324 443 561 679 797 914 *031 148 264 380 496 611 726 841 955 *070 184 297 410 524 636 749 861 973 *084 195 306 417 528 638 748 857 966 *076 184 293 518 642 765 888 *011 133 255 376 497 618 739 859 979 *098 217 336 455 573 691 808 926 *043 159 276 392 507 623 738 852 967 *081 195 309 422 535 647 760 872 984 *095 207 318 428 539 649 759 868 977 *086 195 304 13 1 1.3 2 2.6 3 3.9 4 5.2 5 6.5 6 7.8 7 9.1 8 10.4 9 11.7 12 1 1.2 2 2.4 3 3.6 4 4.8 5 6.0 6 7.2 7 8.4 8 9.6 9 10.8 11 1 1.1 2 2.2 3 3.3 4 4.4 5 5.5 6 6.6 7 7.7 8 8.8 9 9.9 10 1 1.0 2 2.0 3 3.0 4 4.0 5 5.0 6 6.0 7 7.0 8 8.0 9 9.0 726 737 835 846 945 956 *054 *065 163 173 271 282 N. L. o I 2 3 4 5 6 7 8 9 Prop. Parts 0~ 5'= 300" S. 4.68 557 T. 4.68 558 1~ 1'=3660" S. 4.68 555 T. 4.68 562 0 6= 360 557 558 1 2 =3720 555 562 0 7 = 420 557 558 1 3=3780 555 562 -- X1 4 =3840 555 563 0 58 =3480 555 562 1 5 =3900 555 563 0 59 =3540 555 562 1 6 =3960 555 563 1 0 =3600 555 562 1 7 =4020 555 563,~~ ~~~ _,,_, l 37 TABLE I 400-450 N. | L.o I 2 1 4 5 6 | 7 8 9 Prop. Parts 400 60 206 217 228 239 249 260 271 282 293 304, 401 314 325 336 347 358 369 379 390 401 412 402 423 433 444 455 466 477 487 498 509 520 403 531 541 552 563 574 584 595 606 617 627 404 638 649 660 670 681 692 703 713 724 735 405 746 756 767 778 788 799 810 821 831 842 406 853 863 874 885 895 906 917 927 938 949 1 407 959 970 981 991 *002 *013 *023 *034 *045 *055 2 2.2 408 61 066 077 087 098 109 119 130 140 151 162 3. 409 172 183 194 204 215 225 236 247 257 268 4 410 278 289 300 310 321 331 342 352 363 374 5 5.5 411 384 395 405 416 426 437 448 458 469 479 6 6.6 412 490 500 511 521 532 542 553 563 574 584 7 7.7 413 595 606 616 627 637 648 658 669 679 690 8 8.8 414 700 711 721 731 742 752 763 773 784 794 9 9.9 415 805 815 826 836 847 857 868 878 888 899 416 909 920 930 941 951 962 972 982 993 *003 417 62 014 024 034 045 055 066 076 086 097 107 418 118 128 138 149 159 170 180 190 201 211 419 221 232 242 252 263 273 284 294 304 315 420 325 335 346 356 366 377 387 397 408 418 421 428 439 449 459 469 480 490 500 511 521 10 422 531 542 552 562 572 583 593 603 613 624 1 1.0 423 634 644 655 665 675 685 696 706 716 726 2 2.0 424 737 747 757 767 778 788 798 808 818 829 3 3.0 425 839 849 859 870 880 890 900 910 921 931 4 40 426 941 951 961 972 982 992 *002 *012 *022 *033 5 5.0 427 63 043 053 063 073 083 094 104 114 124 134 7 0 428 144 155 165 175 185 195 205 215 225 236. 429 246 256 266 276 286 296 306 317 327 337 9 8.0 9 9.0 430 347 357 367 377 387 397 407 417 428 438 431 448 458 468 478 488 498 508 518 528 538 432 548 558 568 579 589 599 609 619 629 639 433 649 659 669 679 689 699 709 719 729 739 434 749 759 769 779 789 799 809 819 829 839 435 849 859 869 879 889 899 909 919 929 939 436 949 959 969 979 988 998 *008 *018 *028 *038 9 437 64 048 058 068 078 088 098 108 118 128 137 1 0.9 438 147 157 167 177 187 197 207 217 227 237 2 1.8 439 246 256 266 276 286 296 306 316 326 335 3 2.7 440 345 355 365 375 385 395 404 414 424 434 4 3.6 441 444 454 464 473 483 493 503 513 523 532 5 4.5 442 542 552 562 572 582 591 601 611 621 631 6 5.4 443 640 650 660 670 680 689 699 709 719 729 7 6.3 444 738 748 758 768 777 787 797 807 816 826 8 7.2 445 836 846 856 865 875 885 895 904 914 924 9 8 446 933 943 953 963 972 982 992 *002 *011 *021 447 65 031 040 050 060 070 079 089 099 108 118 448 128 137 147 157 167 176 186 196 205 215 449 225 234 244 254 263 273 283 292 302 312 450 321 331 341 350 360 369 379 389 398 408 I N. L. o I I 2 | 3 | 4 5 6 7 8 | 9 Prop. Parts 0~ 6'= 360" S. 4.68 557 T. 4.68 558 1~ 9'=4140" S. 4.68 555 T. 4.68 563 0 7 = 420 557 558 1 10 =4200 554 563 0 8 = 480 557 558 1 11 =4260 554 564 1 12 =4320 554 564 1 6 =3960 555 563 1 13 =4380 554 564 1 7 =4020 555 563 1 14 =4440 554 564 1 8 =4080 555 563 1 15 =4500 554 564 38 TABLE I 450-500 - N. I L. o 2 314151 6 7 8 | 9 Prop. Parts 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 65 321 418 514 610 706 801 896 992 66 087 181 276 370 464 558 652 745 839 932 67 025 117 210 302 394 486 578 669 761 852 943 68 034 124 215 305 395 485 574 664 753 842 931 69 020 108 197 285 373 461 548 636 723 810 897 331 427 523 619 715 811 906 *001 096 191 285 380 474 567 661 755 848 941 034 127 219 311 403 495 587 679 770 861 952 043 133 224 314 404 494 583 673 762 851 940 028 117 205 294 381 469 557 644 732 819 906 341 437 533 629 725 820 916 *011 106 200 295 389 483 577 671 764 857 950 043 136 228 321 413 504 596 688 779 870 961 052 142 233 323 413 502 592 681 771 860 949 037 126 214 302 390 478 566 653 740 827 914 350 447 543 639 734 830 925 *020 115 210 304 398 492 586 680 773 867 960 052 145 237 330 422 514 605 697 788 879 970 061 151 242 332 422 511 601 690 780 869 958 046 135 223 311 399 487 574 662 749 836 923 360 456 552 648 744 839 935 *030 124 219 314 408 502 596 689 783 876 969 062 154 247 339 431 523 614 706 797 888 979 070 160 251 341 431 520 610 699 789 878 966 055 144 232 320 408 496 583 671 758 845 932 I 369 466 562 658 753 849 944 *039 134 229 323 417 511 605 699 792 885 978 071 164 256 348 440 532 624 715 806 897 988 079 169 260 350 440 529 619 708 797 886 975 064 152 241 329 417 504 592 679 767 854 940 379 475 571 667 763 858 954 *049 143 238 332 427 521 614 708 801 894 987 080 173 265 357 449 541 633 724 815 906 997 088 178 269 359 449 538 628 717 806 895 984 073 161 249 338 425 513 601 688 775 862 949 389 485 581 677 772 868 963 *058 153 247 342 436 530 624 717 811 904 997 089 182 274 367 459 550 642 733 825 916 *006 097 187 278 368 458 547 637 726 815 904 993 082 170 258 346 434 522 609 697 784 871 958 398 495 591 686 782 877 973 *068 162 257 351 445 539 633 727 820 913 *006 099 191 284 376 468 560 651 742 834 925 *015 106 196 287 377 467 556 646 735 824 913 *002 090 179 267 355 443 531 618 705 793 880 966 408 504 600 696 792 887 982 *077 172 266 361 455 549 642 736 829 922 *015 108 201 293 385 477 569 660 752 843 934 *024 115 205 296 386 476 565 655 744 833 922 *011 099 188 276 364 452 539 627 714 801 888 975 10 1 1.0 2 2.0 3 3.0 4 4.0 5 5.0 6 6.0 7 7.0 8 8.0 9 9.0 9 1 0.9 2 1.8 3 2.7 4 3.6 5 4.5 6 5.4 7 6.3 8 7.2 9 8.1 8 1 0.8 2 1.6 3 2.4 4 3.2 5 4.0 6 4.8 7 5.6 8 6.4 9 7.2 N.IL o I 2 | 3 4 I 5 6 7 8 9 Prop. Parts 0~ 7'= 420" S. 4.68 557 T. 4.68 558 1 18'=4680" S. 4.68 554 T. 4.68 565 0 8 = 480 557 558 i 19 =4740 554 565 0 9 = 540 557 558 1 20 =4800 554 565 1 21 =4860 553 566 1 15 =4500 554 564 1 22 =4920 553 566 1 16 =4560 554 565 1 23 =4980 553 566 1 17 =4620 554 565 124 =5040 553 566 39 TABLE I 500-550 N. IL. o | I 1 2 | 3 | 4 | 5 | 6 I 7 I 8 I 9 Prop. Parts ~ _. -p 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 69 897 906 984 992 70 070 079 157 165 243 252 329 338 415 424 501 509 586 595 672 680 757 766 842 851 927 935 71 012 020 096 105 181 189 265 273 349 357 433 441 517 525 600 609 684 692 767 775 850 858 933 941 72 016 024 099 107 181 189 263 272 346 354 428 436 509 518 591 599 673 681 754 762 835 843 916 925 997 *006 73 078 086 159 167 239 247 320 328 400 408 480 488 560 568 640 648 719 727 799 807 878 886 957 965 74 036 044 914 *001 088 174 260 346 432 518 603 689 774 859 944 029 113 198 282 366 450 533 617 700 784 867 950 032 115 198 280 362 444 526 607 689 770 852 933 *014 094 175 255 336 416 496 576 656 735 815 894 973 052 923 932 *010 *018 096 105 183 191 269 278 355 364 441 449 526 535 612 621 697 706 783 791 868 876 952 961 037 046 122 130 206 214 290 299 374 383 458 466 542 550 625 634 709 717 792 800 875 883 958 966 041 049 123 132 206 214 288 296 370 378 452 460 534 542 616 624 697 705 779 787 860 868 941 949 *022 *030 102 111 183 191 263 272 344 352 424 432 504 512 584 592 664 672 743 751 823 830 902 910 981 989 060 068 I 940 949.958 s 940 *027 114 200 286 372 458 544 629 714 800 885 969 054 139 223 307 391 475 559 642 725 809 892 975 057 140 222 304 387 469 550 632 713 795 876 957 *038 119 199 280 360 440 520 600 679 759 838 918 997 076 949 *036 122 209 295 381 467 552 638 723 808 893 978 063 147 231 315 399 483 567 650 734 817 900 -983 066 148 230 313 395 477 558 640 722 803 884 965 *046 127 207 288 368 448 528 608 687 767 846 926 *005 084 958 *044 131 217 303 389 475 561 646 731 817 902 986 071 155 240 324 408 492 575 659 742 825 908 991 074 156 239 321 403 485 567 648 730 811 892 973 *054 135 215 296 376 456 536 616 695 775 854 933 *013 092 966 975 *053 *062 140 148 226 234 312 321 398 406 484 492 569 578 655 663 740 749 825 834 910 919 995 *003 079 088 164 172 248 257 332 341 416 425 500 508 584 592 667 675 750 759 834 842 917 925 999 *008 082 090 165 173 247 255 329 337 411 419 493 501 575 583 656 665 738 746 819 827 900 908 981 989 *062 *070 143 151 223 231 304 312 384 392 464 472 544 552 624 632 703 711 783 791 862 870 941 949 *020 *028 099 107 9 1 0.9 2 1.8 3 2.7 4 3.6 5 4.5 6 5.4 7 6.3 8 7.2 9 8.1 8 1 0.8 2 1.6 3 2.4 4 3.2 5 4.0 6 4.8 7 5.6 8 6.4 9 7.2 7 1 0.7 2 1.4 3 2.1 4 2.8 5 3.5 6 4.2 7 4.9 8 5.6 9 6.3 N. L. o 2 | 3 1 4 |5 6 7 8 9 | Prop. Parts 0~ 8'= 480" S. 4.68 557 T. 4.68 558 10 26'=5160" S. 4.68 553 T. 4.68 567 0 9 = 540 557 558 1 27 =5220 553 567 0 10 = 600 557 558 1 28 =5280 553 567 1 29 =5340 553 567 1 23 =4980 553 566 1 30 =5400 553 567 1 24 =5040 553 566 1 31 =5460 552 568 1 25 =5100 553 566 1 32 =5520 552 568 40 TABLE I 550-600 N. L. o I '2 3 4 | 6 7 8 9 I Prop. Parts - ~ ~ ~..-.... --- - ~ L --- L 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590. 591 592 593 594 595 596 597 598 599 600 74 036 115 194 273 351 429 507 586 663 741 819 896 974 75 051 128 205 282 358 435 511 587 664 740 815 891 967 76 042 118 193 268 343 418 492 567 641 716 790 864 938 77 012 085 159 232 305 379 452 525 597 670 743 815 044 123 202 280 359 437 515 593 671 749 827 904 981 059 136 213 289 366 442 519 595 671 747 823 899 974 050 125 200 275 350 425 500 574 649 723 797 871 945 019 093 166 240 313 386 459 532 605 677 750 822 052 131 210 288 367 445 523 601 679 757 834 912 989 066 143 220 297 374 450 526 603 679 755 831 906 982 057 133 208 283 358 433 507 582 656 730 805 879 953 026 100 173 247 320 393 466 539 612 685 757 830 060 139 218 296 374 453 531 609 687 764 842 920 997 074 151 228 305 381 458 534 610 686 762 838 914 989 065 140 215 290 365 440 515 589 664 738 812 886 960 034 107 181 254 327 401 474 546 619 692 764 837 068 147 225 304 382 461 539 617 695 772 850 927 *005 082 159 236 312 389 465 542 618 694 770 846 921 997 072 148 223 298 373 448 522 597 671 745 819 893 967 041 115 188 262 335 408 481 554 627 699 772 076 155 233 312 390 468 547 624 702 780 858 935 *012 089 166 243 320 397 473 549 626 702 778 853 929 *005 080 155 230 305 380 455 530 604 678 753 827 901 975 048 122 195 269 342 415 488 561 634 706 779 851 084 162 241 320 398 476 554 632 710 788 865 943 *020 097 174 251 328 404 481 557 633 709 785 861 937 *012 087 163 238 313 388 462 537 612 686 760 834 908 982 056 129 203 276 349 422 495 568 641 714 786 859 092 170 249 327 406 484 562 640 718 796 873 950 *028 105 182 259 335 412 488 565 641 717 793 868 944 *020 095 170 245 320 395 470 545 619 693 768 842 916 989 063 137 210 283 357 430 503 576 648 721 793 866 099 178 257 335 414 492 570 648 726 803 881 958 *035 113 189 266 343 420 496 572 648 724 800 876 952 *027 103 178 253 328 403 477 552 626 701 775 849 923 997 070 144 217 291 364 437 510 583 656 728 801 873 107 186 265 343 421 500 578 656 733 811 889 966 *043 120 197 274 351 427 504 580 656 732 808 884 959 *035 110 185 260 335 410 485 559 634 708 782 856 930 *004 078 151 225 298 371 444 517 590 663 735 808 880 8 1 0.8 2 1.6 3 2.4 4 3.2 5 4.0 6 4.8 7 5.6 8 6.4 9 7.2 7 1 0.7 2 1.4 3 2.1 4 2.8 5 3.5 6 4.2 7 4.9 8 5.6 9 6.3 844 N. I L. o I 2 | 3 | 4 | 5 6 7 8 1 9 1 Prop. Parts 0~ 9'= 540" S. 4.68 557 T. 4.68 558 1~ 35'=5700" S. 4.68 552 T. 4.68 569 0 10= 600 557 558 1 36 =5760 552 569 1 37 =5820 552 569 1 31 =5460 552 568 1 38 =5880 552 569 1 32 =5520 552 568 1 39 =5940 551 569 1 33 =5580 552 568 1 40 =6000 551 570 1 34 =5640 552 568,, _ 41 TABLE I 600-650 N. IL. o II 2 3 4 5 6 7 8 9 Prop. Parts 600 77 815 822 830 837 844 851 859 866 873 880 601 887 895 902 909 916 924 931 938 945 952 602 960 967 974 981 988 996 *003 *010 *017 *025 603 78 032 039 046 053 061 068 075 082 089 097 604 104 111 118 125 132 140 147 154 161 168 605 176 183 190 197 204 211 219 226 233 240 606 247 254 262 269 276 283 290 297 305 312 1 8 607 319 326 333 340 347 355 362 369 376 383 2. 608 390 398 405 412 419 426 433 440 447 455 3 2 609 462 469 476 483 490 497 504 512 519 526 4 3.2 610 533 540 547 554 561 569 576 583 590 597 5 4.0 611 604 611 618 625 633 640 647 654 661 668 6 4.8 612 675 682 689 696 704 711 718 725 732 739 7 5.6 613 746 753 760 767 774 781 789 796 803 810 8 6.4 614 817 824 831 838 845 852 859 866 873 880 9 7.2 615 888 895 902 909 916 923 930 937 944 951 616 958 965 972 979 986 993 *000 *007 *014 *021 617 79 029 036 043 050 057 064 071 078 085 092 618 099 106 113 120 127 134 141 148 155 162 619 169 176 183 190 197 204 211 218 225 232 620 239 246 253 260 267 274 281 288 295 302 621 309 316 323 330 337 344 351 358 365 372 7 622 379 386 393 400 407 414 421 428 435 442 1 0.7 623 449 456 463 470 477 484 491 498 505 511 2 1.4 624 518 525 532 539 546 553 560 567 574 581 3 2.1 625 588 595 602 609 616 623 630 637 644 650 4 2.8 626 657 664 671 678 685 692 699 706 713 720 6 3. 627 727 734 741 748 754 761 768 775 782 789 6 4.2 628 796 803 810 817 824 831 837 844 851 858 7 4.9 629 865 872 879 886 893 900 906 913 920 927 8 5.6 630 934 941 948 955 962 969 975 982 989 996.3 631 80 003 010 017 024 030 037 044 051 058 065 632 072 079 085 092 099 106 113 120 127 134 633 140 147 154 161 168 175 182 188 195 202 634 209 216 223 229 236 243 250 257 264 271 635 277 284 291 298 305 312 318 325 332 339 636 346 353 359 366 373 380 387 393 400 407 6 637 414 421 428 434 441 448 455 462 468 475 1 0.6 638 482 489 496 502 509 516 523 530 536 543 2 1.2 639 550 557 564 570 577 584 591 598 604 611 3 1.8 640 618 625 632 638 645 652 659 665 672 679 4 2.4 641 686 693 699 706 713 720 726 733 740 747 5 3.0 642 754 760 767 774 781 787 794 801 808 814 6 3.6 643 821 828 835 841 848 855 862 868 875 882 7 4.2 644 889 895 902 909 916 922 929 936 943 949 8 4.8 645 956 963 969 976 983 990 996 *003 *010 *017 9 5.4 646 81 023 030 037 043 050 057 064 070 077 084 647 090 097 104 111 117 124 131 137 144 151 648 158 164 171 178 184 191 198 204 211 218 649 224 231 238 245 251 258 265 271 278 285 650 291 298 305 311 318 325 331 338 345 351 I. N. 1 L. O LI I1 2 1 3 | 4 I, 5 I 6 7 8 9 Prop. Parts 0~ 10'= 600" S. 4.68 557 T. 4.68 558 1~ 44'=6240" S. 4.68 551 T. 4.68 571 0 11 = 660 557 558 1 45 =6300 551 571 1 46 =6360 551 571 1 40 =6000 551 570 1 47 =6420 550 572 1 41 =6060 551 570 1 48 =6480 550 572 1 42 =6120 551 570 1 49 =6540 550 572 1 43 =6180 551 570 42 TABLE I 650-700 N. L. 0 - I] 2 3 1 4 1 5 1 6 - '~ I, I -, I I I -I 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 81 291 358 425 491 558 624 690 757 823 889 954 82 020 086 151 217 282 347 413 478 543 607 672 737 802 866 930 995 83 059 123 187 251 315 378 442 506 569 632 696 759 822 885 948 84 011 073 136 198 261 323 386 448 510 298 365 431 498 564 631 697 763 829 895 961 027 092 158 223 289 354 419 484 549 614 679 743 808 872 937 *001 065 129 193 257 321 385 448 512 575 639 702 765 828 891 954 017 080 142 205 267 330 392 454 516 305 371 438 505 571 637 704 770 836 902 968 033 099 164 230 295 360 426 491 556 620 685 750 814 879 943 *008 072 136 200 264 327 391 455 518 582 645 708 771 835 897 960 023 086 148 211 273 336 398 460 522 311 378 445 511 578 644 710 776 842 908 974 040 105 171 236 302 367 432 497 562 627 692 756 821 885 950 *014 078 142 206 270 334 398 461 525 588 651 715 778 841 904 967 029 092 155 217 280 342 404 466 528 318 385 451 518 584 651 717 783 849 915 981 046 112 178 243 308 373 439 504 569 633 698 763 827 892 956 *020 085 149 213 276 340 404 467 531 594 658 721 784 847 910 973 036 098 161 223 286 348 410 473 535 325 391 458 525 591 657 723 790 856 921 987 053 119 184 249 315 380 445 510 575 640 705 769 834 898 963 *027 091 155 219 283 347 410 474 537 601 664 727 790 853 916 979 042 105 167 230 292 354 417 479 541 331 398 465 531 598 664 730 796 862 928 994 060 125 191 256 321 387 452 517 582 646 711 776 840 905 969 *033 097 161 225 289 353 417 480 544 607 670 734 797 860 923 985 048 111 173 236 298 361 423 485 547 71 8 338 345 405 411 471 478 538 544 604 611 671 677 737 743 803 809 869 875 935 941 *000 *007 066 073 132 138 197 204 263 269 328 334 393 400 458 465 523 530 588 595 653 659 718 724 782 789 847 853 911 918 975 982 *040 *046 104 110 168 174 232 238 296 302 359 366 423 429 487 493 550 556 613 620 677 683 740 746 803 809 866 872 929 935 992 998 055 061 117 123 180 186 242 248 305 311 367 373 429 435 491 497 553 559 I 9 Prop. Parts 351 418 485 551 617 684 750 816 882 948 *014 079 145 210 276 341 406 471 536 601 666 730 795 860 924 988 *052 117 181 245 308 372 436 499 563 626 689 753 816 879 942 *004 067 130 192 255 317 379 442 504 566 I 7 1 0.7 2 1.4 3 2.1 4 2.8 5 3.5 6 4.2 7 4.9 8 5.6 9 6.3 6 1 0.6 2 1.2 3 1.8 4 2.4 5 3.0 6 3.6 7 4.2 8 4.8 9 5.4 N. L. o I 3 1 4 5 6 | 7 1 8 6 9 Prop. Parts 0~ 10'= 600" S. 4.68 557 T. 4.68 558 1 51'= 6660" S. 4.68 550 T. 4.68 573 0 11 = 660 557 558 1 52 =6720 550 573 0 12 = 720 557 558 1 53 =6780 550 573 1 54 =6840 550 573 1 48 =6480 550 572 1 55 =6900 549 574 1 49 =6540 550 572 1 56 =6960 549 574 1 50 =6600 550 572 1 57 =7020 549 574 43 TABLE I 700-750 N. L. o 2 3 4 5 6 8 j 1 1 Prop. Parts 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 I 84 510 572 634 696 757 819 880 942 85 003 065 126 187 248 309 370 431 491 552 612 673 733 794 854 914 974 86 034 094 153 213 273 332 392 451 510 570 629 688 747 806 864 923 982 87 040 099 157 216 274 332 390 448 506.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- --- -- I i 516 522 578 584 640 646 702 708 763 770 825 831 887 893 948 954 009 016 071 077 132 138 193 199 254 260 315 321 376 382 437 443 497 503 558 564 618 625 679 685 739 745 800 806 860 866 920 926 980 986 040 046 100 106 159 165 219 225 279 285 338 344 398 404 457 463 516 522 576 581 635 641 694 700 753 759 812 817 870 876 929 935 988 994 046 052 105 111 163 169 221 227 280 286 338 344 396 402 454 460 512 518 528 590 652 714 776 837 899 960 022 083 144 205 266 327 388 449 509 570 631 691 751 812 872 932 992 052 112 171 231 291 350 410 469 528 587 646 705 764 823 882 941 999 058 116 175 233 291 -349 408 466 523 535 597 658 720 782 844 905 967 028 089 150 211 272 333 394 455 516 576 637 697 757 818 878 938 998 058 118 177 237 297 356 415 475 534 593 652 711 770 829 888 947 *005 064 122 181 239 297 355 413 471 529 541 603 665 726 788 850 911 973 034 095 156 217 278 339 400 461 522 582 643 703 763 824 884 944 *004 064 124 183 243 303.362 -421 -481 540 599 658 717 776 835 894 953 *011 070 128 186 245 303 361 419 477 535 547 609 671 733 794 856 917 979 040 101 163 224 285 345 406 467 528 588 649 709 769 830 890 950 *010 070 130 189 249 308 368 427 487 546 605 664 723 782 841 900 958 *017 075 134 192 251 309 367 425 483 541 553 615 677 739 800 862 924 985 046 107 169 230 291 352 412 473 534 594 655 715 775 836 896 956 *016 076 136 195 255 314 374 433 493 552 611 670 729 788 847 906 964 '023 081 140 198 256 315 373 431 489 547 559 621 683 745 807 868 930 991 052 114 175 236 297 358 418 479 540 600 661 721 781 842 902 962 *022 082 141 201 261 320 380 439 499 558 617 676 735 794 853 911 970 *029 087 146 204 262 320 379 437 495 552 566 628 689 751 813 874 936 997 058 120 181 242 303 364 425 485 546 606 667 727 788 848 908 968 *028 088 147 207 267 326 386 445 504 564 623 682 741 800 859 917 976 *035 093 151 210 268 326 384 442 500 558 7 1 0.7 2 1.4 3 2.1 4 2.8 5 3.5 6 4.2 7 4.9 8 5.6 9 6.3 6 1 0.6 2 1.2 3 1.8 4 2.4 5 3.0 6 3.6 7 4.2 8 4.8 9 5.4 5 1 0.5 2 1.0 3 1.5 4 2.0 5 2.5 6 3.0 7 3.5 8 4.0 9 4.5 N. JL. o | I 3 | 4 2 5 6 1 7 8 9 Prop. Parts 00 11'= 660" S. 4.68 557 T. 4.68 558 10 59'=7140" S. 4.68 549 T. 4.68 575 0 12 = 720 557 558 2 0 =7200 549 575 0 13 = 780 557 558 2 1 =7260 549 575 2 2 =7320 548 576 1 56 =6960 549 574 2 3 =7380 548 576 1 57 =7020 549 574 2 4 =7440 548 576 1 58 =7080 549 575 2 5 =7500 548 577 44 TABLE I 750-800 L. L. o I 2i: | 3 8 4 5:j 6 7 8 9 7 Prop. Parts 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 87 506 564 622 679 737 795 852 910 967 88 024 081 138 195 252 309 366 423 480 536 593 649 705 762 818 874 930 986 89 042 098 154 209 265 321 376 432 487 542 597 653 708 763 818 873 927 982 90 037 091 146 200 255 309 512 570 628 685 743 800 858 915 973 030 087 144 201 258 315 372 429 485 542 598 655 711 767 824 880 936 992 048 104 159 215 271 326 382 437 492 548 603 658 713 768 823 878 933 988 042 097 151 206 260 314 518 576 633 691 749 806 864 921 978 036 093 150 207 264 321 377 434 491 547 604 660 717 773 829 885 941 997 053 109 165 221 276 332 387 443 498 553 609 664 719 774 829 883 938 993 048 102 157 211 266 320 523 581 639 697 754 812 869 927 984 041 098 156 213 270 326 383 440 497 553 610 666 722 779 835 891 947 *003 059 115 170 226 282 337 393 448 504 559 614 669 724 779 834 889 944 998 053 108 162 217 271 325 529 587 645 703 760 818 875 933 990 047 104 161 218 275 332 389 446 502 559 615 672 728 784 840 897 953 *009 064 120 176 232 287 343 398 454 509 564 620 675 730 785 840 894 949 *004 059 113.168 222 276 331 535 593 651 708 766 823 881 938 996 053 110 167 224 281 338 395 451 508 564 621 677 734 790 846 902 958 *014 070 126 182 237 293 348 404 459 515 570 625 680 735 790 845 900 955 *009 064 119 173 227 282 336 541 599 656 714 772 829 887 944 *001 058 116 173 230 287 343 400 457 513 570 627 683 739 795 852 908 964 *020 076 131 187 243 298 354 409 465 520 575 631 686 741 796 851 905 960 *015 069 124 179 233 287 342 547 604 662 720 777 835 892 950 *007 064 121 178 235 292 349 406 463 519 576 632 689 745 801 857 913 969 *025 081 137 193 248 304 360 415 470 526 581 636 691 746 801 856 911 966 *020 075 129 184 238 293 347 552 610 668 726 783 841 898 955 *013 070 127 184 241 298 355 412 468 525 581 638 694 750 807 863 919 975 *031 087 143 198 254 310 365 421 476 531 586 642 697 752 807 862 916 971 *026 080 135 189 244 298 352 558 616 674 731 789 846 904 961 *018 076 133 190 247 304 360 417 474 530 587 643 700 756 812 868 925 981 *037 092 148 204 260 315 371 426 481 537 592 647 702 757 812 867 922 977 *031 086 140 195 249 304 358 6 1 0.6 2 1.2 3 1.8 4 2.4 5 3.0 6 3.6 7 4.2 8 4.8 9 5.4 5 1 0.5 2 1.0 3 1.5 4 2.0 5 2.5 6 3.0 7 3.5 8 4.0 9 4.5 N. I L. o | 2 3 |4 | 5 6 1 7 8 9 | Prop. Parts 0~ i2'= 720" S. 4.68 557 T. 4.68 558 2~ 8'=7680" S. 4.68 547 T. 4.68 578 0 13 = 780 557 558 2 9 =7740 547 578 0 14 = 840 557 558 2 10 =7800 547 578 2 11 =7860 547 579 2 5 =7500 548 577 2 12 =7920 547 579 2 6 =7560 548 577 2 13 =7980 547 579 2 7 =7620 548 577 2 14 =8040 546 579 45 TABLE I 800-850 - N. I L. o |I 2 3 4 5 6 | 7 | 8 1 9 Prop. Parts i,,. _. *.... 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 90 309 363 417 472 526 580 634 687 741 795 849 902 956 91 009 062 116 169 222 275 328 381 434 487 540 593 645 698 751 803 855 908 960 92 012 065 117 169 221 273 324 376 428 480 531 583 634 686 737 788 840 891 942 314 369 423 477 531 585 639 693, 747 800 854 907 961 014 068 121 174 228 281 334 387 440 492 545 598 651 703 756 808 861 913 965 018 070 122 174 226 278 330 381 433 485 536 588 639 691 742 793 845 896 947 320 374 428 482 536 590 644 698 752 806 859 913 966 020 073 126 180 233 286 339 392 445 498 551 603 656 709 761 814 866 48 971 023 075 127 179 231 283 335 387 438 490 542 593 645 696 747 799 850 901 952 325 380 434 488 542 596 650 703 757 811 865 918 972 025 078 132 185 238 291 344 397 450 503 556 609 661 714 766 819 871 924 976 028 080 132 184 236 288 340 392 443 495 547 598 650 701 752 804 855 906 957 331 385 439 493 547 601 655 709 763 816 870 924 977 030 \084 137 190 243 297 350 403 455 508 561 614 666 719 772 824 876 929 981 033 085 137 189 241 293 345 397 449 500 552 603 655 706 758 809 860 911 962 I 336 390 445 499 553 607 660 714 768 822 875 929 982 036 089 142 196 249 302 355 408 461 514 566 619 672 724 777 829 882 934 986 038 091 143 195 247 298 350 402 454 505 557 609 660 711 763 814 865 916 967 342 396 450 504 558 612 666 720 773 827 881 934 988 041 094 148 201 254 307 360 413 466 519 572 624 677 730 782 834 887 939 991 044 096 148 200 252 304 355 407 459 511 562 614 665 716 768 819 870 921 973 347 401 455 509 563 617 671 725 779 832 886 940 993 046 100 153 206 259 312 365 418 471 524 577 630 682 735 787 840 892 944 997 049 101 153 205 257 309 361 412 464 516 567 619 670 722 773 824 875 927 978 352 407 461 515 569 623 677 730 784 838 891 945 998 052 105 158 212 265 318 371 424 477 529 582 635 687 740 793 845 897 950 *002 054 106 158 210 262 314 366 418 469 521 572 624 675 727 778 829 881 932 983 358 412 466 520 574 628 682 736 789 843 897 950 *004 057 110 164 217 270 323 376 429 482 535 587 640 693 745 798 850 903 955 *007 059 111 163 215 267 319 371 423 474 526 578 629 681 732 783 834 886 937 988 6 1 0.6 2 1.2 3 1.8 4 2.4 5 3.0 6 3.6 7 4.2 8 4.8 9 5.4 5 1 0.5 2 1.0 3 1.5 4 2.0 5 2.5 6 3.0 7 3.5 8 4.0 9 4.5 I N. I L. o I |2 3 4 |5 6 7 8 9 | Prop. Parts 0~ 13'= 780" S. 4.68 557 T. 4.68 558 20 16'=8160" S. 4.68 546 T. 4.68 580 0 14 = 840 557 558 2 17 =8220 546 580 0 15 = 900 557 558 2 18 =8280 546 581 2 19 =8340 546 581 2 13 =7980 547 579 2 20 =8400 545 582 2 14 =8040 546 579 2 21 =8460 545 582 2 15 =8100 546 580 2 22 =8520 545 582 I 46 TABLE I 850-900 I N.| L. o2 3 1 4 5 | 6 7 8 91 Prop. Parts,, ~ ~ ~ ~ ~ 57 92 97 93 7 8 8.. I. _ 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 -897 898 899 900 92 942 993 93 044 095 146 197 247 298 349 399 450 500 551 601 651 702 752 802 852 902 952 94 002 052 101 151 201 250 300 349 399 448 498 547 596 645 694 743 792 841 890 939 988 95 036 085 134 182 231 279 328 376 424 947 998 049 100 151 202 252 303 354 404 455 505 556 606 656 707 757 807 857 907 957 007 057 106 156 206 255 305 354 404 453 503 552 601 650 699 748 797 846 895 944 993 041 090 139 187 236 284 332 381 429 952 *003 054 105 156 207 258 308 359 409 460 510 561 611 661 712 762 812 862 912 962 012 062 111 161 211 260 310 359 409 458 507 557 606 655 704 753 802 851 900 949 998 046 095 143 192 240 289 337 386 434 957 *008 059 110 161 212 263 313 364 414 465 515 566 616 666 717 767 817 867 917 967 017 067 116 166 216 265 315 364 414 463 512 562 611 660 709 758 807 856 905 954 *002 051 100 148 197 245 294 342 390 439 962 *013 064 115 166 217 268 318 369 420 470 520 571 621 671 722 772 822 872 922 972 022 072 121 171 221 270 320 369 419 468 517 567 616 665 714 763 812 861 910 959 *007 056 105 153 202 250 299 347 395 444 I 967 973 *018 *024 069 075 120 125 171 176 222 227 273 278 323 328 374 379 425 430 475 480 526 531 576 581 626 631 676 682 727 732 777 782 827 832 877 882 927 932 977 982 027 032 077 082 126 131 176 181 226 231 275 280 325 330 374 379 424 429 473 478 522 527 571 576 621 626 670 675 719 724 768 773 817 822 866 871 915 919 963 968 *012 *017 061 066 109 114 158 163 207 211 255 260 303 308 352 357 400 405 448 453 I 978 *029 080 131 181 232 283 334 384 435 485 536 586 636 687 737 787 837 887 937 987 037 086 136 186 236 285 335 384 433 483 532 581 630 680 729 778 827 876 924 973 *022 071 119 168 216 265 313 361 410 458 983 *034 085 136 186 237 288 339 389 440 490 541 591 641 692 742 792 842 892 942 992 042 091 141 191 240 290 340 389 438 488 537 586 635 685 734 783 832 880 929 978 *027 075 124 173 221 270 318 366 415 463 988 *039 090 141 192 242 293 344 394 445 495 546 596 646 697 747 797 847 897 947 997 047 096 146 196 245 295 345 394 443 493 542 591 640 689 738 787 836 885 934 983 *032 080 129 177 226 274 323 371 419 468 6 1 0.6 2 1.2 3 1.8 4 2.4 5 3.0 6 3.6 7 4.2 8 4.8 9 5.4 5 1 0.5 2 1.0 3 1.5 4 2.0 5 2.5 6 3.0 7 3.5 8 4.0 9 4.5 4 1 0.4 2 0.8 3 1.2 4 1.6 5 2.0 6 2.4 7 2.8 8 3.2 9 3.6 I N. o I I0 2 3 1 4 5 16 7 8 1 9 Prop. Parts 0~0 14'= 840" S. 4.68 557 T. 4.68 558 2~ 25'= 8700" S. 4.68 545 T. 4.68 583 0 15 = 900 557 558 2 26 =8760 544 584 2 27 =8820 544 584 2 21 =8460 545 582 2 28 =8880 544 584 2 22 =8520 545 582 2 29 =8940 544 585 2 23 =8580 545 583 2 30 =9000 544 585 2 24 =8640 545 583 47 TABLE I 900-950 N. L. o I 2 3 1 4 1 5 6 |7 8 9 I Prop. Parts 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 I 95 424 472 521 569 617 665 713 761 809 856 904 952 999 96 047 095 142 190 237 284 332 379 426 473 520 567 614 661 708 755 802 848 895 942 988 97 035 081 128 174 220 267 313 359 405 451 497 543 589 635 681 727 772. I. I. i 429 477 525 574 622 670 718 766 813 861 909 957 *004 052 099 147 194 242 289 336 384 431 478 525 572 619 666 713 759 806 853 900 946 993 039 086 132 179 225 271 317 364 410 456 502 548 594 640 685 731 777 434 482 530 578 626 674 722 770 818 866 914 961 *009 057 104 152 199 246 294 341 388 435 483 530 577 624 670 717 764 811 858 904 951 997 044 090 137 183 230 276 322 368 414 460 506 552 598 644 690 736 782 439 487 535 583 631 679 727 775 823 871 918 966 *014 061 109 156 204 251 298 346 393 440 487 534 581 628 675 722 769 816 862 909 956 *002 049 095 142 188 234 280 327 373 419 465 511 557 603 649 695 740 786 444 492 540 588 636 684 732 780 828 875 923 971 *019 066 114 161 209 256 303 350 398 445 492 539 586 633 680 727 774 820 867 914 960 *007 053 100 146 192 239 285 331 377 424 470 516 562 607 653 699 745 791 448 497 545 593 641 689 737 785 832 880 928 976 *023 071 118 166 213 261 308 355 402 450 497 544 591 638 685 731 778 825 872 918 965 *011 058 104 151 197 243 290 336 382 428 474 520 566 612 658 704 749 795 453 501 550 598 646 694 742 789 837 885 933 980 *028 076 123 171 218 265 313 360 407 454 501 548 595 642 689 736 783 830 876 923 970 *016 063 109 155 202 248 294 340 387 433 479 525 571 617 663 708 754 800 458 506 554 602 650 698 746 794 842 890 938 985 *033 080 128 175 223 270 317 365 412 459 506 553 600 647 694 741 788 834 881 928 974 *021 067 114 160 206 253 299 345 391 437 483 529 575 621 667 713 759 804 463 511 559 607 655 703 751 799 847 895 942 990 *038 085 133 180 227 275 322 369 417 464 511 558 605 652 699 745 792 839 886 932 979 *025 072 118 165 211 257 304 350 396 442 488 534 580 626 672 717 763 809 468 516 564 612 660 708 756 804 852 899 947 995 *042 090 137 185 232 280 327 374 421 468 515 562 609 656 703 750 797 844 890 937 984 *030 077 123 169 216 262 308 354 400 447 493 539 585 630 676 722 768 813 5 1 0.5 2 1.0 3 1.5 4 2.0 5 2.5 6 3.0 7 3.5 8 4.0 9 4.5 4 1. 0.4 2 0.8 3 1.2 4 1.6 5 2.0 6 2.4 7 2.8 8 3.2 9 3.6 N. IL. o I x 2 3 1 4 1 6 7 1 8 9 | Prop. Parts 0~ 15'= 900" S. 4.68 557 T. 4.68 558 2~ 34'=9240" S. 4.68 543 T. 4.68 587 0 16 = 960 557 558 2 35 =9300 543 587 2 36 =9360 543 587 2 30 =9000 544 585 2 37 =9420 542 588 2 31 =9060 544 585 2 38 =9480 542 588 2 32 =9120 543 586 2 39 =9540 542 588 2 33 =9180 543 586 48 TABLE I 950-1000 N. I L. o ) I 2 3 | 4 5 | 6. A;......~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 97 772 818 864 909 955 98 000 046 091 137 182 227 272 318 363 408 453 498 543 588 632 677 722 767 811 856 900 945 989 99 034 078 123 167 211 255 300 344 388 432 476 520 564 607 651 695 739 782 826 870 913 957 00 000 777 823 868 914 959 005 050 096 141 186 232 277 322 367 412 457 502 547 592 637 682 726 771 816 860 905 949 994 038 083 127 171 216 260 304 348 392 436 480 524 568 612 656 699 743 787 830 874 917 961 004 782 827 873 918 964 009 055 100 146 191 236 281 327 372 417 462 507 552 597 641 686 731 776 820 865 909 954 998 043 087 131 176 220 264 308 352 396 441 484 528 572 616 660 704 747 791 835 878 922 965 009 786 832 877 923 968 014 059 105 150 195 241 286 331 376 421 466 511 556 601 646 691 735 780 825 869 914 958 *003 047 092 136 180 224 269 313 357 401 445 489 533 577 621 664 708 752 795 839 883 926 970 013 791 836 882 928 973 019 064 109 155 200 245 290 336 381 426 471 516 561 605 650 695.740 784 829 874 918 963 *007 052 096 140 185 229 273 317 361 405 449 493 537 581 625 669 712 756 800 843 887 930 974 017 795 841 886 932 978 023 068 114 159 204 250 295 340 385 430 475 520 565 610 655 700 744 789 834 878 923 967 *012 056 100 145 189 233 277 322 366 410 454 498 542 585 629 673 717 760 804 848 891 935 978 022 800 845 891 937 982 028 073 118 164 209 254 299 345 390 435 480 525 570 614 659 704 749 793 838 883 927 972 *016 061 105 149 193 238 282 326 370 414 458 502 546 590 634 677 721 765 808 852 896 939 983 026 7 8 804 809 850 855 896 900 941 946 987 991 032 037 078 082 123 127 168 173 214 218 259 263 304 308 349 354 394 399 439 444 484 489 529 534 574 579 619 623 664 668 709 713 753 758 798 802 843 847 887 892 932 936 976 981 *021 *025 065 069 109 114 154 158 198 202 242 247 286 291 330 335 374 379 419 423 463 467 506 511 550 555 594 599 638 642 682 686 726 730 769 774 813 817 856 861 900 904 944 948 987 991 030 035 I 9 Prop. Parts 813 859 905 950 996 041 087 132 177 223 268 313 358 403 448 493 538 583 628 673 717 762 807 851 896 941 985 *029 074 118 162 207 251 295 339 383 427 471 515 559 603 647 691 734 778 822 865 909 952 996 039 5 1 0.5 2 1.0 3 1.5 4 2.0 5 2.5 6 3.0 7 3.5 8 4.0 9 4.5 4 1 0.4 2 0.8 3 1.2 4 1.6 5 2.0 6 2.4 7 2.8 8 3.2 9 3.6 N. JL.o I |2|3 4 5 l 6 1 7 | 8 I 9 Prop. Parts 0~ 15'= 900" S. 4.68 557 T. 4.68 558 2~ 41'= 9660" S. 4.68 542 T. 4.68 589 0 16 = 960 557 558 2 42 = 9720 541 590 0 17 =1020 557 558 2 43 = 9780 541 590 2 44 = 9840 541 590 2 38 =9480 542 588 2 45 = 9900 541 591 2 39 =9540 542 588 2 46 = 9960 541 591 2 40 =9600 542 589 2 47 =10020 540 592 49 TABLE I 1000-1050 N. I L. 0 | I 2 3 | 4 | 5 6 | 7 | 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 000 0000 4341 8677 001 3009 7337 002 1661 5980 003 0295 4605 8912 004 3214 7512 005 1805 6094 006 0380 4660 8937 007 3210 7478 008 1742 6002 009 0257 4509 8756 010 3000 7239 011 1474 5704 9931 012 4154 8372 013 2587 6797 014 1003 5205 9403 015 3598 7788 016 1974 6155 017 0333 4507 8677 018 2843 7005 019 1163 5317 9467 020 3613 7755 021 1893 0434 4775 9111 3442 7770 2093 6411 0726 5036 9342 3644 7941 2234 6523 0808 5088 9365 3637 7904 2168 6427 0683 4934 9181 3424 7662 1897 6127 *0354 4576 8794 3008 7218 1424 5625 9823 4017 8206 2392 6573 0751 4924 9094 3259 7421 1578 5732 9882 4027 8169 2307 0869 5208 9544 3875 8202 2525 6843 1157 5467 9772 4074 8371 2663 6952 1236 5516 9792 4064 8331 2594 6853 1108 5359 9605 3848 8086 2320 6550 *0776 4998 9215 3429 7639 1844 6045 *0243 4436 8625 2810 6991 1168 5342 9511 3676 7837 1994 6147 *0296 4442 8583 2720 1303 5642 9977 4308 8635 2957 7275 1588 5898 *0203 4504 8800 3092 7380 1664 5944 [*0219 4490 8757 3020 7279 1533 5784 *0030 4272 8510 2743 6973 *1198 5420 9637 3850 8059 2264 6465 *0662 4855 9044 3229 7409 1586 5759 9927 4092 8253 2410 6562 *0711 4856 8997 3134 1737 6076 *0411 4741 9067 3389 7706 2019 6328 *0633 4933 9229 3521 7809 2092 6372 *0647 4917 9184 3446 7704 1959 6208 *0454 4696 8933 3166 7396 *1621 5842 *0059 4271 8480 2685 6885 *1082 5274 9462 3647 7827 2003 6176 *0344 4508 8669 2825 6977 *1126 5270 9411 3547 2171 6510 *0844 5174 9499 3821 8138 2451 6759 *1063 5363 9659 3950 8238 2521 6799 *1074 5344 9610 3872 8130 2384 6633 *0878 5120 9357 3590 7818 *2043 6264 *0480 4692 8901 3105 7305 '1501 5693 9881 4065 8245 2421 6593 *0761 4925 9084 3240 7392 *1540 5684 9824 3961 2605 6943 *1277 5607 9932 4253 8569 2882 7190 *1493 5793 *0088 4379 8666 2949 7227 *1501 5771 *0037 4298 8556 2809 7058 *1303 5544 9780 4013 8241 *2465 6685 *0901 5113 9321 3525 7725 *1920 6112 *0300 4483 8663 2838 7010 *1177 5341 9500 3656 7807 *1955 6099 *0238 4374 3039 7377 *1710 6039 *0364 4685 9001 3313 7620 *1924 6223 *0517 4808 9094 3377 7655 *1928 6198 *0463 4724 8981 3234 7483 *1727 5967 *0204 4436 8664 *'2887 7107 *1323 5534 9742 3945 8144 *2340 6531 *0718 4901 9080 3256 7427 *1594 5757 9916 4071 8222 *2369 6513 *0652 4787 8 9 3473 3907 7810 8244 *2143 *2576 6472 6905 *0796 *1228 5116 5548 9432 9863 3744 4174 8051 8481 *2354 *2784 6652 7082 *0947 *1376 5237 5666 9523 9951 3805 4233 8082 8510 *2355 *2782 6624 7051 *0889 *1316 5150 5576 9407 9832 3659 4084 7907 8332 *2151 *2575 6391 6815 *0627 *1050 4859 5282 9086 9509 *3310 *3732 7529 7951 *1744 *2165 5955 6376 *0162 *0583 4365 4785 8564 8984 *2759 *3178 6950 7369 *1137 *1555 5319 5737 9498 9916 3673 4090 7844 8260 *2010 *2427 6173 6589 *0332 *0747 4486 4902 8637 9052 *2784 *3198 6927 7341 *1066 *1479 5201 5614 N. I L. o | I | 2 3 4 s 6 7 8 | 9 2~ 46'= 9960" S. 4.68 541 T. 4.68 591 2~ 51'=10 260" S. 4.68 540 T. 4.68 593 2 47 =10 020 540 592 2 52 =10 320 539 594 2 48 =10080 540 592 2 53 =:10 380 539 594 2 49 =10140 540 592 2 54 =10440 539 595 2 50 = 10 200 540 593 2 55 = 10 500 539 595 50 TABLE I 1050-1100 N. I L. o I 2 1 3 4 5 6 ( 7 8 19 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 I 021 1893 6027 022 0157 4284 8406 023 2525 6639 024 0750 4857 8960 025 3059 7154 026 1245 5333 9416 027 3496 7572 028 1644 5713 9777 029 3838 7895 030 1948 5997 031 0043 4085 8123 032 2157 6188 033 0214 4238 8257 034 2273 6285 035 0293 4297 8298 036 2295 6289 037 0279 4265 8248 038 2226 6202 039 0173 4141 8106 040 2066 6023 9977 041 3927 2307 6440 0570 4696 8818 2936 7050 1161 5267 9370 3468 7563 1654 5741 9824 3904 7979 2051 6119 *0183 4244 8300 2353 6402 0447 4489 8526 2560 6590 0617 4640 8659 2674 6686 0693 4698 8698 2695 6688 0678 4663 8646 2624 6599 0570 4538 8502 2462 6419 *0372 4322 2720 6854 0983 5109 9230 3348 7462 1572 5678 9780 3878 7972 2063 6150 *0233 4312 8387 2458 6526 *0590 4649 8706 2758 6807 0851 4893 8930 2963 6993 1019 5042 9060 3075 7087 1094 5098 9098 3094 7087 1076 5062 9044 3022 6996 0967 4934 8898 2858 6814 *0767 4716 3134 7267 1396 5521 9642 3759 7873 1982 *0190 4288 8382 2472 6558 *0641 4719 8794 2865 6932 *0996 5055 9111 3163 7211 1256 5296 9333 3367 7396 1422 5444 9462 3477 7487 1495 5498 9498 3494 7486 1475 5460 9442 3419 7393 1364 5331 9294 3254 7210 *1162 5111 3547 7680 1808 5933 *0054 4171 8284 2393 6498 *0600 4697 8791 2881 6967 *1049 5127 9201 3272 7339 *1402 5461 9516 3568 7616 1660 5700 9737 3770 7799 1824 5846 9864 3878 7888 1895 5898 9898 3893 7885 1874 5858 9839 3817 7791 1761 5727 9690 3650 7605 *1557 5506 3961 8093 2221 6345 *0466 4582 8695 2804 6909 *1010 5107 9200 3289 7375 *1457 5535 9609 3679 7745 *1808 5867 9922 3973 8020 2064 6104 *0140 4173 8201 2226 6248 *0265 4279 8289 2296 6298 *0297 4293 8284 2272 6257 *0237 4214 8188 2158 6124 *0086 4045 8001 *1952 5900 4374 8506 2634 6758 *0878 4994 9106 3214 7319 *1419 5516 9609 3698 7783 *1865 5942 *0016 4086 8152 *2214 6272 *0327 4378 8425 2468 6508 *0544 4576 8604 2629 6650 *0667 4680 8690 2696 6698 *0697 4692 8683 2671 6655 *0635 4612 8585 2554 6520 *0482 4441 8396 *2347 6295 4787 8919 3046 7170 *1289 5405 9517 3625 7729 *1829 5926 *0018 4107 8192 *2273 6350 *0423 4492 8558 *2620 6678 *0732 4783 8830 2872 6912 *0947 4979 9007 3031 7052 *1068 5081 9091 3096 7098 *1097 5091 9082 3070 7053 *1033 5009 8982 2951 6917 *0878 4837 8791 *2742 6690 5201 9332 3459 7582 *1701 5817 9928 4036 8139 *2239 6335 *0427 4515 8600 *2680 6757 *0830 4899 8964 *3026 7084 *1138 5188 9234 3277 7315 *1350 5382 9409 3433 7453 '1470 5482 9491 3497 7498 *1496 5491 9481 3468 7451 *1431 5407 9379 3348 7313 *1274 5232 9187 *3137 7084 5614 9745 3871 7994 *2113 6228 *0339 4446 8549 *2649 6744 *0836 4924 9008 *3088 7165 1 1237 5306 9371 *3432 7489 *1543 5592 9638 3681 7719 *1754 5785 9812 3835 7855 *1871 5884 9892 3897 7898 *1896 5890 9880 3867 7849 *1829 5804 9776 3745 7709 *1670 5628 9582 *3532 7479 N. I L. o I |2 '3 4 5 6 7 8 | 9 2~ 55'= 10 500" S. 4.68 539 T. 4.68 595 3~ 0'=10 800" S. 4.68 538 T. 4.68 597 2 56 =10560 539 595 3 1 =10860 537 598 2 57 =10620 538 596 3 2 =10920 537 598 2 58 =10680 538 596 3 3 =10980 537 599 2 59=10740 538 597 3 4=11 040 537 599 51 TABLE II. Base of common (Briggs) logarithms = 10. Base of natural (Napierian) logarithms (e) = 2.718281828459 * ~ ~. Modulus of Com. Logs. = logloe = M = 0.4342944819 * *. 1 = log 2.30258509299 = log10= 2.30258509299 ~ ~ ~. logioN = M X logeN. 1 logeN = x logjoN. nr Multiples of M Multiples of To Convert from Corn. to Nat. Logs M To Convert from Nat. to Com. Logs --.. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 0.00 000 000 0.43 429 448 0.86 858 896 1.30 288 345 1.73 717 793 2.17 147 241 2.60 576 689 3.04 006 137 3.47 435 586 3.90 865 034 4.34 294 482 4.77 723 930 5.21 153 378 5.64 582 826 6.08 012 275 6.51 441 723 6.94 871 171 7.38 300 619 7.81 730 067 8.25 159 516 8.68 588 964 9.12 018 412 9.55 447 860 9.98 877 308 10.42 306 757 10.85 736 205 11.29 165 653 11.72 595 101 12.16 024 549 12.59 453 998 13.02 883 446 13.46 312 894 13.89 742 342 14.33 171 790 14.76 601 238 15.20 030 687 15.63 460 135 16.06 889 583 16.50 319 031 16.93 748 479 17.37 177 928 17.80 607 376 18.24 036 824 18.67 466 272 19.10 895 720 19.54 325 169 19.97 754 617 20.41 184 065 20.84 613 513 21.28 042 961 21.71 472 410 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99.100 21.71 472 410 22.14 901 858 22.58 331 306 23.01 760 754 23.45 190 202 23.88 619 650 24.32 049 099 24.75 478 547 25.18 907 995 25.62 337 443 26.05 766 891 26.49 196 340 26.92 625 788 27.36 055 236 27.79 484 684 28.22 914 132 28.66 343 581 29.09 773 029 29.53 202 477 29.96 631 925 30.40 061 373 30.83 490 822 31.26 920 270 31.70 349 718 32.13 779 166 32.57 208 614 33.00 638 062 33.44 067 511 33.87 496 959 34.30 926 407 34.74 355 855 35.17 785 303 35.61 214 752 36.04 644 200 36.48 073 648 36.91 503 096 37.34 932 544 37.78 361 993 38.21 791 441 38.65 220 889 39.08 650 337 39.52 079 785 39.95 509 234 40.38 938 682 40.82 368 130 41.25 797 578 41.69 227 026 42.12 656 474 42.56 085 923 42.99 515 371 43.42 944 819 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 , _ 0.00 000 000 2.30 258 509 4.60 517 019 6.90 775 528 9.21 034 037 11.51 292 546 13.81 551 056 16.11 809 565 18.42 068 074 20.72 326 584 23.02 585 093 25.32 843 602 27.63 102 112 29.93 360 621 32.23 619 130 34.53 877 639 36.84 136 149 39.14 394 658 41.44 653 167 43.74 911 677 46.05 170 186 48.35 428 695 50.65 687 205 52.95 945 714 55.26 204 223 57.56 462 732 59.86 721 242 62.16 979 751 64.47 238 260 66.77 496 770 69.07 755 279 71.38 013 788 73.68 272 298 75.98 530 807 78.28 789 316 80.59 047 825 82.89 306 335 85.19 564 844 87.49 823 353 89.80 081 863 92.10 340 372 94.40 598 881 96.70 857 391 99.01 115 900 101.31 374 409 103.61 632 918 105.91 891 428 108.22 149 937 110.52 408 446 112.82 666 956 115.12 925 465 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 115.12 925 465 117.43 183974 119.73 442 484 122.03 700 993 124.33 959 502 126.64 218 011 128.94 476 521 131.24 735 030 133.54 993 539 135.85 252 049 138. 15 510 558 140.45 769 067 142.76 027 577 145.06 286 086 147.36 544 595 149.66 803 104 151.97 061 614 154.27 320 123 156.57 578 632 158.87 837 142 161.18 095 651 163.48 354 160 165.78 612 670 168.08 871 179 170.39 129 688 172.69 388 197 174.99 646 707 177.29 905 216 179.60 163 725 181.90 422 235 184.20 680 744 186.50 939 253 188.81 197 763 191.11 456 272 193.41 714 781 195.71 973 290 198.02 231 800 200.32 490 309 202.62 748 818 204.93 007 328 207.23 265 837 209.53 524 346 211.83 782 856 214.14 041 365 216.44 299 874 218.74 558 383 221.04 816 893 223.35 075 402 225.65 333 911 227.95 592 421 230.25 850 930 TABLE III FIVE PLACE LOGARITHMS OF TRIGONOMETRIC FUNCTIONS (For explanations, see pages 21 to 24.) Pages 54 to 61 give logarithms of sines and tangents for each 10 seconds from 3~ to 7~ and of cosines and cotangents for each 10 seconds from 83~ to 87~. For values of functions for angles less than 3~ and greater than 87~ the S. and T. method may be used. Pages 62 to 106 give logarithms of sines, tangents, cosines, and cotangents for each minute from 0~ to 90~. FORMULAS FOR THE USE OF S. AND T. (For explanations see page 25.) (1) For a near zero degrees. log sin a = log a" + S. log tan a = log a" + T. log cot a = cpl log a" + cpl T. = cpl log tan a. (2) For a near ninety degrees. log cos a = log (90~ - a)" + S. log cot a = log (90~ - a)" + T. log tan a = cpl log (90~ - a)" + cpl T. = cpl log cot a. log a" = log sin a + cpl S. = log tan a- + cpl T. = cpl log cot a + cpl T. log (90~ - a)" = log cos a + cpl S. = log cot a + cpl T. = cpl log tan a + cpl T. 53 L. cos TABLE III 30 log sin 9.99 ' I o" I 0" 20" 30 40" 50o" 60" I d I Prop. Parts... -,,. I._. 940 940 939 938 938 937 936 936 935 934 934 933 932 932 931 930 929 929 928 927 926 926 925 924 923 923 922 921 920 920 919 918 917 917 916 915 914 913 913 912 911 910 909 909 908 907 906 905 904 904 903 902 901 900 899 898 898 897 896 895 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 8.71 880 8.72 120 359 597 834 8.73 069 303 535 767 997 8.74 226 454 680 906 8.75 130 353 575 795 8.76 015 234 451 667 883 8.77 097 310 522 733 943 8.78 152 360 568 774 979 8.79 183 386 588 789 990 8.80 189 388 585 782 978 8.81 173 367 560 752 944 8.82 134 324 513 701 888 8.83 075 261 446 630 813 996 8.84 177 920 960 160 200 399 439 637 676 873 912 108 147 342 380 574 613 805 844 *035 *073 264 302 491 529 718 755 943 980 167 204 390 427 612 648 832 869 052 088 270 306 487 523 703 739 919 954 133 168 346 381 558 593 768 803 978 *013 187 222 395 430 602 636 808 842 *013 *047 217 251 420 453 622 655 823 856 *023 *056 222 255 421 454 618 651 815 847 *010 *043 205 237 399 431 592 624 784 816 975 *007 166 198 356 387 544 576 732 764 920 951 106 137 292 322 476 507 660 691 844 874 *026 *056 208 238 *000 240 478 716 951 186 419 651 882 *112 340 567 793 *018 241 464 685 905 125 343 559 775 990 204 416 628 838 *048 257 464 671 876 *081 284 487 689 890 *090 289 487 684 880 *075 270 463 656 848 *039 229 419 607 795 982 168 353 538 721 904 *087 268 *040 280 518 755 991 225 458 690 920 *150 378 605 831 *055 279 501 722 942 161 379 595 811 *026 239 452 663 873 *083 291 499 705 910 *115 318 521 722 923 *123 322 519 716 913 *108 302 496 688 880 *071 261 450 639 826 *013 199 384 568 752 935 *117 298 *080 320 558 794 *030 264 497 728 959 *188 416 642 868 *092 316 538 759 979 197 415 631 847 *061 275 487 698 908 *118 326 533 739 945 *149 352 555 756 956 *156 355 552 749 945 *140 334 528 720 912 *103 292 482 670 857 *044 230 415 599 783 965 *147 328 *120 359 597 834 *069 303 535 767 997 *226 454 680 906 *130 353 475 795 *015 234 451 667 883 *097 310 522 733 943 *152 360 568 774 979 *183 386 588 789 990 *189 388 585 782 978 *173 367 560 752 944 *134 324 513 701 888 *075 261 446 630 813 996 *177 358 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10.9 8 7 6 5 940 939 938 938 937 936 936 935 934 934 933 932 932 931 930 929 929 928 927 926 926 925 924 923 923 922 921 920 920 919 918 917 917 916 915 914 913 913 912 911 910 909 909 908 907 906 905 904 904 903 902 901 900 899 898 898 897 896 895 894 40 40 40 40 39 39 39 39 38 38 38 38 38 37 37 37 37 37 36 36 36 36 36 36 35 35 35 35 35 35 34 34 34 34 34 34 34 33 33 33 33 33 32 32 32 32 32 32 32 32 31 31 31 31 31 31 30 30 30 30 40 39 1 4.0 3.9 2 8.0 7.8 3 12.0 11.7 4 16.0 15.6 5 20.0 19.5 6 24.0 23.4 7 28.0 27.3 8 32.0 31.2 9 36.0 35.1 38 37 1 3.8 3.7 2 7.6 7.4 3 11.4 11.1 4 15.2 14.8 5 19.0 18.5 6 22.8 22.2 7 26.6 25.9 8 30.4 29.6 9 34.2 33.3 36 1 3.6 2 7.2 3 10.8 4 14.4 5 18.0 6 21.6 7 25.2 8 28.8 9 32.4 35 34 1 3.5 3.4 2 7.0 6.8 3 10.5 10.2 4 14.0 13.6 5 17.5 17.0 6 21.0 20.4 7 24.5 23.8 8 28.0 27.2 9 31.5 30.6 33 32' 1 3.3 3.2 2 6.6 6.4 3 9.9 9.6 4 13.2 12.8 5 16.5 16.0 6 19.8 19.2 7 23.1 22.4 8 26.4 25.6 9 29.7 28.8 31 30 1 3.1 3.0 2 6.2 6.0 3 9.3 9.0 4 12.4 12.0 5 15.5 15.0 6 18.6 18.0 7 21.7 21.0 8 24.8 24.0 9 27.9 27.0 4 3 2 0 I _ __,.,.,, - I I 6'6o" 50" 20 9.99 I d I Prop. Parts log cos 86~ L. sin 54 TABLE III 3~ log tan I I0" io" 120" 130"1 40" 150" 60 1 d j Prop. Parts 0 8.71 940 980 *020 *060 *100 *141 '*181 59 40 41 40 1 8.72 181 221 261 301 341 380 420 58 40 1 4.1 4.0 2 420 460 500 540 579 619 659 57 40 2 8.2 8.0 3 659 698 738 777 817 856 896 56 40 3 12.3 12.0 4 896 935 975 *014 *053 *093 *132 55 39 4 16.4 16.0 5 8.73 132 171 210 249 288 327 366 54 39 5 20.5 20.0 6 366 405 444 483 522 561 600 53 39 6 24.6 24.0 7 600 638 677 716 754 793 832 52 39 7 28.7 28.0 8 832 870 909 947 986 *024 *063 51 38 8 32.8 32.0 9 8.74 063 101 139 178 216 254 292 50 38 9 36.9 36.0 10 292 330 369 407 445 483 521 49 38 39 38 11 521 559 597 634 672 710 748 48 38 1 3.9 3.8 12 748 786 823 861 899 936 974 47 38 2 7.8 7.6 13 974 *012 *049 *087 *124 *162 *199 46 38 3 11.7 11.4 14 8.75 199 236 274 311 '348 385 423 45 37 4 15.6 15.2 15 423 460 497 534 571 608 645 44 37 5 19.5 19.0 16 645 682 719 756 793 830 867 43 37 6 23.4 22.8 17 867 904 940 977 *014 *051 *087 42 37 7 27.3 26.6 18 8.76 087 124 160 197 233 270 306 41 36 8 31.2 30.4 19 306 343 379 416 452 488 525 40 36 9 35.1 34.2 20 525 561 597 633 669 706 742 39 36 37 36 21 742 778 814 850 886 922 958 38 36 1 3.7 3.6 22 958 994 *030 *065 *101 *137 *173 37 36 2 7.4 7.2 23 8.77 173 208 244 280 315 351 387 36 36 3 11.1 10.8 24 387 422 458 493 529 564 600 35 36 4 14.8 14.4 25 600 635 670 706 741 776 811 34 35 5 18.5 18.0 26 811 847 882 917 952 987 *022 33 35 6 22.2 21.6 27 8.78 022 057 092 127 162 197 232 32 35 7 25.9 25.2 28 232 267 302 337 371 406 441 31 35 8 29.6 28.8 29 441 475 510 545 579 614 649 30 35 9 33.3 32.4 30 649 683 718 752 787 821 855 29 34 35 34 31 855 890 924 958 993 *027 *061 28 34 1 3.5 3.4 32 8.79 061 096 130 164 198 232 266 27 34 2 7.0 6.8 33 266 300 334 368 402 436 470 26 34 3 10.5 10.2 34 470 504 538 572 606 639 673 25 34 4 14.0 13.6 35 673 707 741 774 808 842 875 24 34 5 17.5 17.0 36 875 909 942 976 *009 *043 *076 23 34 6 21.0 20.4 37 8.80 076 110 143 177 210 243 277 22 34 7 24.5 23.8 38 277 310 343 376 409 443 476 21 33 8 28.0 27.2 39 476 509 542 575 608 641 674 20 33 9 31.5 30.6 40 674 707 740 773 806 839 872 19 33 33 32 41 872 905 937 970 *003 *036 *068 18 33 1 3.3 3.2 42 8.81 068 101 134 166 199 232 264 17 33 2 6.6 6.4 43 264 297 329 362 394 427 459 16 32 3 9.9 9.6 44 459 491 524 556 588 621 653 15 32 4 13.2 12.8 45 653 685 717 750 782 814 846 14 32 5 16.5 16.2 46 846 878 910 942 974 *006 *038 13 32 6 318 1922.4 47 8.82 038 070 102 134 166 198 230 12 32 231 2. 48 230 262 293 325 357 389 420 11 32 8 26.4 25.6 49 420 452 484 515 547 579 610 10 329.7 8.8 50 610 642 673 705 736 768 799 9 32 31 30 51 799 831 862 893 925 956 987 8 31 1 3.1 3.0 52 987 *019 *050 *081 *112 *144 *175 7 31 2 6.2 6.0 53 8.83 175 206 237 268 299 330 361 6 31 3 9.3 9.0 54 361 392 423 454 485 516 547 5 31 4 12.4 12.0 55 547 578 609 640 671 701 732 4 31 5 15.5 15.0 56 732 763 794 824 855 886 916 3 31 6 18.6 18.0 57 916 947 978 *008 *039 *069 *100 2 31 7 21.7 21.0 58 8.84 100 130 161 191 222 252 282 1 30 8 24.8 24.0 59 282 313 343 374 404 434 464 0 30 27.9 27.0 1 60" 50" 40" 1 30" 20" |I o | of" | I.' |I. |d Prop. Parts log cot 86~ 55 L. cos TABLE III 4 0 log sin L. -~:;*;-~~ 0" 11J"20" I 4" I 6I 9.99 I I ) 0" I io 2~ 30 i 40" 50 I 6o" I I 894 893 892 891 891 890 889 888 887 886 885 884 883 882 881 880 879 879 878 877 876 875 874 873 872 871 870 869 868 867 866 865 864 863 862 861 860 859 858 857 856 855 854 853 852 851 850 848 847 846 845 844 843 842 841 840 839 838 837 836 0 2 3 4 5 6 7 8 9 10 11 12 13 1 4 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 8.84 358 1 389 539 569 718 748 897 927 8.85 075 105 252 282 429 458 605 634 780 809 955 984 8.86 128 157 301 330 474 502 645 674 816 845 987 *015 8.87 156 185 325 354 494 522 661 689 829 856 995 *023 8.88 161 188 326 353 490 518 654 681 817 845 980 *007 8.89 142 169 304 330 464 491 625 651 784 811 943 970 8.90 102 128 260 286 417 443 574 600 730 756 885 911 8.91 040 066 195 221 349 374 502 528 655 680 807 833 959 984 8.92 110 135 261 286 411 436 561 586 710 735 859 883 8.93 007 031 154 179 301 326 448 472 594 619 740 764 885 909 6o" 419 599 778 957 134 311 488 663 838 *013 186 359 531 703 873 *043 213 382 550 717 884 *050 216 381 545 709 872 *034 196 357 518 678 837 996 154 312 469 626 782 937 092 246 400 553 706 858 *010 161 311 461 611 760 908 056 203 350 497 643 788 933 449 629 808 986 164 341 517 693 867 *042 215 388 560 731 902 *072 241 410 578 745 912 *078 243 408 572 736 899 *061 223 384 545 704 864 *023 181 338 495 652 808 963 118 272 426 579 731 883 *035 186 336 486 636 784 933 081 228 375 521 667 812 957 479 659 838 *016 193 370 546 722 896 *070 244 416 588 760 930 *100 269 438 606 773 940 *106 271 436 600 763 926 *088 250 411 571 731 890 *049 207 364 521 678 834 989 143 298 451 604 757 909 *060 211 361 511 660 809 957 105 253 399 546 691 837 981 509 688 867 *045 223 400 576 751 926 *099 273 445 617 788 958 *128 297 466 634 801 967 *133 298 463 627 790 953 *115 277 438 598 758 917 *075 233 391 548 704 859 *015 169 323 477 630 782 934 *085 236 386 536 685 834 982 130 277 424 570 716 861 *006 539 718 897 *075 252 429 605 780 955 *128 301 474 645 816 987 *156 325 494 661 829 995 *161 326 490 654 817 980 * 142 304 464 625 784 943 *102 260 417 574 730 885 *040 195 349 502 655 807 959 *110 261 411 561 710 859 *007 154 301 448 594 740 885 *030 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 0 u 893 892 891 891 890 889 888 887 886 885 884 883 882 881 880 879 879 878 877 876 875 874 873 872 871 870 869 868 867 866 865 864 863 862 861 860 859 558 857 856 855 854 853 852 851 850 848 847 846 845 844 843 842 841 840 839 838 837 836 834 30 30 30 30 30 30 30 29 29 29 29 29 29 29 29 28 28 28 28 28 28 28 28 28 27 27 27 27 27 27 27 27 26 26 26 26 26 26 26 26 26 26 26 26 26 25 25 25 25 25 25 25 25 25 24 24 24 24 124 24 1 24 Prop. Parts 31 30 1 3.1 3.0 2 6.2 6.0 3 9.3 9.0 4 12.4 12.0 5 15.5 15.0 6 18.6 18.0 7 21.7 21.0 8 24.8 24.0 9 27.9 27.0 29 1 2.9 2 5.8 3 8.7 4 11.6 5 14.5 6 17.4 7 20.3 8 23.2 9 26.1 28 27 1 2.8 2.7 2 5.6 5. 4 3 8.4 8. 1 4 11.2 10.8 5 14.0 13.5 6 16.8 16.2 7 19.6 18.9 8 22.4 21.6 9 25.2 24..3 26 1 2.6 2 5.2 3 7.8 4 10.4 5 13.0 6 15.6 7 18.2 8 20.8 9 23.4 25 24 1 2.5 2.4 2 5.0 4..8 3 7.5 7.2 4 10.0 9.6 5 12.5 12.0 6 15.0 14.4 7 17.5 16.8 8 20.0 19.2 9 22.5 21.6 I I - 1 40" 1 30" 20" 10" I o" I' I 999 I d Prop. Parts log cos 850 L. sin 56 TABLE LUI 40 log tan I I -' _, I I __, I I - ' " I I - - I I I I.L I r%. - I O" I Eol I'I 0 1 2 3 4.5 6 7 8 9 10 I I 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 - 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 8.84 464 646 826 8.85 006 183 363 540 717 893 8.86 069 243 417 591 763 935 8.87 106 277 447 616 785 953 8.88 120 287 453 618 783 948 8.89 111 274 437 598 760 920 8.90 080 240 399 557 715 872 8.91 029 185 340 495 650 803 957 8.92 110 262 414 565 716 866 8.93 016 165 313 462 609 756 903 8.94 049 495 676 856 036 214 392 570 747 922 098 272 447 619 792 964 135 305 475 644 813 981 148 315 481 646 811 975 138 301 464 625 786 947 107 266 425 583 741 898 055 211 366 521 675 829 982 135 287 439 590 741 891 040 190 338 486 634 781 928 074 525 706 886 065 244 422 599 776 952 127 301 475 648 821 992 163 334 503 673 841 *009 176 342 508 674 838 *002 166 328 491 652 813 974 134 293 451 610 767 924 081 236 392 547 701 855 *008 160 313 464 615 766 916 065 214 363 511 658 805 952 098 30~ I 40~~ 555 585 736 766 916 946 095 125 274 304 452 481 629 658 805 835 981 *010 156 185 330 359 504 533 677 706 849 878 *021 *049 192 220 362 390 532 560 701 729 869 897 *037 *065 204 231 370 398 536 563 701 728 866 893 *029 *057 193 220 355 383 518 545 679 706 840 867 -*000 *027 160 187 319 346 478 504 636 662 793 820 950 976 107 133 262 288 418 443 572 598 727 752 880 906 *033 *059 186 211 338 363 489 515 640 665 791 816 941 966 090 115 239 264 388 412 536 560 683 707 830 854 976 *001 122 147 50'' 1 0 I 615 796 976 155 333 511 688 864 *039 214 388 562 734 907 *078 249 419 588 757 925 *092 259 425 591 756 920 *084 247 410 571 733 894 *054 213 372 531 688 846 *002 159 314 469 624 778 931 *084 237 388 540 691 841 991 140 289 437 585 732 879 *025 171 I U I rirup..1:-Urs 646 826 *006 185 363 540 717 893,~069 243 417 591 763 935 * 106 277 447 616 785 953 *120 287 453 618 783 948 274 437 598 760 920 *080 240 399 557 715 872 *029 185 340 495 650 803 957 *110 262 414 565 716 866 *016 165 313 462 609 756 903 *049 195 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 1 4 13 12 11 10 9 8 7 6 5 4 3 2 0 v I 30 30 30 30 30 30 30 29 29 29 29 29 29 29 28 28 28 28 28 28 28 28 28 28 28 28 27 27 27 27 27 27 27 27 26 26 26 26 26 26 26 26 26 26 26 26 25 25 25 25 25 25 25 25 25 24 24 24 24 24 I 31 30 1 3.1 3.0 2 6.2 6.0 3 9.3 9.0 4 12.4 12.0 5 15.5 15.0 6 18.6 18.0 7 21.7 21.0 8 24. 8 24.0 9 27.9 27.0 29 1 2.9 2 5.8 3 8.7 4 11.6 5 14.5 6 17.4 7 20.3 8 23.2 9 26.1 28 1 2.8 2 5.6 3 8.4 4 11.2 5 14.0 6 16.8 7 19.6 8 22.4 9 25.2 27 2.7 5.4 8.1 10.8 13.5 16.2 18.9 21.6 24.3 26 1 2.6 2 5.2 3 7.8 4 10.4 5 13.0 6 15.6 7 18.2 8 20.8 9 23.4 25 1 2.5 2 5.0 3 7.5 4 10.0 5 12.5 6 15.0 7 17.5 8 20.0 9 22.5 24 2.4 4..8 7.2 9.6 12.0 14..4 16.8 19.2 21.6 I - - - 601" 50f 40" 30"j 20" 10" 0" T d I Prop. Parts log cot 850 57 L. cos Table III 50 log sin 9.99 1 / I o/ i 2o0' 30o " 40 50o 60" I I d Prop. Parts 1..,.-.. _ -.. 834 833 832 831 830 829 828 827 825 824 823 822 821 820 819 817 816 815 814 813 812 810 809 808 807 806 804 803 802 801 800 798 797 796 795 793 792 791 790 788 787 786 785 783 782 781 780 778 777 776 775 773 772 771 769 768 767 765 764 763 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 8.94 030 174 317 461 603 746 887 8.95 029 170 310 450 589 728 867 8.96 005 143 280 417 553 689 825 960 8.97 095 229 363 496 629 762 894 8.98 026 157 288 419 549 679 808 937 8.99 066 194 322 450 577 704 830 956 9.00 082 207 332 456 581 704 828 951 9.01 074 196 318 440 561 682 803 054 078 198 222 341 365 484 508 627 651 769 793 911 935 052 076 193 216 333 357 473 496 613 636 752 775 890 913 028 051 166 189 303 326 440 462 576 599 712 735 847 870 982 *005 117 139 251 274 385 407 518 541 651 674 784 806 916 938 048 070 179 201 310 332 441 462 571 592 701 722 830 851 959 980 087 109 216 237 343 365 471 492 598 619 725 746 851 872 977 998 103 123 228 249 353 373 477 498 601 622 725 746 848 869 971 992 094 115 217 237 339 359 460 480 582 602 703 723 823 843 50"| 40" 102 246 389 532 675 817 958 099 240 1380 520 659 798 936 074 212 349 485 621 757 892 *027 162 296 430 563 696 828 960 092 223 354 484 614 744 873 002 130 258 386 513 640 767 893 *019 144 269 394 518 642 766 889 *012 135 257 379 501 622 743 863 126 270 413 556 698 840 982 123 263 403 543 682 821 959 097 234 371 508 644 780 915 *050 184 318 452 585 718 850 982 114 243 375 506 636 765 894 *023 152 280 407 534 661 788 914 *040 165 290 415 539 663 787 910 *033 155 278 399 521 642 763 883 150 294 437 580 722 864 *005 146 287 427 566 705 844 982 120 257 394 531 667 802 937 *072 207 341 474 607 740 872 *004 135 266 397 527 657 787 916 *045 173 301 428 556 682 809 935 *061 186 311 436 560 684 807 930 *053 176 298 420 541 662 783 903 174 317 461 603 746 887 *029 170 310 450 589 728 867 *005 143 280 417 553 689 825 960 *095 229 363 496 629 762 894 *026 157 288 419 549 679 808 937 *066 194 322 450 577 704 830 956 *082 207 332 456 581 704 828 951 *074 196 318 440 561 682 803 923 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 833 832 831 830 829 828 827 825 824 823 822 821 820 819 817 816 815 814 813 812 810 809 808 807 806 804 803 802 801 800 798 797 796 795 793 792 791 790 788 787 786 785 783 782 781 780 778 777 776 775 773 772 771 769 768 767 765 764 763 761 24 24 24 24 24 24 24 24 23 23 23 23 23 23 23 23 23 23 23 23 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 21 21 21 21 21 21 21 21 21 21 21 21 21 21 20 20 20 20 20 20 20 20 20 24 1 2.4 2 4.8 3 7.2 4 9.6 5 12.0 6 14.4 7 16.8 8 19.2 9 21.6 23 1 2.3 2 4.6 3 6.9 4 9.2 5 11.5 6 13.8 7 16.1 8 18.4 9 20.7 22 1 2.2 2 4.4 3 6.6 4 8.8 5 11.0 6 13.2 7 15.4 8 17.6 9 19.8 21 1 2.1 2 4.2 3 6.3 4 8.4 5 10.5 6 12.6 7 14.7 8 16.8 9 18.9 20 1 2.0 2 4.0 3 6.0 4 8.0 5 10.0 6 12.0 7 14.0 8 16.0 9 18.0 I 1 60o 1 30~" 20"| I~ 10" 1' I 9-99 I d Prop. Parts log cos 84" L. sin 58 TABLE III 5~ log tan [ / I 0"/ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 8.94 195 340 485 630 773 917 8.95 060 202 344 486 627 767 908 8.96 047 187 325 464 602 739 877 8.97 013 150 285 421 556 691 825 959 8.98 092 225 358 490 622 753 884 8.99 015 145 275 405 534 662 791 919 9.00 046 174 301 427 553 679 805 930 9.01 055 179 303 427 550 673 796 918 9.02 040 IO/ 20" 30" 40" | 219 244 268 292 365 389 413 437 509 533 557 581 654 678 702 725 797 821 845 869 941 964 988 *012 083 107 131 155 226 249 273 297 368 391 415 439 509 533 556 580 650 674 697 721 791 814 838 861 931 954 977 *001 071 094 117 140 210 233 256 279 349 372 395 418 487 510 533 556 625 648 671 694 762 785 808 831 899 922 945 968 036 059 081 104 172 195 218 240 308 331 353 376 443 466 488 511 578 601 623 646 713 735 758 780 847 869 892 914 981 *003 *025 *048 114 136 159 181 247 269 291 314 380 402 424 446 512 534 556 578 644 666 687 709 775 797 819 841 906 928 950 971 037 058 080 102 167 188 210 232 297 318 340 361 426 448 469 491 555 577 598 620 684 705 727 748 812 834 855 876 940 961 983 *004 068 089 110 131 195 216 237 258 322 343 364 385 448 469 490 511 574 595 616 637 700 721 742 763 826 846 867 888 951 971 992 *013 075 096 117 138 200 220 241 262 324 344 365 386 447 468 489 509 571 591 612 632 694 714 735 755 816 837 857 878 939 959 979 *000 061 081 101 121 316 461 606 749 893 *036 178 320 462 603 744 884 *024 163 302 441 579 717 854 991 127 263 398 533 668 802 936 *070 203 336 468 600 731 862 993 123 253 383 512 641 769 898 *025 153 280 406 532 658 784 909 *034 158 282 406 530 653 776 898 *020 142 340 485 630 773 917 *060 202 344 486 627 767 908 *047 187 325 464 602 739 877 *013 150 285 421 556 691 825 959 *092 225 358 490 622 753 884 *015 145 275 405 534 662 791 919 *046 174 301 427 553 679 805 930 *055 179 303 427 550 673 796 918 *040 162 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 50 60" I d Prop. Parts 59. 2 24 24 24 24 24 24 24 24 24 24 23 23 23 23 23 23 23 23 23 23 23 23 23 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 20 20 20 20 20 20 25 1 2.5 2 5.0 3 7.5 4 10.0 5 12.5 6 15.0 7 17.5 8 20.0 9 22.5 24 1 2.4 2 4.8 3 7.2 4 9.6 5 12.0 6 14.4 7 16.8 8 19.2 9 21.6 23 1 2.3 2 4.6 3 6.9 4 9.2 5 11.5 6 13.8 7 16.1 8 18.4 9 20.7 22 1 2.2 2 4.4 3 6.6 4 8.8 5 11.0 6 13.2 7 15.4 8 17.6 9 19.8 21 1 2.1 2 4.2 3 6.3 4 8.4 5 10.5 6 12.6 7 14.7 8 16.8 9 18.9 20 1 2.0 2 4.0 3 6.0 4 8.0 5 10.0 6 12.0 7 14.0 8 16.0 9 18.0 4 3 2 1 0 I 6 " 50"' 40" 30" 20" 1o" o0" I' d Prop. Parts ] log cot 84~ 59 L. cos TABLE III 6~ log sin 9.99 l | o" I I0 | 20" 30" 40" 50" 60" | I d Prop. Parts A... 3 0 761 760 759 757 756 755 753 752 751 749 748 747 745 744 742 741 740 738 737 736 734 733 731 730 728 727 726 724 723 721 720 718 717 716 714 713 711 710 708 707 705 704 702 701 699 698 696 695 693 692 690 689 687 686 684 683 681 680 678 677 0 1 2 3 4 5 6 7 8 9 I 9.01 923 943 9.02 043 063 163 183 283 302 402 421 520 540 639 658 757 776 874 894 992 *011 I 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 I ).03 109 226 342 458 574 690 805 920 9.04 034 149 262 376 490 603 715 828 940 9.05 052 164 275 386 497 607 717 827 937 9.06 046 155 264 372 481 589 696 804 911 9.07 018 124 231 337 442 548 653 758 863 968 9.08 072 176 280 383 486 128 245 361 478 593 709 824 939 053 168 281 395 508 621 734 847 959 071 182 293 404 515 625 736 845 955 064 173 282 390 499 606 714 821 929 035 142 248 354 460 566 671 776 881 985 089 193 297 400 504 964 *984 083 103 203 223 322 342 441 461 560 579 678 698 796 816 914 933 *031 *050 148 167 265 284 381 400 497 516 613 632 728 747 843 862 958 977 072 091 187 206 300 319 414 433 527 546 640 659 753 772 865 884 977 996 089 108 201 219 312 330 423 441 533 552 644 662 754 772 864 882 973 991 082 101 191 210 300 318 408 426 517 535 624 642 732 750 839 857 946 964 053 071 160 177 266 284 372 390 478 495 583 601 688 706 793 811 898 915 *002 *020 107 124 211 228 314 331 418 435 521 538 *004 123 243 362 481 599 717 835 953 *070 187 303 420 535 651 766 881 996 110 225 338 452 565 678 790 903 *015 126 238 349 460 570 681 791 900 *010 119 228 336 445 553 660 768 875 982 089 195 301 407 513 618 723 828 933 *037 141 245 349 452 555 *024 143 263 382 501 619 737 855 972 *089 206 323 439 555 670 786 901 *015 129 244 357 471 584 697 809 921 *033 145 256 367 478 589 699 809 918 *028 137 246 354 463 571 678 786 893 *000 106 213 319 425 530 636 741 846 950 *055 159 262 366 469 572 *043 163 283 402 520 639 757 874 992 *109 226 342 458 574 690 805 920 *034 149 262 376 490 603 715 828 940 *052 164 275 386 497 607 717 827 937 *046 155 264 372 481 589 696 804 911 *018 124 231 337 442 548 653 758 863 968 *072 176 280 383 486 589 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 760 759 757 756 755 753 752 751 749 748 747 745 744 742 741 740 738 737 736 734 733 731 730 728 727 726 724 723 721 720 718 717 716 714 713 711.710 708 707 705 704 702 701 699 698 696 695 693 692 690 689 687 686 684 683 681 680 678 677 675 20 20 20 20 20 20 20 20 20 20 20 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 17 17 17 17 17 17 21 1 2.1 2 4.2 3 6.3 4 8.4 5 10.5 6 12.6 7 14.7 8 16.8 9 18.9 20 1 2.0 2 4.0 3 6.0 4 8.0 5 10.0 6 12.0 7 14.0 8 16.0 9 18.0 19 1 1.9 2 3.8 3 5.7 4 7.6 5 9.5 6 11.4 7 13.3 8 15.2 9 17.1 18 1 1.8 2 3.6 3 5.4 4 7.2 5 9.0 6 10.8 7 12.6 8 14.4 9 16.2 17 1 1.7 2 3.4 3 5.1 4 6.8 5 8.5 6 10.2 7 11.9 8 13.6 9 15.3 I [ 60o" [ 50" | 40" | 3o" 20"t io, j o" 'I 9.99 | d [ Prop. Parts log cos 83~ L. sin 60 TABLE III 60 log tan I I fI0" 10" 20" 30" 40" 50" / 6" I d I Prop. Parts I 0 1 0 f 12,0' 3ff 4 50 6 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 9.02 162 283 404 525 645 766 885 9.03 005 124 242 361 479 597 714 832 948 9.04 065 181 297 413 528 643 758 873 987 9.05 101 214 328 441 553 666 778 890 9.06 002 113 224 335 445 556 666 775 885 994 9.07 103 211 320 428 536 643 751 858 964 9.08 071 177 283 389 495 600 705 810 182 304 425 545 666 785 905 025 144 262 381 499 616 734 851 968 084 201 317 432 548 663 777 892 *006 120 233.347 460 572 685 797 909 020 132 243 353 464 574 684 793 903 *012 121 229 338 446 554 661 768 875 982 089 195 301 407 512 617 722 827 203 324 445 565 686 805 925 044 163 282 400 518 636 753 871 987 104 220 336 451 567 682 796 911 *025 139 252 365 478 591 703 815 927 039 150 261 372 482 592 702 812 921 *030 139 247 356 464 5/2 679 786 893 *000 106 213 319 424 530 635 740 845 223 344 465 585 706 825 945 064 183 302 420 538 656 773 890 *007 123 239 355 471 586 701 815 930 *044 158 271 384 497 610 722 834 946 057 169 279 390 500 611 720 830 939 *048 157 266 374 482 589 697 804 911 *018 124 230 336 442 547 653 757 862 243 364 485 605 726 845 965 084 203 321 440 558 675 793 910 *026 143 259 374 490 605 720 835 949 *063 177 290 403 516 628 741 853 964 076 187 298 409 519 629 739 848 957 *066 175 284 392 500 607 715 822 929 *035 142 248 354 460 565 670 775 880 263 384 505 625 746 865 985 104 223 341 459 577 695 812 929 *046 162 278 394 509 624 739 854 968 *082 195 309 422 535 647 759 871 983 094 206 316 427 537 647 757 866 976 *085 193 302 410 518 625 733 840 947 *053 160 266 371 477 582 688 792 897 283 404 525 645 766 885 *005 124 242 361 479 597 714 832 948 *065 181 297 413 528 643 758 873 987 *101 214 328 441 553 666 778 890 *002 113 224 335 445 556 666 775 885 994 *103 211 320 428 536 643 751 858 964 *071 177 283 389 495 600 705 810 914 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 20 20 20 20 20 20 20 20 20 20 20 20 20 20 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 1 7 21 1 2.1 2 4.2 3 6.3 4 8.4 5 10.5 6 12.6 7 14..7 8 16.8 9 18.9 20 1 2.0 2 4.0 3 6.0 4 8.0 5 10.0 6 12.0 7 14.0 8 16.0 9 18.0 19 1 1.9 2 3.8 3 5.7 4 7.6 5 9.5 6 11.4 7 13.3 8 15.2 9 17. 1 18 1 1.8 2 3.6 3 5.4 4 7.2 5 9.0 6 10.8 7 12.6 8 14.4 9 16.2 17 1 1.7 2 3.4 3 5.1 4 6.8 5 8.5 6 10.2 7 11.9 8 13.6 9 15.3 9 8 7 6 5 4 3 2 0 u. - -. ___ II6o" Iso" 40" 30"f 20"" 10 " o' o j I I d Prop. Parts I log cot 61 TABLE III 0O 90~ 180~ 270~ "f' I log sin I d, I f cpl S cpl T I I log tan I 0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 1260 1320 1380 1440 1500 1560 1620 1680 1740 1800 1860 1920 1980 2040 2100 2160 2220 2280 2340 2400 2460 2520 2580 2640 2700 2760 2820 2880 2940 3000 3060 3120 3180 3240 3300 3360 3420 3480 3540 3600 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 15 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 6.46 373 6.76 476 6.94 085 7.06 579 7.16 270 7.24 188 7.30 882 7.36 682 7.41 797 7.46 373 7.50 512 7.54 291 7.57 767 7.60 985 7.63 982 7.66 784 7.69 417 7.71 900 7.74 248 7.76 475 7.78 594 7.80 615 7.82 545 7.84 393 7.86 166 7.87 870 7.89 509 7.91 088 7.92 612 7.94 084 7.95 508 7.96 887 7.98 223 7.99 520 8.00 779 8.02 002 8.03 192 8.04 350 8.05 478 8.06 578 8.07 650 8.08 696 8.09 718 8.10 717 8.11 693 8.12 647 8.13 581 8.14 495 8.15 391 8.16 268 8.17 128 8.17 971 8.18 798 8.19 610 8.20 407 8.21 189 8.21 958 8.22 713 8.23 456 8.24 186 30103 17609 12494 9691 7918 6694 5800 5115 4576 4139 3779 3476 3218 2997 2802 2633 2483 2348 2227 2119 2021 1930 1848 1773 1704 1639 1579 1524 1472 1424 1379 1336 1297 1259 1223 1190 1158 1128 1100 1072 1046 1022 999 976 954 934 914 896 877 860 843 827 812 797 782 769 755 743 730 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 443 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 444 5.31 445 5.31 445 5.31 445 5.31 445 5.31 443 5.31 443 5.31 443 5.31 442 5.31 442 5.31 442 5.31 442 5.31 442 5.31 442 5.31 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8.60 662 8.60 973 8.61 282 8.61 589 8.61 894 8.62 196 8.62 497 8.62 795 8.63 091 8.63 385 8.63 678 8.63 968 8.64 256 8.64 543 8.64 827 8.65 110 8.65 391 8.65 670 8.65 947 8.66 223 8.66 497 8.66 769 8.67 039 8.67 308 8.67 575 8.67 841 8.68 104 8.68 367 8.68 627 8.68 886 8.69 144 8.69 400 8.69 654 8.69 907 8.70 159 8.70 409 8.70 658 8.70 905 8.71 151 8.71 395 8.71 638 8.71 880 360 357 355 351 349 346 343 341 337 336 332 330 328 325 323 320 318 316 313 311 309 307 305 302 301 298 296 294 293 290 288 287 284 283 281 279 277 276 274 272 270 269 267 266 263 263 260 259 258 256 254 253 252 250 249 247 246 244 243 242 I 5.31 451 5.31 451 5.31 452 5.31 452 5.31 452 5.31 452 5.31 452 5.31 452 5.31 453 5.31 453 5.31 453 5.31 453 5.31 453 5.31 453 5.31 454 5.31 454 5.31 454 5.31 454 5.31 454 5.31 454 5.31 455 5.31 455 5.31 455 5.31 455 5.31 455 5.31 455 5.31 456 5.31 456 5.31 456 5.31 456 5.31 456 5.31 456 5.31 457 5.31 457 5.31 457 5.31 457 5.31 457 5.31 458 5.31 458 5.31 458 5.31 458 5.31 458 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27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 logsin d log tan cd log cot log cosProp. Parts 8.71 880 240 8.71 940 241 1.28 060 9.99 940 60 241 239 237 235 234 8.72 120 8.72 181 1.27819 9.99 940 59 1 4.0 4.0 4.0 3.9 3.9 8.72 359 238 8.72 420 239 1.27 580 9.99 939 58 3 120 8.0 7.9.8 17.8 8.72 59723 8.72 659 1.27 341 9.99 938 57 4 16.1 15.9 15.8 15.7 15.6 8.72 834 235 8.72 896 237 1.27 104 9.99 938 56 6 24.1 23.9 23.7 23.5 23.4 8.73 06934 8.73 132 2341.26 868 9.99 937 55 7 28.1 27.9 27.6 27.4 27.3 82734132 234 26 868 99 937~55 8 32.1 31.9 31.6 31.3 31.2 8.73 303 232 8.73 366 234 1.26 634 9.99 936 54 9 36.2 35.8 35.6 35.2 35.1 8.73 535 232 8.73 600 232 1.26 400 9.99 936 53 10 40.2 39.8 39.5 39.2 39.0 8.73 767 230 8.73 832 231 1.26 168 9.99 935 52 3o 120.5 119.5 118.5 117.5 117.0 8.73 997 229 8.74 063 22 1.25 937 9.99 934 51 40 160.7 159.3 158.0 156.7 156.0 8.74 226 228 8.74 292 229 1.25 708 9.99 934 50 200.S 199.2 197.5 195.8 195.0 8.74 454 226 8.74 521 227 1.25 479 9.99 933 49 1 3.9 3.8 3.8 3.8 3.7 2 7.7 7.6 7.6 7.5 4 8.74 680 26 8.74 748 226 1.25 252 9.99 932 48 3 11.6 114 11-4 11:2 11.2 8.74 906 224 8.74 974 225 1.25 026 9.99 932 47 4 15.5 15.3 15.1 15.o 14.9 8 75 13024 8 75 199 225 1 24 801 9.99 931 46 5 193 19.1 18.9 188 186 8.7513 223 224 6 23.2 22.9 22.7 22.5 22.3 8 75 353 222 8.75 423 222 1.24 577 9 99 930 45 7 27.1 26.7 26.5 26.2 26.0 8.75 5 220 8.75 645 222 1.24 355 9.99 929 44 98 34.8 34.4 340 338 329.4. 220 222 9 34.8 34.4 34.0 33.8 33.4 8.75 795 220 8.75 867 220 1.24 133 9.99 929 43 10 38.7 38.2 37.8 37.5 37.2 8.76 015 2 8.76 087 219 1.23913 9.9 928 42 30 1160114 511351125115 8.76 234 217 8.76 306 219 1.23 694 9.99 927 41 40 154.7 152.7 151.3 150.0 148.7 - ~~~~~~~50 1193.3 190.8 189.2 187.5 185.8 8.76 451 2 8.76 525 1.23 475 9.99 926 40' 22 13 20 1.215 13 8.76 667 216 8.76 742 216 1.23 258 9.99 926 39 1 3.7 3.7 3.6 3.6 3.6 8.76 883 1 8.76958 1.230429.9992538 2 7.4 7.3 7.2 7.2 7. 214 25 '3 11.1 11.0 10.8 10.8 10.6 8.77 097 213 8.77 173 214 1.22 827 9.99 924 37 4 14.8 14.7 14.5 14.3 14.2 8.77 310 212 8.77 387 213 1.22 613 9.99 923 36 5 18.5 18.3 18.1 17.9 17.8 212 23 '6 22.2 22.0 21.7 21.5 21.3 8.77 522 1 8.77 600 211.22400 9.99 923 35 7 259 25.7 25.3 25.1 24.8 8.77 733 210 8.77 811 211 1.22 189 9.99 922 34 239.3 339 328:9 3282 28.4 8.77 943 209 878 022 210 1.21 978 9.99 921 33 10 37.0 36.7 36.2 35.8 35.5 1t'7~ 9 R7932 91 768 9 999 20320 74.0 73.3 72.3 71.7 71.0 8.78 152 208 8.78 232 209 1.21 768 9.99 920 32 0 111.0 110.0 108.5 107.5 106.5 8.78 360 208 8.78 441 208 1.21 559 9.99 920 31 40 148.0 146.7 144.7 143.3 142.0 8.78 568 206 8.78 649 206 1.21 351 9.99 919 30 2t1 201 26 203 20 1 8.78 774 205 8.78 855 206 1.21145 9.9991829 1 3.5 3. 3.4 3.4 3.1 8.78 979 204 8.79 061 205 1.20939 9.9991728 2 7.0 6.9 6.9 6.8 6.7 8.79 183 203 8.79 266 204 1.20 734 9.99 917 27 4 14.1 13.9 13.7 13.5 13.4 8.79 386 202 8.79 470 203 1.20530 9.99 916 26 5 17.6 17.3 17.2 16.9 16.8 8.79 588 201 8.79.673.6 21.1 20.8 20.6 20.3 20.1 8 79 588 201 8.79 673 202 1.20 327 9.99 915 25 7 24.6 24.3 24.0 23.7 23.4 8 79 789 201 8.79 875 201 1.20 125 9.99 914 24 9 31.6 31.2 30.9 30.4 30.2 8.79 990 199 8.80 076 201 1.19 924 9.99 913 23 10 35.2 34.7 34.3 33.8 33.5 8.80 189 199 8.80 277 199 1.19723 9.99 913 22 20 70.3 69.3 68.7 67.7 67.0 'S8o 476 30 105.5 104.0 103.0 101.5 100.5 8.80 388 197 8.80 476 98 1.19 524 9.99 912 21 40 140.7 138.7 137.3 135.3 134.5 8.80 585 197 8.80 674 198 1.19 326 9.99 911 20 50 175.8 173.3 171.7 169.2 167.5....8 872199 197 195 193 192 8.80 782 96 8.80 872 96 1.19 128 9.99 910 19 1 3.3 3.3 3.2 3.2 3.2 8.80 978 195 8.81 068 196 1.18 932 9.99 909 18 2 6.6 6.6 6.5 6.4 6.4 8.81 173 194 8.81 264 195 1.18 736 9.99 909 17 4 13.3 13.1 13.0 12:9 12:8 8.81 367 193 8.81 459 194 1.18 541 9.99 908 16 5 16.6 16.4 16.2 16.1 16.0 -*~~~' ''' * |~ ~ 7JI6 19 9 19.7 19.5 19.3 19.2 8.81 560 192 8.81 653 193 1.18 347 9.99 907 15 7 23.2 23.0 22.8 22.5 22.4 8 81 752 192 8.81 846 192 1.18 154 999 906 14 8 26.5 26. 2326.0 25.7 25.6 9 29.8 29.6 29.2 29.0 28.8 8.81 944 190 8.82038192 1.179629.99 905 13 10 33.2 32.8 32.5 32.2 32.0 8.82 134 90 8.82 230 190 1.17 770 9.99 904 12 20 66.3 65.7 65.0 64.3 64.0 8.82 324 189 8.82 420 190 1.17 580 9.99 904 11 40 132.7 131.3 130o. 128.7 128.o 8.82 513 188 8.82 610 189 1.17 390 9.99 903 10 50 165.8 164.2 162.5 160.8 160.0 189 187 185 183 181 8.82 701 187 8.82 799 188 1.17201 9.99 902 9 1 3.2 3.1 3.1 3.0 3.0 8.82 888 187 8.82 987 188 1.17013 9.99 901 8 2 6.3 6.2 6.2 6.1 6.0 3 9.4 9.4 9.2 9.2 9.0 8.83 075 186 8.83 175 186 1.16 825 9.99 900 7 4 12.6 12.5 12.3 12.2 12.1 8.83 261 185 8.83 361 186 1.16 639 9.99 899 6 5 15.8 15.6 15.4 15.2 15.1 6 18.9 18.7 18.5 18.3 18.1 8.83 446 184 8.83 547 185 1.16 453 9.99 898 5 7 22.0 21.8 21.6 21.4 21.1 8.83 630 83 8.83 732 184.162689.99 898 4 8 25.2 24.9 24.7 24.4 24.1 9 28.4 28.0 27.8 27.4 27.2 8.83 813 183 8.83 916 184 1.16 084 9.99 897 3 10 31.5 31.2 30.8 30.5 30.2 8 83 996 181 8.84 100 182 1.15 900 9.99 896 2 20 63.0 62.3 61.7 61.0 60.3 8.84 177 181 8.84 282 182 1.15 718 9.99 895 1 94126.0 1245 91235 91225 9120 8.84 358 8.84 464 1.15 536 9.99 894 0 50 157.5 155.8 154.2 152.5 150.8 I log cos d I log cot Icd log tan I log sin I j Prop. Parts 17026 50806 176~ 266~ 356~ 86~ 65 TABLE III 40 94~ 184~ 274~ log sin T d log tcot log cos jProp. Parts 0 8.84 358 181 8.84 464 182 1.15 536 9.99 894 60 is2 181 179 178 177 1 8.84 539 179 8.84 646 180 1.15 354 9.99 893 59 1 3.0 3.0 3.0 3.0 3.0 2 8.84 718 179 8.84 826 180 1.15 174 9.99 892 58 2 96.1 6.0 6.0 59 5.9 3 8.84 897 8.85 006 179 1.14 994 9.99 891 57 4 12.1 12.1 11.9 11.9 11.8 4 8.85 075 1787 8.85 185 178 1.14 815 9.99 891 56 6 18.2 18.1 17.9 17.8 17:7 5 8.85 -252 177 8.85 363 177 1.14 637 9.99 890 55 7 21.2 21.1 20.9 20.8 20.6.... 77178 24.3 24.1 23.9 23.7 23.6 6 8.85 429 176 8.85 540 177 1.14 460 9.99 889 54 9 27.3 27.2 26.8 26.7 26.6 7 8.85 605 175 8.85 717 1.14283 9.99 888 53 10 30.3 30.2 29.8 29.7 29.5 8,85t; 780 8 85 9 1 1 7 07 9 t987 20 60.7 60.3 59.7 59.3 59.0 8 8.85 780 / 175 8.85 893 176 1.14 107 9.99 887 52 30 91.0 90.5 89.5 89.0 88.5 9 8.85 955 173 8.86 06974 1.13 931 9.99 886 51 40 121.3 120.7 119.3 118.7 118.0 '173 174 50 151.7 150.8 149.2 148.3 147.5 10 8.86 128 173 8.86 243 174 1.13757 9.99 885 50 50 176 175 174 173 172 11 8.86 301 73 8.86 417 74 1.13 583 9.99 884 49 1 2.9 2.9 2.9 2.9 2.9 12 8.86 474 17 8.86 591 172 1.13 409 9.99 883 48 23 5.8 8.8 878 5.8 8.7 13 8.86645 171 8.86763 172 1.13 237 9.99 882 47 4 11.7 11.7 11.6 11.5 115 14 8.86 816 171 8.86 935 171 1.13 065 9.99 881 46 6 147.6 147.6 147.4 14.43 147.2 15 8.86987 169 8.87 106 1 1.12 894 9.99 880 45 7 20.5 20.4 20.3 20.2 20.1 8.86 987 1169 *9 87 106 7 2 3.9 16 8.87 156 169 8.87 277 70 1.12 723 9.99 879 44 9 264 262 262 260 258 160.07 1569^Q 1.07277lyn.127 230 ~ yvO/y 448 26.4 26.2 26.1 26.0 25.8 17 8.87 325 169 8.87 447 169 1.12 553 9.99 879 43 10 29.3 29.2 29.0 28.8 28.7 18 8.87 494 167 8.87 616 169 1.12 384 9.99 878 42 230 88.0 5875 870 86 57 86. 19 8.87 661 168 8.87 785 168 1.12 215 9.99 877 41 40 117.3 116.7 116.0 115.3 114.7 20 8.87 829 166 8.87 953 167 1.12 047 9.99 876 40 50 146.7 145.8 145.0 144.2 143.3 21 0.0/0/yi. 0- 166 ~ '171 170 169 168 167 21 8.87 995 166 8.88 120 167 1.11 880 9.99 875 39 1 2.8 2.8 2.8 2.8 2.8 22 8.88 161 165 8.88287 166 1.11 713.9987438 2 5.7 5.7 5.6 5.6 5.6 23 366 11 713 9 99 873 3 8.6 8.5 8.4 8.4 8.4 23 8.88 326164 8.88 453 165 1.11 547 9.99873 37 4 114 113 11.3 11.2 11.1 24 8.88 490 164 8.88 618 165 1.11 382 9.99 872 36 5 14.2 14.2 14.1 14.0 13.9 6 17.1 17.0 16.9 16.8 16.7 25 8.88 654 163 8.88 783 165 1.11 217 9.99871 35 7 20.0 19.8 19.7 19.6 19.5 26 8.88 817 163 8.88 948 163 1.11 052 9.99 870 34 8 22.8 22.7 22.5 22.4 22.3 27 8.88 980 162 8.89 111 163 1.10 889 9.99 869 33 10 28.5 28.3 28.2 28.0 27.8 28 8.89 142 162 8.89 274 163 1.10726 9.99 868 32 20 57.0 56.7 56.3 56.0 55.7 ~o'onoA.6 S-0'27-63 * u/-0nnnQ? 30 85.5 85.0 84.5 84.0 83.5 29 8.89 304 160 8.89 437 161 1.10 563 9.99 867 31 40 114.0 113.3 112.7 112.0 111.3 30 8.89 464 161 8.89 598 162 1.10 402 9.99 866 30 50 142.5 141.7 140.8 140.0 139.2 166 165 164 163 162 31 8.89 625 159 8.89 760 160 1.10 240 9.99 865 29 1 2.8 2.8 2.7 2.7 2.7 32 8.89 784 159 8.89 920 160 1.10080 9.99 864 28 2 5.5 5.5 5.5 5.4 5.4 3360 1.09 920 9. 99 863 27 4 11.1 11.0 10.9 10.9 10.8 338.89 943159 8.90 080 160 1.09 9209.99863 27 4 - 8: 1 - 10. 34 8.90 102 158 8.90 240 159 1. 09 760 9.99 862 26 5 13.8 13.8 13.7 13.6 13.5 6 16.6 16.5 16.4 16.3 16.2 35 8.90 260 157 8.90 399 158 1.09 601 9.99 861 25 7 19.4 19.2 19.1 19.0 18.9 36 8.90417 57 8.90557 58 1.09443 9.99860 24 8 22.1 22.0 21.9 21.7 21.6 9 24.9 24.8 24.6 24.4 24.3 37 8.90 574 1 56 8.90 715 157 1.09 285 9.99 859 23 10 27.7 27.5 27.3 27.2 27.0 38 8.90 730 155 8.90 872 157 1.09 128 9.99 858 22 20 55.3 55.0 54.7 54.3 54.0 39 8.90 885 155 8.91 029 156 1.08 971 9.99 857 21 40 110.7 110.0 109.3 108.7 108:0 40 8.91 040 155 8.91 185 155 1.08 815 9.99 856 20 50 138.3 137.5 136.7 135.8 135.0 41 8.91 195 154 8.91 340 155 1.08 660 9.99 855 19 1 2.7 2.7 2.6 2.6 2.6 42 8.91 349 153 8.91 495 155 1.08 505 9.99 854 18 2 5.4 5.3 5.3 5.3 5.2 43 8.91 502 153 8.91 650 153 1.08 350 9.99 853 17 10.7 107 10. 105 10. 44 8.91 655 152 8.91 803 154 1.08 197 9.99 852 16 5 13.4 13.3 13.2 13.2 13.1 6 16.1 16.0 15.9 15.8 15.7 45 8.91 807 152 8.91 957 153 1.08 043 9.99 851 15 7 18. 18.7 18. 18.4 18.3 46 8.91 959 151 8.92 110 152 1.07 890 9.99 850 14 8 21.5 21.3 21.2 21.1 20.9 47 8.92 110 151 8.92 262 152 1.07 738 99 848 13 26.8 26.7 265 263 26.2 48 8.92 261 150 8.92 414 151 1.07586 9.99 847 1 220 53.7 53.3 53.0 52.7 52.3 49 8.92 411 150 8.92 565 151 1.07 435 9.99 846 11 40 107.3 106.7 106.0 105.3 104.7 50 8.92 561 149 8.92 716 150 1.07 284 9.99 845 10 50 134.2 133.3 132.5 131.7 130.8 51 8.92 710 149 8.92 866 150 1.07 134 9.99 844 9 156 155 154 153 152 52 8.92 859 148 8.93 016 149 1.06 984 9.99 843 8 2 5.2 5:2 5.1 5.1 5.1 53 8.93 007 47 8.93 165 48 1.06 835 9.99 842 7 3 7.8 7.8 7.7 7.6 7.6 147 148 4 10.4 10.3 10.3 10.2 10.1 54 8.93 154 147 8.93 313 149 1.06 687 9.99 841 6 5 13.0 12.9 12.8 12.8 12.7 55 8.93 301 147 8.93 462 147 1.06538 9.99 840 5 7 181. 1. 180 1758 1757 56 8.93 448 146 8.93 609 147 1.06 391 9.99 839 4 8 20.8 20.7 20.5 20.4 20.3 9 3.4 23.2 23.1 230 2208 57 8.93 594 146 8.93 756 147 1.06 244 9.99 838 3 1 23.0 25.8 25.7 235.5 25.3 58 8.93 740 145 8.93 903 146 1.06 097 9.99 837 2 20 52.0 51.7 51.3 51.0 50.7 59 8.93 885 145 8.94 049 146 1.05 951 9.99 836 1 3o 1040 103 102 7 1060 101.3 60 8.94 030 8.94 195 1.05 805 9.99 834 0 50 130.0 129.2 128.3 127.5 126.7 I log cos d log cot c d log tan I log sin I 'I J Prop. Parts 175~ 2650 3556 66 TABLE III 50 95~ 185~ 275~ ' log sin I d logtan c t d logcot locos I Prop. Parts 0 8.94 03044 8.94 195 1.05 805 9.99 834 60 151 149 14s 147 146 1 8.94 174 144 8. 94340 1.05660 9.99 833 59 1 2.5 2.5 2.5 24 2.4 2 8.94317 43 994 485 145 105515 9.99 83258 7.67.4 7.479 4 79 3 3 8.94461 8.94630 1053709.99831 57 4 10.1 9.9 9.9 9.8 9.7 4 8.94 603 142 894 143 105 227 9.99 830 56 5 12.6 12.4 12.3 12.2 12.2. U 143 O4 144 * ~' 6 15.1 14.9 14.8 14.7 14.6 5 8.94 74614 8.94 91714 1.05 083 9.99 829 55 7 17.6 17.4 17.3 17.2 17.0 6 r8.947 8 14100 ' 8 9 06014' '1.04 940 9.9 82 8 20.1 19.9 19.7 19.6 19.5 8.94 887 8.95 060 142 1.04 940 9.99 828 54 9 22.6 22.4 22.2 22.0 21.9 7 8.95 029 141 8.95 202 142 1.04 798 9.99 827 53 l 25.2 24.8 24.7 24.5 24.3 20 50-3 49.7 49.3 49.0 48.7 8 8.95 170 140 8.95 344 142 1.04 656 9.99 825 52 30 75.5 74.5 74.0 73.5 73.0 9 8.95310 140 8.95 486 141 1.04514 9.99 824 51 40 100.7 99.3 98.7 98.0 97.3 - 5 2. 2. 123.3 122.5 121.7 108.95450 39 8.9562740 1 04373 9 99823 50 5 125.8 124.13 12 11 1U0.y.)4.)U139 Q.\f3 OZ/140.U4/3V 990/145 144 143 142 141 11 8.95 589 8.95 767 411.04233 9.99 82249 1 2.4 2.4 2.4 2.4 2.4 12 8.95 728 139 8.95 908 1 1.040929.99821 48 2 72 7. 4.7 4.7: 13 8.95 867 138 8.96 047 140 1.039539.9982047 4 9.7 9.6 9.59.5 9.4 13814 46 9.7 9.6 9.'5 9.5 9.4 148.96 005 13 8.96 187 103 813 9599 819 46 12.1 12.0 11.9 11.8 11.8 14 8.96 005 96138 89 1 1378 19 6 14.5 14.4 14.3 14.2 14.1 15 8.96 143 8.96 325 1.03 675 9.99 817 45 7 16.9 16.8 16.7 16.6 16.4 8 19.3 19.2 19.1 18.9 18.8 16 8.96 280 137 8.96 464 38 1.03536 9.99 816 44 9 21.8 21.6 21.4 21.3 21.2 17 8.96417 36 8.96602 37 1.03398 9.99815 43 10 24.2 24.0 23.8 23.7 23.5 18 8.96553 136 8.96 739 38 1.03 261 9.99 814 42 20 48.3 48.72o 47.7 47.3 4705 13 138 30 72.5 72.0 71.5 71.0 70. 19 8.96689 136 8.96 877 136 1.03 123 9.99813 41 40 96:7 96:0 95:.3 94:7 94:0.... Q t ft0 n 9Q70O t)A 50m120.8 120.0 119.2 118.3 117.5 20 8.96 825 135 8.97 013 13 1.02 987 9.99 812 40 140 o 139 138 137 136 21 8.96 960 135 8.97 150 135 1.02850 9.99 81039 1 2.3 2.3 2.3 2.3 2.3 22 8.97 09534 8.97 285 136.02 715 9.99 809 38 2 4.7 4.6 4.6 4.6 4.5 134 136 ~~~~~~~3 7.0 7.0 6.9 6.8 6.8 23 8.97 229 34 8.97 421 35 1.02 579 9.99 808 37 4 9.3 9.3 9.2 9.1 9.1 24 8.97 36333 8.97556 1.024449.99 807 36 5 11.7 11.6 11.5 11.4 11.3 6 14.0 13.9 13.8 13.7 13.6 25 8.97 496 1338.97 69134 1.02309 9.9980635 7 16.3 16.2 16.1 16.0 159 133.97 25 14 1.2 17 9.9 80 8 18.7 18.5 18.4 18.3 18.1 26 8.97 629 33 8.97 825 34 1.02175 9.9980434 9 21.0 20.8 20.7 20.6 20.4 27 8.97 762 132 8.97 959 133 1.02041 9.99803 33 10 23.3 23.2 23.0 22.8 22.7 20 46.7 46.3 46.0 45.7 45.3 28 8.97 894 132 8.98092 133 1.019089.99802 3220 46746.346.0 45.745.3 29 8.98026 131 8.98 225 133 1.01 775 9.99 801 31 40 93.3 92.:7 92:0 91.:3 90.7 30 8.98 157 131 8.98 358 132 1.01 642 9.99 800 30 50 116.7 115.8 115.0 114.2 113.3 135 134 133 132 131 31 8.98 288 31 8.98 490 32 1.01 510 9.99 79829 1 2.2 2.2 2.2 2.2 2.2 32 8.98 419 130 8.98 622 131 1.01 378 9.99 797 28 2 4.5 4.5 4.4 4.4 4.4 3 6.8 6.7 6.6 6.6 6.6 33 8.98 549 30 8.98 753 31 1.01 247 9.99 79627 4 9.0 8.9 8.9 8.8 18.7 348.98679 129 8.98884 131 1.01 1169.99 795 26 5 11.2 11.2 11.1 11.0 10.9 0 13.5 13.4 13.3 13.2 13.1 35 8.98 808 129 8.99 015 13.00 985 9.99793 25 7 15.8 15.6 15.5 15.4 15.3 36 8.98937 129 8.99 145 30 1.00855 9.99792 24 8 18.0 17.9 17.7 17.6 17-5 9 20.2 20.1 20.0 19.8 19.6 37 8.99 066 128 8.99 275 130 1.00725 9.99 791 23 10 22.5 22.3 22.2 22.0 21.8 38 8.99 194 128 8.99 405 129 1.00 595 9.99 790 22 20 45.0 44.7 44.3 44.0 43.7 301 67.5 67.0 66.5 66.0 65.5 39 8.99 322 128 8.99534 128 1.00466 9.99 788 21 40 90.0 89:3 88.7 88-0 87.3 - ~~~~~~~~~~~~5o 112.5 111.7 11o.8 11o.o 109.2 40 8.99450 127 8.99662 129 1.00338 9.99787 20 501125117110.8110.0109.2.... ~~~~~~~~~~~130 129 128 127 126 41 8.99577 127 8.99791 28 1.00209 9.99786 19 1 2.2 2.2 2.1 2.1 2.1 42 8.99 704 126 8.99919 127 1.00081 9.99 785 18 2 4.3 4.3 4.3 4.2 4.2 43 8.99 830 126 9.00046 128 0.99954 9.99 783 17 3 8.7 64 8 8: 8: 4 8.7 8. 6 8. 5 8.5 8.4 44 8.99956 126 9.00 174 127 0.99826 9.99782 16 5 10.8 10.8 o10.7 10.6 10.5 6 13.0 12.9 12.8 12.7 12.6 45 9.00 082 125 9.00301 126 0.99699 9.99781 15 7 15.2 15.0 14.9 14.8 14.7 46 9.00 207 25 9.00 427 1260.99 573 9.99 780 14 17.3 17.2 17.1 16.9 16.8 9 19.5 19.4 19.2 19.0 18.9 47 9.00 332 124 9.00 553 260.99 447 9.99 778 13 10 21.7 21.5 21.3 21.2 21.0o 48 9.00456 125 9.00 679 126 0.99 321 9.99 777 12 20 43.3 43.0 42.7 42.3 42.0 49 9.00581 123 9.00805 125 0.99 195 9.99 776 11 o8 65.0 64.5 64.o: 63.5 63.07 40 86.7 86.0 85.3 84.7 84.0 50 9.00 704 124 9.00 930 125 0.99 070 9.99 775 1050108.3107.5106.7105.8105.0 - 125 124 123 122 121 51 9.00 828 123 9.01 055 124 0.98 945 9.99 773 9 1 2.1 2.1 2.0 2.0 2.0 52 9.00951 123 9.01 179 124 0.98 821 9.99 772 8 2 4.2 4.1 4.1 4.1 4.0 53 9.01 074 122 9.01 303 24 0.98697 9.99 771 7 3 6.2 6.2 6.2 6.1 6.0 4 8.3 8.3 8.2 8.1 8.1 54 9.01 196 122 9.01 427 123 0.98573 9.99769 6 5 10.4 10.3 10.2 10.2 10.1 550 123 7686 12.5 12.4 12.3 12.2 12.1 55 9.01 318 122 9.01 5501230.984509.99768 5 7 14.6 14.-5 14.4 14.2 1.4.1 56 9.01 440 121 9.01 673123 0.98 327 9.99 767 4 8 16.7 16.5 16.4 16.3 16.1 57 9.01 56' ' 3 9 18.8 18.6 18.4 18.3 18.2 579.01561121 9.01 796 122 0.98 204 9.99 765 310 20:.8 20.7 20.5 20.3 20.2 58 9.01 682 121 9.01 918 122 0.98 082 9.99 764 2 20 41.7 41.3 41.0 40.7 40.3 59 9.01 803 120 9.02 040 122 0.97 960 9.99 763 1 30 62.5 62.0 61.5 61.0 60.5 60 9)yy~~.01 9 23 926 0988. 71 040 81 3.3 82.7 82.0 81.3 80.7 60 9.01 923 9.02 162 0.97 838 9.99 761 0 50 104.2 103.3 102.5 101.7 100.8 I log cos I d log cot I cd logtan I log sin ' Prop. Parts 174~ 264~ 354~ 84~ 67 TABLE III 60 96~ 186~ 2760 ' I log sin d i log tan cd log cot Ilog cos I I Prop. Parts 0 9.01 923 9.02162 0.97838 9.99 761 60 1 9.02 043 20 9.02 2832 0.97 717 9.99 760 59 121 120 119 118 2 9.02 163 0 9.02 404 0. 97596 9.99 759 58 1 2.0 2.0 2.0 2.0 3 9.02 283 120 9.02 525 121 0.97 475 9.99 757 57 2 4.0 4.0 4.0 3.9 4 9.02 402 118 9.02 645 21 0.97 355 9.99 756 56 3 6.0 6.0 6.0 5.9 5 9.02 520 119 9.02 766 119 0.97234 9.99 755 55 4 8. 1 80 7.9 7. 9 6 9.02 639 11 9.02 885 1 0.97 115 9.99 753 54 10.1 10.0 9. 9.8 7 9.02 757 117 9 03 005 119 0.96995 9.99 752 53 76 2.1 14.0 1319 13.8 8 9.02 874 1 9.03 124 0.96 876 9.99 751 52 7 14.1 14.0 13.9 13.7 9 9.02 992 117 9.03 242 119 0.96 758 9.99 749 51 8 16.2 16.0 159 1.7 109.03 109 117 9.03 361 0.96 639 9.99 748 50 10 202 20.0 198 19.7 11 9.03 226 116 9.03 479 118 0.96 521 9.99 747 49 20 40.3 40.0 39.7 39.3 12 9.03 342 116 9.03 597 117 0.96 403 9.99 745 48 30 60.5 60.0 59.5 59.0 13 9.03 458 116 9.03 714 118 0.96 286 9.99 744 47 40 80.7 80.0 79.3 78.7 14 9.03 574 116 9.03 832 116 0.96 168 9.99 742 46 50 100.8 100.0 99.2 98.3 15 9.03 690 115 9.03 948 117 0.96052 9.99 741 45 16 9.03 805 115 9.04 065 116 0.95 935 9.99 740 44 117 116 115 114 17 9.03 920 114 9.04 181 116 0.95 819 9.99 738 43 1 2.0 1.9 1.9 1.9 18 9.04 034 115 9.04 297 116 0.95 703 9.99 737 42 2 3.9 3.9 3.8 3.8 19 9.04 149 113 9.04 413 115 0.95 587 9.99 736 41 3 5.8 5.8 5.8 5.7 20 9.04 262 114 9.04 528 115 0.95 472 9.99 734 40 4 7.8 7.7 7.7 7.6 21 9.04 376 119.04 643115 0.95357 9.99733399.6 9.5 22 9.04490 113 9.04 758 115 0.95 242 9.99 731 38 6 11.7 11.6 11.5 11.4 23 9.04 603 112 9.04 873 114 0.95 127 9.99 730 37 8 16 1.5 1534 15. 24 9.04 715 113 9.04 987 11 0.95 013 9.99 728 36 17 6 17 4 17 2 17 1 11459 17.6 17.4 17.2 17.1 25 9.04 828 112 9.05 101 113 0.94 899 9.99 727 35 10 19.5 19.3 19.2 19.0 26 9.04 940 112 9.05 214 114 0.94 786 9.99 726 34 20 39.0 38.7 38.3 38.0 27 9.05 052 112 9.05 328 113 0.94672 9.99 724 33 30 58.5 58.0 57'.5 57.0 28 9.05 164 I 9.05 441 112 0.94 559 9.99 723 32 40 78.0 77.3 76.7 76.0 29 9.05 275 111 9.05 553 113 0.94 447 9.99 721 31 50 97.5 96.7 95.8 95.0 30 9.05 386 111 9.05 666 112 0.94334 9.99720 30 31 9.05 497 110 9.05 778 112 0.94 222 9.99 718 29 113 112 111 110 32 9.05 607 110 9.05 890 112 0.94 110 9.99717 28 1 1.9 1.9 1.8 1.8 33 9.05 717 110 9.06 002 11 0.93 998 9.99 716 27 2 3.8 3.7 3.7 3.7 34 9.05 827 110 9.06 113 111 0.93 887 9.99 714 26 3 5.6 5.6 5.6 5.5 35 9.05 937 109 9.06 224 11 0.93 776 9.99 713 25 5 9 4 79 4 72 9 36 9.06 046 109 9.06 335 110 0.93 665 9.99 711 24 6 113 112 111 110 37 9.06 155 109 9.06 445 I 0.93 555 9.99 710 23 7 1323 1.2 130 12 38 9.06 264 108 9.06 556 110 0.93 444 9.99 708 22 8 15 1.19 1438 1427 39 9.06 372 109 9.06 666 109 0.93 334 9.99 707 21 9 1750 1648 1646 1645 40 9.06 481 108 9.06775 110 0.93 225 9.99 705 20 10 18.8 18.7 18.5 18.3 41 9.06 589 107 9.06 885 109 0.93 115 9.99 704 19 20 37.7 37.3 37.0 36.7 42 9.06 696 108 9.06 994 109 0.93 006 9.99 702 18 30 56.5 56.0 55.5 55.0 43 9.06 804 107 9.07 103 108 0.92 897 9.99 701 17 40 75.3 74.7 74.0 73.3 44 9.06 911 107 9.07 211 109 0.92 789 9.99 699 16 50 94.2 93.3 92.5 91-.7 45 9.07 018 106 9.07 320 108 0.92 680 9.99 698 15 46 9.07 124 107 9.07 428 108 0.92 572 9.99 696 14 109 108 107 106 47 9.07 231 106 9.07 536 107 0.92 464 9.99 695 13 1 1.8 1.8 1.8 1.8 48 9.07 337 105 9.07 643 108 0.92 357 9.99 693 12 2 3.6 3.6 3.6 3.5 49 9.07 442 106 9.07 751 107 0.92 249 9.99 692 11 3 5.4 5.4 5.4 5.3 50 9.07548 105 9.07 858 106 0.92 142 9.99 690 10 5 93 721 719 878 51 9.07 653 105 9.07 964 107 0.92 036 9.99 689 9.1 90. 0. 52 9.07 758 105 9.08 071 106 0.91 929 9.99 687 8 6 0.9 10.8 10.7 10.6 53 9.07 863 105 9.08 177 106 0.91 823 9.99 686 7 8 14. 1442 14 3 4 1. 54 9.07 968 104 9.08 283 106 0.91 717 9.99 684 6 9 14.5 14.4 14.3 14.1 55 9.08 072 104 9.08389 106 0.91 611 9.99 683 5 10 18.2 18.0 17.8 17.7 56 9.08 176 104 9.08 495 105 0.91 505 9.99 681 4 20 36.3 36.0 35.7 35.3 57 9.08 280 103 9.08 600 105 0.91 400 9.99 680 3 30 54.5 54.0 53.5 53.0 58 9.08 383 103 9.08 705 105 0.91 295 9.99 678 2 40 72.7 72.0 71.3 70.7 59 9.08 486 103 9.08 810 104 0.91 190 9.99 677 1 50 90.8 90.0 89.2 88.3 60 9.08 589 9.08 914 0.91 086 9.99 675 0 I log cos d log cot c d log tan I log sin I Prop. Parts 13 260330806 173~ 263~ 353~ 83" 68 TABLE III 70 97~ 1870 277~ I log sin I d log tan cdl log cot log cos I Prop. Parts 0 1 2 3 4 5 6 7. 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.08 589 9.08 692 9.08 795 9.08 897 9.08 999 9.09 101 9.09 202 9.09 304 9.09 405 9.09 506 9.09 606 9.09 707 9.09 807 9.09 907 9.10 006 9.10 106 9.10 205 9.10 304 9.10 402 9.10 501 9.10 599 9.10 697 9.10 795 9.10 893 9.10 990 9.11 087 9.11 184 9.11 281 9.11 377 9.11 474 9.11 570 9.11 666 9.11 761 9.11 857 9.11 952 9.12 047 9.12 142 9.12 236 9.12 331 9.12 425 9.12 519 9.12 612 9.12 706 9.12 799 9.12 892 9.12 985 9.13 078 9.13 171 9.13 263 9.13 355 9.13 447 9.13 539 9.13 630 9.13 722 9.13 813 9.13 904 9.13 994 9.14 085 9.14 175 9.14 266 9.14 356 103 103 102 102 102 101 102 101 101 100 101 100 100 99 100 99 99 98 99 98 98 98 98 97 97 97 97 96 97 96 96 95 96 95 95 95 94 95 94 94 93 94 93 93 93 93 93 92 92 92 92 91 92 91 91 90 91 90 91 9C 9.08 914 9.09 019 9.09 123 9.09 227 9.09 330 9.09 434 9.09 537 9.09 640 9.09 742 9.09 845 9.09 947 9.10 049 9.10 150 9.10 252 9.10 353 9.10 454 9.10 555 9.10 656 9.10 756 9.10 856 9.10 956 9.11 056 9.11 155 9.11 254 9.11 353 9.11 452 9.11 551 9.11 649 9.11 747 9.11 845 9.11 943 9.12 040 9.12 138 9.12 235 9.12 332 9.12 428 9.12 525 9.12 621 9.12 717 9.12 813 9.12 909 9.13 004 9.13 099 9.13 194 9.13 289 9.13 384 9.13 478 9.13 573 9.13 667 9.13 761 9.13 854 9.13 948 9.14 041 9.14 134 9.14 227 9.14 320 9.14 412 9.14 504 9.14 597 9.14 688 9.14 780 105 104 104 103 104 103 103 102 103 102 102 101 102 101 101 101 101 100 100 100 100 99 99 99 99 99 98 98 98 98 97 98 97 97 96 97 96 96 96 96 95 95 95 95 95 94 95 94 94 93 94 93 93 93 93 92 92 93 91 92 0.91 086 0.90 981 0.90 877 0.90 773 0.90 670 0.90 566 0.90 463 0.90 360 0.90 258 0.90 155 0.90 053 0.89 951 0.89 850 0.89 748 0.89 647 0.89 546 0.89 445 0.89 344 0.89 244 0.89 144 0.89 044 0.88 944 0.88 845 0.88 746 0.88 647 0.88 548 0.88 449 0.88 351 0.88 253 0.88 155 0.88 057 0.87 960 0.87 862 0.87 765 0.87 668 0.87 572 0.87 475 0.87 379 0.87 283 0.87 187 0.87 091 0.86 996 0.86 901 0.86 806 0.86 711 0.86 616 0.86 522 0.86 427 0.86 333 0.86 239 0.86 146 0.86 052 0.85 959 0.85 866 0.85 773 0.85 680 0.85 588 0.85 496 0.85 403 0.85 312 0.85 220 9.99 675 9.99 674 9.99 672 9.99 670 9.99 669 9.99 667 9.99 666 9.99 664 9.99 663 9.99 661 9.99 659 9.99 658 9.99 656 9.99 655 9.99 653 9.99 651 9.99 650 9.99 648 9.99 647 9.99 645 9.99 643 9.99 642 9.99 640 9.99 638 9.99 637 9.99 635 9.99 633 9.99 632 9.99 630 9.99 629 9.99 627 9.99 625 9.99 624 9.99 622 9.99 620 9.99 618 9.99 617 9.99 615 9.99 613 9.99 612 9.99 610 9.99 608 9.99 607 9.99 605 9.99 603 9.99 601 9.99 600 9.99 598 9.99 596 9.99 595 9.99 593 9.99 591 9.99 589 9.99 588 9.99 586 9.99 584 9.99 582 9.99 581 9.99 579 9.99 577 9.99 575 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 2 3 A 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 2.0 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 105 1.8 3.5 5.2 7.0 8.8 10.5 12.2 14.0 15.8 17.5 35.0 52.5 70.0 87.5 101 1.7 3.4 5.0 6.7 8.4 10.1 11.8 13.5 15.2 16.8 33.7 50.5 67.3 84.2 97 1.6 3.2 4.8 6.5 8.1 9.7 11.3 12.9 14.6 16.2 32.3 48.5 64.7 80.8 93 1.6 3.1 4.6 6.2 7.8 9.3 10.8 12.4 14.0 15.5 31.0 46.5 62.0 77.5 104 103 102 1.7 1.7 1.7 3.5 3.4 3.4 5.2 5.2 5.1 6.9 6.9 6.8 8.7 8.6 8.5 10.4 10.3 10.2 12.1 12.0 11.9 13.9 13.7 13.6 15.6 15.4 15.3 17.3 17.2 17.0 34.7 34.3 34.0 52.0 51.5 51.0 69.3 68.7 68.0 86.7 85.8 85.0 100 99 98 1.7 1.6 1.6 3.3 3.3 3.3 5.0 5.0 4.9 6.7 6.6 6.5 8.3 8.2 8.2 10.0 9.9 9.8 11.7 11.6 11.4 13.3 13.2 13.1 15.0 14.8 14.7 16.7 16.5 16.3 33.3 33.0 32.7 50.0 49.5 49.0 66.7 66.0 65.3 83.3 82.5 81.7 96 95 94 1.6 1.6 1.6 3.2 3.2 3.1 4.8 4.8 4.7 6.4 6.3 6.3 8.0 7.9 7.8 9.6 9.5 9.4 11.2 11.1 11.0 12.8 12.7 12.5 14.4 14.2 14.1 16.0 15.8 15.7 32.0 31.7 31.3 48.0 47.5 47.0 64.0 63.3 62.7 80.0 79.2 78.3 92 91 90 1.5 1.5 1.5 3.1 3.0 3.0 4.6 4.6 4.5 6.1 6.1 6.0 7.7 7.6 7.5 9.2 9.1 9.0 10.7 10.6 10.5 12.3 12.1 12.0 13.8 13.6 13.5 15.3 15.2 15.0 30.7 30.3 30.0 46.0 45.5 45.0 61.3 60.7 60.0 76.7 75.8 75.0 I I log cos I d log cot cd log tan I log sin I ' I Prop. Parts 172~ 262'0 352~ 82~ 69 TABLE III 80 980 1880 2780 Prop. Parts I' log sin I d I log tanI cdlI log cotI log cosI I 0 2 3 4 5 6 7 8 9 10 I11 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9. 14 356 9. 14 445 9. 14 53 5 9. 14 624 9.14 714 9. 14 803 9. 14 891 9. 1I4 980 9. 15 069 9. 15 157 9. 15 245 9. 15 333 9. 15 421 9. 15 508 9. 15 596 9. 15 683 9. 15 770 9. 15 857 9. 15 944 9. 16 030 9. 16 116 9. 16 203 9. 16 289 9. 16 374 9. 16 460 9. 16 545 9. 16 631 9. 16 71 6 9. 16 801 9. 16 886 9.1 6970 9. 17 055 9.1 7 139 9. 17 223 9. 17 307 9. 17 391 9. 17 474 9. 17 558 9. 17 641 9. 17 724 9. 17 807 9. 17 890 9. 17 973 9. 18 055 9. 18 1 37 9. 18 220 9. 18 302 9. 18 383 9. 18 465 9. 18 547 9. 18 628 9. 18 709 9. 18 790 9. 18 871 9. 18 952 9. 19 033 9. 19 113 9. 19 193 9. 19 273 9. 19 353 9. 19 433 89 90 89 90 89 88 89 89 88 88 88 88 87 88 87 87 87 87 86 86 87 86 85 86 85 86 85 85 85 84 85 84 84 84 84 83 84 83 83 83 83 83 82 82 83 82 8 1 82 82 8 1 8 1 8 1 8 1 8 1 8 1 80 80 80 80 80 9. 14 780 9. 14 872 9. 14 963 9. 15 054 9. 15 145 9. 15 236 9. 15 327 9. 15417 9. 15 508 9. 15 598 9. 15 688 9. 15 777 9. 15 867 9. 15 956 9. 16 046 9. 16 1 35 9. 16 224 9. 16 312 9. 16 401 9. 16 489 9. 16 577 9. 16 665 9. 16 753 9. 16 841 9. 16 928 9. 17 016 9. 17 103 9. 17 190 9. 17 277 9. 17 363 9. 17 450 9. 17 536 9. 17 622 9. 17 708 9. 17 794 9. 17 880 9. 17 965 9. 18 051 9. 18 1 36 9. 18 221 9. 18 306 9. 18 391 9. 1 8475 9. 18 560 9. 18 644 9. 18 728 9. 18 812 9. 18 896 9. 18 979 9. 19 063 9. 19 146 9. 19 229 9. 19 312 9. 19 395 9. 19 478 9. 19 561 9. 19 643 9. 19 725 9. 19 807 9. 19 889 9.19 971 92 9 1 9 1 9 1 91 91 90 9 1 90 90 89 90 89 90 89 89 88 89 88 88 88 88 88 87 88 87 87 87 86 87 86 86 86 86 86 85 86 85 85 85 85 84 85 84 84 84 84 83 84 83 83 83 83 83 8 3 82 82 82 82 82 0. 85 220 0. 85 128 0. 85 037 0. 84 946 0.84 855 0. 84 764 0. 84 673 0.84 583 0.84 492 0. 84 402 0. 84 312 6,84 223 0.84 1 33 0. 84 044 0. 83 954 0. 83 865 0. 83 776 0. 83 688 0. 83 599 0. 83 511 0. 83 423 0. 83 335 0. 83 247 0. 83 159 0. 83 072 0. 82 984 0. 82 897 0. 82 810 0. 82 723 0. 82 637 0. 82 550 0. 82 464 0. 82 378 0.82 292 0. 82 206 0. 82 120 0. 82 035 0. 81 949 0. 81 864 0. 81 779 0. 81 694 0. 81 609 0. 81 525 0. 81 440 0. 81 356 0. 81 272 0. 81 188 0. 81 104 0. 81 021 0. 80 937 0. 80 854 0. 80 771 0. 80 688 0. 80 605 0. 80 522 0. 80 439 0. 80 357 0. 80 275 0. 80 193 0. 80 111 0. 80 029 9.99 575 9.99 574 9.99 572 9. 99 570 9.99 568 9. 99 566 9.99 565 9.99 563 9. 99 561 9. 99 559 9. 99 557 9. 99 556 9. 99 554 9. 99 552 9. 99 550 9.99 548 9.99 546 9. 99 545 9. 99 543 9.99 541 9. 99 539 9. 99 537 9.99 535 9. 99 533 9.99 532 9.99 530 9.99 528 9.99 526 9. 99 524 9. 99 522 9.99 520 9. 99 518 9.99 51 7 9.99 51 5 9.99 513 9.99 511 9.99 509 9.99 507 9.99 505 9. 99 503 9. 99 501 9.99 499 9. 99 497 9. 99 495 9. 99 494 9. 99 492 9. 99 490 9. 99 488 9.99 486 9.99 484 9. 99 482 9. 99 480 9. 99 478 9. 99 476 9. 99 474 9.99 472 9. 99 470 9. 99 468 9. 99 466 9. 99 464 9.99 462 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 92 1 1.5 2 3.1 3 4. 6 4 6.1 5, 7. 7 6 9.2 7 10. 7 8 12. 3 9 13. 8 1 0 1.5.3 20 30. 7 30 46.0 40 61.3 50 76. 7 89 1 1.5 2 3. 0 3 4. 4 4 5.9 5 7.4 6 8.9 7 10.4 8 11. 9 9 13.4 1 0 14. 8 20 29. 7 30 44.5 40 59. 3 50 74.2 86 1 1.4 2 2. 9 3 4. 3 4 5. 7 5 7. 2 6 8. 6 7 10. 0 8 11. 5 9 12. 9 1 0 14. 3 20 28. 7 30 43. 0 40 57. 3 50 71.7 83 1 1.4 2 2. 8 3 4.2 4 5.5 5 6. 9 6 8. 3 7 9. 7 8 11.1 9 12. 4 1 0 13. 8 20 27. 7 30 41.5 40 55. 3 50 69. 2 91 1.5 3.0 4. 6 6. 1 7. 6 9. 1 10. 6 12. 1 13. 6 15. 2 30. 3 45.5 60. 7 75.8 88 1.5 2. 9 4.4 5. 9 7. 3 8. 8 10. 3 11.7 13.2 14. 7 29. 3 44.0 58. 7 73. 3 85 1.4 2. 8 4.2 5. 7 7. 1 8.5 9. 9 11. 3 12. 8 14. 2 28. 3 42. 5 56. 7 70. 8 82 1. 4 2. 7 4. 1 5.5 6.8 8.2 9. 6 10. 9 12. 3 13. 7 27. 3 41. 0 54. 7 68. 3 90 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13. 5 15.0 30. 0 45.0 60.0 75.0 87 1.4 2. 9 4.4 5. 8 7. 2 8. 7 10.2 11.6 13.0 14.5 29.0 43.5 58.0 72.5 84 1.4 2. 8 4. 2 5. 6 7. 0 8.4 9.8 11.2 12. 6 14. 0 28.0 42.0 56. 0 70.0 81 1. 4 2. 7 4. 0 5.4 6.8 8. 1 9.4 10. 8 12. 2 13. 5 27.0 40:5 54. 0 67.5 Ilog Cos I djI log cot I cd I log tanI log sin III Prop. Parts 1710 261 31 81 7 171" 2610 3510 8 10 70 TABLE III 90 99~ 189~ 279~ I ' log sin I d I log tan I c d I log cot I log cos T I Prop. Parts _~ ~ ~ ~~~~99 42 6 I I I 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.19 433 9.19513 9.19 592 9.19 672 9.19 751 9.19 830 9.19 909 9.19 988 9.20 067 9.20 145 9.20 223 9.20 302 9.20 380 9.20 458 9.20 535 9.20 613 9.20 691 9.20 768 9.20 845 9.20 922 9.20 999 9.21 076 9.21 153 9.21 229 9.21 306 9.21 382 9.21 458 9.21 534 9.21 610 9.21 685 9.21 761 9.21 836 9.21 912 9.21 987 9.22 062 9.22 137 9.22 211 9.22 286 9.22 361 9.22 435 9.22 509 9.22 583 9.22 657 9.22 731 9.22 805 9.22 878 9.22 952 9.23 025 9.23 098 9.23 171 9.23 244 9.23 317 9.23 390 9.23 462 9.23 535 9.23 607 9.23 679 9.23 752 9.23 823 9.23 895 9.23 967 80 79 80 79 79 79 79 79 78 78 79 78 78 77 78 78 77 77 77 77 77 77 76 77 76 76 76 76 75 76 75 76 75 75 75 74 75 75 74 74 74 74 74 74 73 74 73 73 73 73 73 73 72 73 72 72 73 71 72 72 9.19 971 9.20 053 9.20 134 9.20 216 9.20 297 9.20 378 9.20 459 9.20 540 9.20 621 9.20 701 9.20 782 9.20 862 9.20 942 9.21 022 9.21 102 9.21 182 9.21 261 9.21 341 9.21 420 9.21 499 9.21 578 9.21 657 9.21 736 9.21 814 9.21 893 9.21 971 9.22 049 9.22 127 9.22 205 9.22 283 9.22 361 9.22 438 9.22 516 9.22 593 9.22 670 9.22 747 9.22 824 9.22 901 9.22 977 9.23 054 9.23 130 9.23 206 9.23 283 9.23 359 9.23 435 9.23 510 9.23 586 9.23 661 9.23 737 9.23 812 9.23 887 9.23 962 9.24 037 9.24 112 9.24 186 9.24 261 9.24 335 9.24 410 9.24 484 9.24 558 9.24 632 82 81 82 81 81 81 81 81 80 81 80 80 80 80 80 79 80 79 79 79 79 79 78 79 78 78 78 78 78 78 77 78 77 77 77 77 77 76 77 76 76 77 76 76 75 76 75 76 75 75 75 75 75 74 75 74 75 74 74 74 0.80 029 0.79 947 0.79 866 0.79 784 0.79 703 0.79 622 0.79 541 0.79 460 0.79 379 0.79 299 0.79 218 0.79 138 0.79 058 0.78 978 0.78 898 0.78 818 0.78 739 0.78 659 0.78 580 0.78 501 0.78 422 0.78 343 0.78 264 0.78 186 0.78 107 0.78 029 0.77 951 0.77 873 0.77 795 0.77 717 0.77 639 0.77 562 0.77 484 0.77 407 0.77 330 0.77 253 0.77 176 0.77 099 0.77 023 0.76 946 0.76 870 0.76 794 0.76 717 0.76 641 0.76 565 0.76 490 0.76 414 0.76 339 0.76 263 0.76 188 0.76 113 0.76 038 0.75 963 0.75 888 0.75 814 0.75 739 0.75 665 0.75 590 0.75 516 0.75 442 0.75 368 9.99 462 9.99 460 9.99 458 9.99 456 9.99 454 9.99 452 9.99 450 9.99 448 9.99 446 9.99 444 9.99 442 9.99 440 9.99 438 9.99 436 9.99 434 9.99 432 9.99 429 9.99 427 9.99 425 9.99 423 9.99 421 9.99 419 9.99 417 9.99 415 9.99 413 9.99 411 9.99 409 9.99 407 9.99 404 9.99 402 9.99 400 9.99 398 9.99 396 9.99 394 9.99 392 9.99 390 9.99 388 9.99 385 9.99 383 9.99 381 9.99 379 9.99 377 9.99 375 9.99 372 9.99 370 9.99 368 9.99 366 9.99 364 9.99 362 9.99 359 9.99 357 9.99 355 9.99 353 9.99 351 9.99 348 9.99 346 9.99 344 9.99 342 9.99 340 9.99 337 *9.99 335 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 80 1.3 2.7 4.0 5.3 6.7 8.0 9.3 10.7 12.0 13.3 26.7 40.0 53.3 i 66.7 76 1.3 2.5 3.8 5.1 6.3 7.6 8.9 10.1 11.4 12.7 25.3 38.0 50.7 63.3 79 78 77 1.3 1.3 1.3 2.6 2.6 2.6 4.0 3.9 3.8 5.3 5.2 5.1 6.6 6.5 6.4 7.9 7.8 7.7 9.2 9.1 9.0 10.5 10.4 10.3 11.8 11.7 11.6 13.2 13.0 12.8 26.3 26.0 25.7 39.5 39.0 38.5 52.7 52.0 51.3 65.8 65.0 64.2 75 74 1.2 1.2 2.5 2.5 3.8 3.7 5.0 4.9 6.2 6.2 7.5 7.4 8.8 8.6 10.0 9.9 11.2 11.1 12.5 12.3 25.0 24.7 37.5 37.0 50.0 49.3 62.5 61.7 73 1.2 2.4 3.6 4.9 6.1 7.3 8.5 9.7 11.0 12.2 24.3 36.5 48.7 60.8 2 0.0 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.7 1.0 1.3 1.7 72 71 1.2 1.2 2.4 2.4 3.6 3.6 4.8 4.7 6.0 5.9 7.2 7.1 8.4 8.3 9.6 9.5 10.8 10.6 12.0 11.8 24.0 23.7 36.0 35.5 48.0 47.3 60.0 59.2 3 0.0 0.1 0.2 0.2 0.2 0.3 0.4 0.4 0.4 0.5 1.0 1.5 2.0 2.5 I I log cos I d I log cot Ic d c log tan I log sin | ' | Prop. Parts 170~ 260~ 350~ 80~ 71 TABLE III 10~ 100~ 190~ 280~ ' log sin d J 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.23 967 9.24 039 9.24 110 9.24 181 9.24 253 9.24 324 9.24 395 9.24 466 9.24 536 9.24 607 9.24 677 9.24 748 9.24 818 9.24 888 9.24 958 9.25 028 9.25 098 9.25 168 9.25 237 9.25 307 9.25 376 9.25 445 9.25 514 9.25 583 9.25 652 9.25 721 9.25 790 9.25 858 9.25 927 9.25 995 9.26 063 9.26 131 9.26 199 9.26 267 9.26 335 9.26 403 9.26 470 9.26 538 9.26 605 9.26 672 9.26 739 9.26 806 9.26 873 9.26 940 9.27 007 9.27 073 9.27 140 9.27 206 9.27 273 9.27 339 9.27 405 9.27 471 9.27 537 9.27 602 9.27 668 9.27 734 9.27 799 9.27 864 9.27 930 9.27 995 9.28 060 72 71 71 72 71 71 71 70 71 70 71 70 70 70 70 70 70 69 70 69 69 69 69 69 69 69 68 69 68 68 68 68 68 68 68 67 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 65 66 66 65 65 66 65 65 log tan c d 9.24 6327 9.24 706 9.24 779 9.24 853 9.24 926 9.25 0007 9.25 073 9.25 146 9.25 2197 9.25 292 3 9.25 365 72 9.25 43773 9.25 510 72 9.25 5827 9.25 655 72 9.25 727 72 9.25 799 72 9.25 871 72 9.25 943 72 9.26 015 71 9.26 086 72 9.26 158 71 9.26 229 72 9.26 301 71 9.26 372 71 9.26 443 71 9.26 514 71 9.26 585 70 9.26 655 71 9.26 726 71 9.26 797 70 9.26 867 70 9.26 937 71 9.27 008 70 9.27 078 70 9.27 148 70 9.27 218 70 9.27 288 69 9.27 357 70 9.27 427 69 9.27 496 70 9.27 566 69 9.27 635 69 9.27 704 69 9.27 773 69 9.27 842 69 9.27 911 69 9.27 980 69 9.28 049 68 9.28 117 69 9.28 186 68 9.28 254 69 9.28 323 68 9.28 391 68 9.28 459 68 9.28 527 68 9.28 595 67 9.28 662 68 9.28 730 68 9.28 798 67 9.28 865 0:75 368 0.75 294 0.75 221 0.75 147 0.75 074 0.75 000 0.74 927 0.74 854 0.74 781 0.74 708 0.74 635 0.74 563 0.74 490 0.74 418 0.74 345 0.74 273 0.74 201 0.74 129 0.74 057 0.73 985 0.73 914 0.73 842 0.73 771 0.73 699 0.73 628 0.73 557 0.73 486 0.73 415 0.73 345 0.73 274 0.73 203 0.73 133 0.73 063 0.72 992 0.72 922 0.72 852 0.72 782 0.72 712 0.72 643 0.72 573 0.72 504 0.72 434 0.72 365 0.72 296 0.72 227 0.72 158 0.72 089 0.72 020 0.71 951 0.71 883 0.71 814 0.71 746 0.71 677 0.71 609 0.71 541 0.71 473 0.71 405 0.71 338 0.71 270 0.71 202 0.71 135 9.99 335 9.99 333 9.99 331 9.99 328 9.99 326 9.99 324 9.99 322 9.99 319 9.99 317 9.99 315 9.99 313 9.99 310 9.99 308 9.99 306 9.99 304 9.99 301 9.99 299 9.99 297 9.99 294 9.99 292 9.99 290 9.99 288 9.99 285 9.99 283 9.99 281 9.99 278 9.99 276 9.99 274 9.99 271 9.99 269 9.99 267 9.99 264 9.99 262 9.99 260 9.99 257 9.99 255 9.99 252 9.99 250 9.99 248 9.99 245 9.99 243 9.99 241 9.99 238 9.99 236 9.99 233 9.99 231 9.99 229 9.99 226 9.99 224 9.99 221 9.99 219 9.99 217 9.99 214 9.99 212 9.99 209 9.99 207 9.99 204 9.99 202 9.99 200 9.99 197 9.99 195 2 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 2 2 3 2 3 2 2 3 2 3 2 3 2 2 3 2 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3. 2 1 O log cot I log cos d Prop. Parts 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 74 1.2 2.5 3.7 4.9 6.2 7.4 8.6 9.9 11.1 12.3 24.7 37.0 49.3 61.7 71 1.2 2.4 3.6 4.7 5.9 7.1 8.3 9.5 10.6 11.8 23.7 35.5 47.3 59.2 68 1.1 2.3 3.4 4.5 5.7 6.8 7.9 9.1 10.2 11.3 22.7 34.0 45.3 56.7 73 1.2 2.4 3.6 4.9 6.1 7.3 8.5 9.7 11.0 12.2 24.3 36.5 48.7 60.8 70 1.2 2.3 3.5 4.7 5.8 7.0 8.2 9.3 10.5 11.7 23.3 35.0 46.7 58.3 67 1.1 2.2 3.4 4.5 5.6 6.7 7.8 8.9 10.0 11.2 22.3 33.5 44.7 55.8 72 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 12.0 24.0 36.0 48.0 60.0 69 1.2 2.3 3.4 4.6 5.8 6.9 8.0 9.2 10.4 11.5 23.0 34.5 46.0 57.5 66 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11.0 22.0 33.0 44.0 55.0 'e I log cos I d log cot c d l,og tan I log sin I d ' Prop. Parts 169~ 259~ 349~ 79~ 72 TABLE III 11~ 101~ 191~ 281~ I log sin I d I log tan cd[ log cot I log cos I dI I Prop. Parts....~~~~ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.28 060 9.28 125 9.28 190 9.28 254 9.28 319 9.28 384 9.28 448 9.28 512 9.28 577 9.28 641 9.28 705 9.28 769 9.28 833 9.28 896 9.28 960 9.29 024 9.29 087 9.29 150 9.29 214 9.29 277 9.29 340 9.29 403 9.29 466 9.29 529 9.29 591 9.29 654 9.29 716 9.29 779 9.29 841 9.29 903 9.29 966 9.30 028 9.30 090 9.30 151 9.30 213 9.30 275 9.30 336 9.30 398 9.30 459 9.30 521 9.30 582 9.30 643 9.30 704 9.30 765 9.30 826 9.30 887 9.30 947 9.31 008 9.31 068 9.31 129 9.31 189 9.31 250 9.31 310 9.31 370 9.31 430 9.31 490 9.31 549 9.31 609 9.31 669 9.31 728 9.31 788 65 65 64 65 65 64 64 65 64 64 64 64 63 64 64 63 63 64 63 63 63 63 63 62 63 62 63 62 62 63 62 62 61 62 62 61 62 61 62 61 61 61 61 61 61 60 61 60 61 60 61 60 60 60 60 59 60 60 59 60 9.28 865 9.28 933 9.29 000 9.29 067 9.29 134 9.29 201 9.29 268 9.29 335 9.29 402 9.29 468 9.29 535 9.29 601 9.29 668 9.29 734 9.29 800 9.29 866 9.29 932 9.29 998 9.30 064 9.30 130 9.30 195 9.30 261 9.30 326 9.30 391 9.30 457 9.30 522 9.30 587 9.30 652 9.30 717 9.30 782 9.30 846 9.30 911 9.30 975 9.31 040 9.31 104 9.31 168 9.31 233 9.31 297 9.31 361 9.31 425 9.31 489 9.31 552 9.31 616 9.31 679 9.31 743 9.31 806 9.31 870 9.31 933 9.31 996 9.32 059 9.32 122 9.32 185 9.32 248 9.32 311 9.32 373 9.32 436 9.32 498 9.32 561 9.32 623 9.32 685 9.32 747 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 66 66 66 65 66 65 65 66 65 65 65 65 65 64 65 64 65 64 64 65 64 64 64 64 63 64 63 64 63 64 63 63 63 63 63 63 63 62 63 62 63 62 62 62 0.71 135 0.71 067 0.71 000 0.70 933 0.70 866 0.70 799 0.70 732 0.70 665 0.70 598 0.70 532 0.70 465 0.70 399 0.70 332 0.70 266 0.70 200 0.70 134 0.70 068 0.70 002 0.69 936 0.69 870 0.69 805 0.69 739 0.69 674 0.69 609 0.69 543 0.69 478 0.69 413 0.69 348 0.69 283 0.69 218 0.69 154 0.69 089 0.69 025 0.68 960 0.68 896 0.68 832 0.68 767 0.68 703 0.68 639 0.68 575 0.68 511 0.68 448 0.68 384 0.68 321 0.68 257 0.68 194 0.68 130 0.68 067 0.68 004 0.67 941 0.67 878 0.67 815 0.67 752 0.67 689 0.67 627 0.67 564 0.67 502 0.67 439 0.67 377 0.67 315 0.67 253 9.99 195 9.99 192 9.99 190 9.99 187 9.99 185 9.99 182 9.99 180 9.99 177 9.99 175 9.99 172 9.99 170 9.99 167 9.99 165 9.99 162 9.99 160 9.99 157 9.99 155 9.99 152 9.99 150 9.99 147 9.99 145 9.99 142 9.99 140 9.99 137 9.99 135 9.99 132 9.99 130 9.99 127 9.99 124 9.99 122 9.99 119 9.99 117 9.99 114 9.99 112 9.99 109 9.99 106 9.99 104 9.99 101 9.99 099 9.99 096 9.99 093 9.99 091 9.99 088 9.99 086 9.99 083 9.99 080 9.99 078 9.99 075 9.99 072 9.99 070 9.99 067 9.99 064 9.99 062 9.99 059 9.99 056 9.99 054 9.99 051 9.99 048 9.99 046 9.99 043 9.99 040 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 | 3 2 3 2 3 3 2 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 I 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 65 1.1 2.2 3.2 4.3 5.4 6.5 7.6 8.7 9.8 10.8 21.7 32.5 43.3 54.2 62 1.0 2.1 3.1 4. 1 5.2 6.2 7.2 8.3 9.3 10.3 20.7 31.0 41.3 51.7 59 1.0 2.0 3.0 3.9 4.9 5.9 6.9 7.9 8.8 9.8 19.7 29.5 39.3 49.2 64 63 1.1 1.0 2.1 2.1 3.2 3.2 4.3 4.2 5.3 5.2 6.4 6.3 7.5 7.4 8.5 8.4 9.6 9.4 10.7 10.5 21.3 21.0 32.0 31.5 42.7 42.0 53.3 52.5 61 60 1.0 1.0 2.0 2.0 3.0 3.0 4.1 4.0 5.1 5.0 6.1 6.0 7.1 7.0 8.1 8.0 9.2 9.0 10.2 10.0 20.3 20.0 30.5 30.0 40.7 40.0 50.8 50.0 3 2 0.0 0.0 0.1 0.1 0.2 0.1 0.2 0.1 0.2 0.2 0.3 0.2 0.4 0.2 0.4 0.3 0.4 0.3 0.5 0.3 1.0 0.7 1.5 1.0 2.0 1.3 2.5 1.7 5 4 3 2 1 0 I log cos I d j log cot c d log tan I log sin I d j ' I Prop. Parts 168~ 258~ 348~ 78" 73 TABLE III 12~ 102~ 192" 282~ I I 7 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18.19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 log sin 9.31 788 9.31 847 9.31 907 9.31 966 9.32 025 9.32 084 9.32 143 9.32 202 9.32 261 9.32 319 9.32 378 9.32 437 9.32 495 9.32 553 9.32 612 9.32 670 9.32 728 9.32 786 9.32 844 9.32 902 9.32 960 9.33 018 9.33 075 9.33 133 9.33 190 9.33 248 9.33 305 9.33 362 9.33 420 9.33 477 9.33 534 9.33 591 9.33 647 9.33 704 9.33 761 9.33 818 9.33 874 9.33 931 9.33 987 9.34 043 9.34 100 9.34 156 9.34 212 9.34 268 9.34 324 9.34 380 9.34 436 9.34 491 9.34 547 9.34 602 9.34 658 9.34 713 9.34 769 9.34 824 9.34 879 9.34 934 9.34 989 9.35 044 9.35 099 9.35 154 9.35 209 Cd 59 60 59 59 59 59 59 59 58 59 59 58 58 59 58 58 58 58 58 58 58 57 58 57 58 57 57 58 57 57 57 56 57 57 57 56 57 56 56 57 56 56 56 56 56 56 55 56 55 56 55 56 55 55 55 55 55 55 55 55 log tan 9.32 747 9.32 810 9.32 872 9.32 933 9.32 995 9.33 057 9.33 119 9.33 180 9.33 242 9.33 303 9.33 365 9.33 426 9.33 487 9.33 548 9.33 609 9.33 670 9.33 731 9.33 792 9.33 853 9.33 913 9.33 974 9.34 034 9.34 095 9.34 155 9.34 215 9.34 276 9.34 336 9.34 396 9.34 456 9.34 516 9.34 576 9.34 635 9.34 695 9.34 755 9.34 814 9.34 874 9.34 933 9.34 992 9.35 051 9.35 111 9.35 170 9.35 229 9.35 288 9.35 347 9.35 405 9.35 464 9.35 523 9.35 581 9.35 640 9.35 698 9.35 757 9.35 815 9.35 873 9.35 931 9.35 989 9.36 047 9.36 105 9.36 163 9.36 221 9.36 279 9.36 336 cd log cot Ilog cos I d Prop. Parts - ~ ~~~~ ~~ ---,, -- 63 0.67 253 62 0.67 190 60.67 128 6 0.67 067 62 0.67 005 62 0.66 943 60.66 881 62 0.66 820 61 0.66 758 62 0.66 697 61 0.66 635 6 0.66 574 6 0.66 513 60.66 452 6 0.66 391 61 0.66 330 1 0.66 269 61 0.66 208 60 0.66 147 60.66 087 60 0.66 026 1 0.65 966 60 0.65 905 60 0.65 845 61 0.65 785 60 0.65 724 60 0.65 664 60 0.65 604 60 0.65 544 60 0.65 484 59 0.65 424 60 0.65 365 60 0.65 305 59 0.65 245 60 0.65 186 59 0.65 126 59 0.65 067 59 0.65 008 60 0.64 949 59 0.64 889 59 0.64 830 59 0.64 771 59 0.64 712 58 0.64 653 59 0.64 595 59 0.64 536 58 0.64 477 59 0.64 419 58 0.64 360 59 0.64 302 58 0.64 243 58 0.64 185 58 0.64 127 58 0.64 069 58 0.64 011 58 0.63 953 58 0.63 895 58 0.63 837 58 0.63 779 57 0.63 721 0.63 664 9.99 040 9.99 038 9.99 035 9.99 032 9.99 030 9.99 027 9.99 024 9.99 022 9.99 019 9.99 016 9.99 013 9.99 011 9.99 008 9.99 005 9.99 002 9.99 000 9.98 997 9.98 994 9.98 991 9.98 989 9.98 986 9.98 983 9.9'8 980 9.98 978 9.98 975 9.98 972 9.98969 9.98 967 9.98 964 9.98 961 9.98 958 9.98 955 9.98 953 9.98 950 9.98 947 9.98 944 9.98 941 9.98 938 9.98 936 9.98 933 9.98 930 9.98 927 9.98 924 9.98 921 9.98 919 9.98 916 9.98 913 9.98 910 9.98 907 9.98 904 9.98 901 9.98 898 9.98 896 9.9'8 893 9.98 890 9.98 887 9.98 884 9.98 881 9.98 878 9.98 875 9.98 872 2 3 3 2 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 63 1.0 2.1 3.2 4.2 5.2 6.3 7.4 8.4 9.4 10.5 21.0 31.5 42.0 52.5 60 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 57 1.0 1.9 2.8 3.8 4.8 5.7 6.6 7.6 8.6 9.5 19.0 28.5 38.0 47.5 62 1.0 2.1 3.1 4.1 5.2 6.2 7.2 8.3 9.3 10.3 20.7 31.0 41.3 51.7 59 1.0 2.0 3.0 3.9 4.9 5.9 6.9 7.9 8.8 9.8 19.7 29.5 39.3 49.2 56 0.9 1.9 2.8 3.7 4.7 5.6 6.5 7.5 8.4 9.3 18.7 28.0 37.3 46.7 61 1.0 2.0 3.0 4.1 5.1 6.1 7.1 8.1 9.2 10.2 20.3 30.5 40.7 50.8 58 1.0 1.9 2.9 3.9 4.8 5.8 6.8 7.7 8.7 9.7 19.34 29.0 38.7 48.3 55 0.9 1.8 2.8 3.7 4.6 5.5 6.4 7.3 8.2 9.2 18.3 27.5 36.7 45.8 6 5 4 2 2 1 0 _,. I _ logcos d log cot c d log tan I log sin d | ' I Prop. Parts 167~ 257~ 347~ 77~ 74 TABLE III 13~ 1030 1930 283~ Prop. Parts ' I log sin d | log tan Icd log cot log cos d 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.35 209 9.35 263 9.35 318 9.35 373 9.35 427 9.35 481 9.35 536 9.35 590 9.35 644 9.35 698 9.35 752 9.35 806 9.35 860 9.35 914 9.35 968 9.36 022 9.36 075 9.36 129 9.36 182 9.36 236 9.36 289 9.36 342 9.36 395 9.36 449 9.36 502 9.36 555 9.36 608 9.36 660 9.36 713 9.36 766 9.36 819 9.36 871 9.36 924 9.36 976 9.37 028 9.37 081 9.37 133 9.37 185 9.37 237 9.37 289 9.37 341 9.37 393 9.37 445 9.37 497 9.37 549 9.37 600 9.37 652 9.37 703 9.37 755 9.37 806 9.37 858 9.37 909 9.37 960 9.38 011 9.38 062 9.38 113 9.38 164 9.38 215 9.38 266 9.38 317 9.38 368 log cos 54 55 55 54 54 55 54 54 54 54 54 54 54 54 54 53 54 53 54 53 53 53 54 53 53 53 52 53 53 53 52 53 52 52 53 52 52 52 52 52 52 52 52 52 51 52 51 52 51 52 51 51 51 51 51 51 51 51 51 51 9.36 336 9.36 394 9.36 452 9.36 509 9.36 566 9.36 624 9.36 681 9.36 738 9.36 795 9.36 852 9.36 909 9.36 966 9.37 023 9.37 080 9.37 137 9.37 193 9.37 250 9.37 306 9.37 363 9.37 419 9.37 476 9.37 532 9.37 588 9.37 644 9.37 700 9.37 756 9.37 812 9.37 868 9.37 924 9.37 980 9.38 035 9.38 091 9.38 147 9.38 202 9.38 257 9.38 313 9.38 368 9.38 423 9.38 479 9.38 534 9.38 589 9.38 644 9.38 699 9.38 754 9.38 808 9.38 863 9.38 918 9.38 972 9.39 027 9.39 082 9.39 136 9.39 190 9.39 245 9.39 299 9.39 353 9.39 407 9.39 461 9.39 515 9.39 569 9.39 623 9.39 677 58 58 57 57 58 57 57 57 57 57 57 57 57 57 56 57 56 57 56 57 56 56 56 56 56 56 56 56 56 55 56 56 55 55 56 55 55 56 55 55 55 55 55 54 55 55 54 55 55 54 54 55 54 54 54 54 54 54 54 54 0.63 664 0.63 606 0.63 548 0.63 491 0.63 434 0.63 376 0.63 319 0.63 262 0.63 205 0.63 148 0.63 091 0.63 034 0.62 977 0.62 920 0.62 863 0.62 807 0.62 750 0.62 694 0.62 637 0.62 581 0.62 524 0.62 468 0.62 412 0.62 356 0.62 300 0.62 244 0.62 188 0.62 132 0.62 076 0.62 020 0.61 965 0.61 909 0.61 853 0.61 798 0.61 743 0.61 687 0.61 632 0.61 577 0.61 521 0.61 466 0.61 411 0.61 356 0.61 301 0.61 246 0.61 192 0.61 137 0.61 082 0.61 028 0.60 973 0.60 918 0.60 864 0.60 810 0.60 755 0.60 701 0.60 647 0.60 593 0.60 539 0.60 485 0.60 431 0.60 377 0.60 323 9.98 872 9.98 869 9.98 867 9.98 864 9.98 861 9.98 858 9.98 855 9.98 852 9.98 849 9.98 846 9.98 843 9.98 840 9.98 837 9.98 834 9.98 831 9.98 828 9.98 825 9.98 822 9.98 819 9.98 816 9.98 813 9.98 810 9.98 807 9.98 804 9.98 801 9.98 798 9.98 795 9.98 792 9.98 789 9.98 786 9.98 783 9.98 780 9.98 777 9.98 774 9.98 771 9.98 768 9.98 765 9.98 762 9.98 759 9.98 756 9.98 753 9.98 750 9.98 746 9.98 743 9.98 740 9.98 737 9.98 734 9.98 731 9.98 728 9.98 725 9.98 722 9.98 719 9.98 715 9.98 712 9.98 709 9.98 706 9.98 703 9.98 700 9.98 697 9.98 694 9.98 690 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 4 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 2 3 4 5 6 7 8 9 10 20 30 40 50 2 3 4 5 6 7 8 9 10 20 30 40 50 57 1.0 1.9 2.8 3.8 4.8 5.7 6.6 7.6 8.6 9.5 19.0 28.5 38.0 47.5 54 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 18.0 27.0 36.0 45.0 56 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 18. 28. 37. 46. 0. 1. 2. 3. 4. 5. 6. 7. 8. 8. 17. 26. 35. 44. 3 55 9 0.9 9 1.8 8 2.8 7 3.7 7 4.6 6 5.5 5 6.4 5 7.3 4 8.2 3 9.2 7 18.3 0 27.5 3 36.7 7 45.8 1 52 9 0.9 8 1.7 6 2.6 5 3.5 4 4.3 3 5.2 2 6.1 1 6.9 0 7.8 8 8.7 7 17.3 5 26.0 3 34.7 2 43.3 3 2 0.0 0.0 0.1 0.1 0.2 0.1 0.2 0.1 0.2 0.2 0.3 0.2 0.4 0.2 0.4 0.3 0.4 0.3 0.5 0.3 1.0 0.7 1.5 1.0 2.0 1.3 2.5 1.7 51 4 0.8 0.1 1.7 0.1 2.6 0.2 * 3.4 0.3 4.2 0.3 5.1 0.4 6.0 0.5 6.8 0.5 7.6 0.6 8.5 0.7 17.0 1.3 25.5 2.0 34.0 2.7 42.5 3.3 I d I log cot cd log tan I log sin d ' - Prop. Parts 166~ 256~ 346~ 76~ 75 TABLE III 14~ 104~ 194~ 284~ I log sin d log tan cd log cot log os d( I Prop. Parts 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.38 368 9.38 418 9.38 469 9.38 519 9.38 570 9.38 620 9.38 670 9.38 721 9.38 771 9.38 821 9.38 871 9.38 921 9.38 971 9.39 021 9.39 071 9.39 121 9.39 170 9.39 220 9.39 270 9.39 319 9.39 369 9.39 418 9.39 467 9.39 517 9.39 566 9.39 615 9.39 664 9.39 713 9.39 762 9.39 811 9.39 860 9.39 909 9.39 958 9.40 006 9.40 055 9.40 103 9.40 152 9.40 200 9.40 249 9.40 297 9.40 346 9.40 394 9.40 442 9.40 490 9.40 538 9.40 586 9.40 634 9.40 682 9.40 730 9.40 778 9.40 825 9.40 873 9.40 921 9.40 968 9.41 016 9.41 063 9.41 111 9.41 158 9.41 205 9.41 252 9.41 300 log cos l 50 51 50 51 50 50 51 50 50 50 50 50 50 50 50 49 50 50 49 50 49 49 50 49 49 49 49 49 49 49 49 49 48 49 48 49 48 49 48 49 48 48 48 48 48 48 48 48 48 47 48 48 47 48 47 48 47 47 47 48 9.39 677 9.39 731 9.39 785 9.39 838 9.39 892 9.39 945 9.39 999 9.40 052 9.40 106 9.40 159 9.40 212 9.40 266 9.40 319 9.40 372 9.40 425 9.40 478 9.40 531 9.40 584 9.40 636 9.40 689 9.40 742 9.40 795 9.40 847 9.40 900 9.40 952 9.41 005 9.41 057 9.41 109 9.41 161 9.41 214 9.41 266 9.41 318 9.41 370 9.41 422 9.41 474 9.41 526 9.41 578 9.41 629 9.41 681 9.41 733 9.41 784 9.41 836 9.41 887 9.41 939 9.41 990 9.42 041 9.42 093 9.42 144 9.42 195 9.42 246 9.42 297 9.42 348 9.42 399 9.42 450 9.42 501 9.42 552 9.42 603 9.42 653 9.42 704 9.42 755 9.42 805 54 54 53 54 53 54 53 54 53 53 54 53 53 53 53 53 53 52 53 53 53 52 53 52 53 52 52 52 53 52 52 52 52 52 52 52 51 52 52 51 52 51 52 51 51 52 51 51 51 51 51 51 51 51 51 51 50 51 51 50 0.60 323 0.60 269 0.60 215 0.60 162 0.60 108 0.60 055 0.60 001 0.59 948 0.59 894 0.59 841 0.59 788 0.59 734 0.59 681 0.59 628 0.59 575 0.59 522 0.59 469 0.59 416 0.59 364 0.59 311 0.59 258 0.59 205 0.59 153 0.59 100 0.59 048 0.58 995 0.58 943 0.58 891 0.58 839 0.58 786 0.58 734 0.58 682 0.58 630 0.58 578 0.58 526 0.58 474 0.58 422 0.58 371 0.58 319 0.58 267 0.58 216 0.58 164 0.58 113 0.58 061 0.58 010 0.57 959 0.57 907 0.57 856 0.57 805 0.57 754 0.57 703 0.57 652 0.57 601 0.57 550 0.57 499 0.57 448 0.57 397 0.57 347 0.57 296 0.57 245 0.57 195 9.98 690 9.98 687 9.98 684 9.98 681 9.98 678 9.98 675 9.98 671 9.98 668 9.98 665 9.98 662 9.98 659 9.98 656 9.98 652 9.98 649 9.98 646 9.98 643 9.98 640 9.98 636 9.98 633 9.98 630 9.98 627 9.98 623 9.98 620 9.98 617 9.98 614 9.98 610 9.98 607 9.98 604 9.98 601 9.98 597 9.98 594 9.98 591 9.98 588 9.98 584 9.98 581 9.98 578 9.98 574 9.98 571 9.98 568 9.98 565 9.98 561 9.98 558 9.98 555 9.98 551 9.98 548 9.98 545 9.98 541 9.98 538 9.98 535 9.98 531 9.98 528 9.98 525 9.98 521 9.98 518 9.98 515 9.98 511 9.98 508 9.98 505 9.98 501 9.98 498 9.98 494 3 3 3 3 3 4 3 3 3 3 3 4 3 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 4 3 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 4 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 6 5 4 3 2 1 0 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 2 3 4 5 6 7 8 9 10 20 30 40 50 54 53 0.9 0.9 1.8 1.8 2.7 2.6 3.6 3.5 4.5 4.4 5.4 5.3 6.3 6.2 7.2 7.1 8.1 8.0 9.0 8.8 18.0 17.7 27.0 26.5 36.0 35.3 45.0 44.2 51 50 0.8 0.8 1.7 1.7 2.6 2.5 3.4 3.3 4.2 4.2 5.1 5.0 6.0 5.8 6.8 6.7 7.6 7.5 8.5 8.3 17.0 16.7 25.5 25.0 34.0 33.3 42.5 41.7 48 47 0.8 0.8 1.6 1.6 ( 2.4 2.4 ( 3.2 3.1 ( 4.0 3.9 ( 4.8 4.7 ( 5.6 5.5 ( 6.4 6.3 ( 7.2 7.0 8.0 7.8 ( 16.0 15.7 1 24.0 23.5 32.0 31.3 40.0 39.2 2 52 0.9 1.7 2.6 3.5 4.3 5.2 6.1 6.9 7.8 8.7 17.3 26.0 34.7 43.3 49 0.8 1.6 2.4 3.3 4.1 4.9 5.7 6.5 7.4 8.2 16.3 24.5 32.7 40.8 4 3 ).1 0.0 ).1 0.1 ).2 0.2 ).3 0.2 ).3 0.2 ).4 0.3 ).5 0.4 ).5 0.4 ).6 0.4 ).7 0.5 1.3 1.0 Z.0 1.5 Z.7 2.0 3.3 2.5 _I I! I! d I log cot c d log tan log sin d ' Prop. Parts I~~~~~~~~~ I 165~ 255~ 345~ 75~ 76 TABLE III 15~ 105~ 195~ 285~ - ' I log sin Id log tan | c d | log cot I log cos I d i I Prop. Parts 1 I I 0 1 2 3 I 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.41 300 9.41 347 9.41 394 9.41 441 9.41 488 9.41 535 9.41 582 9.41 628 9.41 675 9.41 722 9.41 768 9.41 815 9.41 861 9.41 908 9.41 954 9.42 001 9.42 047 9.42 093 9.42 140 9.42 186 9.42 232 9.42 278 9.42 324 9.42 370 9.42 416 9.42 461 9.42 507 9.42 553 9.42 599 9.42 644 9.42 690 9.42 735 9.42 781 9.42 826 9.42 872 9.42 917 9.42 962 9.43 008 9.43 053 9.43 098 9.43 143 9.43 188 9.43 233 9.43 278 9.43 323 9.43 367 9.43 412 9.43 457 9.43 502 9.43 546 9.43 591 9.43 635 9.43 680 9.43 724 9.43 769 9.43 813 9.43 857 9.43 901 9.43 946 9.43 990 9.44 034 79.42 805 9.42 856 79.42 906 479.42 957 9.43 007 479.43 057 9.43 108 49.43 158 9.43 208 469.43 258 479.43 308 46 9.43 358 79.43 408 46 9.43 458 47 9.43 508 469.43 558 469.43 607 79.43 657 46 9.43 707 46 9.43 756 46 9.43 806 469.43 855 46 9.43 905 46 9.43 954 45 9.44 004 46 9.44 053 46 9.44 102 46 9.44 151 45 9.44 201 46 9.44 250 45 9.44 299 46 9.44 348 45 9.44 397 46 9.44 446 45 A9.44 495 45 9.44 544 46 9.44 592 45 9.44 641 45 9.44 690 45 9.44 738 45 9.44 787 45 9.44 836 45 9.44 884 45 9.44 933 44 9.44 981 45 9.45 029 45 9.45 078 5 9.45 126 44 9.45 174 45 9.45 222 44 9.45 271 45 9.45 319 44 9.45 367 45 9.45 415 44 9.45 463 44 9.45 511 44 9.45 559 45 9.45 606 44 9.45 654 44 9.45 702 9.45 750 51 50 51 50 50 51 50 50 50 50 50 50 50 50 50 49 50 50 49 50 49 50 49 50 49 49 49 50 49 49 49 49 49 49 49 48 49 49 48 49 49 48 49 48 48 49 48 48 48 49 48 48 48 48 48 48 47 48 48 48 0.57 195 0.57 144 0.57 094 0.57 043 0.56 993 0.56 943 0.56 892 0.56 842 0.56 792 0.56 742 0.56 692 0.56 642 0.56 592 0.56 542 0.56 492 0.56 442 0.56 393 0.56 343 0.56 293 0.56 244 0.56 194 0.56 145 0.56 095 0.56 046 0.55 996 0.55 947 0.55 898 0.55 849 0.55 799 0.55 750 0.55 701 0.55 652 0.55 603 0.55 554 0.55 505 0.55 456 0.55 408 0.55 359 0.55 310 0.55 262 0.55 213 0.55 164 0.55 116 0.55 067 0.55 019 0.54 971 0.54 922 0.54 874 0.54 826 0.54 778 0.54 729 0.54 681 0.54 633 0.54 585 0.54 537 0.54 489 0.54 441 0.54 394 0.54 346 0.54 298 0.54 250 9.98 494 9.98 491 9.98 488 9.98 484 9.98 481 9.98 477 9.98 474 9.98 471 9.98 467 9.98 464 9.98 460 9.98 457 9.98 453 9.98 450 9.98 447 9.98 443 9.98 440 9.98 436 9.98 433 9.98 429 9.98 426 9.98 422 9.98 419 9.98 415 9.98 412 9.98 409 9.98 405 9.98 402 9.98 398 9.98 395 9.98 391 9.98 388 9.98 384 9.98 381 9.98 377 9.98 373 9.98 370 9.98 366 9.98 363 9.98 359 9.98 356 9.98 352 9.98 349 9.98 345 9.98 342 9.98 338 9.98 334 9.98 331 9.98 327 9.98 324 9.98 320 9.98 317 9.98 313 9.98 309 9.98 306 9.98 302 9.98 299 9.98 295 9.98 291 9.98 288 9.98 284.. g, 3 3 4 3 4 3 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 4 3 4 3 4 3 4 3 4 3 4 4 3 4 3 4 3 4 4 3 4 3 4 4 3 4 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 51 0.8 1.7 2.6 3.4 4.2 5.1 6.0 6.8 7.6 8.5 17.0 25.5 34.0 42.5 48 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 16.0 24.0 32.0 40.0 50 0.8 1.7 2.5 3.3 4.2 5.0 5.8 6.7 7.5 8.3 16.7 25.0 33.3 41.7 47 0.8 1.6 2.4 3.1 3.9 4.7 5.5 6.3 7.0 7.8 15.7 23.5 31.3 39.2 49 0.8 1.6 2.4 3.3 4.1 4.9 5.7 6.5 7.4 8.2 16.3 24.5 32.7 40.8 46 0.8 1.5 2.3 3.1 3.8 4.6 5.4 6.1 6.9 7.7 15.3 23.0 30.7 38.3 45 44 4 3 0.8 0.7 0.1 0.0 1.5 1.5 0.1 0.1 2.2 2.2 0.2 0.2 3.0 2.9 0.3 0.2 3.8 3.7 0.3 0.2 4.5 4.4 0.4 0.3 5.2 5.1 0.5 0.4 6.0 5.9 0.5 0.4 6.8 6.6 0.6 0.4 7.5 7.3 0.7 0.5 15.0 14.7 1.3 1.0 22.5 22.0 2.0 1.5 30.0 29.3 2.7 2.0 37.5 36.7 3.3 2.5 I I _,: - ---- - - i I --- L I log cos d I log cot c d log tanI ogsin i d I I _ Prop. Parts 164~ 254~ 344~ 74~ 77 TABLE III 16~ 106~ 196~ 286~ log sin d log tan cd log cot log cos I d I I Prop. Parts. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.44 03444 9.44 07844 9.44 12244 9.44 16644 9.44 210 4 9.44 253 44 9.44 29744 9.44341 4 9.44 385 4 9.44 428 4 9.44 47244 9.44 516 4 9.44 559 43 9.44 602 4 9.44 64643 9.44 6894 9.44 733 4 9.44 776 9.44 819 3 9.44 8624 43 9.44 905 43 9.44 94844 9.44 992 43 9.45 035 42 9.45 07743 9.45 12043 9.45 16343 9.45 20643 9.45 24943 9.45 292 42 9.45 334 43 9.45 377 42 9.45 419 43 9.45 462 42 9.45 504 43 9.45 547 42 9.45 589 43 9.45 632 42 9.45 674 42 9.45 716 42 9.45 758 43 9.45 801 42 9.45 843 42 9.45 885 42 9.45 927 42 9.45 969 42 9.46 011 42 9.46 053 42 9.46 095 41 9.46 136 42 9.46 178 42 9.46 220 42 9.46 262 41 9.46 303 42 9.46 345 41 9.46 386 42 9.46 428 41 9.46 469 42 9.46511 41 9.46 552 42 9.46 594 logcos d I 9.45 750 9.45 797 9.45 845 9.45 892 9.45 940 9.45 987 9.46 035 9.46 082 9.46 130 9.46 177 9.46 224 9.46 271 9.46 319 9.46 366 9.46 413 9.46 460 9.46 507 9.46 554 9.46 601 9.46 648 9.46 694 9.46 741 9.46 788 9.46 835 9.46 881 9.46 928 9.46 975 9.47 021 9.47 068 9.47 114 9.47 160 9.47 207 9.47 253 9.47 299 9.47 346 9.47 392 9:47 438 9.47 484 9.47 530 9.47 576 9.47 622 9.47 668 9.47 714 9.47 760 9.47 806 9.47 852 9.47 897 9.47 943 9.47 989 9.48 035 9.48 080 9.48 126 9.48 171 9.48 217 9.48 262 9.48 307 9.48 353 9.48 398 9.48 443 9.48 489 9.48 534 47 48 47 48 47 48 47 48 47 47 47 48 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 46 46 47 46 46 47 46 46 46 46 46 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 45 46 45 45 46 45 I 0.54 250 0.54 203 0.54 155 0.54 108 0.54 060 0.54 013 0.53 965 0.53 918 0.53 870 0.53 823 0.53 776 0.53 729 0.53 681 0.53 634 0.53 587 0.53 540 0.53 493 0.53 446 0.53 399 0.53 352 0.53 306 0.53 259 0.53 212 0.53 165 0.53 119 0.53 072 0.53 025 0.52 979 0.52 932 0.52 886 0.52 840 0.52 793 0.52 747 0.52 701 0.52 654 0.52 608 0.52 562 0.52 516 0.52 470 0.52 424 0.52 378 0.52 332 0.52 286 0.52 240 0.52 194 0.52 148 0.52 103 0.52 057 0.52 011 0.51 965 0.51 920 0.51 874 0.51 829 0.51 783 0.51 738 0.51 693 0.51 647 0.51 602 0.51 557 0.51 511 0.51 466 9.98 284 9.98 281 9.98 277 9.98 273 9.98 270 9.98 266 9.98 262 9.98 259 9.98 255 9.98 251 9.98 248 9.98 244 9.98 240 9.98 237 9.98 233 9.98 229 9.98 226 9.98 222 9.98 218 9.98 215 9.98 211 9.98 207 9.98 204 9.98 200 9.98 196 9.98 192 9.98 189 9.98 185 9.98 181 9.98 177 9.98 174 9.98 170 9.98 166 9.98 162 9.98 159 9.98 155 9.98 151 9.98 147 9.98 144 9.98 140 9.98 136 9.98 132 9.98 129 9.98 125 9.98 121 9.98 117 9.98 113 9.98 110 9.98 106 9.98 102 9.98 098 9.98 094 9.98 090 9.98 087 9.98 083 9.98 079 9.98 075 9.98 071 9.98 067 9.98 063 9.98 060 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 4 3 4 4 4 4 4 3 4 4 4 4 4 4 3 4~ I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 48 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 16.0 24.0 32.0 40.0 45 0.8 1.5 2.2 3.0 3.8 4.5 5.2 6.0 6.8 7.5 15.0 22.5 30.0 37.5 42 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 14.0 21.0 28.0 35.0 47 46 0.8 0.8 1.6 1.5 2.4 2.3 3.1 3.1 3.9 3.8 4.7 4.6 5.5 5.4 6.3 6.1 7.0 6.9 7.8 7.7 15.7 15.3 23.5 23.0 31.3 30.7 39.2 38.3 44 43 0.7 0.7 1.5 1.4 2.2 2.2 2.9 2.9 3.7 3.6 4.4 4.3 5.1 5.0 5.9 5.7 6.6 6.4 7.3 7.2 14.7 14.3 22.0 21.5 29.3 28.7 36.7 35.8 41 4 3 0.70.1 0.0 1.4 1.1 0.1 2.0 0.2 0.2 2.7 0.3 0.2 3.4 0.3 0.2 4.1 0.4 0.3 4.8 0.5 0.4 5.5 0.5 0.4 6.2 0.6 0.4 6.8 0.7 0.5 13.7 1.3 1.0 20.5 2.0 1.5 27.3 2.7 2.0 34.2 3.3 2.5 1 2 3 4 5 6 7 8 9 10 20 30 40 50 7 6 5 4 3 2 1 0 I log cot c d log tan I log sin I d | ' j Prop. Parts 163~ 253~ 343~ 73~ 78 TABLE III 17~ 107~ 197~ 287~ ' I log sin d logtan I cd logcot I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.46 594 9.46 635 9.46 676 9.46 717 9.46 758 9.46 800 9.46 841 9.46 882 9.46 923 9.46 964 9.47 005 9.47 045 9.47 086 9.47 127 9.47 168 9.47 209 9.47 249 9.47 290 9.47 330 9.47 371 9.47 411 9.47 452 9.47 492 9.47 533 9.47 573 9.47 613 9.47 654 9.47 694 9.47 734 9.47 774 9.47 814 9.47 854 9.47 894 9.47 934 9.47 974 9.48 014 9.48 054 9.48 094 9.48 133 9.48 173 9.48 213 9.48 252 9.48 292 9.48 332 9.48 371 9.48 411 9.48 450 9.48 490 9.48 529 9.48 568 9.48 607 9.48 647 9.48 686 9.48 725 9.48 764 9.48 803 9.48 842 9.48 881 9.48 920 9.48 959 9.48 998 41 41 41 41 42 41 41 41 41 41 40 41 41 41 41 40 41 40 41 40 41 40 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 39 40 40 39 40 40 39 40 39 40 39 39 39 40 39 39 39 39 39 39 39 39 39 9.48 534 9.48 579 9.48 624 9.48 669 9.48 714 9.48 759 9.48 804 9.48 849 9.48 894 9.48 939 9.48 984 9.49 029 9.49 073 9.49 118 9.49 163 9.49 207 9.49 252 9.49 296 9.49 341 9.49 385 9.49 430 9.49 474 9.49 519 9.49 563 9.49 607 9.49 652 9.49 696 9.49 740 9.49 784 9.49 828 9.49 872 9.49 916 9.49 960 9.50 004 9.50 048 9.50 092 9.50 136 9.50 180 9.50 223 9.50 267 9.50 311 9.50 355 9.50 398 9.50 442 9.50 485 9.50 529 9.50 572 9.50 616 9.50 659 9.50 703 9.50 746 9.50 789 9.50 833 9.50 876 9.50 919 9.50 962 9.51 005 9.51 048 9.51 092 9.51 135 9.51 178 45 45 45 45 45 45 45 45 45 45 45 44 45 45 44 45 44 45 44 45 44 45 44 44 45 44 44 44 44 44 44 44 44 44 44 44 44 43 44 44 44 43 44 43 44 43 44 43 44 43 43 44 43 43 43 43 43 44 43 43 0.51 466 0.51 421 0.51 376 0.51 331 0.51 286 0.51 241 0.51 196 0.51 151 0.51 106 0.51 061 0.51 016 0.50 971 0.50 927 0.50 882 0.50 837 0.50 793 0.50 748 0.50 704 0.50 659 0.50 615 0.50 570 0.50 526 0.50 481 0.50 437 0.50 393 0.50 348 0.50 304 0.50 260 0.50 216 0.50 172 0.50 128 0.50 084 0.50 040 0.49 996 0.49 952 0.49 908 0.49 864 0.49 820 0.49 777 0.49 733 0.49 689 0.49 645 0.49 602 0.49 558 0.49 515 0.49 471 0.49 428 0.49 384 0.49 341 0.49 297 0.49 254 0.49 211 0.49 167 0.49 124 0.49 081 0.49 038 0.48 995 0.48 952 0.48 908 0.48 865 0.48 822 log cos 9.98 060 9.98 056 9.98 052 9.98 048 9.98 044 9.98 040 9.98 036 9.98 032 9.98 029 9.98 025 9.98 021 9.98 017 9.98 013 9.98 009 9.98 005 9.98 001 9.97 997 9.97 993 9.97 989 9.97 986 9.97 982 9.97 978 9.97 974 9.97 970 9.97 966 9. 7 962 9.97 958 9.97 954 9.97 950 9.97 946 9.97 942 9.97 938 9.97 934 9.97 930 9.97 926 9.97 922 9.97 918 9.97 914 9.97 910 9.97 906 9.97 902 9.97 898 9.97 894 9.97 890 9.97 886 9.97 882 9.97 878 9.97 874 9.97 870 9.97 866 9.97 861 9.97 857 9.97 853 9.97 849 9.97 845 9.97 841 9.97 837 9.97 833 9.97 829 9.97 825 9.97 821 d I|I Prop. Parts 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 4 4 4 4 4 4 4 4 4 4 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 45 0.8 1.5 2.2 3.0 3.8 4.5 5.2 6.0 6.8 7.5 15.0 22.5 30.0 37.5 42 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 14.0 21.0 28.0 35.0 39 0.6 1.3 2.0 2.6 3.2 3.9 4.6 5.2 5.8 6.5 13.0 19.5 26.0 32.5 I I I I I 44 0.7 1.5 2.2 2.9 3.7 4.4 5-1 5.9 6.6 7.3 14.7 22.0 29.3 36.7 41 40 0.7 0.7 1.4 1.3 2.0 2.0 2.7 2.7 3.4 3.3 4.1 4.0 4.8 4.7 5.5 5.3 6.2 6.0 6.8 6.7 13.7 13.3 20.5 20.0 27.3 26.7 34.2 33.3 5 4 3 0.1 0.1 0.0 0.2 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.2 0.4 0.3 0.2 0.5 0.4 0.3 0.6 0.5 0.4 0.7 0.5 0.4 0.8 0.6 0.4 0.8 0.7 0.5 1.7 1.3 1.0 2.5 2.0 1.5 3.3 2.7 2.0 4.2 3.3 2.5 43 0.7 1.4 2.2 2.9 3.6 4.3 5.0 5.7 6.4 7.2 14.3 21.5 28.7 35.8 _ _ I log cos I d log cot c dl log tan log sin d Prop. Parts 162~ 252~ 342~ 72~ 79 TABLE III 18~ 108~ 198~ 288~ I _ II I _Il t _ I, di I 1-r t r 1 ( I 1- I I I_ rr PartcI SII,- " - I v I. I -- I "b - I ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.48 998 9.49 037 9.49 076 9.49 115 9.49 153 9.49 192 9.49 231 9.49 269 9.49 308 9.49 347 9.49 385 9.49 424 9.49 462 9.49 500 9.49 539 9.49 577 9.49 615 9.49 654 9.49 692 9.49 730 9.49 768 9.49 806 9.49 844 9.49 882 9.49 920 9.49 958 9.49 996 9.50 034 9.50 072 9.50 110 9.50 148 9.50 185 9.50 223 9.50 261 9.50 298 9.50 336 9.50 374 9.50 411 9.50 449 9.50 486 9.50 523 9.50 561 9.50 598 9.50 635 9.50 673 9.50 710 9.50 747 9.50 784 9.50 821 9.50 858 9.50 896 9.50 933 9.50 970 9.51 007 9.51 043 9.51 080 9.51 117 9.51 154 9.51 191 9.51 227 9.51 264 39 39 39 38 39 39 38 39 39 38 39 38 38 39 38 38 39 38 38 38 38 38 38 38 38 38 38 38 38 38 37 38 38 37 38 38 37 38 37 37 38 37 37 38 37 37 37 37 37 38 37 37 37 36 37 37 37 37 36 37 9.51 178 9.51 221 9.51 264 9.51 306 9.51 349 9.51 392 9.51 435 9.51 478 9.51 520 9.51 563 9.51 606 9.51 648 9.51 691 9.51 734 9.51 776 9.51 819 9.51 861 9.51 903 9.51 946 9.51 988 9.52 031 9.52 073 9.52 115 9.52 157 9.52 200 9.52 242 9.52 284 9.52 326 9.52 368 9.52 410 9.52 452 9.52 494 9.52 536 9.52 578 9.52 620 9.52 661 9.52 703 9.52 745 9.52 787 9.52 829 9.52 870 9.52 912 9.52 953 9.52 995 9.53 037 9.53 078 9.53 120 9.53 161 9.53 202 9.53 244 9.53 285 9.53 327 9.53 368 9.53 409 9.53 450 9.53 492 9.53 533 9.53 574 9.53 615 9.53 656 9.53 697 43 43 42 43 43 43 43 42 43 43 42 43 43 42 43 42 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 42 42 42 41 42 42 42 42 41 42 41 42 42 41 42 41 41 42 41 42 41 41 41 42 41 41 41 41 41 0.48 822 0.48 779 0.48 736 0.48 694 0.48 651 0.48 608 0.48 565 0.48 522 0.48 480 0.48 437 0.48 394 0.48 352 0.48 309 0.48 266 0.48 224 0.48 181 0.48 139 0.48 097 0.48 054 0.48 012 0.47 969 0.47 927 0.47 885 0.47 843 0.47 800 0.47 758 0.47 716 0.47 674 0.47 632 0.47 590 0.47 548 0.47 506 0.47 464 0.47 422 0.47 380 0.47 339 0.47 297 0.47 255 0.47 213 0.47 171 0.47 130 0.47 088 0.47 047 0.47 005 0.46 963 0.46 922 0.46 880 0.46 839 0.46 798 0.46 756 0.46 715 0.46 673 0.46 632 0.46 591 0.46 550 0.46 508 0.46 467 0.46 426 0.46 385 0.46 344 0.46 303 9.97 821 9.97 817 9.97 812 9.97 808 9.97 8044 9.97 800 9.97 796 9.97 792 9.97 788 9.97 784 9.97 779 4 9.97 775 4 9.97 771 9.97 767 4 9.97 763 9.97 759 5 9.97 754 4 9.97 750 9.97 746 9.97 7424 9.97 7384 9.97 734 5 9.97 729 4 9.97 7254 9.97 721 4 9.97 7174 9.97 713 9.97 708 4 9.97 704 4 9.97 700 4 9.97 696 9.97 691 4 9.97 687 4 9.97 683 4 9.97 679 5 9.97 674 4 9.97 670 4 9.97 666 4 9.97 662 5 9.97 657 4 9.97 653 4 9.97 649 4 9.97 645 5 9.97 640 4 9.97 636 4 9.97 632 4 9.97 628 5 9.97 623 4 9.97 619 4 9.97 615 5 9.97 610 4 9.97 606 4 9.97 602 5 9.97 597 4 9.97 593 4 9.97 589 5 9.97 584 4 9.97 580 4 9.97 576 5 9.97 571 4 9.97 567 I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 I I V.. I u ma I 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 43 0.7 1.4 2.2 2.9 3.6 4.3 5.0 5.7 6.4 7.2 14.3 21.5 28.7 35.8 39 0.6 1.3 2.0 2.6 3.2 3.9 4.6 5.2 5.8 6.5 13.0 19.5 26.0 32.5 42 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 14.0 21.0 28.0 35.0 38 0.6 1.3 1.9 2.5 3.2 3.8 4.4 5.1 5.7 6.3 12.7 19.0 25.3 31.7 41 0.7 1.4 2.0 2.7 3.4 4.1 4.8 5.5 6.2 6.8 13.7 20.5 27.3 34.2 37 0.6 1.2 1.8 2.5 3.1 3.7 4.3 4.9 5.6 6.2 12.3 18.5 24.7 30.8 36 5 4 1 0.6 0.1 0.1 2 1.2 0.2 0.1 3 1.8 0.2 0.2 4 2.4 0.3 0.3 5 3.0 0.4 0.3 6 3.6 0.5 0.4 7 4.2 0.6 0.5 8 4.8 0.7 0.5 9 5.4 0.8 0.6 10 6.0 0.8 0.7 20 12.0 1.7 1.3 30 18.0 2.5 2.0 40 24.0 3.3 2.7 50 30.0 4.2 3.3 I 6 5 4 3 2 1 0 I I I: I I - - log cos d log cot cd log tan I I - log sin d I Prop. Parts - - I - - 161~ 251~ 341~ 71~ 80 TABLE III 19~ 109~ 199~ 289~ Prop. Parts log sin I d I log tan c d log cot log cos I d 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.51 264 9.51 301 9.51 338 9.51 374 9.51 411 9.51 447 9.51 484 9.51 520 9.51 557 9.51 593 9.51 629 9.51 666 9.51 702 9.51 738 9.51 774 9.51 811 9.51 847 9.51 883 9.51 919 9.51 955 9.51 991 9.52 027 9.52 063 9.52 099 9.52 135 9.52 171 9.52 207 9.52 242 9.52 278 9.52 314 9.52 350 9.52 385 9.52 421 9.52 456 9.52 492 9.52 527 9.52 563 9.52 598 9'.52 634 9.52 669 9.52 705 9.52 740 9.52 775 9.52 811 9.52 846 9.52 881 9.52 916 9.52 951 9.52 986 9.53 021 9.53 056 9.53 092 9.53 126 9.53 161 9.53 196 9.53 231 9.53 266 9.53 301 9.53 336 9.53 370 9.53 405 37 37 36 37 36 37 36 37 36 36 37 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 35 36 35 36 35 36 35 36 35 35 36 35 35 35 35 35 35 35 36 34 35 35 35 35 35 35 34 35 9.53 697 9.53 738 9.53 779 9.53 820 9.53 861 9.53 902 9.53 943 9.53 984 9.54 025 9.54 065 9.54 106 9.54 147 9.54 187 9.54 228 9.54 269 9.54 309 9.54 350 9.54 390 9.54 431 9.54 471 9.54 512 9.54 552 9.54 593 9.54 633 9.54 673 9.54 714 9.54 754 9.54 794 9.54 835 9.54 875 9.54 915 9.54 955 9.54 995 9.55 035 9.55 075 9.55 115 9.55 155 9.55 195 9.55 235 9.55 275 9.55 315 9.55 355 9.55 395 9.55 434 9.55 474 9.55 514 9.55 554 9.55 593 9.55 633 9.55 673 9.55 712 9.55 752 9.55 791 9.55 831 9.55 870 9.55 910 9.55 949 9.55 989 9.56 028 9.56 067 9.56 107 41 41 41 41 41 41 41 41 40 41 41 40 41 41 40 41 40 41 40 41 40 41 40 40 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 40 40 40 39 40 40 40 39 40 40 39 40 39 40 39 40 39 40 39 39 40 0.46 303 0.46 262 0.46 221 0.46 180 0.46 139 0.46 098 0.46 057 0.46 016 0.45 975 0.45 935 0.45 894 0.45 853 0.45 813 0.45 772 0.45 731 0.45 691 0.45 650 0.45 610 0.45 569 0.45 529 0.45 488 0.45 448 0.45 407 0.45 367 0.45 327 0.45 286 0.45 246 0.45 206 0.45 165 0.45 125 0.45 085 0.45 045 0.45 005 0.44 965 0.44 925 0.44 885 0.44 845 0.44 805 0.44 765 0.44 725 0.44 685 0.44 645 0.44 605 0.44 566 0.44 526 0.44 486 0.44 446 0.44 407 0.44 367 0.44 327 0.44 288 0.44 248 0.44 209 0.44 169 0.44 130 0.44 090 0.44 051 0.44 011 0.43 972 0.43 933 0.43 893 9.97 567 9.97 563 9.97 558 9.97 554 9.97 550 9.97 545 9.97 541 9.97 536 9.97 532 9.97 528 9.97 523 9.97 519 9.97 515 9.97 510 9.97 506 9.97 501 9.97 497 9.97 492 9.97 488 9.97 484 9.97 479 9.97 475 9.97 470 9.97 466 9.97 461 9.97 457 9.97 453 9.97 448 9.97 444 9.97 439 9.97 435 9.97 430 9.97 426 9.97 421 9.97 417 9.97 412 9.97 408 9.97 403 9.97 399 9.97 394 9.97 390 9.97 385 9.97 381 9.97 376 9.97 372 9.97 367 9.97 363 9.97 358 9.97 353 9.97 349 9.97 344 9.97 340 9.97 335 9.97 331 9.97 326 9.97 322 9.97 317 9.97 312 9.97 308 9.97 303 9.97 299 4 5 4 4 5 4 5 4 4 5 4 4 5 4 5 4 5 4 4 5 4 5 4 5 4 4 5 4 S 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 41 1 0.7 2 1.4 3 2.0 4 2.7 5 3.4 6 4.1 7 4.8 8 5.5 9 6.2 10 6.8 20 13.7 30 20.5 40 27.3 50 34.2 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 37 0.6 1.2 1.8 2.5 3.1 3.7 4.3 4.9 5.6 6.2 12.3 18.5 24.7 30.8 34 0.6 1.1 1.7 2.3 2.8 3.4 4.0 4.5 5.1 5.7 11.3 17.0 22.7 28.3 40 39 0.7 0.6 1.3 1.3 2.0 2.0 2.7 2.6 3.3 3.2 4.0 3.9 4.7 4.6 5.3 5.2 6.0 5.8 6.7 6.5 13.3 13.0 20.0 19.5 26.7 26.0 33.3 32.5 36 35 0.6 0.6 1.2 1.2 1.8 1.8 2.4 2.3 3.0 2.9 3.6 3.5 4.2 4.1 4.8 4.7 5.4 5.2 6.0 5.8 12.0 11.7 18.0 17.5 24.0 23.3 30.0 29.2 5 4 0.1 0.1 0.2 0.1 0.2 0.2 0.3 0.3 0.4 0.3 0.5 0.4 0.6 0.5 0.7 0.5 0.8 0.6 0.8 0.7 1.7 1.3 2.5 2.0 3.3 2.7 4.2 3.3 I log cos I d log cot cdl log tan I log sin d | ' I Prop. Parts 160~ 250~ 340~ 70~ 81 TABLE III 20~ 110~ 200~ 290~ I log sin d I log tan Icd log cot I log Cos I d j. Prop. Parts 7 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.53 405 35 9.53 440 35 9.53 475 3 9.53 509 3 9.53 544 3 34 9.53 578 3 9.53 613 3 9.53 647 3 9.53 682 3 9.53 716 3 9.53 751 3 9.53 785 3 9.53 819 3 9.53 854 3 9.53 888 3 9.53 922 3 9.53 957 3 9.53 991 3 9.54 025 34 9.54 059 3 9.54 093 3 9.54 127 3 9.54 161 3 9.54 195 3 9.54 229 3 9.54 263 3 9.54 297 3 9.54 331! 3 9.54 365 3 9.54 399 34 9.54 433 33 9.54 466 34 9.54 500 34 9.54 534 33 9.54 567 34 9.54 601 34 9.54 635 33 9.54 668 34 9.54 702 33 9.54 735 34 9.54 769 33 9.54 802 34 9.54 836 33 9.54 869 34 9.54 903 33 9.54 936 33 9.54 969 34 9.55 003 33 9.55 036 33 9.55 069 33 9.55 102 34 9.55 136 33 9.55 169 33 9.55 202 33 9.55 235 33 9.55 268 33 9.55 301 33 9.55 334 33 9.55 367 33 9.55 400 33 9.55 433 9.56 107 9.56 146 9.56 185 9.56 224 9.56 264 9.56 303 9.56 342 9.56 381 9.56 420 9.56 459 9.56 498 9.56 537 9.56 576 9.56 615 9.56 654 9.56 693 9.56 732 9.56 771 9.56 810 9.56 849 9.56 887 9.56 926 9.56 965 9.57 004 9.57 042 9.57 081 9.57 120 9.57 158 9.57 197 9.57 235 9.57 274 9.57 312 9.57 351 9.57 389 9.57 428 9.57 466 9.57 504 9.57 543 9.57 581 9.57 619 9.57 658 9.57 696 9.57 734 9.57 772 9.57 810 9.57 849 9.57 887 9.57 925 9.57 963 9.58 001 9.58 039 9.58 077 9.58 115 9.58 153 9.58 191 9.58 229 9.58 267 9.58 304 9.58 342 9.58 380 9.58 418 39 39 39 40 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 38 39 39 39 38 39 39 38 39 38 39 38 39 38 39 38 38 39 38 38 39 38 38 38 38 39 38 38 38 38 38 38 38 38 38 38 38 37 38 38 38 0.43 893 0.43 854 0.43 815 0.43 776 0.43 736 0.43 697 0.43 658 0.43 619 0.43 580 0.43 541 0.43 502 0.43 463 0.43 424 0.43 385 0.43 346 0.43 307 0.43 268 0.43 229 0.43 190 0.43 151 0.43 113 0.43 074 0.43 035 0.42 996 0.42 958 0.42 919 0.42 880 0.42 842 0.42 803 0.42 765 0.42 726 0.42 688 0.42 649 0.42 611 0.42 572 0.42 534 0.42 496 0.42 457 0.42 419 0.42 381 0.42 342 0.42 304 0.42 266 0.42 228 0.42 190 0.42 151 0.42 113 0.42 075 0.42 037 0.41 999 0.41 961 0.41 923 0.41 885 0.41 847 0.41 809 0.41 771 0.41 733 0.41 696 0.41 658 0.41 620 0.41 582 9.97 299 9.97 294 9.97 289 9.97 285 9.97 280 9.97 276 9.97 271 9.97 266 9.97 262 9.97 257 9.97 252 9.97 248 9.97 243 9.97 238 9.97 234 9.97 229 9.97 224 9.97 220 9.97 215 9.97 210 9.97 206 9.97 201 9.97 196 9.97 192 9.97 187 9.97 182 9.97 178 9.97 173 9.97 168 9.97 163 9.97 159 9.97 154 9.97 149 9.97 145 9.97 140 9.97 135 9.97 130 9.97 126_ 9.97 121 9.97 116 9.97 111 9.97 107 9.97 102 9.97 097 9.97 092 9.97 087 9.97 083 9.97 078 9.97 073 9.97 068 9.97 063 9.97 059 9.97 054 9.97 049 9.97 044 9.97 039 9.97 035 9.97 030 9.97 025 9.97 020 9.97 015 5 5 4 5 4 -5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 5 4 5 5 4 5 5 S 4 5 5 5 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 40 0.7 1.3 2.0 2.7 3.3 4.0 4.7 5.3 6.0 6.7 13.3 20.0 26.7 33.3 39 0.6 1.3 2.0 2.6 3.2 3.9 4.6 5.2 5.8 6.5 13.0 19.5 26.0 32.5 35 0.6 1.2 1.8 2.3 2.9 3.5 4.1 4.7 5.2 5.8 11.7 17.5 23.3 29.2 37 1 0.6 2 1.2 3 1.8 4 2.5 5 3.1 6 3.7 7 4.3 8 4.9 9 5.6 10 6.2 20 12.3 30 18.5 40 24.7 50 30.8 38 0.6 1.3 1.9 2.5 3.2 3.8 4.4 5.1 5.7 6.3 12.7 19.0 25.3 31.7 34 0.6 1.1 1.7 2.3 2.8 3.4 4.0 4.5 5 1 5.7 11.3 17.0 22.7 28.3 4 0.1 0.1 0.2 0.3 0.3 0.4 0.5 0.5 0.6 0.7 1.3 2.0 2.7 3.3 4 5 5 5 5 4 5 5 5 5 4 5 5 5 5 4 5 S 5 5 I 2 3 4 5 6 7 8 9 10 20 30 40 50 33 5 0.6 0.1 1.1 0.2 1.6 0.2 2.2 0.3 2.8 0.4 3.3 0.5 3.8 0.6 4.4 0.7 5.0 0.8 5.5 0.8 11.0 1.7 16.5 2.5 22.0 3.3 27.5 4.2.. _ - I log I d log cot Icd log tan log sin I d ' I Prop. Parts 159~ 249~ 339~ 690 82 TABLE III 21~ 111~ 201~ 291~ I log sin 1d I log tan Icdj log cot I log cos I d I I jProp. Parts O~~~,.., I-~ —~ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.55 433 9.55 466 9.55 499 9.55 532 9.55 564 9.55 597 9.55 630 9.55 663 9.55 695 9.55 728 9.55 761 9.55 793 9.55 826 9.55 858 9.55 891 9.55 923 9.55 956 9.55 988 9.56 021 9.56 053 9.56 085 9.56 118 9.56 150 9.56 182 9.56 215 9.56 247 9.56 279 9.56 311 9.56 343 9.56 375 9.56 408 9.56 440 9.56 472 9.56 504 9.56 536 9.56 568 9.56 599 9.56 631 9.56 663 9.56 695 9.56 727 9.56 759 9.56 790 9.56 822 9.56 854 9.56 886 9.56 917 9.56 949 9.56 980 9.57 012 9.57 044 9.57 075 9.57 107 9.57 138 9.57 169 9.57 201 9.57 232 9.57 264 9.57 295 9.57 326 9.57 358 33 33 33 32 33 33 33 32 33 33 32 33 32 33 32 33 32 33 32 32 33 32 32 33 32 32 32 32 32 33 32 32 32 32 32 31 32 32 32 32 32 31 32 32 32 31 32 31 32 32 31 32 31 31 32 31 32 31 31 32 9.58 418 9.58 455 9.58 493 9.58 531 9.58 569 9.58 606 9.58 644 9.58 681 9.58 719 9.58 757 9.58 794 9.58 832 9.58 869 9.58 907 9.58 944 9.58 981 9.59 019 9.59 056 9.59 094 9.59 131 9.59 168 9.59 205 9.59 243 9.59 280 9.59 317 9.59 354 9.59 391 9.59 429 9.59 466 9.59 503 9.59 540 9.59 577 9.59 614 9.59 651 9.59 688 9.59 725 9.59 762 9.59 799 9.59 835 9.59 872 9.59 909 9.59 946 9.59 983 9.60 019 9.60 056 9.60 093 9.60 130 9.60 166 9.60 203 9.60 240 9.60 276 9.60 313 9.60 349 9.60 386 9.60 422 9.60 459 9.60 495 9.60 532 9.60 568 9.60 605 9.60 641 37 38 38 38 37 38 37 38 38 37 38 37 38 37 37 38 37 38 37 37 37 38 37 37 37 37 38 37 37 37 37 37 37 37 37 37 37 36 37 37 37 37 36 37 37 37 36 37 37 36 37 36 37 36 37 36 37 36 37 36 0.41 582 0.41 545 0.41 507 0.41 469 0.41 431 0.41 394 0.41 356 0.41 319 0.41 281 0.41 243 0.41 206 0.41 168 0.41 131 0.41 093 0.41 056 0.41 019 0.40 981 0.40 944 0.40 906 0.40 869 0.40 832 0.40 795 0.40 757 0.40 720 0.40 683 0.40 646 0.40 609 0.40 571 0.40 534 0.40 497 0.40 460 0.40 423 0.40 386 0.40 349 0.40 312 0.40 275 0.40 238 0.40 201 0.40 165 0.40 128 0.40 091 0.40 054 0.40 017 0.39 981 0.39 944 0.39 907 0.39 870 0.39 834 0.39 797 0.39 760 0.39 724 0.39 687 0.39 651 0.39 614 0.39 578 0.39 541 0.39 505 0.39 468 0.39 432 0.39 395 0.39 359 9.97 015 9.97 010 9.97 005 9.97 001 9.96 996 9.96 991 9.96 986 9.96 981 9.96 976 9.96 971 9.96 966 9.96 962 9.96 957 9.96 952 9.96 947 9.96 942 9.96 937 9.96 932 9.96 927 9.96 922 9.96 917 9.96 912 9.96 907 9.96 903 9.96 898 9.96 893 9.96 888 9.96 883 9.96 878 9.96 873 9.96 868 9.96 863 9.96 858 9.96 853 9.96 848 9.96 843 9.96 838 9.96 833 9.96 828 9.96 823 9.96 818 9.96 813 9.96 808 9.96 803 9.96 798 9.96 793 9.96 788 9.96 783 9.96 778 9.96 772 9.96 767 9.96 762 9.96 757 9.96 752 9.96 747 9.96 742 9.96 737 9.96 732 9.96 727 9.96 722 9.96 717 I 5 5 4 5 5 5 5 5 5 5 4 5 5 5 5 5 5 5 5 5 5 5 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 5 5 5 5 5 5 5 5 5 5 5 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 38 37 0.6 0.6 1.3 1.2 1.9 1.8 2.5 2.5 3.2 3.1 3.8 3.7 4.4 4.3 5.1 4.9 5.7 5.6 6.3 6.2 12.7 12.3 19.0 18.5 25.3 24.7 31.7 30.8 33 32 0.6 0.5 1.1 1.1 1.6 1.6 2.2 2.1 2.8 2.7 3.3 3.2 3.8 3.7 4.4 4.3 5.0 4.8 5.5 5.3 11.0 10.7 16.5 16.0 22.0 21.3 27.5 26.7 6 5 0.1 0.1 0.2 0.2 0.3 0.2 0.4 0.3 0.5 0.4 0.6 0.5 0.7 0.6 0.8 0.7 0.9 0.8 1.0 0.8 2.0 1.7 3.0 2.5 4.0 3.3 5.0 4.2 I - 36 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 12.0 18.0 24.0 30.0 31 0.5 1.0 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.2 10.3 15.5 20.7 25.8 4 0.1 0.1 0.2 0.3 0.3 0.4 0.5 0.5 0.6 0.7 1.3 2.0 2.7 3.3 I log cos I d I log cot Icd log tan I log sin dl ' Prop. Parts 15028088,608 158~ 248~ 338~ 68~ 83 TABLE III 22~ 112~ 2020 292~ I log sin d I log tan c d log cot I log cos d I Prop. Parts 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.57 358 9.57 389 9.57 420 9.57 451 9.57 482 9.57 514 9.57 545 9.57 576 9.57 607 9.57 638 9.57 669 9.57 700 9.57 731 9.57 762 9.57 793 9.57 824 9.57 855 9.57 885 9.57 916 9.57 947 9.57 978 9.58 008 9.58 039 9.58 070 9.58 101 9.58 131 9.58 162 9.58 192 9.58 223 9.58 253 9.58 284 9.58 314 9.58 345 9.58 375 9.58 406 9.58 436 9.58 467 9.58 497 9.58527 9.58 557 9.58 588 9.58 618 9.58 648 9.58 678 9.58 709 9.58 739 9.58 769 9.58 799 9.58 829 9.58 859 9.58 889 9.58 919 9.58 949 9.58 979 9.59 009 9.59 039 9.59 069 9.59 098 9.59 128 9.59 158 9.59 188 31 31 31 31 32 31 31 31 31 31 31 31 31 31 31 31 30 31 31 31 30 31 31 31 30 31 30 31 30 31 30 31 30 31 30 31 30 30 30 31 30 30 30 31 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 9.60 641 9.60 677 9.60 714 9.60 750 9.60 786 9.60 823 9.60 859 9.60 895 9.60 931 9.60 967 9.61 004 9.61 040 9.61 076 9.61 112 9.61 148 9.61 184 9.61 220 9.61 256 9.61 292 9.61 328 9.61 364 9.61 400 9.61 436 9.61 472 9.61 508 9.61 544 9.61 579 9.61 615 9.61 651 9.61 687 9.61 722 9.61 758 9.61 794 9.61 830 9.61 865 9.61 901 9.61 936 9.61 972 9.62 008 9.62 043 9.62 079 9.62 114 9.62 150 9.62 185 9.62 221 9.62 256 9.62 292 9.62 327 9.62 362 9.62 398 9.62 433 9.62 468 9.62 504 9.62 539 9.62 574 9.62 609 9.62 645 9.62 680 9.62 715 9.62 750 9.62 785 36 0.39 359 7 0.39 323 36 0.39 286 36 0.39250 3 0.39 214 37 36 0.39 177 36 0.39 141 36 0.39 105 36 0.39 069 37 0.39 033 36 0.38 996 36 0.38 960 36 0.38 924 36 0.38 888 36 0.38 852 36 0.38 816 36 0.38 780 36 0.38 744 36 0.38 708 36 0.38 672 36 0.38 636 36 0.38 600 36 0.38 564 36 0.38 528 36 0.38 492 35 0.38 456 36 0.38 421 36 0.38 385 36 0.38 349 35 0.38313 36 0.38 278 36 0.38 242 36 0.38 206 35 0.38 170 36 0.38 135 35 0.38 099 36 0.38 064 36 0.38 028 35 0.37 992 36 0.37 957 35 0.37 921 36 0.37 886 35 0.37 850 36 0.37 815 35 0.37 779 36 0.37 744 35 0.37 708 35 0.37 673 36 0.37 638 35 0.37 602 35 0.37 567 36 0.37 532 35 0.37 496 35 0.37 461 35 0.37 426 36 0.37 391 35 0.37 355 35 0.37 320 35 0.37 285 35 0.37 250 0.37 215 9.96 717 9.96 711 9.96 706 9.96 701 9.96 696 9.96 691 9.96 686 9.96 681 9.96 676 9.96 670 9.96 665 9.96 60 9.96 655 9.96 650 9.9-6 645 9.96 640 9.96 634 9.96 629 9.96 624 9.96 619 9.96 614 9.96 608 9.96 603 9.96 598 9.96 593 9.96 588 9.96 582 9.96 577 9.96 572 9.96 567 9.96 562 9.96 556 9.96 551 9.96 546 9.96 541 9.96 535 9.96 530 9.96 525 9.96 520 9.96 514 9.96 509 9.96 504 9.96 498 9.96 493 9.96 488 9.96 483 9.96 477 9.96 472 9.96 467 9.96 461 9.96 456 9.96 451 9.96 445 9.96 440 9.96 435 9.96 429 9.96 424 9.96 419 9.96 413 9.96 408 9.96 403 6 5 5 5 5 5 5 5 6 5 5 5 5 5 5 6 5 5 5 5 6 5 5 5 5 6 5 5 5 5 6 5 5 5 6 5 5 5 6 5 5 6 5 5 5 6 5 5 6 5 5 6 5 5 6 5 5 6 5 5 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 I I 37 36 0.6 0."6 1.2 1.2 1.8 1.8 2.5 2.4 3.1 3.0 3.7 3.6 4.3 4.2 4.9 4.8 5.6 5.4 6.2 6.0 12.3 12.0 18.5 18.0 4.7 24.0 30.8 30.0 32 31 0.5 0.5 1.1 1.0 1.6 1.6 2.1 2.1 2.7 2.6 3.2 3.1 3.7 3.6 4.3 4.1 4.8 4.6 5.3 5.2 10.7 10.3 16.0 15.5!1.3 20.7 26.7 25.8 29 6 0.5 0.1 1.0 0.2 1.4 0.3 1.9 0.4 2.4 0.5 2.9 0.6 3.4 0.7 3.9 0.8 4.4 0.9 4.8 1.0 9.7 2.0 14.5 3.0 19.3 4.0 24.2 5.0 35 0.6 1.2 1.8 2.3 2.9 3.5 4.1 4.7 5.2 5.8 11.7 17.5 23.3 29.2 30 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10.0 15.0 20.0 25.0 5 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.8 1.7 2.5 3.3 4.2 I II. I I log cos Id log cot jI d l ogsin J' Prop. Parts 157~ 247~ 337~ 67" 84 TABLE III 23~ 113~ 203~ 293~ log sin d log tan |cdl logcot T log cos d 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.59 188 9.59 218 9.59 247 9.59 277 9.59 307 9.59 336 9.59 366 9.59 396 9.59 425 9.59 455 9.59 484 9.59 514 9.59 543 9.59 573 9.59 602 9.59 632 9.59 661 9.59 690 9.59 720 9.59 749 9.59 778 9.59 808 9.59 837 9.59 866 9.59 895 9.59 924 9.59 954 9.59 983 9.60 012 9.60 041 9.60 070 9.60 099 9.60 128 9.60 157 9.60 186 9.60 215 9.60 244 9.60 273 9.60 302 9.60 331 9.60 359 9.60 388 9.60 417 9.60 446 9.60 474 9.60 503 9.60 532 9.60 561 9.60 589 9.60 618 9.60 646 9.60 675 9.60 704 9.60 732 9.60 761 9.60 789 9.60 818 9.60 846 9.60 875 9.60 903 9.60 931.1 30 29 30 30 29 30 30 29 30 29 30 29 30 29 30 29 29 30 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 28 29 29 29 28 29 28 29 29 28 29 28 29 28 29 28 28 I I I I I 0 I 9.62 785 9.62 820 9.62 855 9.62 890 9.62 926 9.62 961 9.62 996 9.63 031 9.63 066 9.63 101 9.63 135 9.63 170 9.63 205 9.63 240 9.63 275 9.63 310 9.63 345 9.63 379 9.63 414 9.63 449 9.63 484 9.63 519 9.63 553 9.63 588 9.63 623 9.63 657 9.63 692 9.63 726 9.63 761 9.63 796 9.63 830 9.63 865 9.63 899 9.63 934 9.63 968 9.64 003 9.64 037 9.64 072 9.64 106 9.64 140 9.64 175 9.64 209 9.64 243 9.64 278 9.64 312 9.64 346 9.64 381 9.64 415 9.64 449 9.64 483 9.64 517 9.64 552 9.64 586 9.64 620 9.64 654 9.64 688 9.64 722 9.64 756 9.64 790 9'. 64 824 9.64 858 35 35 35 36 35 35 35 35 35 34 35 35 35 35 35 35 34 35 35 35 35 34 35 35 34 35 34 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 0.37 215 0.37 180 0.37 145 0.37 110 0.37 074 0.37 039 0.37 004 0.36 969 0.36 934 0.36 899 0.36 865 0.36 830 0.36 795 0.36 760 0.36 725 0.36 690 0.36 655 0.36 621 0.36 586 0.36 551 0.36 516 0.36 481 0.36 447 0.36 412 0.36 377 0.36 343 0.36 308 0.36 274 0.36 239 0.36 204 0.36 170 0.36 135 0.36 101 0.36 066 0.36 032 0.35 997 0.35 963 0.35 928 0.35 894 0.35 860 0.35 825 0.35 791 0.35 757 0.35 722 0.35 688 0.35 654 0.35 619 0.35 585 0.35 551 0.35 517 0.35 483 0.35 448 0.35 414 0.35 380 0.35 346 0.35 312 0. 35 278 0.35 244 0.35 210 0.35 176 0.35 142 9.96 403 9.96 397 9.96 392 9.96 387 9.96 381 9.96 376 9.96 370 9.96 365 9.96 360 9.96 354 9.96 349 9.96 343 9.96 338 9.96 333 9.96 327 9.96 322 9.96 316 9.96 311 9.96 305 9.96 300 9.96 294 9.96 289 9.96 284 9.96 278 9.96 273 9.96 267 9.96 262 9.96 256 9.96 251 9.96 245 9.96 240 9.96 234 9.96 229 9.96 223 9.96 218 9.96 212 9.96 207 9.96 201 9.96 196 9.96 190 9.96 185 9.96 179 9.96 174 9.96 168 9.96 162 9.96 157 9.96 151 9.96 146 9.96 140 9.96 135 9.96 129 9.96 123 9.96 118 9.96 112 9.96 107 9.96 101 9.96 095 9.96 090 9.96 084 9.96 079 9.96 073 6 5 5 6 5 6 5 5 6 5 5 5 6 5 6 5 6 5 6 5 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 6 5 6 5 6 5 6 6 5 6 5 6 6 5 6 5 6 I I Prop. Parts I -60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 36 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 12.0 18.0 24.0 30.0 30 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10.0 15.0 20.0 25.0 1 ( 2 ( 3 ( 4 ( 5 ( 35 0.6 1.2 1.8 2.3 2.9 3.5 4.1 4.7 5.2 5.8 11.7 17.5 23.3 29.2 29 0.5 1.0 1.4 1.9 2.4 2.9 3.4 3.9 4.4 4.8 9.7 14.5 19.3 24.2 34 0.6 1.1 1.7 2.3 2.8 3.4 4.0 4.5 5.1 5.7 11.3 17.0 22.7 28.3 28 0.5 0.9 1.4 1.9 2.3 2.8 3.3 3.7 4.2 4.7 9.3 14.0 18.7 23.3 6 7 8 9 10 20 30 40 50 ( ( ( ( 6 5 ).1 0.1 ).2 0.2 ).3 0.2 ).4 0.3 ).5 0.4 ).6 0.5 ).7 0.6 ).8 0.7 ).9 0.8 1.0 0.8 2.0 1.7 3.0 2.5 4.0 3.3 5.0 4.2 7 6 5 4 3 2 1 0 I I log cos I djl log cot [cd log tan I log sin I d ' I Prop. Parts 156~ 246~ 336~ 66~ 85 TABLE III 24~ 114~ 204~ 294~ - I' log sin I d log tan Icd| log cot I log cos I d I Prop. Parts 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.60 931 9.60 960 9.60 988 9.61 016 9.61 045 9.61 073 9.61 101 9.61 129 9.61 158 9.61 186 9.61 214 9.61 242 9.61 270 9.61 298 9.61 326 9.61 354 9.61 382 9.61 411 9.61 438 9.61 466 9.61 494 9.61 522 9.61 550 9.61 578 9.61 606 9.61 634 9.61 662 9.61 689 9.61 717 9.61 745 9.61 773 9.61 800 9.61 828 9.61 856 9.61 883 9.61 911 9.61 939 9.61 966 9.61 994 9.62 021 9.62 049 9.62 076 9.62 104 9.62 131 9.62 159 9.62 186 9.62 214 9.62 241 9.62 268 9.62 296 9.62 323 9.62 350 9.62 377 9.62 405 9.62 432 9.62 459 9.62 486 9.62 513 9.62 541 9.62 568 9.62 595' 29 28 28 29 28 28 28 29 28 28 28 28 28 28 28 28 29 27 28 28 28 28 28 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 27 28 27 28 27 28 27 28 27 27 28 27 27 27 28 27 27 27 27 28 27 27 9.64 858 9.64 892 9.64 926 9.64 960 9.64 994 9.65 028 9.65 062 9.65 096 9.65 130 9.65 164 9.65 197 9.65 231 9.65 265 9.65 299 9.65 333 9.65 366 9.65 400 9.65 434 9.65 467 9.65 501 9.65 535 9.65 568 9.65 602 9.65 636 9.65 669 9.65 703 9.65 736 9.65 770 9.65 803 9.65 837 9.65 870 9.65 904 9.65 937 9.65 971 9.66 004 9.66 038 9.66 071 9.66 104 9.66 138 9.66 171 9.66 204 9.66 238 9.66 271 9.66 304 9.66 337 9.66 371 9.66 404 9.66 437 9.66 470 9.66 503 9.66 537 9.66 570 9.66 603 9.66 636 9.66 669 9.66 702 9.66 735 9.66 768 9.66 801 9.66 834 9.66 867 34 34 34 34 34 34 34 34 34 33 34 34 34 34 33 34 34 33 34 34 33 34 34 33 34 33 34 33 34 33 34 33 34 33 34 33 33 34 33 33 34 33 33 33 34 33 33 33 33 34 33 33 33 33 33 33 33 33 33 33 0.35 142 0.35 108 0.35 074 0.35 040 0.35 006 0.34 972 0.34 938 0.34 904 0.34 870 0.34 836 0.34 803 0.34 769 0.34 735 0.34 701 0.34 667 0.34 634 0.34 600 0.34 566 0.34 533 0.34 499 0.34 465 0.34 432 0.34 398 0.34 364 0.34 331 0.34 297 0.34 264 0.34 230 0.34 197 0.34 163 0.34 130 0.34 096 0.34 063 0.34 029 0.33 996 0.33 962 0.33 929 0.33 896 0.33 862 0.33 829 0.33 796 0.33 762 0.33 729 0.33 696 0.33 663 0.33 629 0.33 596 0.33 563 0.33 530 0.33 497 0.33 463 0.33 430 0.33 397 0.33 364 0.33 331 0.33 298 0.33 265 0.33 232 0.33 199 0.33 166 0.33 133 9.96 073 9.96 067 9.96 062 9.96 056 9.96 050 9.96 045 9.96 039 9.96 034 9.96 028 9.96 022 9.96 017 9.96 011 9.96 005 9.96 000 9.95 994 9.95 988 9.95 982 9.95 977 9.95 971 9.95 965 9.95 960 9.95 954 9.95 948 9.95 942 9.95 937 9.95 931 9.95 925 9.95 920 9.95 914 9.95 908 9.95 902 9.95 897 9.95 891 9.95 885 9.95 879 9.95 873 9.95 868 9.95 862 9.95 856 9.95 850 9.95 844 9.95 839 9.95 833 9.95 827 9.95 821 9.95 815 9.95 810 9.95 804 9.95 798 9.95 792 9.95 786 9.95 780 9.95 775 9.95 769 9.95 763 9.95 757 9.95 751 9.95 745 9.95 739 9.95 733 9.95 728 6 5 6 6 5 6 5 6 6 5 6 6 5 6 6 6 5 6 6 5 6 6 6 5 6 6 5 6 6 6 5 6 6 6 6 5 6 6 6 6 5 6 6 6 6 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8. 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 34 0.6 1.1 1.7 2.3 2.8 3.4 4.0 4.5 5.1 5.7 11.3 17.0 22.7 28.3 1 2 3 4 5 6 7 8 9 10 20 30 40 50 29 0.5 1.0 1.4 1.9 2.4 2.9 3.4 3.9 4.4 4.8 9.7 14.5 19.3 24.2 28 0.5 0.9 1.4 1.9 2.3 2.8 3.3 3.7 4.2 4.7 9.3 14.0 18.7 23.3 27 0.4 0.9 1.4 1.8 2.2 2.7 3.2 3.6 4.0 4.5 9.0 13.5 18.0 22.5 33 0.6 1.1 1.6 2.2 2.8 3.3 3.8 4.4 5.0 5.5 11.0 16.5 22.0 27.5 5 6 6 6 6 6 5 6 6 6 6 6 6 6 5 2 3 4 5 6 7 8 9 10 20 30 40 50 6 5 0.1 0.1 0.2 0.2 0.3 0.2 0.4 0.3 0.5 0.4 0.6 0.5 0.7 0.6 0.8 0.7 0.9 0.8 1.0 0.8 2.0 1.7 3.0 2.5 4.0 3.3 5.0 4.2 I S.... I logcos logcot c d logtan j logsin ld I ' I Prop. Parts 1550 2450 3350 650 86~~~~~~~~~~~~~~~~~~~~~, 155~ 245~ 335~ 65~ 86 TABLE III 25~ 115~ 205~ 295 ' log sin I d I log tan cd log cot log cos d I I Prop. Parts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.62 595 9.62 622 9.62 649 9.62 676 9.62 703 9.62 730 9.62 757 9.62 784 9.62 811 9.62 838 9.62 865 9.62 892 9.62 918 9.62 945 9.62 972 9.62 999 9.63 026 9.63 052 9.63 079 9.63 106 9.63 133 9.63 159 9.63 186 9.63 213 9.63 239 9.63 266 9.63 292 9.63 319 9.63 345 9.63 372 9.63 398 9.63 425 9.63 451 9.63 478 9.63 504 9.63 531 9.63 557 9.63 583 9.63 610 9.63 636 9.63 662 9.63 689 9.63 715 9.63 741 9.63 767 9.63 794 9.63 820 9.63 846 9.63 872 9.63 898 9.63 924 9.63 950 9.63 976 9.64 002 9.64 028 9.64 054 9.64 080 9.64 106 9.64 132 9.64 158 9.64 184 27 27 27 27 27 27 27 27 27 27 27 26 27 27 27 27 26 27 27 27 26 27 27 26 27 26 27 26 27 26 27 26 27 26 27 26 26 27 26 26 27 26 26 26 27 9.66 867 9.66 900 9.66 933 9.66 966 9.66 999 9.67 032 9.67 065 9.67 098 9.67 131 9.67 163 9.67 196 9.67 229 9.67 262 9.67 295 9.67 327 9.67 360 9.67 393 9.67 426 9.67 458 9.67 491 9.67 524 9.67 556 9.67 589 9.67 622 9.67 654 9.67 687 9.67 719 9.67 752 9.67 785 9.67 817 9.67 850 9.67 882 9.67 915 9.67 947 9.67 980 9.68 012 9.68 044 9.68 077 9.68 109 9.68 142 9.68 174 9.68 206 9.68 239 9.68 271 9.68 303 9.68 336 9.68 368 9.68 400 9.68 432 9.68 465 9.68 497 9.68 529 9.68 561 9.68 593 9.68 626 9.68 658 9.68 690 9.68 722 9.68 754 9.68 786 9.68 818 33 33 33 33 33 33 33 33 32 33 33 33 33 32 33 33 33 32 33 33 32 33 33 32 33 32 33 33 32 33 32 33 32 33 32 32 33 32 33 32 32 33 32 32 33 32 32 32 33 32 32 32 32 33 32 32 32 32 32 32 0.33 133 0.33 100 0.33 067 0.33 034 0.33 001 0.32 968 0.32 935 0.32 902 0.32 869 0.32 837 0.32 804 0.32 771 0.32 738 0.32 705 0.32 673 0.32 640 0.32 607 0.32 574 0.32 542 0.32 509 0.32 476 0.32 444 0.32 411 0.32 378 0.32 346 0.32 313 0.32 281 0.32 248 0.32 215 0.32 183 0.32 150 0.32 118 0.32 085 0.32 053 0.32 020 0.31 988 0.31 956 0.31 923 0.31 891 0.31 858 0.31 826 0.31 794 0.31 761 0.31 729 0.31 697 0.31 664 0.31 632 0.31 600 0.31 568 0.31 535 0.31 503 0.31 471 0.31 439 0.31 407 0.31 374 0.31 342 0.31 310 0.31 278 0.31 246 0.31 214 0.31 182 9.95 728 9.95 722 9.95 716 9.95 710 9.95 704 9.95 698 9.95 692 9.95 686 9.95 680 9.95 674 9.95 668 9.95 663 9.95 657 9.95 651 9.95 645 9.95 639 9.95 633 9.95 627 9.95 621 9.95 615 9.95 609 9.95 603 9.95 597 9.95 591 9.95 585 9.95 579 9.95 573 9.95 567 9.95 561 9.95 555 9.95 549 9.95 543 9.95 537 9.95 531 9.95 525 9.95 519 9.95 513 9.95 507 9.95 500 9.95 494 9.95 488 9.95 482 9.95 476 9.95 470 9.95 464 9.95 458 9.95 452 9.95 446 9.95 440 9.95 434 9.95 427 9.95 421 9.95 415 9.95 409 9.95 403 9.95 397 9.95 391 9.95 384 9.95 378 9.95 372 9.95 366 6 6 6 6 6 6 6 6 6 6 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 7 6 6 6 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 I 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 3 2. 2. 3. 3. 4. 5. 5. 11. 16. 22. 27. 2' 0. 0. 1. 1. 2. 2. 3. 3. 4. 4. 9. 13. 18. 22. 3 32 6 0.5 1 1.1 6 1.6 2 2.1 8 2.7 3 3.2 8 3.7 4 4.3 0 4.8 5 5.3 0 10.7 5 16.0 0 21.3 5 26.7 7 26 4 0.4 9 0.9 4 1.3 8 1.7 2 2.2 7 2.6 2 3.0 6 3.5 0 3.9 5 4.3 0 8.7 5 13.0 0 17.3 5 21.7 6 5 0.1 0.1 0.2 0.2 0.3 0.2 0.4 0.3 0.5 0.4 0.6 0.5 0.7 0.6 0.8 0.7 0.9 0.8 1.0 0.8 2.0 1.7 3.0 2.5 4.0 3.3 5.0 4.2 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 1 2 3 4 5 6 7 8 9 10 20 30 40 50 7 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 2.3 3.5 4.7 5.8 L i log cos I d I log cot Icdl log tan I log sin'1 d I I Prop. Parts 1540 2440 3340 64~~ ~ 87 154~ 244~ 334~ 64~ 87 TABLE III 26~ 116~ 206~ 296~ ' log sin d | log tan I cd log cot I log cos d | Prop. Parts 0 9.64 184 26 9.68 818 32 0.31 182 9.95 366 6 60 1 9.64 210 9.68 850 320.31 150 9.95 360 59 2 9.64 236 26 9.68 882 32 0.31 118 9.95 354 6 58 3 9.64 262 26 9.68 914 32 0.31 086 9.95 348 57 4 9.64 288 2 9.68 946 32 0.31 054 9.95 341 7 56 5 9.64 313 25 9.68 978 32 0.31 022 9.95 335 6 55 32 3 6 9.64 339 26 9.69 010 32 0.30 990 9.95 329 6 54 1 0.5 0. 7 9.64 365 26 9.69 042 32 0.30 958 9.95 323 6 53 2 1. 1.0 8 9.64 391 26 9.69 074 32 0.30926 9.95 317 52 2.1 2.1 9 9.64 417 25 9.69 106 32 0.30 894 9.95 310 51 2.- 2.6 10 9.64 442 9.69 138 32 0.30 862 9.95 304 6 50 6 3.2 3.1 11 9.64 468 26 9.69 170 32 0.30 830 9 95 298 6 49 7 3.7 3.6 12 9.64 494 25 9.69 202 32 0.30 798 9.95 292 6 48 8 4.3 4.1 13 9.64 519 26 9.69 234 32 0.30 766 9.95 286 47 9 4.8 4.6 14 9.64 545 26 9.69 266 32 0.30 734 9.95 279 6 46 10 5.3 5.2 15 9.64 571 25 9.69 298 31 0.30 702 9.95 273 645 20 10.7 10.3 16 9.64 596 26 9.69 329 32 0.30 671 9.95 267 44 30 16.0 15.5 17 9.64 622 25 9.69 361 320.30 639 9.95 261 43 40 21.3 20.7 18 9.64 647 26 9.69 393 32 0.30 607 9.95 254 6 42 50 26.7 25.8 19 9.64 673 25 9.69 425 32 0.30 575 9.95 248 6 41 20 9.64 698 26 9.69 457 31 0.30543 9.95 242 6 40 21 9.64 724 25 9.69 488 32 0.30512 9.95 236 39 22 9.64 749 26 9.69 520 32 0.30480 9.95 229 6 38 23 9.64 775 25 9.69 552 32 0.30 448 9.95 223 6 3726 25 24 24 9.64 800 26 9.69 584 31 0.30 416 9.95 217 6 36 1 0.4 0.4 0.4 25 9.64 826 25 9.69 615 32 0.30 385 9.95 211 735 2 0 9 0 8 0 8 26 9.64 851 26 9.69 647 32 0.30 353 9.95 204 34 3 1 3 1.2 1.2 27 9.64 877 9.69 679 31 0.30 321 9.95 198 33 4 1.7 1.7 1.6 28 9.64 902 25 9.69 710 32 0.30 290 9.95 192 32 5 2.2 2.1 2.0 29 9.64 927 26 9.69 742 32 0.30 258 9.95 185 6 31 6 2.6 2.5 2.4 30 9.64 953 5 9.69 774 31 0.30 226 9.95 179 6 30 7 3.0 2.9 2.8 31 9.64 978 2 9.69 805 32 0.30 1953 9.95 173 6 29 8 3.5 3.3 3.2 32 9.65 003 26 9.69 837 31 0.30 163 9.95 167 28 9 3.9 3.8 3.6 33 9.65 029 25 9.69 868 32 0.30 132 9.95 160 6 27 10 4.3 4.2 4.0 34 9.65 054 9.69 900 32 0.30 100 9.95 154 26 20 8.7 8.3 8.0 35 9.65 079 25 9.69 932 31 0.30 068 9.95 148 25 30 13.0 12.5 12.0 36 9.65 104 25 9.69 963 31 0.30 037 9.95 141 7 24 40 17.3 16.7 16.0 37 9.65 130 26 9.69 995 31 0. 30 005 9.95 135 6 23 50 21.7 20.8 20.0 38 9.65 155 25 9.70 026 32 0.29 974 9.95 129 22 39 9.65 180 25 9.70 058 31 0.29 942 9.95 122 6 21 40 9.65 205 25 9.70 089 32 0.29 911 9.95 116 20 41 9.65 230 25 9.70 1211 0. 29 879 9.95 110 19 42 9.65 255 26 9. 70 152 32 029 848 9.95 103 18 7 6 43 9.65 281 2 9.70 184 30.29 8169.95097 17 1 0 0 1 44 9.65 306 25 9.70215 32 0.29785 9.95090 6 16 2 0.2 0 2 45 9.65 331 25 9.70 247 31 0.29 753 9.95 084 615 3 0.4 0.3 46 9.65 356 9.70 278 31 0.29 722 9.95 078 14 4 0.5 0.4 47 9.65 381 25 9.70 309 32 0.29 691 9.95 071 6 13 5 0.6 0.5 48 9.65 406 25 9.70 341 31 0.29 659 9.95 06 6 12 6 0.7 0.6 49 9.65 431 25 9.70 372 32 0.29 628 9.95 059 11 7 0.8 0.7 50 9.65 456 9.70 404 0.29 596 9.95 052 10 8 0.9 0.8 51 9.65 481 25 9.70 435 31 0.29565 9.95 046 9 9 1.0 0.9 52 9.65 506 25 9.70 466 32 0.29 534 9.95 039 7 8 10 1.2 1.0 53 9.65 531 25 9.70 498 3 0.29 502 9.95 033 6 7 20 2.3 2.0 54 9.65 556 25 9.70 529 31 0.29 471 9.95 027 6 6 40 4.57 4.0 55 9.65 580 29.70 560 32 0.29440 9.95 020 6 5 50 58 5 0 56 9.65 605 25 9.70 592 31 0.29408 9.95 014 4 5 57 9.65 630 9.70 623 0.29377 9.95 007 3 58 9.65 655 25 9.70 654 31 0.29 346 9.95 001 6 2 59 9.65 680 25 9.70 685 32 0.29 315 9.94 995 6 1 60 9.65 705 9.70 717 0.29283 9.94 988 0 | log cos |d log cot | cd log tan I log sin I d I ' Prop. Parts 1530 243~ 333~ 63~ 88 TABLE III 27~ 117~ 207~ 297~ I I I log sin I d log tan cd log cot J log cos I d I Prop. Parts 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.65 705 9.65 729 9.65 754 9.65 779 9.65 804 9.65 828 9.65 853 9.65 878 9.65 902 9.65 927 9.65 952 9.65 976 9.66 001 9.66 025 9.66 050 9.66 075 9.66 099 9.66 124 9.66 148 9.66 173 9.66 197 9.66 221 9.66 246 9.66 270 9.66 295 9.66 319 9.66 343 9.66 368 9.66 392 9.66 416 9.66 441 9.66 465 9.66 489 9.66 513 9.66 537 9.66 562 9.66 586 9.66 610 9.66 634 9.66 658 9.66 682 9.66 706 9.66 731 9.66 755 9.66 779 9.66 803 9.66 827 9.66 851 9.66 875 9.66 899 9.66'922 9.66 946 9.66 970 9.66 994 9.67 018 9.67 042 9.67 066 9.67 090 9.67 113 9.67 137 9.67 161 I 24 25 25 25 24 25 25 24 25 25 24 25 24 25 25 24 25 24 25 24 24 25 24 25 24 24 25 24 24 25 24 24 24 24 25 24 24 24 24 24 24 25 24 24 24 24 24 24 24 23 24 24 24 24 24 24 24 23 24 24 9.70 717 9.70 748 9.70 779 9.70 810 9.70 841 9.70 873 9.70 904 9.70 935 9.70 966 9.70 99'7 9.71 028 9.71 059 9.71 090 9.71 121 9.71 153 9.71 184 9.71 215 9.71 246 9.71 277 9.71 308 9.71 339 9.71 370 9.71 401 9.71 431 9.71 462 9.71 493 9.71 524 9.71 555 9.71 586 9.71 617 9.71 648 9.71 679 9.71 709 9.71 740 9.71 771 9.71 802 9.71 833 9.71 863 9.71 894 9.71 925 9.71 955 9.71 986 9.72 017 9.72 048 9.72 078 9.72 109 9.72 140 9.72 170 9.72 201 9.72 231 9.72 262 9.72 293 9.72 323 9.72 354 9.72 384 9.72 415 9.72 445 9.72 476 9.72 506 9.72 537 9.72 567 I 31 31 31 31 32 31 31 31 31 31 31 31 31 32 31 31 31 31 31 31 31 31 30 31 31 31 31 31 31 31 31 30 31 31 31 31 30 31 31 30 31 31 31 30 31 31 '30 31 30 31 31 30 31 30 31 30 31 30 31 30 0.29 283 0.29 252 0.29 221 0.29 190 0.29 159 0.29 127 0.29 096 0.29 065 0.29 034 0.29 003 0.28 972 0.28 941 0.28 910 0.28 879 0.28 847 0.28 816 0.28 785 0.28 754 0.28 723 0.28 692 0.28 661 0.28 630 0.28 599 0.28 569 0.28 538 0.28 507 0.28 476 0.28 445 0.28 414 0.28 383 0.28 352 0.28 321 0.28 291 0.28 260 0.28 229 0.28 198 0.28 167 0.28 137 0.28 106 0.28 075 0.28 045 0.28 014 0.27 983 0.27 952 0.27 922 0.27 891 0.27 860 0.27 830 0.27 799 0.27 769 0.27 738 0.27 707 0.27 677 0.27 646 0.27 616 0.27 585 0.27 555 0.27 524 0.27 494 0.27 463 0.27 433 9.94 988 9.94 982 9.94 975 9.94 969 9.94 962 9.94 956 9.94 949 9.94 943 9.94 936 9.94 930 9.94 923 9.94 917 9.94 911 9.94 904 9.94 898 9.94 891 9.94 885 9.94 878 9.94 871 9.94 865 9.94 858 9.94 852 9 94 845 9o94 839 9.94 832 9.94 826 9.94 819 9.94 813 9.94 806 9.94 799 9.94 793 9.94 786 9.94 780 9.94 773 9.94 767 9.94 760 9.94 753 9.94 747 9.94 740 9.94 734 9.94 727 9.94 720 9.94 714 9.94 707 9.94 700 9.94 694 9.94 687 9.94 680 9.94 674 9.94 667 9.94 660 9.94 654 9.94 647 9.94 640 9.94 634 9.94 627 9.94 620 9.94 614 9.94 607 9.94 600 9.94 593 6 7 6 7 6 7 I I 6 7 6 7 6 6 7 6 7 6 7 7 6 7 6 7 6 7 6 7 6 7 7 6 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 7 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 I 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 32 0.5 1.1 1.6 2.1 2.7 3.2 3.7 4.3 4.8 5.3 10.7 16.0 21.3 26.7 25 0.4 0.8 1.2 1.7 2.1 2.5 2.9 3.3 3.8 4.2 8.3 12.5 16.7 20.8 31 0.5 1.0 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.2 10.3 15.5 20.7 25.8 24 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 8.0 12.0 16.0 20.0 30 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10.0 15.0 20.0 25.0 23 0.4 0.8 1.2 1.5 1.9 2.3 2.7 3.1 3.4 3.8 7.7 11.5 15.3 19.2 1 2 3 4 5 6 7 8 9 10 20 30 40 50 7 6 0.1 0.1 0.2 0.2 0.4 0.3 0.5 0.4 0.6 0.5 0.7 0.6 0.8 0.7 0.9 0.8 1.0 0.9 1.2 1.0 2.3 2.0 3.5 3.0 4.7 4.0 5.8 5.0 I I - I Ilogcos dI logcot lcdl log tan log sin I d Prop. Parts I 152~ 242~ 3320 62" 89 TABLE III 28~ 118~ 208~ 298~ log sin I d I logtan I c d Ilog cot I logcos d I Prop. Parts....... -...... P 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.67 161 9.67 185 9.67 208 9.67 232 9.67 256 9.67 280 9.67 303 9.67 327 9.67 350 9.67 374 9.67 398 9.67 421 9.67 445 9.67 468 9.67 492 9.67 515 9.67 539 9.67 562 9.67 586 9.67 609 9.67 633 9.67 656 9.67 680 9.67 703 9.67 726 9.67 750 9.67 773 9.67 796 9.67 820 9.67 843 9.67 866 9.67 890 9.67 913 9.67 936 9.67 959 9.67 982 9.68 006 9.68 029 9.68 052 9.68 075 9.68 098 9.68 121 9.68 144 9.68 167 9.68 190 9.68 213 9.68 237 9.68 260 9.68 283 9.68 305 9.68 328 9.68 351 9.68 374 9.68 397 9.68 420 9.68 443 9.68 466 9.68 489 9.68 512 9.68 534 9.68 557 24 23 24 24 24 23 24 23 24 24 23 24 23 24 23 24 23 24 23 24 23 24 23 23 24 23 23 24 23 23 24 23 23 23 23 24 23 23 23 23 23 23 23 23 23 24 23 23 22 23 23 23 23 23 23 23 23 23 22 23 9.72 567 9.72 598 9.72 628 9.72 659 9.72 689 9.72 720 9.72 750 9.72 780 9.72 811 9.72 841 9.72 872 9.72 902 9.72 932 9.72 963 9.72 993 9.73 023 9.73 054 9.73 084 9.73 114 9.73 144 9.73 175 9.73 205 9.73 235 9.73 265 9.73 295 9.73 326 9.73 356 9.73 386 9.73 416 9.73 446 9.73 476 9.73 507 9.73 537 9.73 567 9.73 597 9.73 627 9.73 657 9.73 687 9.73 717 9.73 747 9.73 777 9.73 807 9.73 837 9.73 867 9.73 897 9.73 927 9.73 957 9.73 987 9.74 017 9.74 047 9.74 077 9.74 107 9.74 137 9.74 166 9.74 196 9.74 226 9.74 256 9.74 286 9.74 316 9.74 345 9.74 375 31 30 31 30 31 30 30 31 30 31 30 30 31 30 30 31 30 30 30 31 30 30 30 30 31 30 30 30 30 30 31 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 30 30 29 30 0.27 433 0.27 402 0.27 372 0.27 341 0.27 311 0.27 280 0.27 250 0.27 220 0.27 189 0.27 159 0.27 128 0.27 098 0.27 068 0.27 037 0.27 007 0.26 977 0.26 946 0.26 916 0.26 886 0.26 856 0.26 825 0.26 795 0.26 765 0.26 735 0.26 705 0.26 674 0.26 644 0.26 614 0.26 584 0.26 554 0.26 524 0.26 493 0.26 463 0.26 433 0.26 403 0.26 373 0.26 343 0.26 313 0.26 283 0.26 253 0.26 223 0.26 193 0.26 163 0.26 133 0.26 103 0.26 073 0.26 043 0.26 013 0.25 983 0.25 953 0.25 923 0.25 893 0.25 863 0.25 834 0.25 804 0.25 774 0.25 744 0.25 714 0.25 684 0.25 655 0.25 625 9.94 593 9.94 587 9.94 580 9.94 573 9.94 567 9.94 560 9.94 553 9.94 546 9.94 540 9.94 533 9.94 526 9.94 519 9.94 513 9.94 506 9.94 499 9.94 492 9.94 485 9.94 479 9.94 472 9.94 465 9.94 458 9.94 451 9.94 445 9.94 438 9.94 431 9.94 424 9.94 417 9.94 410 9.94 404 9.94 397 9.94 390 9.94 383 9.94 376 9.94 369 9.94 362 9.94 355 9.94 349 9.94 342 9.94 335 9.94 328 9.94 321 9.94 314 9.94 307 9.94 300 9.94 293 9.94 286 9.94 279 9.94 273 9.94 266 9.94 259 9.94 252 9.94 245 9.94 238 9.94 231 9.94 224 9.94 217 9.94 210 9.94 203 9.94 196 9.94 189 9.94 182 6 7 7 6 7 7 7 6 7 7 7 6 7 7 7 7 6 7 7 7 7 6 7 7 7 7 7 6 7 7 7 7 7 7 7 6 7 7 7 7 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 31 05 1.0 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.2 10.3 15.5 20.7 25.8 24 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 8.0 12.0 16.0 20.0 30 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10.0 15.0 20.0 25.0 23 0.4 0.8 1.2 1.5 1.9 2.3 2.7 3.1 3.4 3.8 7.7 11.5 15.3 19.2 29 0.5 1.0 1.4 1.9 2.4 2.9 3.4 3.9 4.4 4.8 9.7 14.5 19.3 24.2 22 0.4 0.7 1.1 1.5 1.8 2.2 2.6 2.9 3.3 3.7 7.3 11.0 14.7 18.3 7 7 7 7 7 7 6 7 7 7 7 7 7 7 7 7 7 7 7 7 1 2 3 4 5 6 7 8 9 10 20 30 40 50 7 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 2.3 3.5 4.7 5.8 6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 | log cos d | log cot Icd logtan I log sin | d ' Prop. Parts,~O21 309 151~ 241 331 61~ 90 TABLE III 29~ 119~ 209~ 299~ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I log sin d log tan cd log cot l s I d I Prop. Parts 9 68 557 9.68 580 9.68 603 9.68 625 9.68 648 9.68 671 9.68 694 9.68 716 9.68 739 9.68 762 9.68 784 9.68 807 9.68 829 9.68 852 9.68 875 9.68 897 9.68 920 9:68 942 9.68 965 9.68 987 9.69 010 9.69 032 9.69 055 9.69 077 9.69 100 9.69 122 9.69 144 9.69 167 9.69 189 9.69 212 9.69 234 9.69 256 9.69 279 9.69 301 9.69 323 9.69 345 9.69 368 9.69 390 9.69 412 9.69 434 9.69 456 9.69 479 9.69 501 9.69 523 9.69 545 9.69 567 9.69 589 9.69 611 9.69 633 9.69 655 9.69 677 9.69 699 9.69 721 9.69 743 9.69 765 9.69 787 9.69 809 9.69 831 9.69 853 9.69 875 9.69 897 I 23 23 22 23 23 23 22 23 23 22 23 22 23 23 22 23 22 23 22 23 22 23 22 23 22 22 23 22 23 22 22 23 22 22 22 23 22 22 22 22 23 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 9.74 375 9.74 405 9.74 435 9.74 465 9.74 494 9.74 524 9.74 554 9.74 583 9.74 613 9.74 643 9.74 673 9.74 702 9.74 732 9.74 762 9.74 791 9.74 821 9.74 85.1 9.74 880 9.74 910 9.74 939 9.74 969 9.74 998 9.75 028 9.75 058 9.75 087 9.75 117 9.75 146 9.75 176 9.75 205 9.75 235 9.75 264 9.75 294 9.75 323 9.75 353 9.75 382 9.75 411 9.75 441 9.75 470 9.75 500 9.75 529 9.75 558 9.75 588 9.75 617 9.75 647 9.75 676 9.75 705 9.75 735 9.75 764 9.75 793 9.75 822 9.75 852 9.75 881 9.75 910 9.75 939 9.75 969 9.75 998 9.76 027 9.76 056 9.76 086 9.76 115 9.76 144 I I I 30 30 30 29 30 30 29 30 30 30 29 30 30 29 30 30 29 30 29 30 29 30 30 29 30 29 30 29 30 29 30 29 30 29 29 30 29 30 29 29 30 29 30 29 29 30 29 29 29 30 29 29 29 30 29 29 29 30 29 29 0.25 625 0.25 595 0.25 565 0.25 535 0.25 506 0.25 476 0.25 446 0.25 417 0.25 387 0.25 357 0.25 327 0.25 298 0.25 268 0.25 238 0.25 209 0.25 179 0.25 149 0.25 120 0.25 090 0.25 061 0.25 031 0.25 002 0.24 972 0.24 942 0.24 913 0.24 883 0.24 854 0.24 824 0.24 795 0.24 765 0.24 736 0.24 706 0.24 677 0.24 647 0.24 618 0.24 589 0.24 559 0.24 530 0.24 500 0.24 471 0.24 442 0.24 412 0.24 383 0.24 353 0.24 324 0.24 295 0.24 265 0.24 236 0.24 207 0.24 178 0.24 148 0.24 119 0.24 090 0.24 061 0.24 031 0.24 002 0.23 973 0.23 944 0.23 914 0.23 885 0.23 856 9.94 182 9.94 175 9.94 168 9.94 161 9.94 154 9.94 147 9.94 140 9.94 133 9.94 126 9.94 119 9.94 112 9.94 105 9.94 098 9.94 090 9.94 083 9.94 076 9.94 069 9.94 062 9.94 055 9.94 048 9.94 041 9.94 034 9.94 027 9.94 020 9.94 012 9.94 005 9.93 998 9.93 991 9.93 984 9.93 977 9.93 970 9.93 963 9.93 955 9.93 948 9.93 941 9.93 934 9.93 927 9.93 920 9.93 912 9.93 905 9.93 898 9.93 891 9.93 884 9.93 876 9.93 869 9.93 862 9.93 855 9.93 847 9.93 840 9.93 833 9.93 826 9.93 819 9.93 811 9.93 804 9.93 797 9.93 789 9.93 782 9.93 775 9.93 768 9.93 760 9.93 753 7 7 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 8 7 7 7 7 7 8 7 7 7 7 8 7 7 7 8 7 7 7 7 8 7 7 8 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50!30 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10.0 15.0 20.0 25.0 23 1 0.4 2 0.8 3 1.2 4 1.5 5 1.9 6 2.3 7 2.7 8 3.1 9 3.4 10 3.8 20 7.7 30 11.5 40 15.3 50 19.2 29 0.5 1.0 1.4 1.9 2.4 2.9 3.4 3.9 4.4 4.8 9.7 14.5 19.3 24.2 22 0.4 0.7 1.1 1.5 1.8 2.2 2.6 2.9 3.3 3.7 7.3 11.0 14.7 18.3 7 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 2.3 3.5 4.7 5.8 1 2 3 4 5 6 7 8 9 10 20 30 40 50 8 0.1 0.3' 0.4 0.5 0.7 0.8 0.9 1.1 1.2 1.3 2.7 4.0 5.3 6.7 7 7 7 8 7 I I I I log cos d j log cot c Icd log tan log sin d ' Prop. Parts 150~ 240~ 330~ 60~ 91 TABLE III 30~ 120~ 210~ 300~ Prop. Parts log sin I d j log tan c dI log cot log cos I d 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.69 897 9.69 919 9.69 941 9.69 963 9.69 984 9.70 006 9.70 028 9.70 050 9.70 072 9.70 093 9.70 115 9.70 137 9.70 159 9.70 180 9.70 202 9.70 224 9.70 245 9.70 267 9.70 288 9.70 310 9.70 332 9.70 353 9.70 375 9.70 396 9.70 418 9.70 439 9.70 461 9.70 482 9.70 504 9.70 525 9.70 547 9.70 568 9.70 590 9.70 611 9.70 633 9.70 654 9.70 675 9.70 697 9.70 718 9.70 739 9.70 761 9.70 782 9.70 803 9.70 824 9.70 846 9.70 867 9.70 888 9.70 909 9.70 931 9.70 952 9.70 973 9.70 994 9.71 015 9.71 036 9.71 058 9.71 079 9.71 100 9.71 121 9.71 142 9.71 163 9.71 184 22 22 22 21 22 22 22 22 21 22 22 22 21 22 22 21 22 21 22 22 21 22 21 22 21 22 21 22 21 22 21 22 21 22 21 21 22 21 21 22 21 21 21 22 21 21 21 22 21 21 21 21 21 22 21 21 21 21 21 21 9.76 144 9.76 173 9.76 202 9.76 231 9.76 261 9.76 290 9.76 319 9.76 348 9.76 377 9.76 406 9.76 435 9.76 464 9.76 493 9.76 522 9.76 551 9.76 580 9.76 609 9.76 639 9.76 668 9.76 697 9.76 725 9.76 754 9.76 783 9.76 812 9.76 841 9.76 870 9.76 899 9.76 928 9.76 957 9.76 986 9.77 015 9.77 044 9.77 073 9.77 101 9.77 130 9.77 159 9.77 188 9.77 217 9.77 246 9.77 274 9.77 303 9.77 332 9.77 361 9.77 390 9.77 418 9.77 447 9.77 476 9.77 505 9.77 533 9.77 562 9.77 591 9.77 619 9.77 648 9.77 677 9.77 706 9.77 734 9.77 763 9.77 791 9.77 820 9.77 849 9.77 877 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 30 29 29 28 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 29 29 28 29 29 29 29 28 29 29 29 28 29 29 28 29 29 29 28 29 28 29 29 28 0.23 856 0.23 827 0.23 798 0.23 769 0.23 739 0.23 710 0.23 681 0.23 652 0.23 623 0.23 594 0.23 565 0.23 536 0.23 507 0.23 478 0.23 449 0.23 420 0.23 391 0.23 361 0.23 332 0.23 303 0.23 275 0.23 246 0.23 217 0.23 188 0.23 159 0.23 130 0.23 101 0.23 072 0.23 043 0.23 014 0.22 985 0.22 956 0.22 927 0.22 899 0.22 870 0.22 841 0.22 812 0.22 783 0.22 754 0.22 726 0.22 697 0.22 668 0.22 639 0.22 610 0.22 582 0.22 553 0.22 524 0.22 495 0.22 467 0.22 438 0.22 409 0.22 381 0.22 352 0.22 323 0.22 294 0.22 266 0.22 237 0.22 209 0.22 180 0.22 151 0.22 123 9.93 753 9.93 746 9.93 738 9.93 731 9.93 724 9.93 717 9.93 709 9.93 702 9.93 695 9.93 687 9.93 680 9.93 673 9.93 665 9.93 658 9.93 650 9.93 643 9.93 636 9.93 628 9.93 621 9.93 614 9.93 606 9.93 599 9.93 591 9.93 584 9.93 577 9.93 569 9.93 562 9.93 554 9.93 547 9.93 539 9.93 532 9.93 525 9.93 517 9.93 510 9.93 502 9.93 495 9.93 487 9.93 480 9.93 472 9.93 465 9.93 457 9.93 450 9.93 442 9.93 435 9.93 427 9.93 420 9.93 412 9.93 405 9.93 397 9.93 390 9.93 382 9.93 375 9.93 367 9.93 360 9.93 352 9.93 344 9.93 337 9.93 329 9.93 322 9.93 314 9.93 307 7 8 7 7 7 8 7 7 8 7 7 8 7 8 7 7 8 7 7 8 7 8 7 7 8 7 8 7 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 8 7 8 7 8 7 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 2 3 4 5 6 7 8 9 10 20 30 40 50 30 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10.0 15.0 20.0 25.0 29 0.5 1.0 1.4 1.9 2.4 2.9 3.4 3.9 4.4 4.8 9.7 14.5 19.3 24.2 28 0.5 0.9 1.4 1.9 2.3 2.8 3.3 3.7 4.2 4.7 9.3 14.0 18.7 23.3 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 22 0.4 0.7 1.1 1.5 1.8 2.2 2.6 2.9 3.3 3.7 7.3 11.0 14.7 18.3 21 0.4 0.7 1.0 1.4 1.8 2.1 2.4 2.8 3.2 3.5 7.0 10.5 14.0 17.5 8 7 0.1 0.1 0.3 0.2 0.4 0.4 0.5 0.5 0.7 0.6 0.8 0.7 0.9 0.8 1.1 0.9 1.2 1.0 1.3 1.2 2.7 2.3 4.0 3.5 5.3 4.7 6.7 5.8 Ilog cos d I log cot |lcd log tan I log sin I d f I Prop. Parts j4029039 509 149~ 239~ 329~ 59~ 92 TABLE III 31~ 121~ 2110 301~ ' I log sin d log tan I cd I log cot I I I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.71 184 9.71 205 9.71 226 9.71 247 9.71 268 9.71 289 9.71 310 9.71 331 9.71 352 9.71 373 9.71 393 9.71 414 9.71 435 9.71 456 9.71 477 9.71 498 9.71 519 9.71 539 9.71 560 9.71 581 9.71 602 9.71 622 9.71 643 9.71 664 9.71 685 9.71 705 9.71 726 9.71 747 9.71 767 9.71 788 9.71 809 9.71 829 9.71 850 9.71 870 9.71 891 9.71 911 9.71 932 9.71 952 9.71 973 9.71 994 9.72 014 9.72 034 9.72 055 9.72 075 9.72 096 9.72 116 9.72 137 9.72 157 9.72 177 9.72 198 9.72 218 9.72 238 9.72 259 9.72 279 9.72 299 9.72 320 9.72 340 9.72 360 9.72 381 9.72 401 9.72 421 21 21 21 21 21 21 21 21 21 20 21 21 21 21 21 21 20 21 21 21 20 21 21 21 20 21 21 20 21 21 20 21 20 21 20 21 20 21 21 20 20 21 20 21 20 21 20 20 21 20 20 21 20 20 21 20 20 21 20 20 9.77 877 9.77 906 9.77 935 9.77 963 9.77 992 9.78 020 9.78 049 9.78 077 9.78 106 9.78 135 9.78 163 9.78 192 9.78 220 9.78 249 9.78 277 9.78 306 9.78 334 9.78 363 9.78 391 9.78 419 9.78 448 9.78 476 9.78 505 9.78 533 9.78 562 9.78 590 9.78 618 9.78 647 9.78 675 9.78 704 9.78 732 9.78 760 9.78 789 9.78 817 9.78 845 9.78 874 9.78 902 9.78 930 9.78 959 9.78 987 9.79 015 9.79 043 9.79 072 9.79 100 9.79 128 9.79 156 9.79 185 9.79 213 9.79 241 9.79 269 9.79 297 9.79 326 9.79 354 9.79 382 9.79 410 9.79 438 9.79 466 9.79 495 9.79 523 9.79 551 9.79 579 I 29 29 28 29 28 29 28 29 29 28 29 28 29 28 29 28 29 28 28 29 28 29 28 29 28 28 29 28 29 28 28 29 28 28 29 28 28 29 28 28 28 29 28 28 28 29 28 28 28 28 29 28 28 28 28 28 29 28 28 28 0.22 123 0.22 094 0.22 065 0.22 037 0.22 008 0.21 980 0.21 951 0.21 923 0.21 894 0.21 865 0.21 837 0.21 808 0.21 780 0.21 751 0.21 723 0.21 694 0.21 666 0.21 637 0.21 609 0.21 581 0.21 552 0.21 524 0.21 495 0.21 467 0.21 438 0.21 410 0.21 382 0.21 353 0.21 325 0.21 296 0.21 268 0.21 240 0.21 211 0.21 183 0.21 155 0.21 126 0.21 098 0.21 070 0.21 041 0.21 013 0.20 985 0.20 957 0.20 928 0.20 900 0.20 872 0.20 844 0.20 815 0.20 787 0.20 759 0.20 731 0.20 703 0.20 674 0.20 646 0.20 618 0.20 590 0.20 562 0.20 534 0.20 505 0.20 477 0.20 449 0.20 421 log cos d 9.93 307 9.93 299 9.93 291 9.93 284 9.93 276 9.93 269 9.93 261 9.93 253 9.93 246 8 9.93 238 8 9.93 230 9.93 223 9.93 215 9.93 207 9.93 200 8 9.93 192 9.93 1847 9.93 177 9.93 169 9.93 161 9.93 154 9.93 146 9.93 138 9.93 131 8 9.93 123 8 9.93 115 9.93 108 8 9.93 100 8 9.93 092 8 9.93 084 9.93 077 9.93 069 8 9.93 061 8 9.93 053 9.93 046 8 9.93 038 8 9.93 030 8 9.93 022 8 9.93 014 7 9.93 007 8 9.92 999 8 9.92 991 8 9.92 983 7 9.92 976 8 9.92 968 8 9.92 960 8 9.92 952 8 9.92 944 8 9.92 936 7 9 92 929 8 9.92 921 8 9.92 913 8 9.92 905 8 9.92 897 8 9.92 889 8 9.92 881 7 9.92 874 8 9.92 866 8 9.92 858 8 9.92 850 8 9.92 842 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 I I Prop. Parts 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 29 28 0.5 0.5 1.0 0.9 1.4 1.4 1.9 1.9 2.4 2.3 2.9 2.8 3.4 3.3 3.9 3.7 4.4 4.2 4.8 4.7 9.7 9.3 14.5 14.0 19.3 18.7 24.2 23.3 21 20 0.4 0.3 0.7 0.7 1.0 1.0 1.4 1.3 1.8 1.7 2.1 2.0 2.4 2.3 2.8 2.7 3.2 3.0 3.5 3.3 7.0 6.7 10.5 10.0 14.0 13.3 17.5 16.7 8 7 0.1 0.1 0.3 0.2 0.4 0.4 0.5 0.5 0.7 0.6 0.8 0.7 0.9 0.8 1.1 0.9 1.2 1.0 1.3 1.2 2.7 2.3 4.0 3.5 5.3 4.7 6.7 5.8 7 6 5 4 3 2 1 0 I I log cos I d I log cot ] cdl log tan [ log sin I d | ' I Prop. Parts 1480 sld 2380o 3280i 4d8 93 148~ 238~ 328~ 58~ 93 TABLE III 32~ 122~ 212~ 302~ I log sin d I log tan cd log cot log cos d I Prop. Parts 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.72 421 9.72 441 9.72 461 9.72 482 9.72 502 9.72 522 9.72 542 9.72 562 9.72 582 9.72 602 9.72 622 9.72 643 9.72 663 9.72 683 9.72 703 9.72 723 9.72 743 9.72 763 9.72 783 9.72 803 9.72 823 9.72 843 9.72 863 9.72 883 9.72 902 9.72 922 9.72 942 9.72 962 9.72 982 9.73 002 9.73 022 9.73 041 9.73 061 9.73 081 9.73 101 9.73 121 9.73 140 9.73 160 9.73 180 9.73 200 9.73 219 9.73 239 9.73 259 9.73 278 9.73 298 9.73 318 9.73 337 9.73 357 9.73 377 9.73 396 9.73 416 9.73 435 9.73 455 9.73 474 9.73 494 9.73 513 9.73 533 9.73 552 9.73 572 9.73 591 9.73 611 20 20 21 20 20 20 20 20 20 20 21 20 20 20 20 20 20 20 20 20 20 20 20 19 20 20 20 20 20 20 19 20 20 20 20 19 20 20 20 19 20 20 19 20 20 19 20 20 19 20 19 20 19 20 19 20 19 20 19 20 9.79 579 9.79 607 9.79 635 9.79 663 9.79 691 9.79 719 9.79 747 9.79 776 9.79 804 9.79 832 9.79 860 9.79 888 9.79 916 9.79 944 9.79 972 9.80 000 9.80 028 9.80 056 9.80 084 9.80 112 9.80 140 9.80 168 9.80 195 9.80 223 9.80 251 9.80 279 9.80 307 9.80 335 9.80 363 9.80 391 9.80 419 9.80 447 9.80 474 9.80 502 9.80 530 9.80 558 9.80 586 9.80 614 9.80 642 9.80 669 9.80 697 9.80 725 9.80 753 9.80 781 9.80 808 9.80 836 9.80 864 9.80 892 9.80 919 9.80 947 9.80 975 9.81 003 9.81 030 9.81 058 9.81 086 9.81 113 9.81 141 9.81 169 9.81 196 9.81 224 9.81 252 28 0.20 421 0.20 393 28 0.20 365 28 0.20 337 28 0.20 309 0.20 281 29 0.20 253 28 0.20 224 28 0.20 196 28 0.20 168 28 0.20 140 28 0.20 112 28 0.20 084 20.20 056 28 0.20 028 0.20 000 28 0.19 972 28 0.19 944 0.19 916 28 0.19 888 28 0. 19 860 7 0.19 832 28 0.19805 28 0.19 777 28 0.19749 28 0.19 721 28 0.19693 28 0.19 665 28 0.19637 28 0.19 609 28 0.19581 27 0.19 553 28 0.19526 28 0.19 498 28 0.19 470 28 0.19442 28 0.19 414 28 0.19 386 27 0.19 358 28 0.19 331 28 0.19 303 28 0.19275 28 0.19 247 27 0.19 219 28 0.19 192 28 0.19 164 28 0.19 136 27 0.19 108 28 0.19 081 28 0.19 053 28 0.19 025 27 0.18997 28 0.18 970 28 0.18 942 27 0.18914 28 0.18 887 28 0.18859 27 0.18 831 28 0.18 804 28 0.18 776 0. 18 748 9.92 842 9.92 834 9.92 826 9.92 818 9.92 810 9.92 803 9.92 795 9.92 787 9.92 779 9.92 771 9.92 763 9.92 755 9.92 747 9.92 739 9.92 731 9.92 723 9.92 715 9.92 707 9.92 699 9.92 691 9.92 683 9.92 675 9.92 667 9.92 659 9.92 651 9.92 643 9.92 635 9.92 627 9.92 619 9.92 611 9.92 603 9.92 595 9.92 587 9.92 579 9.92 571 9.92 563 9.92 555 9.92 546 9.92 538 9.92 530 9.92 522 9.92 514 9.92 506 9.92 498 9.92 490 9.92 482 9.92 473 9.92 465 9.92 457 '9.92 449 9.92 441 9.92 433 9.92 425 9.92 416 9.92 408 9.92 400 9.92 392 9.92 384 9.92 376 9.92 367 9.92 359 8 8 8 8 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 8 8 8 8 8 8 8 8 9 8 8 8 8 8 8 9 8 8 8 8 8 9 8 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 29 0.5 1.0 1.4 1.9 2.4 2.9 3.4 3.9 4.4 4.8 9.7 14.5 19.3 24.2 21 0.4 0.7 1.0 1.4 1.8 2.1 2.4 2.8 3.2 3.5 7.0 10.5 14.0 17.5 28 0.5 0.9 1.4 1.9 2.3 2.8 3.3 3.7 4.2 4.7 9.3 14.0 18.7 23.3 20 0.3 0.7 1.0 1.3 1.7 2.0 2.3 2.7 3.0 3.3 6.7 10.0 13.3 16.7 27 0.4 0.9 1.4 1.8 2.2 2.7 3.2 3.6 4.0 4.5 9.0 13.5 18.0 22.5 19 0.3 0.6 1.0 1.3 1.6 1.9 2.2 2.5 2.8 3.2 6.3 9.5 12.7 15.8 9 8 7 0.2 0.1 0.1 0.3 0.3 0.2 0.4 0.4 0.4 0.6 0.5 0.5 0.8 0.7 0.6 0.9 0.8 0.7 1.0 0.9 0.8 1.2 1.1 0.9 1.4 1.2 1.0 1.5 1.3 1.2 3.0 2.7 2.3 4.5 4.0 3.5 6.0 5.3 4.7 7.5 6.7 5.8 I log cos I d log cot cd log tan log sin d ' I Prop. Parts 147~ 237~ 327~ 57~ 94 TABLE III 33~ 123~ 213~ 303~ r ' I log sin d c r10 LO U, d|I log cot I log cos I dl Prop. Parts,, I I --- — -, I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.73 611 9.73 630 9.73 650 9.73 669 9.73 689 9.73 708 9.73 727 9.73 747 9.73 766 9.73 785 9.73 805 9.73 824 9.73 843 9.73 863 9.73 882 9.73 901 9.73 921 9.73 940 9.73 959 9.73 978 9.73 997 9.74 017 9.74 036 9.74 055 9.74 074 9.74 093 9.74 113 9.74 132 9.74 151 9.74 170 9.74 189 9.74 208 9.74 227 9.74 246 9.74 265 9.74 284 9.74 303 9.74 322 9.74 341 9.74 360 9.74 379 9.74 398 9.74 417 9.74 436 9.74 455 9.74 474 9.74 493 9.74 512 9.74 531 9.74 549 9.74 568 9.74 587 9.74 606 9.74 625 9.74 644 9.74 662 9.74 681 9.74 700 9.74 719 9.74 737 9.74 756 19 20 19 20 19 19 20 19 19 20 19 19 20 19 19 20 19 19 19 19 20 19 19 19 19 20 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18 19 IS 1i In 1I I 9.81 252 9.81 279 9.81 307 9.81 335 9.81 362 9.81 390 9.81 418 9.81 445 9.81 473 9.81 500 9.81 528 9.81 556 9.81 583 9.81 611 9.81 638 9.81 666 9.81 693 9.81 721 9.81 748 9.81 776 9.81 803 9.81 831 9.81 858 9.81 886 9.81 913 9.81 941 9.81 968 9.81 996 9.82 023 9.82 051 9.82 078 9.82 106 9.82 133 9.82 161 9.82 188 9.82 215 9.82 243 9.82 270 9.82 298 9.82 325 9.82 352 9.82 380 9.82 407 9.82 435 9.82 462 9.82 489 9.82 517 9.82 544 9.82 571 9.82 599 9.82 626 9.82 653 9.82 681 9.82 708 9.82 735 9.82 762 9.82 790 9.82 817 9.82 844 9.82 871 9.82 899 27 28 28 27 28 28 27 28 27 28 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 27 28 27 28 27 27 28 27 28 27 27 28 27 27 28 27 27 28 27 27 27 2E 27 2i 2E 2^ I 0.18 748 0.18 721 0.18 693 0.18 665 0.18 638 0.18 610 0.18 582 0.18 555 0.18 527 0.18 500 0.18 472 0. 18 444 0.18 417 0.18 389 0.18 362 0.18 334 0.18 307 0.18 279 0.18 252 0.18 224 0.18 197 0.18 169 0.18 142 0.18 114 0.18 087 0.18 059 0. 18 032 0.18 004 0.17 977 0.17 949 0. 17 922 0.17 894 0.17 867 0.17 839 0.17 812 0.17 785 0.17 757 0.17 730 0.17 702 0.17 675 0.17 648 0.17 620 0.17 593 0.17 565 0.17 538 0.17 511 0.17483 0.17 456 0. 17 429 0.17 401 0.17 374 0.17 347 0.17 319 0.17 292 0.17 265 0.17 238 0.17 210 0.17 183 70.17 156 0.17 129 0.17 101 9.92 359 9.92 351 9.92 343 9.92 335 9.92 326 9.92 318 9.92 310 9.92 302 9.92 293 9.92 285 9.92 277 9.92 269 9.92 260 9.92 252 9.92 244 9.92 235 9.92 227 9.92 219 9.92 211 9.92 202 9.92 194 9.92 186 9.92 177 9.92 169 9.92 161 9.92 152 9.92 144 9.92 136 9.92 127 9.92 119 9.92 111 9.92 102 9.92 094 9.92 086 9.92 077 9.92 069 9.92 060 9.92 052 9.92 044 9.92 035 9.92 027 9.92 018 9.92 010 9.92 002 9.91 993 9.91 985 9.91 976 9.91 968 9.91 959 9.91 951 9.91 942 9.91 934 9.91 925 9.91 917 9.91 908 9.91 900 9.91 891 9.91 883 9.91 874 9.91 866 9.91 857 8 8 8 9 8 8 8 9 8 8 8 9 8 8 9 8 8 8 9 8 8 9 8 8 9 8 8 9 8 8 9 8 8 9 8 9 8 8 9 8 9 8 8 9 8 I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 20 30 40 50 28 0.5 0.9 1.4 1.9 2.3 2.8 3.3 3.7 4.2 4.7 9.3 14.0 18.7 23.3 I 27 0.4 0.9 1.4 1.8 2.2 2.7 3.2 3.6 4.0 4.5 9.0 13.5 18.0 22.5 1 2 3 4 5 6 7 8 9 10 20 30 40 50 20 0.3 0.7 1.0 1.3 1.7 2.0 2.3 2.7 3.0 3.3 6.7 10.0 13.3 16.7 19 0.3 0.6 1.0 1.3 1.6 1.9 2.2 2.5 2.8 3.2 6.3 9.5 12.7 15.8 18 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 6.0 9.0 12.0 15.0 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 2 3 4 5 6 7 8 9 10 20 30 40 50 9 0.2 0.3 0.4 0.6 0.8 0.9 1.0 1.2 1.4 1.5 3.0 4.5 6.0 7.5 8 0.1 0.3 0.4 0.5 0.7 0.8 0.9 1.1 1.2 1.3 2.7 4.0 5.3 6.7 I I I 0 I I I ~ _~~~~~~~~~ 1 I logcos d log cot cd log tan I log sin I d ' I Prop. Parts I 146~ 236~ 326~ 56" TABLE III 34~ 124~ 214~ 304~ Ilog sin d Ilog tan cd log cot Ilog cos d Prop. Parts 9.7 756 1 ----- 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.74 756 9.74 775 9.74 794 9.74 812 9.74 831 9.74 950 9.74 868 9.74 887 9.74 906 9.74 924 9.74 943 9.74 961 9.74 980 9.74 999 9.75 017 9.75 036 9.75 054 9.75 073 9.75 091 9.75 110 9.75 128 9.75 147 9.75 165 9.75 184 9.75 202 9.75 221 9.75 239 9.75 258 9.75 276 9.75 294 9.75 313 9.75 331 9.75 350 9.75 368 9.75 386 9.75 405 9.75 423 9.75 441 9.75 459 9.75 478 9.75 496 9.75 514 9.75 533 9.75 551 9.75 569 9.75 587 9.75 605 9.75 624 9.75 642 9.75 660 9.75 678 9.75 696 9.75 714 9.75 733 9.75 751 9.75 769 9.75 787 9.75 805 9.75 823 9.75 841 9.75 859 19 19 18 19 19 18 19 19 18 19 18 19 19 18 19 18 19 18 19 18 19 18 19 18 19 18 19 18 18 19 18 19 18 18 19 18 18 18 19 18 18 19 18 18 18 18 19 18 18 18 18 18 19 18 18 18 18 18 18 9.82 899 9.82 926 9.82 953 9.82 980 9.83 008 9.83 035 9.83 062 9.83 089 9.83 117 9.83 144 9.83 171 9.83 198 9.83 225 9.83 252 9.83 280 9.83 307 9.83 334 9.83 361 9.83 388 9.83 415 9.83 442 9.83 470 9.83 497 9.83 524 9.83 551 9.83 578 9.83 605 9.83 632 9.83 659 9.83 686 9.83 713 9.83 740 9.83 768 9.83 795 9.83 822 9.83 849 9.83 876 9.83 903 9.83 930 9.83 957 9.83 984 9.84 011 9.84 038 9.84 065 9.84 092 9.84 119 9.84 146 9.84 173 9.84 200 9.84 227 9.84 254 9.84 280 9.84 307 9.84 334 9.84 361 9.84 388 9.84 415 9.84 442 9.84 469 9.84 496 9.84 523 27 27 27 28 27 27 27 28 27 27 27 27 27 28 27 27 27 27 27 27 28 27 27 27 27 27 27 27 27 27 27 28 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 26 27 27 27 27 27 27 27 27 27 0.17 101 0.17 074 0.17 047 0.17 020 0.16 992 0.16 965 0.16 938 0.16 911 0.16 883 0.16 856 0.16 829 0.16 802 0.16 775 0.16 748 0.16 720 0.16 693 0.16 666 0.16 639 0.16 612 0.16 585 0.16 558 0.16 530 0.16 503 0.16 476 0.16 449 0. 16 422 0. 16 395 0.16 368 0.16 341 0.16 314 0.16 287 0.16 260 0.16 232 0.16 205 0.16 178 0.16 151 0.16 124 0.16 097 0. 16 070 0.16 043 0.16 016 0.15 989 0.15 962 0.15 935 0. 15 908 0.15 881 0.15 854 0.15 827 0.15 800 0.15 773 0.15 746 0.15 720 0.15 693 0.15 666 10.15 639 0. 15 612 0. 15 585 0.15 558 0. 15 531 0.15 504 0.15 477 9.91 857 9.91 849 9.91 840 9.91 832 9.91 823 9.91 815 9.91 806 9.91 798 9.91 789 9.91 781 9.91 772 9.91 763 9.91 755 9.91 746 9.91 738 9.91 729 9.91 720 9.91 712 9.91 703 9.91 695 9.91 686 9.91 677 9.91 669 9.91 660 9.91 651 9.91 643 9.91 634 9.91 625 9.91 617 9.91 608 9.91 599 9.91 591 9.91 582 9.91 573 9.91 565 9.91 556 9.91 547 9.91 538 9.91 530 9.91 521 9.91 512 9.91 504 9.91 495 9.91 486 9.91 477 9.91 469 9.91 460 9.91 451 9.91 442 9.91 433 9.91 425 9.91 416 9.91 407 9.91 398 9.91 389 9.91 381 9.91 372 9.91 363 9.91 354 9.91 345 9.91 336 8 9 8 9 8 9 8 9 8 9 9 8 9 8 9 9 8 9 8 9 9 8 9 9 8 9 9 8 9 9 8 9 9 8 9 9 9 8 9 9 8 9 9 9 8 9 9 9 9 8 9 9 9 9 8 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 2 3 4 5 6 7 8 9 10 20 30 40 50 28 0.5 0.9 1.4 1.9 2.3 2.8 3.3 3.7 4.2 4.7 9.3 14.0 18.7 23.3 27 0.4 0.9 1.4 1.8 2.2 2.7 3.2 3.6 4.0 4.5 9.0 13.5 18.0 22.5 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4.3 8.7 13.0 17.3 21.7 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 19 0.3 0.6 1.0 1.3 1.6 1.9 2.2 2.5 2.8 3.2 6.3 9.5 12.7 15.8 18 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 6.0 9.0 12.0 15.0 9 8 0.2 0.1 0.3 0.3 0.4 0.4 0.6 0.5 0.8 0.7 0.9 0.8 1.0 0.9 1.2 1.1 1.4 1.2 1.5 1.3 3.0 2.7 4.5 4.0 6.0 5.3 7.5 6.7 9 9 9 9 9 I log cos I d log cot c d log tan log sin I ' | Prop. Parts 145~ 235~ 325~ 55~ 96 TABLE III 35~ 125~ 215~ 305~ ' I log sin I d logtan lcd log cot I log os d Prop. Parts 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 [45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.75 859 9.75 877 9.75 895 9.75 913 9.75 931 9.75 949 9.75 967 9.75 985 9.76 003 9.76 021 9.76 039 9.76 057 9.76 075 9.76 093 9.76 111 9.76 129 9.76 146 9.76 164 9.76 182 9.76 200 9.76 218 9.76 236 9.76 253 9.76 271 9.76 289 9.76 307 9.76 324 9.76 342 9.76 360 9.76 378 9.76 395 9.76 413 9.76 431 9.76 448 9.76 466 9.76 484 9.76 501 9.76 519 9.76 537 9.76 554 9.76 572 9.76 590 9.76 607 9.76 625 9.76 642 9.76 660 9.76 677 9.76 695 9.76 712 9.76 730 9.76 747 9.76 765 9.76 782 9.76 800 9.76 817 9.76 835 9.76 852 9.76 870 9.76 887 9.76 904 9.76 922 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 17 18 18 18 18 18 17 18 18 18 17 18 18 18 17 18 18 17 18 18 17 18 18 17 18 18 17 18 17 18 17 18 17 18 17 18 17 18 17 18 9.84 523 9.84 550 9.84 576 9.84 603 9.84 630 9.84 657 9.84 684 9.84 711 9.84 738 9.84 764 9.84 791 9.84 818 9.84 845 9.84 872 9.84 899 9.84 925 9.84 952 9.84 979 9.85 006 9.85 033 9.85 059 9.85 086 9.85 113 9.85 140 9.85 166 9.85 193 9.85 220 9.85 247 9.85 273 9.85 300 9.85 327 9.85 354 9.85 380 9.85 407 9.85 434 9.85 460 9.85 487 9.85 514 9.85 540 9.85 567 9.85 594 9.85 620 9.85 647 9.85 674 9.85 700 9.85 727 9.85 754 9.85 780 9.85 807 9.85 834 9.85 860 9.85 887 9.85 913 9.85 940 9.85 967 9.85 993 9.86 020 9.86 046 9.86 073 9.86 100 9.86 126 27 26 27 27 27 27 27 27 26 27 27 27 27 27 26 27 27 27 27 26 27 27 27 26 27 27 27 26 27 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 26 27 27 26 27 26 27 27 26 0.15 477 0.15 450 0.15 424 0.15 397 0.15 370 0.15 343 0.15 316 0.15 289 0.15 262 0.15 236 0.15 209 0.15 182 0.15 155 0.15 128 0.15 101 0.15 075 0.15 048 0.15 021 0.14 994 0.14 967 0.14 941 0.14 914 0.14 887 0. 14 860 0.14 834 0.14 807 0.14 780 0. 14 753 0.14 727 0.14 700 0.14 673 0. 14 646 0.14 620 0.14 593 0.14 566 0.14 540 0.14 513 0.14 486 0.14 460 0. 14 433 0.14 406 0. 14 380 0.14 353 0. 14 326 0.14 300 0.14 273 0.14 246 0.14 220 0.14 193 0.14 166 0.14 140 0.14 113 0.14 087 0.14 060 0.14 033 0.14 007 0.13 980 0.13 954 0.13 927 0.13 900 0.13 874 9.91 336 9.91 328 9.91 319 9.91 310 9.91 301 9.91 292 9.91 283 9.91 274 9.91 266 9.91 257 9.91 248 9.91 239 9.91 230 9.91 221 9.91 212 9.91 203 9.91 194 9.91 185 9.91 176 9.91 167 9.91 158 9.91 149 9.91 141 9.91 132 9.91 123 9.91 114 9.91 105 9.91 096 9.91 087 9.91 078 9.91 069 9.91 060 9.91 051 9.91 042 9.91 033 9.91 023 9.91 014 9.91 005 9.90 996 9.90 987 9.90 978 9.90 969 9.90 960 9.90 951 9.90 942 9.90 933 9.90 924 9.90 915 9.90 906 9.90 896 9.90 887 9.90 878 9.90 869 9.90 860 9.90 851 9.90 842 9.90 832 9.90 823 9.90 814 9.90 805 9.90 796 8 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 10 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 27 0.4 0.9 1.4 1.8 2.2 2.7 3.2 3.6 4.0 4.5 9.0 13.5 18.0 22.5 18 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 6.0 9.0 12.0 1 15.0 1 17 0.3 0.6 0.8 1.1 1.4 1.7 2.0 2.3 2.6 2.8 5.7 8.5 1.3 14.2 10 0.2 0.3 0.5 0.7 0.8 1.0 1.2 1.3 1.5 1.7 3.3 5.0 6.7 8.3 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4.3 8.7 13.0 17.3 21.7 9 10 20 30 40 50 9 9 9 9 9 9 9 9 9 9 9 9 9 10 9 9 9 9 9 9 10 9 9 9 9 1 2 3 4 5 6 7 8 9 10 20 30 40 50 9 8 0.2 0.1 0.3 0.3 0.4 0.4 0.6 0.5 0.8 0.7 0.9 0.8 1.0 0.9 1.2 1.1 1.4 1.2 1.5 1.3 3.0 2.7 4.5 4.0 6.0 5.3 7.5 6.7 17 18 17 17 18 5 4 3 2 1 0 I log cos d I log cot Icd log tan log sin d l I Prop. Parts 144~ 234~ 324~ 54~ 97 TABLE III 36~ 126~ 216~ 306~ 1 log d log tani cd logtan ogcot I log cos d I Prop. Parts 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.76 922 9.76 939 9.76 957 9.76 974 9.76 991 9.77 009 9.77 026 9.77 043 9.77 061 9.77 078 9.77 095 9.77 112 9.77 130 9.77 147 9.77 164 9.77 181 9.77 199 9.77 216 9.77 233 9.77 250 9.77 268 9.77 285 9.77 302 9.77 319 9.77 336 9.77 353 9.77 270 9.77 387 9.77 405 9.77 422 9.77 439 9.77 456 9.77 473 9.77 490 9.77 507 9.77 524 9.77 541 9.77 558 9.77 575 9.77 592 9.77 609 9.77 626 9.77 643 9.77 660 9.77 677 9.77 694 9.77 711 9.77 728 9.77 744 9.77 761 9.77 778 9.77 795 9.77 812 9.77 829 9.77 846 9.77 862 9.77 879 9.77 896 9.77 913 9.77 930 9.77 946 I 17 18 17 17 18 17 17 18 17 17 17 18 17 17 17 18 17 17 17 18 17 17 17 17 17 I 17 17 18 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 16 17 17 17i 17 17 17 16 17 17 17 17 16 9.86 126 9.86 153 9.86 179 9.86 206 9.86 232 9.86 259 9.86 285 9.86 312 9.86 338 9.86 365 9.86 392 9.86 418 9.86 445 9.86 471 9.86 498 9.86 524 9.86 551 9.86 577 9.86 603 9.86 630 9.86 656 9.86 683 9.86 709 9.86 736 9.86 762 9.86 789 9.86 815 9.86 842 9.86 868 9.86 894 9.86 921 9.86 947 9.86 974 9.87 000 9.87 027 9.87 053 9.87 079 9.87 106 9.87 132 9.87 158 9.87 185 9.87 211 9.87 238 9.87 264 9.87 290 9.87 317 9.87 343 9.87 369 9.87 396 9.87 422 9.87 448 9.87 475 9.87 501 9.87 527 9.87 554 9.87 580 9.87 606 9.87 633 9.87 659 9.87 685 9.87 711 27 26 27 26 27 26 27 26 27 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 26 27 26 26 27 26 27 26 26 27 26 26 27 26 26 27 2C 26 27 26 26 26 27 25 I 0.13 874 0.13 847 0.13 821 0.13 794 0.13 768 0.13 741 0.13 715 0.13 688 0.13 662 0.13 635 0.13 608 0.13 582 0.13 555 0.13 529 0.13 502 0.13 476 0.13 449 0.13 423 0.13 397 0.13 370 0.13 344 0.13 317 0.13 291 0.13264 0.13 238 0.13 211 0.13 185 0.13 158 0.13 132 0.13 106 0.13 079 0.13 053 0.13 026 0.13 000 0.12 973 0.12 947 0.12 921 0.12 894 0.12 868 0.12 842 0.12 815 0.12 789 0.12 762 0.12 736 0.12 710 0.12 683 0.12 657 0.12 631 0.12 604 0.12 578 0.12 552 0.12 525 0.12 499 0. 12 473 0.12 446 0.12 420 0.12 394 0.12 367 0.12 341 0.12 315 0.12 289 9.90 796 9.90 787 9.90 777 9.90 768 9.90 759 9.90 750 9.90 741 9.90 731 9.90 722 9.90 713 9.90 704 9.90 694 9.90 685 9.90 676 9.90 667 9.90 657 9.90 648 9.90 639 9.90 630 9.90 620 9.90 611 9.90 602 9.90 592 9.90 583 9.90 574 9.90 565 9.90 555 9.90 546 9.90 537 9.90 527 9.90 518 9.90 509 9.90 499 9.90 490 9.90 480 9.90 471 9.90 462 9.90 452 9.90 443 9.90 434 9.90 424 9.90 415 9.90 405 9.90 396 9.90 386 9.90 377 9.90 368 9.90 358 9.90 349 9.90 339 9.90 330 9.90 320 9.90 311 9.90 301 9.90 292 9.90 282 9.90 273 9.90 263 9.90 254 9.90 244 9.90 235 9 10 9 9 9 9 10 9 9 9 10 9 9 9 10 9 9 9 10 9 9 10 9 9 9 10 9 9 10 9 9 10 9 10 9 9 10 9 9 10 9 10 9 10 9 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 2 3 4 5 6 7 8 9 10 20 30 40 50 18 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 6.0 9.0 12.0 15.0 17 0.3 0.6 0.8 1.1 1.4 1.7 2.0 2.3 2.6 2.8 5.7 8.5 11.3 14.2 16 0.3 0.5 0.8 1.1 1.3 1.6 1.9 2.1 2.4 2.7 '5.3 8.0 10.7 13.3 1 2 3 4 5 6 7 8 9 10 20 30 40 50 27 0.4 0.9 1.4 1.8 2.2 2.7 3.2 3.6 4.0 4.5 9.0 13.5 18.0 22.5 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4.3 8.7 13.0 17.3 21.7 9 10 9 10 9 10 9 10 9 10 9 10 9 10 9 1 2 3 4 5 6 7 8 9 10 20 30 40 50 10 9 0.2 0.2 0.3 0.3 0.5 0.4 0.7 0.6 0.8 0.8 1.0 0.9 1.2 1.0 1.3 1.2 1.5 1.4 1.7 1.5 3.3 3.0 5.0 4.5 6.7 6.0 8.3 7.5 I I log cos d | log cot cd I log tan | log sin | d | ' | Prop. Parts 143~ 233~ 323~ 53~ 98 TABLE III 37~ 127~ 217~ 307~ log sin d log tan lcd I log cot log cos d I Prop. Parts 0 1 2 3 4 0 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.77 946 9.77 963 9.77 980 9.77 997 9.78 013 9.78 030 9.78 047 9.78 063 9.78 080 9.78 097 9.78 113 9.78 130 9.78 147 9.78 163 9.78 180 9.78 197 9.78 213 9.78 230 9.78 246 9.78 263 9.78 280 9.78 296 9.78 313 9.78 329 9.78 346 9.78 362 9.78 379 9.78 395 9.78 412 9.78 428 9.78 445 9.78 461 9.78 478 9.78 494 9.78 510 9.78 527 9.78 543 9.78 560 9.78 576 9.78 592 9.78 609 9.78 625 9.78 642 9.78 658 9.78 674 9.78 691 9.78 707 9.78 723 9.78 739 9.78 756 9.78 772 9.78 788 9.78 805 9.78 821 9.78 837 9.78 853 9.78 869 9.78 886 9.78 902 9.78 918 9.78 934 17 17 17 16 17 17 16 17 17 16 17 17 16 17 17 16 17 16 17 17 16 17 16 17 16 17 16 17 16 17 16 17 16 16 17 16 17 16 16 17 16 17 16 16 17 16 16 16 17 16 16 17 16 16 16 9.87 711 9.87 738 9.87 764 9.87 790 9.87 817 9.87 843 9.87 869 9.87 895 9.87 922 9.87 948 9.87 974 9.88 000 9.88 027 9.88 053 9.88 079 9.88 105 9.88 131 9.88 158 9.88 184 9.88 210 9.88 236 9.88 262 9.88 289 9.88 315 9.88 341 9.88 367 9.88 393 9.88 420 9.88 446 9.88 472 9.88 498 9.88 524 9.88 550 9.88 577 9.88 603 9.88 629 9.88 655 9.88 681 9.88 707 9.88 733 9.88 759 9.88 786 9.88 812 9.88 838 9.88 864 9.88 890 9.88 916 9.88 942 9.88 968 9.88 994 9.89 020 9.89 046 9.89 073 9.89 099 9.89 125 9.89 151 9.89 177 9.89 203 9.89 229 9.89 255 9.89 281 27 26 26 27 26 26 26 27 26 26 26 27 26 26 26 26 27 26 26 26 26 27 26 26 26 26 27 26 26 26 26 26 27 26 26 26 26 26 26 26 27 26 26 26 26 26 26 26 26 26 26 27 26 26 26 26 26 26 26 26 0.12 289 0.12 262 0.12 236 0.12 210 0.12 183 0.12 157 0.12 131 0.12 105 0.12 078 0.12 052 0.12 026 0.12 000 0.11 973 0.11 947 0.11 921 0.11 895 0.11 869 0.11 842 0.11 816 0.11 790 0.11 764 0.11 738 0.11 711 0.11 685 0.11 659 0.11 633 0.11 607 0.11 580 0.11 554 0.11 528 0.11 502 0.11 476 0.11 450 0.11 423 0.11 397 0.11 371 0.11 345 0.11 319 0.11 293 0.11 267 0.11 241 0.11 214 0.11 188 0.11 162 0.11 136 0.11 110 0.11 084 0.11 058 0.11 032 0.11 006 0.10 980 0.10 954 0.10 927 0.10 901 0.10 875 0.10 849 0.10 823 0.10 797 0.10 771 0.10 745 0.10 719 9.90 235 9.90 225 9.90 216 10 9.90 206 9.90 197 10 9.90 187 9 9.90 178 1 9.90 168 9.90 159 1 9.90 149 10 9.90 139 9.90 130 10 9.90 120 9.90 111 0 9.90 101 10 9.90 091 9.90 082 i0 9.90 072 9.90 063 10 9.90 053 10 9.90 043 9 9.90 034 10 9.90 024 10 9.90 014 9 9.90 005 10 9.89 995 1 9.89 985 9 9.89 976 10 9.89 966 10 9.89 956 9 9.89 947 10 9.89 937 10 9.89 927 9 9.89 918 10 9.89 908 10 9.89 898 10 9.89 888 9 9.89 879 10 9.89 869 10 9.89 859 10 9.89 849 9 9.89 840 10 9.89 830 10 9.89 820 10 9.89 810 9 9.89801 10 9.89 791 10 9.89 781 10 9.89 771 10 9.89 761 9 9.89 752 10 9.89 742 10 9.89 732 10 9.89 722 10 9.89 712 10 9.89 702 9 9.89 693 10 9.89 683 10 9.89 673 10 9.89 663 10 9.89 653 log sin d 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 27 26 0.4 0.4 0.9 0.9 1.4 1.3 1.8 1.7 2.2 2.2 2.7 2.6 3.2 3.0 3.6 3.5 4.0 3.9 4.5 4.3 9.0 8.7 13.5 13.0 18.0 17.3 22.5 21.7 17 16 0.3 0.3 0.6 0.5 0.8 0.8 1.1 1.1 1.4 1.3 1.7 1.6 2.0 1.9 2.3 2.1 2.6 2.4 2.8 2.7 5.7 5.3 8.5 8.0 11.3 10.7 14.2 13.3 10 9 0.2 0.2 0.3 0.3 0.5 0.4 0.7 0.6 0.8 0.8 1.0 0.9 1.2 1.0 1.3 1.2 1.5 1.4 1.7 1.5 3.3 3.0 5.0 4.5 6.7 6.0 8.3 7.5 16 17 16 16 16 0 I I I log cos I d I log cot I cd log tan I I I Prop. Parts 142~ 232~ 322~ 52~ 99 TABLE III 38~ 128~ 218~ 308~ I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 56 55 51 5$ 5S I log sin d Ilog tan lcd log cot log cos I d JI Prop. Parts I I I I I I } I I I I ) ) I 9.78 934 9.78 950 9.78 967 9.78 983 9.78 999 9.79 015 9.79 031 9.79 047 9.79 063 9.79 079 9.79 095 9.79 111 9.79 128 9.79 144 9.79 160 9.79 176 9.79 192 9.79 208 9.79 224 9.79 240 9.79 256 9.79 272 9.79 288 9.79 304 9.79 319 9.79 335 9.79 351 9.79 367 9.-79 383 9.79 399 9.79 415 9.79 431 9.79 447 9.79 463 9.79 478 9.79 494 9.79 510 9.79 526 9.79 542 9.79 558 9.79 573 9.79 589 9.79 605 9.79 621 9.79 636 9.79 652 9.79 668 9.79 684 9.79 699 9.79 715 9.79 731 9.79 746 9.79 762 9.79 778 9.79 793 9.79 809 9.79 825 9.79 840 9.79 856 9.79 872 9.79 887 16 17 16 16 16 16 16 16 16 16 16 17 16 16 16 16 16 16 16 16 16 16 16 15 16 16 16 16 16 16 16 16 16 15 16 16 16 16 16 15 16 16 16 15 16 16 16 15 16 16 15 16 16 15 16 16 15 16 16 15 9.89 281 9.89 307 9.89 333 9.89 359 9.89 385 9.89 411 9.89 437 9.89 463 9.89 489 9.89 515 9.89 541 9.89 567 9.89 593 9.89 619 9.89 645 9.89 671 9.89 697 9.89 723 9.89 749 9.89 775 9.89 801 9.89 827 9.89 853 9.89 879 9.89 905 9.89 931 9.89 957 9.89 983 9.90 009 9.90 035 9.90 061 9.90 086 9.90 112 9.90 138 9.90 164 9.90 190 9.90 216 9.90 242 9.90 268 9.90 294 9.90 320 9.90 346 9.90 371 9.90 397 9.90 423 9.90 449 9.90 475 9.90 501 9.90 527 9.90 553 9.90 578 9.90 604 9.90 630 9.90 656 9.90 682 9.90 708 9.90 734 9.90 759 9.90 785 9.90 811 9.90 837 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 26 26 26 25 26 26 26 26 26 26 25 26 26 26 0.10 719 0.10 693 0.10 667 0.10 641 0.10 615 0.10 589 0.10 563 0.10 537 0.10 511 0.10 485 0.10 459 0.10 433 0.10 407 0.10 381 0.10 355 0.10 329 0.10 303 0.10 277 0.10 251 0.10 225 0.10 199 0.10 173 0.10 147 0.10 121 0.10 095 0.10 069 0.10 043 0.10017 0.09 991 0.09 965 0.09 939 0.09 914 0.09 888 0.09 862 0.09 836 0.09 810 0.09 784 0.09 758 0.09 732 0.09 706 0.09 680 0.09 654 0.09 629 0.09 603 0.09 577 0.09 551 0.09 525 0.09 499 0.09 473 0.09 447 0.09 422 0.09 396 0.09 370 0.09 344 0.09 318 0.09 292 0.09 266 0.09 241 0.09 215 0.09 189 0.09 163 9.89 653 9.89 643 9.89 633 9.89 624 9.89 614 9.89 604 9.89 594 9.89 584 9.89 574 9.89 564 9.89 554 9.89 544 9.89 534 9.89 524 9.89 514 9.89 504 9.89 495 9.89 485 9.89 475 9.89 465 9.89 455 9.89 445 9.89 435 9.89 425 9.89 415 9.89 405 9.89 395 9.89 385 9.89 375 9.89 364 9.89 354 9.89 344 9.89 334 9.89 324 9.89 314 9.89 304 9.89 294 9.89 284 9.89 274 9.89 264 9.89 254 9.89 244 9.89 233 9.89 223 9.89 213 9.89 203 9.89 193 9.89 183 9.89 173 9.89 162 9.89 152 9.89 142 9.89 132 9.89 122 9.89 112 9.89 101 9.89 091 9.89 081 9.89 071 9.89 060 9.89 050 10 I0 9 10 10 10 10 10 10 10 10 10 10 10 10 9 10 10 10 10 10 10 10 10 10 10 10 10 11 10 10 10 10 10 10 10 10 10 10 10 10 11 10 10 10 10 10 10 11 10 10 10 10 10 11 10 10 10 11 10 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 20 30 40 50 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4.3 8.7 13.0 17.3 21.7 25 0.4 0.8 1.2 1.7 2.1 2.5 2.9 3.3 3.8 4.2 8.3 12.5 16.7 20.8 1 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 17 0.3 0.6 0.8 1.1 1.4 1.7 2.0 2.3 2.6 2.8 5.7 8.5 11.3 14.2 16 0.3 0.5 0.8 1.1 1.3 1.6 1.9 2.1 2.4 2.7 5.3 8.0 10.7 13.3 15 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0 2.2 2.5 5.0 7.5 10.0 12.5 11 10 9 0.2 0.2 0.2 0.4 0.3 0.3 0.6 0.5 0.4 0.7 0.7 0.6 0.9 0.8 0.8 1.1 1.0 0.9 1.3 1.2 1.0 1.5 1.3 1.2 1.6 1.5 1.4 1.8 1.7 1.5 3.7 3.3 3.0 5.5 5.0 4.5 7.3 6.7 6.0 9.2 8.3 7.5 I 0 I log cos I d log cot Icd log tan log sin d I ' I Prop. Parts 141~ 231~ 321~ 51~ 100 TABLE III 390 1290 2190 3090 'I log sin I d I log tan Ic d log cot I log cos I d I Prop. Parts 0 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I 9.79 887 9.79 903 9.79 918 9.79 934 9.79 950 9.79 965 9.79 981 9.79 996 9.80 012 9.80 027 9.80 043 9.80 058 9.80 074 9.80 089 9.80 105 9.80 120 9.80 136 9.80 151 9.80 166 9.80 182 9.80 197 9.80 213 9.80 228 9.80 244 9.80 259 9.80 274 9.80 290 9.80 305 9.80 320 9.80 336 9.80 351 9.80 366 9.80 382 9.80 397 9.80 412 9.80 428 9.80 443 9.80 458 9.80 473 9.80 489 9.80 504 9.80 519 9.80 534 9.80 550 9.80 565 9.80 580 9.80 595 9.80 610 9.80 625 9.80 641 9.80 656 9.80 671 9.80 686 9.80 701 9.80 716 9.80 731 9.80 746 9.80 762 9.80 777 9.80 792 9.80 807 I ~ ~ ~ ~ ~ ~ ~ ~ 16 15 16 16 15 16 15 16 15 16 15 16 15 16 15 16 is 15 16 1 5 16 15 16 15 15 16 15 15 16 15 15 16 15 1 5 16 15 15 15 16 15 15 15 16 15 15 15 15 15 16 15 15 15 15 15 15 9.90 837 9.90 863 9.90 889 9.90 914 9.90 940 9.90 966 9.90 992 9.91 018 9.91 043 9.91 069 9.91 095 9.91 121 9.91 147 9.91 172 9.91 198 9.91 224 9.91 250 9.91 276 9.91 301 9.91 327 9.91 353 9.91 379 9.91 404 9.91 430 9.91 456 9.91 482 9.91 507 9.91 533 9.91 559 9.91 585 9.91 610 9.91 636 9.91 662 9.91 688 9.91 713 9.91 739 9.91 765 9.91 791 9.91 816 9.91 842 9.91 868 9.91 893 9.91 919 9.91 945 9.91 971 9.91 996 9.92 022 9.92 048 9.92 073 9.92 099 9.92 125 9.92 150 9.92 176 9.92 202 9.92 227 9.92`253 9.92 279 9.92 304 9.92 330 9.92 356 9.92 381 26 26 25 26 26 26 26 25 26 26 26 26 25 26 26 26 26 25 26 26 26 25 26 26 26 25 26 26 26 25 26 26 26 25 26 26 26 25 26 26 25 26 26 26 25 26 26 25 26 26 25 26 26 25 26 26 25 26 26 25 0.09 163 0.09 137 0.09 111 0.09 086 0.09 060 0.09 034 0.09 008 0.08 982 0.08 957 0.08 931 0.08 905 0.08 879 0.08 853 0.08 828 0.08 802 0.08 776 0.08 750 0.08 724 0.08 699 0.08_673 0.08 647 0.08 621 0.08 596 0.08 570 0.08 544 0.08 518 0.08 493 0.08 467 0.08 441 0.08 415 0.08 390 0.08 364 0.08 338 0.08 312 0.08 287 0.08 261 0.08 235 0.08 209 0.08 184 0.08 158 0.08 132 0.08 107 0.08 081 0.08 055 0.08 029 0.08 004 0.07 978 0.07 952 0.07 927 0.07 901 0.07 875 0.07 850 0.07 824 0.07 798 0.07 773 0.07 747 0.07 721 0.07 696 0.07 670 0.07 644 0.07 619 9.89 050 9.89 040 9.89 030 9.89 020 9.89 009 9.88 999 9.88 989 9.88 978 9.88 968 9.88 958 9.88 948 9.88 937' 9.88 927 9.88 917 9.88 906 9.88 896 9.88 886 9.88 875 9.88 865 9.88 855 9.88 844 9.88 834 9.88 824 9.88 813 9.88 803 9.88 793 9.88 782 9.88 772 9.88 761 9.88 751 9.88 741 9.88 730 9.88 720 9.88 709 9.88 699 9.88 688 9.88 678 9.88 668 9.88 657 9.88 647 9. 88 63~6 9.88 626 9.38 615 9.88 605 9.8.8 594 9.88 584 9.88 573 9.88 563 9.88 552 9.88 542 9.88 531 9.88 521 9.88 510 9.88 499 9.88 489 9.88 478 9.88 468 9.88 457 9.88 447 9.88 436 9.88 425 10 10 10 11 10 10 11 10 10 10 11 10 10 11 10 10 II 10 10 11 10 10 11 10 10 11 10 11 10 10 11 10 11 10 11 10 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 11 10 11 10 11 10 11 11 60 59 58 57 56 55 5 4 53 52 51 50 49 48 4 7 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 1 4 13 12 11 10 9 8 7 6 5 4 3 2 0 0 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 2 3 4 5 6 7 8 9 10 20 30 40 50 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4..3 8.7 13.0 17.3 21.7 16 0.3 0.5 0.8 1.1 1.3 1.6 1.9 2.1 2.4 2.7 5..3 8.0 10.7 13.3 11 0.2 0.4 0.6 0.7 0.9 1.1 1.3 1.5 1.6 1.8 3.7 5.5 7.3 9.2 25 0.4 0.8 1.2 1.7 2.1 2.5 2.9 3.3 3.8 4.2 8.3 12.5 16.7 20.8 15 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0 2.2 2.5 5.0 7.5 10.0 12.5 10 0.2 0.3 0. 5 0.7 0.8 1.0 1.2 1.3 1.5 1.7 3.3 5.0 6.7 8.3 15 16 15 15 15 - ~ I ~ ~ I log cos I d I log cot I ' log tanI log sin I d I, I Prop. Parts 1400 2300 3200 50 0 101 TABLE III 400 1300 2200 3100 'I log sin d I-log tan cd log cot logCos I d, Prop.Parts 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.80 807 9.80 822 9.80 837 9.80 852 9.80 867 9.80 882 9.80 897 9.80 912 9.80 927 9.80 942 9.80 957 9.80 972 9.80 987 9.81 002 9.81 017 9.81 032 9.81 047 9.81 061 9.81 076 9.81 091 9.81 106 9.81 121 9.81 136 9.81 151 9.81 166 9.81 180 9.81 195 9.81 210 9.81 225 9.81 240 9.81 254 9.81 269 9.81 284 9.81 299 9.81 314 9.81 328 9.81 343 9.81 358 9.81 372 9.81 387 9.81 402 9.81 417 9.81 431 9.81 446 9.81 461 9.81 475 9.81 490 9.81 505 9.81 519 9.81 534 9.81 549 9.81 563 9.81 578 9.81 592 9.81 607 9.81 622 9.81 636 9.81 651 9.81 665 9.81 680 9.81 694 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 14 15 15 15 15 15 15 15 1 4 1 5 1 5 1 5 15 14 1 5 1 5 1 5 1 5 1 4 1 5 15~ 1 4 1 5 1 5 1 5 1 4 1 5 1 5 1 4 1 5 1 5 1 4 1 5 1 5 1 4 1 5 1 4 1 5 1 5 1 4 1 5 1 4 1 5 1 4 9.92 381 9.92 407 9.92 433 9.92 458 9.92 484 9.92 510 9.92 535 9.92 561 9.92 587 9.92 612 9.92 638 9.92 663 9.92 689 9.92 715 9.92 740 9.92 766 9.92 792 9.92 817 9.92 843 9.92 868 9.92 894 9.92 920 9.92 945 9.92 971 9.92 996 9.93 022 9.93 048 9.93 073 9.93 099 9.93 124 9.93 150 9.93 175 9.93 201 9.93 227 9.93 252 9.93 278 9.93 303 9.93 329 9.93 354 9.93 380 9.93 406 9.93 431 9,.93 457 9.93 482 9.93 508 9.93 533 9.93 559 9.93 584 9.93 610 9.93 636 9.93 661 9.93 687 9.93 712 9.93 738 9.93 763 9.93 789 9.93 814 9.93 840 9.93 865 9.93 891 9.93 916 26 26 25 26 26 25 26 26 25 26 25 26 26 25 26 26 25 26 25 26 26 25 26 25 26 26 25 26 25 26 25 z 26 26 25 26 25 26 25 26 26 25 26 25 26 25 26 25 26 26 25 26 25 26 25 26 25 26 25 0.07 619 0.07 593 0.07 567 0. 07 542 0.07 516 0.07 490 0. 07 465 0.07 439 0.07 413 0.07 388 0.07 362 0. 07 337 0.07 311 0.07 285 0.07 260 0.07 234 0. 07 208 0.07 183 0.07 157 0. 07 132 0.07 106 0. 07 080 0.07 055 0.07 029 0. 07 004 0. 06 978 0. 06 952 0. 06 927 0.06 901 0. 06 876 0.06 850 0. 06 825 0.06 799 0.06 773 0. 06 748 0.06 722 0. 06 697 0.06 671 0. 06 646 0.06 620 0.06 594 0.06 569 0.06 543 0.06 518 0.06 492 0.06 467 0.06 441 0.06 416 0.06 390 0.06 364 0. 06 339 0.06 313 0.06 288 0.06 262 0.06 237 0.06 211 0.06 186 0. 06 160 0.06 135 0. 06 109 0.06 084 9.88 425 9.88 415 9.88 404 9.88 394 9.88 383 9.88 372 9.88 362 9.88 351 9.88 340 9.88 330 9. 88 319 9.88 308 9.88 298 9.88 287 9.88 276 9.88 266 9.88 255 9.88 244 9.88 234 9.88 223 9.88 212 9.88 201 9.88 191 9.88 180 9.88 169 9.88 158 9.88 148 9.88 137 9.88 126 9.88 115 9.88 105 9.88 094 9.88 083 9.88 072 9.88 061 9.88 051 9.88 040 9.88 029 9.88 018 9.88 007 9.87 996 9.87 985 9.87 975 9.87 964 9.87 953 9.87 942 9.87 931 9.87 920 9.87 909 9.87 898 9.87 887 9.87 877 9.87 866 9.87 855 9.87 844 9.87 833 9.87 822 9.87 811 9.87 800 9.87 789 9.87 778 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 11I 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 60 59 58 57 56 55 54 53 52 51' 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 3!5 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4.3 8.7 13.0 17.3 21.7 15 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0 2.2 2.5 5.0 7.5 10.0 12.5 25 0.4 0.8 1.2 1.7 2.1 2.5 2.9 3.3 3.8 4.2 8.3 12.5 16.7 20.8 14 0.2 0.5 0.7 0.9 1.2 1.4 1.6 1.9 2.1 2.3 4.7 7.0 9.3 11.7 I It 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 11 10 0.2 0.2 0.4 0.3 0.6 0.5 0.7 0.7 0.9 0.8 1.1 1.0 1.3 1.2 1.5 1.3 1.6 1.5 1.8 1.7 3.7 3.3 5.5 5.0 7.3 6.7 9.2 8.3 I log cos I d log cot I cdl log tan I logsin I dl ' I Prop. Parts 1390 229~ 3190 490 102 TABLE III 41 0 1310 2210 3110 I Prop. Parts IIlog sin I d Ilog tan Ic dI log cotI log Cos Id 0 2 3 4 5 6 7 8 9 1o I1I 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 2 1 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60 9. 81 694 9. 81 709 9. 81 723 9. 81 738 9. 81 752 9. 81 767 9. 81 781 9. 81 796 9. 81 81 0 9. 81 825 9. 81 839 9. 81 854 9. 81 868 9. 81 882 9. 81 897 9. 81 911 9. 81 926 9. 81 940 9. 81 955 9. 81 969 19. 81 983 9. 81 998 9. 82 012 9. 82 026 9. 82 041 9. 82 05 9. 82 069 9. 82 084 9. 82 098 9. 82 112 9. 82 126 9. 82 141 9. 82 155 9. 82 169 9. 82 184 9. 82 198 9. 82 212 9.82 226 9. 82 240 9.82 255 9. 82 269 9. 82 283 9.82 297 9. 82 311 9.82 326 9.82 340 9. 82 354 9. 82 368 9. 82 382 9. 82 396 9. 82 410 9. 82 424 9. 82 439 9. 82 453 9.82 467 9. 82 481 9. 82 495 9-. 82 509 9. 82 523 9. 82 537 9. 82 551 1 5 1 4 1 5 1 4 1 5 1 4 1 5 1 4 1 5 1 4 1 5 1 4 1 4 1 5 1 4 1 5 1 4 1 5 1 4 1 4 1 5 1 4 1 4 1 5 1 4 1 4 1 5 1 4 1 4 1 4 1 5 1 4 1 4 1 5 14, 1 4 1 4 1 4 1 5 1 4 1 4 1 4 1 4 1 5 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 5 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 9. 93 916 9. 93 942 9.93 967 9. 93 993 9.94 01 8 9. 94 044 9. 94 069 9. 94 095 9. 94 120 9. 94 146 9. 94 171 9. 94 197 9. 94 222 9. 94 248 9. 94 273 9. 94 299 9. 94 324 9. 94 350 9. 94 375 9. 94 401 9. 94 426 9. 94 452 9. 94 477 9. 94 503 9. 94 52 8 9. 94 554 9. 94 579 9. 94 604 9. 94 630 9. 94 655 9. 94 681 9. 94 706 9. 94 732 9. 94 757 9. 94 783 9. 94- 808 9. 94 834 9. 94 859 9.94 884 9) 4 910 9. 94 935 9.94 961 9. 94 986 9. 95 012 9.95 037 9. 95 062 9. 95 088 9. 95 113 9.95 139. 9.95 164 9. 95 190 9. 95 21 5 9. 95 240 9.95 266 9.95 291 9.95 31 7 9. 95 342 9. 95 368 9. 95 393 9. 95 418 9. 95 444 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 25' 26 25 26 25 26 25 26 25 26 25 25 26 25 26 25 26 25 25 26 25 26 25 26 25 25 26 25 26 25 26 25.25 26 0.06 084 0. 06 058 0.06 033 0. 06 007 0.05 982 0. 05 956 0. 05 931 0. 05'905 0.05 880 0.05 854 0. 05 829 0. 05 803 0.05 778 0. 05 752 0. 05 727 0. 05 701 0. 05 676 0.05 650 0.05 625 0.05 599 0. 05 574 0. 05 548 0. 05 523 0. 05 497 0. 05 472 0. 05 446 0.05 421 0. 05 396 0. 05 370 0. 05 345 0. 05 319 0. 05 294 0.05 268 0.05 243 0.05 217 0. 05 192 0.05 166 0.05 141 0.05 116 0.05 090 0.05 065 0.05 039 0.05 014 0.04 988 0.04 963 0.04 938 0.04 912 0. 04 887 0.04 861 0.04 836 0.04 810 0.04 785 0.04 760 0.04 734 0.04 709 0. 04 683 0.04 658 0. 04 632 0. 04 607 0.04 582 0. 04 556 9. 87 778 9. 87 767 9. 87 756 9. 87 745 9. 87 734 9. 87 723 9. 87 712 9. 87 701 9. 87 690 9. 87 679 9. 87 668 9. 87 657 9. 87 646 9. 87 635 9. 87 624 9. 87 613 9. 87 601 9. 87 590 9. 87 579 9. 87 568 9. 87 557 9. 87 546 9. 87 535 9.87 524 9.87 51 3 9. 87 501 9. 87 490 19. 87 479 9. 87 468 9. 87 457 9. 87 446 9. 87 434 9. 87 423 9. 87 412 9. 87 401 9. 87 390 9. 87 378 9. 87 367 9. 87 356 9. 87 345 9. 87 334 9. 87 322 9. 87 311 9. 87 300 9. 87 288 9. 87 277 9. 87 266 9.87 255 9. 87 243 9.87 232 9. 87 221 9.87 209 9. 87 198 9. 87 1 87 9. 87 1 75 9. 87 164 9. 87 153 9. 87 141 9.87 130 9. 87 119 9. 87 107 11I 11I 11I 1H 11I 11I 11I 11I 11I 11I 11I 11I 11I 11I 11I 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 I I I I I 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 2 1 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 2 3 4 5 6 7 8 9 1 0 20 30 40 50 2 3 4 5 6 7 8 9 1 0 20 30 40 50 2 3 4 5 6 7 8 9 1 0 20 30 40 50 26 0.4 0. 9 1.3 1.7 2. 2 2. 6 3.0 3. 5 3.9 4. 3 8. 7 13.0 17. 3 21.7 15 0. 2 0.5 0. 8 1.0 1. 2 1. 5 1. 8 2. 0 2. 2 2. 5 5.0 7. 5 10.0 12.5 12 0. 2 0.4 0. 6 0.8 1.0 1. 2 1.4 1. 6 1.8 2.0 4.0 6.0 8.0 10.0 25 0. 4 0.8 1.2 1.7 2. 1 2. 5 2. 9 3. 3 3. 8 4. 2 8. 3 12. 5 16. 7 20. 8 14 0. 2 0.5 0. 7 0. 9 1.2 1.4 1. 6 1. 9 2. 1 2. 3 4.7 7.0 9. 3 11.7 0.2 0.4 0. 6 0.7 0.9 1.1 1.3 1.5 1.6 1.8 3. 7 5.5 7. 3 9.2 I11 12 1 1 1 1 1 2 1 1 1 1 11~ 11I 11I 11: I I~ log cosI d Ilog cotI cd I log tan j log sin I d I I I Prop. Parts 1380 2280 3180 480 103 TABLE III 42~ 132~ 222~ 312~ 'I log sin | d, | log tan I c d log cot I log cos | d I Prop. Parts A 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.82 551 9.82 565 9.82 579 9.82 593 9.82 607 9.82 621 9.82 635 9.82 649 9.82 663 9.82 677 9.82 691 9.82 705 9.82 719 9.82 733 9.82 747 9.82 761 9.82 775 9.82 788 9.82 802 9.82 816 9.82 830 9.82 844 9.82 858 9.82 872 9.82 885 9.82 899 9.82 913 9.82 927 9.82 941 9.82 955 9.82 968 9.82 982 9.82 996 9.83 010 9.83 023 9.83 037 9.83 051 9.83 065 9.83 078 9.83 092 9.83 106 9.83 120 9.83 133 9.83 147 9.83 161 9.83 174 9.83 188 9.83 202 9.83 215 9.83 229 9.83 242 9.83 256 9.83 270 9.83 283 9.83 297 9.83 310 9.83 324 9.83 338 9.83 351 9.83 365 9.83 378 log cos 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 13 14 14 14 14 14 14 13 14 14 14 14 14 13 14 14 14 13 14 14 14 13 14 14 14 13 14 14 13 14 14 13 14 13 14 14 13 14 13 14 14 13 14 13 9.95 444 9.95 469 9.95 495 9.95 520 9.95 545 9.95571 9.95 596 9.95 622 9.95 647 9.95 672 9.95 698 9.95 723 9.95 748 9.95 774 9.95 799 9.95 825 9.95 850 9.95 875 9.95 901 9.95 926 9.95 952 9.95 977 9.96 002 9.96 028 9.96 053 9.96 078 9.96 104 9.96 129 9.96 155 9.96 180 9.96 205 9.96 231 9.96 256 9.96 281 9.96 307 9.96 332 9.96 357 9.96 383 9.96 408 9.96 433 9.96 459 9.96 484 9.96 510 9.96 535 9.96 560 9.96 586 9.96 611 9.96 636 9.96 662 9.96 687 9.96 712 9.96 738 9.96 763 9.96 788 9.96 814 9.96 839 9.96 864 9.96 890 9.96 915 9.96 940 9.96 966 25 26 25 25 26 25 26 25 25 26 25 25 26 25 26 25 25 26 25 26 25 25 26 25 25 26 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 I 0.04 556 0.04 531 0.04 505 0.04 480 0.04 455 0.04 429 0.04 404 0.04 378 0.04 353 0.04 328 0.04 302 0.04 277 0.04 252 0.04 226 0.04 201 0.04 175 0.04 150 0.04 125 0.04 099 0.04 074 0.04 048 0.04 023 0.03 998 0.03 972 0.03 947 0.03 922 0.03 896 0.03 871 0.03 845 0.03 820 0.03 795 0.03 769 0.03 744 0.03 719 0.03 693 0.03 668 0.03 643 0.03 617 0.03 592 0.03 567 0.03 541 0.03 516 0.03 490 0.03 465 0.03 440 0.03 414 0.03 389 0.03 364 0.03 338 0.03 313 0.03 288 0.03 262 0.03 237 0.03 212 0.03 186 0.03 161 0.03 136 0.03 110 0.03 085 0.03 060 0.03 034. _ I 9.87 107 9.87 096 9.87 085 9.87 073 9.87 062 9.87 050 9.87 039 9.87 028 9.87 016 9.87 005 9.86 993 9.86 982 9.86 970 9.86 959 9.86 947 9.86 936 9.86 924 9.86 913 9.86 902 9.86 890 9.86 879 9.86 867 9.86 855 9.86 844 9.86 832 9.86 821 9.86 809 9.86 798 9.86 786 9.86 775 9.86 763 9.86 752 9.86 740 9.86 728 9.86 717 9.86 705 9.86 694 9.86 682 9.86 670 9.86 659 9.86 647 9.86 635 9.86 624 9.86 612 9.86 600 9.86 589 9.86 577 9.86 565 9.86 554 9.86 542 9.86 530 9.86 518 9.86 507 9.86 495 9.86 483 9.86 472 9.86 460 9.86 448 9:86 436 9.86 425 9.86 413 11 11 12 11 12 11 11 12 11 12 11 12 11 12 11 12 11 11 12 11 12 12 11 12 11 12 11 12 11 12 11 12 12 11 12 11 12 12 11 12 12 11 12 12 11 12 12 11 12 12 12 11 12 12 11 12 12 12 11 12 II 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 -40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 2 3 4 5 6 7 8 9 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4.3 8.7 13.0 17.3 21.7 14 0.2 0.5 0.7 0.9 1.2 1.4 1.6 1.9 2.1 2.3 4.7 7.0 9.3 11.7 12 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 4.0 6.0 8.0 10.0 25 0.4 0.8 1.2 1.7 2.1 2.5 2.9 3.3 3.8 4.2 8.3 12.5 16.7 20.8 13 0.2 0.4 0.6 0.9 1.1 1.3 1.5 1.7 2.0 2.2 4.3 6.5 8.7 10.8 11 0.2 0.4 0.6 0.7 0.9 1.1 1.3 1.5 1.6 1.8 3.7 5.5 7.3 9.2 -1I d j log cot c d logtan | log sin I d j I Prop. Parts 137 227 3 104 137~ 227~ 317~ 47~ 104 TABLE III 43~ 133~ 223~ 313~ I log sin d log tan c d log cot log cos d I Prop. Parts I t ---I I - I I s 7 r ' 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.83 378 9.83 392 9.83 405 9.83 419 9.83 432 9.83 446 9.83 459 9.83 473 9.83 486 9.83 500 9.83 513 9.83 527 9.83 540 9.83 554 9.83 567 9.83 581 9.83 594 9.83 608 9.83 621 9.83 634 9.83 648 9.83 661 9.83 674 9.83 688 9.83 701 9.83 715 9.83 728 9.83 741 9.83 755 9.83 768 9.83 781 9.83 795 9.83 808 9.83 821 9.83 834 9.83 848 9.83 861 9.83 874 9.83 887 9.83 901 9.83 914 9.83 927 9.83 940 9.83 954 9.83 967 9.83 980 9.83 993 9.84 006 9.84 020 9.84 033 9.84 046 9.84 059 9.84 072 9.84 085 9.84 098 9.84 112 9.84 125 9.84 138 9.84 151 9.84 164 9.84 177 14 13 14 13 14 13 14 13 14 13 14 13 14 13 14 13 14 13 13 14 13 13 14 13 14 13 13 14 13 13 14 13 13 13 14 13 13 13 14 13 13 13 14 13 13 13 13 14 13 13 13 13 13 13 14 13 13 13 13 13 9.96 966 9.96 991 9.97 016 9.97 042 9.97 067 9.97 092 9.97 118 9.97 143 9.97 168 9.97 193 9.97 219 9.97 244 9.97 269 9.97 295 9.97 320 9.97 345 9.97 371 9.97 396 9.97 421 9.97 447 9.97 472 9.97 497 9.97 523 9.97 548 9.97 573 9.97 598 9.97 624 9.97 649 9.97 674 9.97 700 9.97 725 9.97 750 9.97 776 9.97 801 9.97 826 9.97 851 9.97 877 9.97 902 9.97 927 9.97 953 9.97978 9.98 003 9.98 029 9.98 054 9.98 079 9.98 104 9.98 130 9.98 155 9.98 180 9.98 206 9.98 231 9.98 256 9.98 281 9.98 307 9.98 332 9.98 357 9.98 383 S9.98 408 9.98 433 9.98 458 9.98 484 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 25 26 0.03 034 0.03 009 0.02 984 0.02 958 0.02 933 0.02 908 0.02 882 0.02 857 0.02 832 0.02 807 0.02 781 0.02 756 0.02 731 0.02 705 0.02 680 0.02 655 0.02 629 0.02 604 0.02 579 0.02 553 0.02 528 0.02 503 0.02 477 0.02 452 0.02 427 0.02 402 0.02 376 0.02 351 0.02 326 0.02 300 0.02 275 0.02 250 0.02 224 0.02 i99 0.02 174 0.02 149 0.02 123 0.02 098 0.02 073 0.02 047 0.02 022 0.01 997 0.01 971 0.01 946 0.01 921 0.01 896 0.01 870 0.01 845 0.01 820 0.01 794 0.01 769 0.01 744 0.01 719 0.01 693 0.01 668 0,.01 643 0.01 617 0.01 592 0.01 567 0.01 542 0.01 516 9.86 413 9.86 401 9.86 389 9.86 377 9.86 366 9.86 354 9.86 342 9.86 330 9.86 318 9.86 306 9.86 295 9.86 283 9.86 271 9.86 259 9.86 247 9.86 235 9.86 223 9.86 211 9.86 200 9.86 188 9.86 176 9.86 164 9.86 152 9.86 140 9.86 128 9.86 116 9.86 104 9.86 092 9.86 080 9.86 068 9.86 056 9.86 044 9.86 032 9.86 020 9.86 008 9.85 996 9.85 984 9.85 972 9.85 960 9.85 948 9.85 936 9.85 924 9.85 912 9.85 900 9.85 888 9.85 876 9.85 864 9.85 851 9.85 839 9.85 827 9.85 815 9.85 803 9.85 791 9.85 779 9.85 766 9.85 754 9.85 742 9.85 730 9.85 718 9.85 706 9.85 693 12 12 12 11 12 12 12 12 12 11 12 12 12 12 12 12 12 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 12 12 12 12 12 12 13 12 12 12 12 12 13 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4.3 8.7 13.0 17.3 21.7 25 0.4 0.8 1.2 1.7 2.1 2.5 2.9 3.3 3.8 4.2 8.3 12.5 16.7 20.8 14 13 0.2 0.2 0.5 0.4 0.7 0.6 0.9 0.9 1.2 1.1 1.4 1.3 1.6 1.5 1.9 1.7 2.1 2.0 2.3 2.2 4.7 4.3 7.0 6.5 9.3 8.7 11.7 10.8 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 20 30 40 50 12 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 4.0 6.0 8.0 10.0 11 0.2 0.4 0.6 0.7 0.9 1.1 1.3 1.5 1.6 1.8 3.7 5.5 7.3 9.2 I logcos Id log cot Icd log tan logsin d ' I Prop. Parts 136~ 226~ 316~ 46~ TABLE III 44~ 134~ 224~ 314~ I ' log sin d log tan c d log cot log cos |d Prop. Parts I I 2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.84 177 9.84 190 9.84 203 9.84 216 9.84 229 9.84 242 9.84 255 9.84 269 9.84 282 9.84 295 9.84 308 9.84 321 9.84 334 9.84 347 9.84 360 9.84 373 9.84 385 9.84 398 9.84 411 9.84 424 9.84 437 9.84 450 9.84 463 9.84 476 9.84 489 9.84 502 9.84 515 9.84 528 9.84 540 9.84 553 9.84 566 9.84 579 9.84 592 9.84 605 9.84 618 9.84 630 9.84 643 9.84 656 9.84 669 9.84 682 9.84 694 9.84 707 9.84 720 9.84 733 9.84 745 9.84 758 9.84 771 9.84 784 9.84 796 9.84 809 9.84 822 9.84 835 9.84 847 9.84 860 9.84 873 9.84 885 9.84 898 9.84 911 9.84 923 9.84 936 9.84 949 13 13 13 13 13 13 14 13 13 13 13 13 13 13 13 12 13 13 13 13 13 13 13 13 13 13 13 12 13 13 13 13 13 13 12 13 13 13 13 12 13 13 13 12 13 13 13 12 13 13 13 12 13 13 12 13 13 12 13 13 9.98 484 9.98 509 9.98 534 9.98 560 9.98 585 9.98 610 9.98 635 9.98 661 9.98 686 9.98 711 9.98 737 9.98 762 9.98 787 9.98 812 9.98 838 9.98 863 9.98 888 9.98 913 9.98 939 9.98 964 9.98 989 9.99 015 9.99 040 9.99 065 9.99 090 9.99 116 9.99 141 9.99 166 9.99 191 9.99 217 9.99 242 9.99 267 9.99 293 9.99 318 9.99 343 9.99 368 9.99 394 9.99 419 9.99 444 9.99 469 9.99 495 9.99 520 9.99 545 9.99 570 9.99 596 9.99 621 9.99 646 9.99 672 9.99 697 9.99 722 9.99 747 9.99 773 9.99 798 9.99 823 9.99 848 9.99 874 9.99 899 9.99 924 9.99 949 9.99 975 0.00 000 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 25 26 25 0.01 516 0.01 491 0.01 466 0.01 440 0.01 415 0.01 390 0.01 365 0.01 339 0.01 314 0.01 289 0.01 263 0.01 238 0.01 213 0.01 188 0.01 162 0.01 137 0.01 112 0.01 087 0.01 061 0.01 036 0.01 011 0.00 985 0.00 960 0.00 935 0.00 910 0.00 884 0.00 859 0.00 834 0.00 809 0.00 783 0.00 758 0.00 733 0.00 707 0.00 682 0.00 657 0.00 632 0.00 606 0.00 581 0.00 556 0.00 531 0.00 505 0.00 480 0.00 455 0.00 430 0.00 404 0.00 379 0.00 354 0.00 328 0.00 303 0.00 278 0.00 253 0.00 227 0.00 202 0.00 177 0.00 152 0.00 126 0.00 101 0.00 076 0.00 051 0.00 025 0.00 000 9.85 693 9.85 681 9.85 669 9.85 657 9.85 645 9.85 632 9.85 620 9.85 608 9.85 596 9.85 583 9.85 571 9.85 559 9.85 547 9.85 534 9.85 522 9.85 510 9.85 497 9.85 485 9.85 473 9.85 460 9.85 448 9.85 436 9.85 423 9.85 411 9.85 399 9.85 386 9.85 374 9.85 361 9.85 349 9.85 337 9.85 324 9.85 312 9.85 299 9.85 287 9.85 274 9.85 262 9.85 250 9.85 237 9.85 225 9.85 212 9.85 200 9.85 187 9.85 175 9.85 162 9.85 150 9.85 137 9.85 125 9.85 112 9.85 100 9.85 087 9.85 074 9.85 062 9.85 049 9.85 037 9.85 024 9.85 012 9.84 999 9.84 986 9.84 974 9.84 961 9.84 949 12 12 12 12 13 12 12 12 13 12 12 12 13 12 12 13 12 12 13 12 12 13 12 12 13 12 13 12 12 13 12 13 12 13 12 12 13 12 13 12 13 12 13 12 13 12 13 12 13 13 12 13 12 13 12 13 13 12 13 12 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 26 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.5 3.9 4.3 8.7 13.0 17.3 21.7 25 0.4 0.8 1.2 1.7 2.1 2.5 2.9 3.3 3.8 4.2 8.3 12.5 16.7 20.8 2 3 4 5 6 7 8 9 10 20 30 40 50 14 0.2 0.5 0.7 0.9 1.2 1.4 1.6 1.9 2.1 2.3 4.7 7.0 9.3 11.7 13 0.2 0.4 0.6 0.9 1.1 1.3 1.5 1.7 2.0 2.2 4.3 6.5 8.7 10.8 12 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 4.0 6.0 8.0 10.0; log cos d | log cot c d ilog tan log sin d J ' Prop. Parts 135~ 225~ 315~ 106 TABLE IV NATURAL TRIGONOMETRIC FUNCTIONS Of angles for each minute from 0~ to 90~, correct to five significant figures (For explanation, see page 27.) 900 1800 2700 00 TABLE IV 1~ 91~ 181~ 271o i. I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I I sin tan cot Icos I.00000 029 058 087 116.00145 175 204 233 262.00291 320 349 378 407.00436 465 495 524 553.00582 611 640 669 698.00727 756 785 814 844.00873 902 931 960.01018 047 076 105 134.01164 193 222 251 280.01309 338 367 396 425.01454 483 513 542 571.01600 629 658 687 716.01745.00000 029 058 087 116.00145 175 204 233 262.00291 320 349 378 407.00436 465 495 524 553.00582 611 640 669 698.00727 756 785 815 844.00873 902 931 960.00989.01018 047 076 105 135.01164 193 222 251 280.01309 338 367 396 425.01455 484 513 542 571.01600 629 658 687 716.01746 00 3437.7 1718.9 1145.9 859.44 687.55 572.96 491.11 429.72 381.97 343.77 312.52 286.48 264.44 245.55 229.18 214.86 202.22 190.98 180.93 171.89 163.70 156.26 149.47 143.24 137.51 132.22 127.32 122.77 118.54 114.59 110.89 107.43 104.17 101.11 98.218 95.489 92.908 90.463 88.144 85.940 83.844 81.847 79.943 78.126 76.390 74.729 73.139 71.615 70.153 68.750 67.402 66.105 64.858 63.657 62.4,99 61.383 60.306 59.266 58.261 57.290 1.0000 000 000 000 000 1.0000 000 000 000 000 1.0000.99999 999 999 999.99999 999 999 999 998.99998 998 998 998 998.99997 997 997 997 996.99996 996 996 995 995.99995 995 994 994 994.99993 993 993 992 992.99991 991 991 990 990.99989 989 989 988 988.99987 987 986 986 985.99985 I s i - 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 I I sin I tan cot I cos I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.01745 774 803 832 862.01891 920 949.01978.02007.02036 065 094 123 152.02181 211 240 269 298.02327 356 385 414 443.02472 501 530 560 589.02618 647 676 705 734.02763 792 821 850 879.02908 938 967.02996.03025.03054 083 112 141 170.03199 228 257 286 316.03345 374 403 432 461.03490.01746 775 804 833 862.01891 920 949.01978.02007.02036 066 095 124 153.02182 211 240 269 298.02328 357 386 415 444.02473 502 531 560 589.02619 648 677 706 735.02764 793 822 851 881.02910 939 968.02997.03026.03055 084 114 143 172.03201 230 259 288 317.03346 376 405 434 463.03492 57.290 56.351 55.442 54.561 53.709 52.882 52.081 51.303 50.549 49.816 49.104 48.412 47.740 47.085 46.449 45.829 45.226 44.639 4,4.066 43.508 42.964 42.433 41.916 41.411 40.917 40.436 39.965 39.506 39.057 38.618 38.188 37.769 37.358 36.956 36.563 36.178 35.801 35.431 35.070 34.715 34.368 34.027 33.694 33.366 33.045 32.730 32.421 32.118 31.821 31.528 31.242 30.960 30.683 30.412 30.145 29.882 29.624 29.371 29.122 28.877 28.636.99985 984 984 983 983.99982 982 981 980 980.99979 979 978 977 977.99976 976 975 974 974.99973 972 972 971 970.99969 969 968 967 966.99966 965 964 963 963.99962 961 960 959 959.99958 957 956 955 954.99953 952 952 951 950.99949 948 947 946 945.99944 943 942 941 940.99939 I I I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5. 4 3 2 1 5 4 3 2 1 0 0 Cos cot tan sin cos I cot tan sin I 179~ 269~ 359~ 89~ 108 88~ 1780 268~ 358~ 92 182~ 272~ 2~ TABLE IV 3~ 93~ 183~ 273~ ' I sin tan I cot cos I I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.03490 519 548 577 606.03635 664 693 723 752.03781 810 839 868 897.03926 955.03984.04013 042.04071 100 129 159 188.04217. 246 275 304 333.04362 391 420 449 478.04507 536 565 594 623.04653 682 711 740 769.04798 827 856 885 914.04943.04972.05001 030 059.05088 117 146 175 205.05234.03492 521 550 579 609.03638 667 696 725 754.03783 812 842 871 900.03929 958.03987.04016 046.04075 104 133 162 191 04220 250 279 308 337.04366 395 424 454 483.04512 541 570 599 628.04658 687 716 745 774.04803 833 862 891 920.04949.04978.05007 037 066.05095 124 153 182 212.05241 28.636.399 28.166 27.937.712 27.490.271 27.057 26.845.637 26.432.230 26.031 25.835.642 25.452.264 25.080 24.898.719 24.542.368.196 24.026 23.859 23.695.532.372.214 23.058 22.904.752.602.454.308 22.164 22.022 21.881.743.606 21.470.337.205 21.075 20.946 20.819.693.569.446.325 20.206 20.087 19.970.855.740 19.627.516.405.296.188 19.081.99939 938 937 936 935.99934 933 932 931 930.99929 927 926 925 924.99923 922 921 919 918.99917 916 915 913 912.99911 910 909 907 906.99905 904 902 901 900.99898 897 896 894 893.99892 890 889 888 886.99885 883 882 881 879.99878 876 875 873 872.99870 869 867 866 864.99863 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 I IW! I sin tan I cot 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.05234 263 292 321 350.05379 408 437 466 495.05524 553 582 611 640.05669 698 727 756 785.05814 844 873 902 931.05960.05989.06018 047 076.06105 134 163 192 221.06250 279 308 337 366.06395 424 453 482 511.06540 569 598 627 656.06685 714 743 773 802.06831 860 889 918 947.06976.05241 270 299 328 357.05387 416 445 474 503.05533 562 591 620 649.05678 708 737 766 795.05824 854 883 912 941.05970.05999.06029 058 087.06116 145 175 204 233.06262 291 321 350 379.06408 438 467 496 525.06554 584 613 642 671.06700 730 759 788 817.06847 876 905 934 963.06993 19.081 18.976.871.768.666 18.564.464.366.268.171 18.075 17.980.886.793.702 17.611.521.431.343.256 17.169 17.084 16.999.915.832 16.750.668.587.507.428 16.350.272.195.119 16.043 15.969.895.821.748.676 15.605.534.464.394.325 15.257.189.122 15.056 14.990 14.924.860.795.732.669 14.606.544.482.421'.361 14.301 Cos.99863 60 861 59 860 58 858 57 857 56.99855 55 854 54 852 53 851 52 849 51.99847 50 846 49 844 48 842 47 841 46.99839 45 838 44 836 43 834 42 833 41.99831 40 829 39 827 38 826 37 824 36.99822 35 821 34 819 33 817 32 815 31.99813 30 812 29. 810 28 808 27 806 26.99804 25 803 24 801 23 799 22 797 21.99795 20 793 19 792 18 790 17 788 16.99786 15 784 14 782 13 780 12 778 11.99776 10 774 9 772 8 770 7 768 6.99766 5 764 4 762 3 760 2 758 1.99756 0 0 - I cos I cot tan sin ' I: - I cos cot tan I sin | ' Ir 177~ 267~ 357~ 87 109 86~ 176~ 266~ 356~ 940 1840 2740 40TAL IV5 9018020 TABLE IV 50 950 1850 2750 f I sin I tan [ cot I cos I [/ I sin I tan I cot I Cos I 0 2 3 4 5 6 7 8 9 10 11I 1 2 1 3 14 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 51 52 54'3 55 56 57 58 59 60.06976.07005 034 063 092.07121 150 179 208 237.07266 295 324 353 382.07411 440 469 498 527.07556 585 614 643 672.0770 1 730 759 788 81 7.07846 875 904 933 962.0799 1.08020 049 078 107.08136 165 194 223 252.0828 1 310 339 368 397.08426 455 484 513 542.0857 1 600 629 658 687.08716.06993.07022 051 080 110.0713 9 168 197 227 256.07285 314 344 373 402.0743 1 461 490 519 548.07578 607 636 665 695.07724 753 782 812 841.07870 899 929 958.07987.08017 046 075 104 134.08 163 192 221 251 280.08309 339 368 397 427.08456 485 514 544 573.08602 632 661 690 720.08749 14. 301.241 * 182 * 124.065 14.008 13. 951.894.838.782 13. 727.672.617.563.510 13.457.404. 352.300.248 13. 197 * 146 13.046 12. 996 12. 947.898.850.801.754 12. 706.659.612.566.520 12. 474.429.384. 339.295 12. 251.207.163.120. 077 12.035 11.992.950.909.867 11.826.785.745.705. 664 11. 625. 585.546.507.468 11. 430.99756 754 752 750 748.99746 744 742 740 738.99736 734 731 729 727.99725 723 721 719 716.997 14 712 710 708 705.99703 701 699 696 694.99692 689 687 685 683.99680 678 67 6 673 671.99668 666 664 661 659.99657 654 652 649 647.99644 642 639 637 635.99632 630 627 625 622.996 19 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 4 1 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 2 1 20 1 9 1 8 1 7 1 6 15 14 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 10 2 3 4 5 6 7 8 9 10 1 1 129 1 3 1 6 1 7 1 8 1 9 20 2 1 22 23 24 25 26 27 2 8 29 30 31 32 33 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60.08716 745 774 803 831.08860 889 918 947.08976.09005 034 063 092 121.09 150 179 208 237 266.09295 324 353 382 411.09440 469 498 527 556.09585 614 642 671 700.09729 758 787 816 845.09874 903 932 961.09990.10019 048 077 106 135.10164 192 221 250 279.10308 337 366 395 424.10453.08749 778 807 837 866.08895 925 954.08983.09013.09042 071 101 130 159.09 189 218 247 277 306.09335 365 394 423 453'.09482 511 541 570 600.09629 658 688 717 746.09776 805 834 864 893.09923 952.09981.10011 040.10069 099 128 158 187.10216 246 275 305 334.10363 393 422 452 481.10510 11. 430.392.354.316.279 11.242.205.168.132.0U95 11. 059 11.024 1 0. 988.953.918 10. 883.848.814.780.746 10. 712.678.64 5.612.579 10. 546.514.481.449.417 10. 385.354.322.291.260 10. 229.199.168.138,108 10. 078.048 10. 019 9. 9893.960 1 9. 9310.902 1. 8734.8448.8164 9. 7882.7601. 7322.7044.6768 9. 6493. 6220.5949. 5679.5411 9. 5144.99619 617 614 612 609.99607 604 602 599 596.99594 591 588 586 583.99580 578 575 572 570.99567 564 562 559 556.99553 551 548 545 542.99540 537 534 531 528.99526 523 520 51 7 514.99511 508 506 503 500.99497 494 491 488 485.99482 479 476 473 470.99467 464 461 458 455.99452 60 59 58 57 56 54 53 52 5 1 50 49 48 47 46 45 44 43 42 4 1 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 I2 1 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 Cos ] cot Itan I sin II I Cos I cot I tan I sin I I 1750 2650'3550 8 50 110 j750 26503550 850 110 ~~~840 1740 2640 3540 96~ 186~ 276~ 60 TABLE IV 7~ 97~ 187~ 277~ I I sin tan I cot cos sin tan I cot I cos 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 52 53 54 55 56 57 58 59 60.10453 482 511 540 569.10597 626 655 684 713.10742 771 800 829 858.10887 916 945.10973.11002.11031 060 089 118 147.11176 205 234 263 291.11320 349 378 407 436.11465 494 523 552 580.11609 638 667 696 725.11754 783 812 840 869.11898 927 956.11985.12014.12043 071 100 129 158.12187 10510 540 569 599 628.10657 687 716 746 775.10805 834 863 893 922.10952.10981.11011 040 070.11099 128 158 187 217.11246 276 305 335 364.11394 423 452 482 511.11541 570 600 629 659.11688 718 747 777 806.11836 865 895 924 954.11983.12013 042 072 101.12131 160 190 219 249.12278 9.5144.4878.4614.4352.4090 9.3831.3572.3315.3060.2806 9.2553.2302.2052.1803.1555 9.1309.1065.0821.0579.0338 9.0098 8.9860.9623.9387.9152 8.8919.8686.8455.8225.7996 8.7769.7542.7317.7093.6870 8.6648.6427.6208.5989.5772 8.5555.5340.5126.4913.4701 8.4490.4280.4071.3863.3656 8.3450.3245.3041.2838.2636 8.2434.2234.2035.1837.1640 8.1443.99452 449 446 443 440.99437 434 431 428 424.99421 418 415 412 409.99406 402 399 396 393.99390 386 383 380 377.99374 370 367 364 360.99357 354 351 347 344.99341 337 334 331 327.99324 320 317 314 310.99307 303 300 297 293.99290 286 283 279 276.99272 269 265 262 258.99255 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.12187 216 245 274 302.12331 360 389 418 447.12476 504 533 562 591.12620 649 678 706 735.12764 793 822 851 880.12908 937 966.12995.13024.13053 081 110 139 168.13197 226 254 283 312.13341 370 399 427 456.13485 514 543 572 600.13629 658 687 716 744.13773 802 831 860 889.13917.12278 308 338 367 397.12426 456 485 515 544.12574 603 633 662 692.12722 751 781 810 840.12869 899 929 958.12988.13017 047 076 106 136.13165 195 224 254 284.13313 343 372 402 432.13461 491 521 550 580.13609 639 669 698 728.13758 787 817 846 876.13906 935 965.13995.14024.14054 8.1443.1248.1054.0860.0667 8.0476.0285 8.0095 7.9906.9718 7.9530.9344.9158.8973.8789 7.8606 8424.8243.8062.7882 7.7704.7525.7348.7171.6996 7.6821.6647.6473.6301.6129 7.5958.5787.5618.5449.5281 7.5113.4947.4781.4615.4451 7.4287.4124.3962.3800.3639 7.3479.3319.3160.3002.2844 7.2687.2531.2375.2220.2066 7.1912.1759.1607.1455.1304 7.1154.99255 251 248 244 240.99237 233 230 226 222.99219 215 211 208 204.99200 197 193 189 186.99182 178 175 171 167.99163 160 156 152 148.99144 141 137 133 129.99125 122 118 114 110.99106 102 098 094 091.99087 083 079 075 071.99067 063 059 055 051.99047 043 039 035 031.99027 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30. 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 cos cot j tan sin | ' I os co t I tan sin I ' 173~ 263~ 353~ 83~ 111 82~ 172~ 262~ 352~ 98~ 188~ 278~ 80 TABLE IV 9~ 99~ 189~ 2790 ' sin I tan I cot cos I sin I tan I cot | cos I.. - - -,, 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I.13917 946.13975.14004 033.14061 090 119 148 177.14205 234 263 292 320.14349 378 407 436 464.14493 522 551 580 608.14637 666 695 723 752.14781 810 838 867 896.14925 954.14982 *15011 040.15069 097 126 155 184.15212 241 270 299 327.15356 385 414 442 471.15500 529 557 586 615.15643.14054 084 113 143 173.14202 232 262 291 321.14351 381 410 440 470.14499 529 559 588 618.14648 678 707 737 767.14796 826 856 886 915.14945.14975.15005 034 064.15094 124 153 183 213.15243 272 302 332 362.15391 421 451 481 511.15540 570 600 630 660.15689 719 749 779 809.15838 7.1154.99027.1004 023.0855 019.0706 015.0558 011 7.0410.99006.0264.99002 7.0117.98998 6.9972 994.9827 990 6.9682.98986.9538 982.9395 978.9252 973.9110 969 6.8969.98965.8828 961.8687 957.8548 953.8408 948 6.8269.98944.8131 940.7994 936.7856 931.7720 927 6.7584.98923.7448 919.7313 914.7179 910.7045 906 6.6912.98902.6779 897.6646 893.6514 889.6383 884 6.6252.98880.6122 876.5992 871.5863 867.5734 863 6.5606.98858.5478 854.5350 849.5223 845.5097 841 6.4971.98836.4846 832.4721 827.4596 823.4472 818 6.4348.98814.4225 809.4103 805.3980 800.3859 796 6.3737.98791.3617 787.3496 782.3376 778.3257 773 6.3138.98769 tan sin 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.15643 672 701 730 758.15787 816 845 873 902.15931 959.15988.16017 046.16074 103 132 160 189.16218 246 275 304 333.16361 390 419 447 476.16505 533 562 591 620.16648 677 706 734 763.16792 820 849 878 906.16935 964 16992.17021 050.17078 107 136 164 193.17222 250 279 308 336.17365.15838 868 898 928 958.15988.16017 047 077 107.16137 167 196 226 256.16286 316 '346 376 405.16435 465 495 525 555.16585 615 645 674 704.16734 764 794 824 854.16884 914 944.16974.17004.17033 063 093 123 153.17183 213 243 273 303.17333 363 393 423 453.17483 513 543 573 603.17633 6.3138.3019.2901.2783.2666 6.2549.2432.2316.2200.2085 6.1970.1856.1742.1628.1515 6.1402.1290.1178.1066.0955 6.0844.0734.0624.0514.0405 6.0296.0188 6.0080 5.9972.9865 5.9758.9651.9545.9439.9333 5.9228.9124.9019 8915.8811 5.8708.8605.8502.8400.8298 5.8197.8095.7994.7894.7794 5.7694.7594.7495.7396.7297 5.7199.7101.7004.6906.6809 5.6713.98769 764 760 755 751.98746 741 737 732 728.98723 718 714 709 704.98700 695 690 686 681.98676 671 667 662 657.98652 648 643 638 633.98629 624 619 614 609.98604 600 595 590 585.98580 575 570 565 561.98556 551 546 541 536.98531 526 521 516 511.98506 501 496 491 486.98481 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 3,6 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19.18 17 16 15 14, 13 12 11 10 9 8 7 6 5 4 3 2 1 0 I cos cot T II I cos ot tan sin I ' 171~ 261~ 351~ 81~ [12 800 170~ 260~ 350~ 100~ 190~ 280~ 1 0 TABLE IV 11~ o101 191~ 281~! r sin I tan I cot cos r I 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.17365 393 422 451 479.17508 537 565 594 623.17651 680 708 737 766.17794 823 852 880 909.17937 966.17995.18023 052.18081 109 138 166 195.18224 252 281 309 338.18367 395 424 452 481.18509 538 567 595 624.18652 681 710 738 767.18795 824 852 881 910.18938 967.18995.19024 052.19081.17633 663 693 723 753.17783 813 843 873 903.17833 963.17993.18023 053.18083 113 143 173 203.18233 263 293 323 353.18384 414 444 474 504.18534 564 594 624 654.18684 714 745 775 805.18835 865 895 925 955.18986.19016 046 076 106.19136 166 197 227 257.19287 317 347 378 408.19438 5.6713.6617.6521.6425.6329 5.6234.6140.6045.5951.5857 5.5764.5671.5578.5485.5393 5.5301.5209.5118.5026.4936 5.4845.4755.4665.4575.4486 5.4397.4308.4219.4131.4043 5.3955.3868.3781.3694.3607 5.3521.3435.3349.3263.3178 5.3093.3008.2924.2839.2755 5.2672.2588.2505.2422.2339 5.2257.2174.2092.2011.1929 5.1848.1767.1686.1606.1526 5.1446.98481 476 471 466 461.98455 450 445 440 435.98430 425 420 414 409.98404 399 394 389 383.98378 373 368 362 357.98352 347 341 336 331.98325 320 315 310 304.98299 294 288 283 277.98272 267 261 256 250.98245 240 234 229 223.98218 212 207 201 196.98190 185 179 174 168.98163 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 3 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 I r sin I tan I cot Icos r 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.19081 109 138 167 195.19224 252 281 309 338.19366 395 423 452 481.19509 538 566 595 623.19652 680 709 737 766.19794 823 851 880 908.19937 965.19994.20022 051.20079 108 136 165 193.20222 250 279 307 336.20364 393 421 450 478.20507 535 563 592 620.20649 677 706 734 763.20791.19438 468 498 529 559.19589 619 649 680 710.19740 770 801 831 861.19891 921 952.19982.20012.20042 073 103 133 164.20194 224 254 285 315.20345 376 406 436 466.20497 527 557 588 618.20648 679 709 739 770.20800 830 861 891 921.20952.20982.21013 043 073.21104 134 164 195 225.21256 5.1446.1366.1286.1207.1128 5.1049.0970.0892.0814.0736 5.0658.0581.0504.0427.0350 5.0273.0197.0121 5.0045 4.9969 4.9894.9819.9744.9669.9594 4.95'20.9446.9372.9298.9225 4.9152.9078.9006.8933.8860 4.8788.8716.8644.8573.8501 4.8430.8359.8288.8218.8147 4.8077.8007.7937.7867.7798 4.7729.7659.7591.7522.7453 4.7385.7317.7249.7181.7114 4.7046.98163 157 152 146 140.98135 129 124 118 112.98107 101 096 090 084.98079 073 067 061 056.98050 044 039 033 027.98021 016 010.98004.97998.97992 987 981 975 969.97963 958 952 946 940.97934 928 922 916 910.97905 899 893 887 881.97875 869 863 857 851.97845 839 833 827 821.97815 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 _ -! cos - I I cot Itan | sin - I I D - I I I cos cot Itan I sin I ' 1690 2590 349~ 79~ 113 78~ 168~ 258~ 348~ 102~ 192~ 2820 12~ TABLE TV _ _ 13~ 103~ 193~ 283~ l-I -' 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 sin tan.20791.21256 820 286 848 316 877 347 905 377.20933.21408 962 438.20990 469.21019 499 047 529.21076.21560 104 590 132 621 161 651 189 682.21218.21712 246 743 275 773 303 804 331 834.21360.21864 388 895 417 925 445 956 474.21986.21502.22017 530 047 559 078 587 108 616 139.21644.22169 672 200 701 231 729 261 758 292.21786.22322 814 353 843 383 871 414 899 444.21928.22475 956 505.21985 536.22013 567 041 597.22070.22628 098 658 126 689 155 719 183 750.22212.22781 240 811 268 842 297 872 325.903.22353.22934 382 964 410.22995 438.23026 467 056.22495.23087 cot I cos.I 4.7046.6979.6912.6845.6779 4.6712.6646.6580.6514.6448 4.6382 6317.6252.6187.6122 4.6057.5993.5928.5864.5800 4.5736.5673.5609.5546.5483 4.5420.5357.5294.5232.5169 4.5107.5045.4983.4922.4860 4.4799.4737.4676.4615.4555 4.4494.4434.4373.4313.4253 4.4194.4134.4075.4015.3956 4.3897.3838.3779.3721.3662 4.3604.3546.3488.3430.3372 4.3315.97815 809 803 797 791.97784 778 772 766 760.97754 748 742 735 729.97723 717 711 705 698.97692 686 680 673 667.97661 655 648 642 636.97630 623 617 611 604.97598 592 585 579 573.97566 560 553 547 541.97534 528 521 515 508.97502 496 489 483 476.97470 463 457 450 444.97437 I I I 1 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O! I ~_ -~ 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I - sin I tan I cot cos i - L I-~~~~~~~~~~~~~~~~~~~~~~~~~~~.22495 523 552 580 608.22637 665 693 722 750.22778 807 835 863 892.22920 948.22977.23005 033.23062 090 118 146 175.23203 231 260 288 316.23345 373 401 429 458.23486 514 542 571 599.23627 656 684 712 740.23769 797 825 853 882.23910 938 966.23995.24023.24051 079 108 136 164.24192.23087 117 148 179 209.23240 271 301 332 363.23393 424 455 485 516.23547 578 608 639 670.23700 731 762 793 823.23854 885 916 946.23977.24008 039 069 100 131.24162 193 223 254 285.24316 347 377 408 439.24470 501 532 562 593.24624 655 686 717 747.24778 809 840 871 902.24933 4.3315.3257.3200.3143.3086 4.3029.2972.2916.2859.2803 4.2747.2691.2635.2580.2524 4.2468.2413.2358.2303.2248 4.2193.2139.2084.2030.1976 4.1922.1868.1814.1760.1706 4.1653.1600.1547.1493.1441 4.1388.1335.1282.1230.1178 4.1126.1074.1022.0970.0918 4.0867.0815.0764.0713.0662 4.0611.0560.0509.0459.0408 4.0358.0308.0257.0207.0158 4.0108.97437 430 424 417 411.97404 398 391 384 378.97371 365 358 351 345.97338 331 325 318 311.97304 298 291 284 278.97271 264 257 251 244.97237 230 223 217 210.97203 196 189 182 176.97169 162 155 148 141.97134 127 120 113 106.97100 093 086 079 072.97065 058 051 044 037.97030 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 I I cos cot I tan _ sin - I I I I L cos cot I tan I sin I' 167~ 257~ 3470 77~ 114 76~ 166~ 256~ 346~ 1040 1940 2840 140 TABLE IV 150 1050 1950 2850 I I sin I tan I cot I cos I-~ I I I sin I tan I cot I Cos I 10 2 3 4 S 6 7 8 9 10 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 2 1 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I.24 192 220 249 277 305.24333 362 390 418 446.24474 503 531 559 587.24615 644 672 700 728.24756 784 813 841 869.24897 925 954.24982.250 10.25038 066 094 122 151.25179 207 235 263 291.25320 348 376 404 432.25460 488. 516 545 573.2560 1 629 657 685 713.2574 1 769 798 826 854.25882..24933 964.24995.'25026 056.25087 118 149 180 211.25242 273 304 335 366.25397 428 459 490 521.25552 583 614 645 676.25707 738 769 800 831.25862 893 924 955.25986.26017 048 079 110 141.26172 203 235 266 297.26328 359 390 421 452.26483 515 546 577 608.26639 670 701 733 764.26795 4. 0108.0058 4. 0009 3. 9959.9910 3. 9861.9812.9763.9714.9665 3.9617.9568.9520.9471.9423 3. 9375.9327.9279.9232.9184 3. 9136.9089.9042.8995.8947 3. 8900. 8854.8807.8760.8714 3. 8667.862 1. 8575.8528.8482 3. 8436.839 1.8345.8299.8254 3. 8208.8163.8118.8073.8028 3. 7983. 7938.7893.7848.7804 3. 7760. 7715.7671.7627. 7583 3. 7539.7495. 745 1.7408.7364 3. 7321.97030 023 015 008.9700 1.96994 987 980 973 966.96959 952 945 937 930.96923 916 909 902 894.96887 880 873 866 858.9685 1 844 837 829 822.96815 807 800 793 786.96778 771 764 756 749 96742 734 727 719 712.96705 697 690 682 675.96667 660 653 645 638.96630 623 615 608 600.96593 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44, 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 10 2 3 4 5 6 7 8 9 10 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 2 1 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.25882 910 938 966.25994.26022 050 079 107 135.26163 191 219 247 275.26303 331 359 387 415.26443 471 500 528 556.26584 612 640 668 696.26724 752 780 808 836.26864 892 920 948.26976.27004 032 060 088 116.27 144 172 200 228 256.27284 312 340 368 396.27424 452 480 508 536.27564.26795 826 857 888 920.2695 1.26982.27013 044 076.27 107 138 169 201 232.27263 294 326 357 388.274 19 451 482 513 545.27576 607 638 670 701.27732 764 795 826 858.27889 921 952.27983.280 15.28046 077 109 140 1 72.28203 234 266 297 329.28360 391 423 454 486.2851 7 549 580 612 643.28675 3. 7321. 727 7.7234. 7191.7148 3. 7105.7062.7019.6976.6933 3. 6891.6848.6806.6764.6722 3. 6680.6638.6596.6554.6512 3. 6470.6429.6387.6346.6305 3. 6264.6222.6181.6140.6100 3. 6059.6018.5978.5937.5897 3. 5856.5816.5776.5736.5696 3. 5656.5616. 5576.5536.5497 3. 5457.5418. 5379.5339.5300 3. 5261.5222.5183.5144.5105 3. 5067.5028.4989.495 1.4912 3. 4874.96593 '585 578 570 562.96555 547 540 532 524.965 17 509 502 494 486.96479 471 463 456 448.96440 433 425 417 410.96402 394 386 379 371.96363 355 347 340 332.96324 316 308 301 293.96285 277 269 261 253.96246 238 230 222 214.96206 198 190 182 174.96166 158 150 142 134.96126 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 5 4 3 2 0 7 I cos I cot I tan I sin I I I Cos I cot I tan I sin I I 1660 2550 3450 750 115 740 1640 2540 3440 ' 106~ 196~ 286~ 16~ TABLE IV 17~ 107~ 197~ 287~ - - 0 1 2 3 4 I I sin I, tan cot 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.27564 592 620 648 676.27704 731 759 787 815.27843 871 899 927 955.27983.28011 039 067 095.28123 150 178 206 234.28262 290 318 346 374.28402 429 457 485 513.28541 569 597 625 652.28680 708 736 764 792.28820 847 875 903 931.28959.28987.29015 042 070.29098 126 154 182 209.29237.28675 3.4874 706.4836 738.4798 769.4760 801.4722 l I cos I -.28832 864 895 927 958.28990.29021 053 084 116.29147 179 210 242 274.29305 337 368 400 432.29463 495 526 558 590.29621 653 685 716 748.29780 811 843 875 906.29938.29970.30001 033 065.30097 128 160 192 224.30255 287 319 351 382.30414 446 478 509 541.30573 3.4684.4646.4608.4570.4533 3.4495.4458.4420.4383.4346 3.4308.4271.4234.4197.4160 3.4124.4087.4050.4014.3977 3.3941.3904.3868.3832.3796 3.3759.3723.3687.3652.3616 3.3580.3544.3509.3473.3438 3.3402.3367.3332.3297.3261 3.3226.3191.3156.3122.3087 3.3052.3017.2983.2948.2914 3.2879.2845.2811.2777.2743 3.2709.96126 118 110 102 094.96086 078 070 062 054.96046 037 029 021 013.96005.95997 989 981 972.95964 956 948 940 931.95923 915 907 898 890.95882 874 865 857 849.95841 832 824 816 807.95799 791 782 774 766.95757 749 740 732 724.95715 707 698 690 681.95673 664 656 647 639.95630 I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12. 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 -! I sin I tan.29237 265 293 321 348.29376 404 432 460 487.29515 543 571 599 626.29654 682 710 737 765.29793 821 849 876 904.29932 960.29987.30015 043.30071 098 126 154 182.30209 237 265 292 320.30348 376 403 431 459.30486 514 542 570 597.30625 653 680 708 736.30763 791 819 846 874.30902.30573 605 637 669 700.30732 764 796 828 860.30891 923 955.30987.31019.31051 083 115 147 178.31210 242 274 306 338.31370 402 434 466 498.31530 562 594 626 658.31690 722 754 786 818.31850 882 914 946.31978.32010 042 074 106 139.32171 203 235 267 299.32331 363 396 428 460.32492 I I cot cos 3.2709.95630.2675 622.2641 613.2607 605.2573 596 3.2539.95588.2506 579.2472 571.2438 562.2405 554 3.2371.95545.2338 536.2305 528.2272 519.2238 511 3.2205.95502.2172 493.2139 485.2106 476.2073 467 3.2041.95459.2008 450.1975 441.1943 433.1910 424 3.1878.95415.1845 407.1813 398.1780 389.1748 380 3.1716.95372.1684 363.1652 354.1620 345.1588 337 3.1556.95328.1524 319..1492 310.1460 301.1429 293 3.1397.95284.1366 275.1334 260.1303 257.1271 248 3.1240.95240.1209 231.1178 222.1146 213.1115 204 3.1084.95195.1053 186.1022 177.0991 168.0961 159 3.0930.95150.0899 142.0868 133.0838 124.0807 115 3.0777.95106 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 '23 22 21 20 19:18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 i I - I I I cos I cot I tan I I I I sin I ' I I - I I cos I cot I tan I sin I - 6 163~ 253~ 343~ 73~ 116 72~ 162~ 252~ 342~ 108~ 198~ 288~ 18~ TABLE IV 19~ 109~ 199~ 2890 I sin tan cot I cos I / I sin I tan I cot cos I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.30902.32492 13.0777 929 524.0746 957 556.0716.30985 588.0686.31012 621.0655.31040.32653 3.0625 068 685.0595 095 717.0565 123 749.0535 151 782.0505.31178.32814 3.0475 206 846.0445 233 878.0415 261 911.0385 289 943.0356.31316.32975 3.0326 344.33007.0296 372 040.0267 399 072.0237 427 104.0208.31454.33136 3.0178 482 169.0149 510 201.0120 537 233.0090 565 266.0061.31593.33298 3.0032 620 330 3.0003 648 363 2.9974 675 395.9945 703 427.9916.31730.33460 2.9887 758 492.9858 786 524.9829 813 557.9800 841 589.9772.31868.33621 2.9743 896 654.9714 923 686.9686 951 718.9657.31979 751.9629.32006 33783 2.9600 034 816.9572 061 848.9544 089 881.9515 116 913.9487.32144.33945 2.9459 171.33978.9431 199.34010.9403 227 043.9375 254 075.9347.32282.34108 2.9319 309 140.9291 337 173.9263 364 205.9235 392 238.9208.32419.34270 2.9180 447 303.9152 474 335.9125 502 368.9097 529 400.9070.32557.34433 2.9042.95106 097 088 079 070.95061 052 043 033 024.95015.95006.94997 988 979.94970 961 952 943 933.94924 915 906 897 888.94878 869 860 851 842.94832 823 814 805 795.94786 777 768 758 749.94740 730 721 712 702.94693 684 674 665 656.94646 637 627 618 609.94599 590 580 571 561.94552 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.32557 584 612 639 667.32694 722 749 777 804.32832 859 887 914 942.32969.32997.33024 051 079.33106 134 161 189 216.33244 271 298 326 353.33381 408 436 463 490.33518 545 573 600 627.33655 682 710 737 764.33792 819 846 874 901.33929 956.33983.34011 038.34065 093 120 147 175.34202.34433 465 498 530 563.34596 628 661 693 726.34758 791 824 856 889.34922 954.34987.35020 052.35085 118 150 183 216.35248 281 314 346 379.35412 445 477 510 543.35576 608 641 674 707.35740 772 805 838 871.35904 937.35969.36002 035.36068 101 134 167 199.36232 265 298 331 364.36397 2.9042.94552.9015 542.8987 533.8960 523.8933 514 2.8905.94504.8878 495.8851 485.8824 476.8797 466 2.8770.94457.8743 447.8716 438.8689 428.8662 418 2.8636.94409.8609 399.8582 390.8556 380.8529 370 2.8502.94361.8476 351.8449 342.8423 332.8397 322 2.8370.94313.8344 303.8318 293.8291 284.8265 274 2.8239.94264.8213 254.8187 245.8161 235.8135 225 2.8109.94215.8083 206.8057 196.8032 186.8006 176 2.7980.94167.7955 157.7929 147.7903 137.7878 127 2.7852.94118.7827 108.7801 098.7776 088.7751 078 2.7725.94068.7700 058.7675 049.7650 039.7625 029 2.7600.94019.7575.94009.7550.93999.7525 989.7500 979 2.7475.93969 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45.44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -~. I. I cos I cot tan I sin ' I cos cot tan | sin | 161~ 251 341~ 7 1~ 117 70~ 160~ 250~ 340~ 1100 2000 2900 200TAL IV201021090 TABLE IV 210 1110 2010 291" I I I I 10 2 3 4 5 6 7 8 9 10 11I 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 I41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 i sin tan Icot Cos I.34202 229 257 284 311.34339 366 393 421 448.34475 503 530 557 584.34612 639 666 694 721.34748 775 803 830 857.34884 912 939 966.34993.3502 1 048 075 102 130.35 157 184 211 239 266.35293 320 347 375 402.35429 456 484 511 538.35565 592 619 647 674.3570 1 728 755 782 810.35837.36397 430 463 496 529.36562 595 628 661 694.36727 760 793 826 859.36892 925 958.3699 1.37024.37057 090 123 157 190.37223 256 289 322 355.37388 422 455 488 521.37554 588 621 654 687.37720 754 787 820 853.37887 920 953.37986.38020.38053 086 120 153 186.38220 253 286 320 353.38386 2. 7475.7450.7425.7400.7376 2. 7351.7326.7302.7277.7253 2. 7228.7204.71 79.7155.7130 2. 7106.7082.7058. 7034.7009 2. 6985.696 1.6937.6913. 6889 2. 6865. 6841.6818.6794.6770 2. 6746.6723.6699.6675.6652 2. 6628.6605.658 1.6558.6534 2. 6511. 6488.6464.6441.6418 2. 6395.637 1.6348.6325.6302 2. 6279.6256.6233.6210.6187 2.6165. 6142.6119.6096. 6074 2. 6051.93969 959 949 939 929.93919 909 899 889 879.93869 859 849 839 829.938 19 809 799 789 779.93769 759 748 738 728.937 18 708 698 688 677.93667 657 647 637 626.93616 606 596 585 575.93565 555 544 534 524.935 14 503 493 483 472.93462 452 441 431 420.934 10 400 389 379 368.93358 I '59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5, 4 3 2 0 I f I 10 2 3 4 65 7 8 9 10 1 2 13 1 4 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I sin tan I cot I Cos F.I.35837 864 891 918 945.35973.36000 027 054 08 1.36108 1 35 162 190 21]7.36244 271 298 325 352.36379 406 434 461 488.365 15 542 569 596 623.36650 677 704 731 758.36785 812 839 867 894.36921 948.36975.37002 029.37056 083 110 137 164.37191 218 245 272 299.37326 353 380 407 434.3746 1.38386 420 453 487 520.38553 587 620 654 687.38721 754 787 821 854.38888 921 955.38988.39022.39055 089 122 156 190.39223 257 290 324 357.3939 1 425 458 492 526.39559 593 626 660 694.39727 761 795 829 862.39896 930 963.39997.4003 1.40065 098 132 166 200.40234 267 301 335 369.40403 2. 6051.6028. 6006.5983.5961 2. 5938.5916.5893. 587 1.5848 2. 5826.5804.5782. 5759.5737 2. 5715.5693.567 1.5649. 5627 2. 5605.5583.5561.5539.55 17 2. 5495. 5473.5452.5430.5408 2.5386.5365.5343.5322.5300 2. 5279.5257.5236. 5214.5193 2. 5172. 5150.5129.5108.5086 2. 5065.5044.5023.5002.498 1 2. 4960.4939.4918.4897.4876 2.4855.4834.4813.4792.4772 2.4751.93358 348 337 327 316.93306 295 285 274 264.93253 243 232 222 211.93201 190 180 169 159.93 148 137 127 116 106.93095 084 074 063 052.93042 031 020.930 10.92999.92988 978 967 956 945.92935 924 913 902 892.9288 1 870 859 849 838.92827 816 805 794 784.92773 762 751 740 729.927 18 60 59 58 57 54 53 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 2 1 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 I. i Cos I cot I tan _]Isin i If i Cos I cot I tan I sin I _ 1590 2490 3390 690 11 8 50 4038 118 680 1580 2480 3380 1120 2020 2920 220 TABLE IV 1I I 230 1130 2030 2930 I in I tan I cot I Cos Isin I tan I cot I cos 0 2 3 4 -5 6 7 8 9 10 11I 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 2 1 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 '38' 3 9 40 4 1 42 43 44 45 46 47 48 49 50 5 1 52 5 3 54 55 5 6 57 5 8 5 9 60.37461 488 515 542 569.37595 622 649 676 703.37730 757 784 811 838.37865 892 919 946 973.37999.38026 053 080 107.38134 161 188 215 241.38268 295 322 349 376.38403 430 456 483 510.38537 564 591 617 644.3867 1 698 725 752 778.38805 832 859 886 912.38939 966.38993.39020 046.39073.40403 2. 4751 436.4730 470.4709 504.4689 538.4668.40572 2.4648 606,.4627 640.4606 674.4586 707.4566.40741 2. 4545 775. 4525 809.4504 843.4484 877.4464.40911 2.4443 945.4423.40979.4403.41013. 4383 047.4362.41081 2.4342 115.4322 149.4302 183.4282 217.4262.41251 2.4242 285.4222 319.4202 353.4182 387.4162.41421 2.4142 455.4122 490.4102 524.4083 558.4063.41592 2.4043 626.4023 660.4004 694.3984 728.3964.41763 2. 3945 797.3925 831.3906 865. 3886 899.3867.41933 2. 3847.41968.3828.42002.3808 036.3789 070.3770.42105 2. 3750 139.3731 173. 3712 207.3693 242.3673.42276 2. 3654 310.3635 345.361 6 379.3597 413. 3578.42447 2. 3559.927 18 707 697 686 675.92664 653 642 631 620.92609 598 587 576 565.92554 543 532 521 510.92499 488 477 466 455.92 444 432 421 410 399.92388 377 366 355 343.92332 321 310 299 287.92276 265 254 243 231.92220 209 198 186 1 75.92 164 152 141 130 119.92 107 096 085 073 062.92050 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 2 1 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 * I I 0 2 3 4 5' 6 7 8 9 10 1 2 1 3 1 4 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60.39073.42447 100 482 127 516 153 551 180 585.39207.42619 234 654 260 688 287 722 314 757.39341.42791 367 826 394 860 421 894 448 929.39474.42963 501. 42998 528.43032 555 067 581 101.39608.43136 635 170 661 205 688 239 715 274.39741.43308 768 343 795 378 822 412 848 447.39875.43481 902 516 928 550 955 585.39982 620.40008.43654 035 689 062 724 088 758 115 793.40141.43828 168 862 195 897 221 932 248.43966.40275.44001 301 036 328 071 355 105 381 140.40408.44175 434 210 461 244 488 279 514 314.40541.44349 567 384 594 418 621 453 647 488 '.40674.44523 Cos cot 2.3559.92050.3539 039.3520 028.3501 016.3483. 92005 2.3464.91994.3445 982.3426 971.3407 959.3388 948 2. 3369.91936. 3351 925.3332 914. 331 3 902.3294 891 2. 3276.91879.3257 868.3238 856.3220 845.3201 833 2.3183.91822.3164 810.3146 799. 3127 787.3109 775 2. 3090.91764.3072 752. 3053 741.3035 729.3017 718 2. 2998.91706.2980 694.2962 683.2944 671.2925 660 2.2907. 91648.2889 636.2871 625.2853 613.2835 601 2. 281 7. 91590.2799 578. 2781 566.2763 555. 2745 54 2. 2727.9153.2709 519.2691 508.2673 496. 2655 484 2.2637.91472. 2620 461.2602 449.2584 437.2566 425 2.2549. 91414.2531 402.2513 390.2496 37 8.2478 366 2.2460 i.91335 tan sin 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 -27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 I I Cos I cot I tan [ sin I I __j i il 1570 2470 3370 67' 119 119 ~~660 1560 2460 3360 1140 204~ 294~ 24~ I 'ABLE IV 2 5~ 115~ 205~ 295~ - I r sin I tan cot I cos I! I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 l.40674 700 727 753 780.40806 833 860 886 913.40939 966.40992.41019 045.41072 098 125 151 178.41204 231 257 284 310.41337 363 390 416 443.41469 496 522 549 575.41602 628 655 681 707.41734 760 787 813 840.41866 892 919 945 972.41998.42024 051 077 104.42130 156 183 209 235.42262.44523 558 593 627 662.44697 732 767 802 837.44872 907 942.44977.45012.45047 082 117 152 187.45222 257 292 327 362.45397 432 467 502 538.45573 608 643 678 713.45748 784 819 854 889.45924 960.45995.46030 065.46101 136 171 206 242.46277 312 348 383 418.46454 489 525 560 595.46631 2.2460.2443.2425.2408.2390 2.2373.2355.2338.2320.2303 2.2286.2268.2251.2234.2216 2.2199.2182.2165.2148.2130 2.2113.2096.2079.2062.2045 2.2028.2011.1994.1977.1960 2.1943.1926.1909.1892.1876 2.1859.1842.1825.1808.1792 2.1775.1758.1742.1725.1708 2.1692.1675.1659.1642.1625 2.1609.1592.1576.1560.1543 2.1527.1510.1494.1478.1461 2. 1445.91355 343 331 319 307.91295 283 272 260 248.91236 224 212 200 188.91176 164 152 140 128.91116 104 092 080 068.91056 044 032 020.91008.90996 984 972 960 948.90936 924 911 899 887.90875 863 851 839 826.90814 802 790 778 766.90753 741 729 717 704.90692 680 668 655 643.90631 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I I.42262 288 315 341 367.42394 420 446 473 499.42525 552 578 604 631.42657 683 709 736 762.42788 815 841 867 894.42920 946 972.42999.43025.43051 077 104 130 156.43182 209 235 261 287.43313 340 366 392 418.43445 471 497 523 549.43575 602 628 654 680.43706 733 759 785 811.43837 sin tan cot cos.46631 2.1445.90631 666.1429 618 702.1413 606 737.1396 594 772.1380 582.46808 2.1364.90569 843.1348 557 879.1332 545 914.1315 532 950.1299 520.46985 2.1283.90507.47021.1267 495 056.1251 483 092.1235 470 128.1219 458.47163 2.1203.90446 199.1187 433 234.1171 421 270.1155 408 305.1139 396.47341 2.1123.90383 377.1107 371 412.1092 358 448.1076 346 483.1060 334.47519 2.1044.90321 555.1028 309 590.1013 296 626 0997 284 662.0981 271.47698 2.0965.90259 733.0950 246 769.0934 233 805.0918 221 840.0903 208.47876 2.0887.90196 912.0872 183 948.0856 171.47984.0840 158.48019.0825 146.48055 2.0809.90133 091.0794 120 127.0778 108 163.0763 095 198.0748 082.48234 2.0732.90070 270.0717 057 306.0701 045 342.0686 032 378.0671 019.48414 2.0655.90007 450.0640.89994 486.0625 981 521.0609 968 557.0594 956.48593 2.0579.89943 629.0564 930 665.0549 918 701.0533 905 737.0518 892.48773 2.0503.89879 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 I I I -. cos cot tan sin ' 120 120 - I cos [ cot I tan sin -, S 155~ 245 ~ 335~ 65~ ~ 64~ 154~ 244~ 334~ 116~ 206~ 296~ 26~ TABLE IV 27~ 117~ 207~ 297~ I sin tan cot 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.43837 863 889 916 942.43968.43994.44020 046 072.44098 124 151 177 203.44229 255 281 307 333.44359 385 411 437 464.44490 516 542 568 594.44620 646 672 698 724.44750 776 802 828 854.44880 906 932 958.44984.45010 036 062 088 114.45140 166 192 218 243.45269 295 321 347 373.45399.48773 809 845 881 917.48953.48989.49026 062 098.49134 170 206 242 278.49315 351 387 423 459.49495 532 568 604 640.49677 713 749 786 822.49858 894 931.49967.50004.50040 076 113 149 185.50222 258 295 331 368.50404 441 477 514 550.50587 623 660 696 733.50769 806 843 879 916.50953 2.0503.0488.0473.0458.0443 2.0428.0413.0398.0383.0368 2.0353.0338.0323.0308.0293 2.0278.0263.0248.0233.0219 2.0204.0189.0174.0160.0145 2.0130.0115.0101.0086.0072 2.0057.0042.0028 2.0013 1.9999 1.9984.9970.9955.9941.9926 1.9912.9897.9883.9868.9854 1.9840.9825.9811.9797.9782 1.9768.9754.9740 9725.9711 1.9697.9683.9669.9654.9640 1.9626 cos.89879 867 854 841 828.89816 803 790 777 764.89752 739 726 713 700.89687 674 662 649 636.89623 610 597 584 571.89558 545 532 519 506.89493 480 467 454 441.89428 415 402 389 376.89363 350 337 324 311.89298 285 272 259 245.89232 219 206 193 180.89167 153 140 127 114.89101 ' Isin I tan I cot cos I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 24 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.45399 425 451 477 503.45529 554 580 606 632.45658 684 710 736 762.45787 813 839 865 891.45917 942 968.45994.46020.46046 072 097 123 149.46175 201 226 252 278.46304 330 355 381 407.46433 458 484 510 536.46561 587 613 639 664.46690 716 742 767 793.46819 844 870 896 921.46947.50953.50989.51026 063.099.51136 173 209 246 283.51319 356 393 430 467.51503 540 577 614 651.51688 724 761 798 835.51872 909 946.51983.52020.52057 094 131 168 205.52242 279 316 353 390.52427 464 501 538 575.52613 650 687 724 761.52798 836 873 910 947.52985.53022 059 096 134.53171 1.9626.9612.9598.9584.9570 1.9556.9542.9528.9514.9500 1.9486.9472.9458.9444.9430 1.9416.9402.9388.9375.9361 1.9347.9333.9319.9306.9292 1.9278.9265.9251.9237.9223 1.9210.9196.9183.9169.9155 1.9142.9128.9115.9101.9088 1.9074.9061.9047.9034.9020 1.9007.8993.8980.8967.8953 1.8940.8927.8913.8900.8887 1.8873.8860.8847.8834.8820 1.8807.89101 087 074 061 048.89035 021.89008.88995 981.88968 955 942 928 915.88902 888 875 862 848.88835 822 808 795 782.88768 755 741 728 715.88701 688 674 661 647.88634 620 607 593 580.88566 553 539 526 512.88499 485 472 458 445.88431 417 404 390 377.88363 349 336 322 308.88295 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 O I cos cot tan sin I ' I Cos cot tan I sin | ' j530 ~~~~~- - - 24 0 3 3 3 20 1 2 4 0 3 2 153~ 243~ 333~ 63~ 121 62~ 152~ 242~ 332~ 1180 2080 2980 280TAL IV20 1929090 TABLE IV 290 1190 2090 2990 I 0 2 3 4 5 6 7 8 9 10 I1 1 2 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I I sin I.46947 973.46999.47024 050.47076 101 127 153 178.47204 229 255 281.306.47332 358 383 409 434.47460 486 511 537 562.47588 614 639 665 690.47716 741 767 793 818.47844 869 895 920 946.4797 1.47997.48022 048 073.48099 124 150 175 201.48226 252 277 303 328.48354 379 405 430 456.4848 1 tan cot.53171 1.8807 208.87 94 246.878 1 283.8768 320.8755.53358 1.8741 395. 8728 432.8715 470.8702 507.8689.53545 1. 8676 582.8663 620.8650 657.8637 694.8624.53732 1. 8611 769.8598 807. 8585 844.8572 882.8559.53920 1.8546 957.8533.53995.8520.54032.8507.4070.8495 145.8469 183.8456 220.8443 258.8430.54296 1.8418 333.8405 371.8392 409.8379 446.8367.54484 1. 8354 522.8341 560.8329 597.8316 635.8303.54673 1.8291 711.8278 748.8265 786.8253 824.8240.54862 1. 8228 900.8215 938.8202.54975.8190.55013.8177.55051 1.8165 089.8152 127.8140 165. 8127 203.8115.55241 1.8103 279.8090 317.8078 355.8065 393.8053.55431 1.8040 Cos I II sin.88295 281 267 254 240.8-8226 21 3 199 185 172.88 158 144 130 117 103.88089 075 062 048 034.88020.88006.87993 979 965.8795 1 937 923 909 896.87882 868 854 8 40 826.878 12 798 784 770 756.87743 729 715 701 687.87673 659 645 631 617.87603 589 575 561 546.87532 518 504 490 476.87462 60 59 58 57 56 55 54 5 3 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 0 2 3 4 6 7 8 9 10 1 1 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.4848 1 506 532 557 583.48608 634 659 684 710.48735 761 786 811 8-37.48862 888 913 938 964.48989.49014 040 065 090.49116 141 166 192 217.49242 268 293 318 344.49369 394 419 445 470.49495 521 546 571.596.49622 647 672 697 723.49748 773 798 824 849.49874 899 924 950.49975.50000 tan [ cot 1 Cos.55431 1.8040.87462 60 469.8028 448 59 507.8016 434 58 545.8003 420 57 583.7991 406 56.55621 1.7979.87391 55 659.7966 377 54 697.7954 363 53 736.7942 349 52 774.7930 335 51.55812 1.7917.87321 50 850.7905 306 49 888.7893 292 48 926.7881 278 47.55964.7868 264 46.56003 1.7856.87250 45 041.7844 235 44 079.7832 221 43 117.7820 207 42 156.7808 193 41.56194 1.7796.87178 40 232.7783 164 39 270.7771 150 38 309.7759 136 37 347.7747 121 36.56385 1.7735.87107 35 424.7723 093 34 462.7711 079 33 501.7699 064 32 539.7687 050 3 1.56577 1.7675.87036 30 616.7663 021 29 654.7651.87007 28 693.7639.86993 27 731 7627 978 26.56769 1.7615.86964 25 808.7603 949 24 846.7591 935 23 885.7579 921 22 923.7567 906 2 1.56962 1.7556.86892 20.57000.7544 878 19 039.7532 863 18 078.7520 849 17 116.7508 834 16.57155 1.7496.86820 15 193.7485 805 14 232.7473 791 13 271.7461 777 12 309.7449 762 11.57348 1.7437.86748 10 386.7426 733 9 425.7414 719 8 464.7402 704 7 503. 7391 690 6.57541 1.7379. 86675 5 580.7367 661 4 619.7355 646 3 657.7344 632 2 696.7332 617 1.57735 1.7321.86603 0 cot tan sin I I. Cos I cot I tan I sin I I Cos 151- 2410 331o 610 122 1S10 2410 3310 610 122 ~600 1500 2400 3300 120~ 210~ 300~ 30~ TABLE IV 3 1~ 121~ 211 ~ 301~ sin I tan s cot I cos 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 / 50 51 52 53 54 55 56 57 58 59 60.50000 025 050 076 101.50126 151 176 201 227.50252 277 302 327 352.50377 403 428 453 478.50503 528 553 578 603.50628 654 679 704 729.50754 779 804 829 854.50879 904 929 954.50979.51004 029 054 079 104.51129 154 179 204 229.51254 279 304 329 354.51379 404 429 454 479.51504.57735 774 813 851 890.57929.57968.58007 046 085.58124 162 201 240 279.58318 357 396 435 474.58513 552 591 631 670.58709 748 787 826 865.58905 944.58983.59022 061.59101 140 179 218 258.59297 336 376 415 454.59494 533 573 612 651.59691 730 770 809 849.59888 928.59967.60007 046.60086 1.7321.7309.7297.7286.7274 1.7262.7251.7239.7228.7216 1.7205.7193.7182.7170.7159 1.7147.7136.7124.7113.7102 1.7090.7079.7067.7056.7045 1.7033.7022.7011.6999.6988 1.6977.6965.6954.6943.6932 1.6920.6909.6898.6887.6875 1.6864.6853.6842.6831.6820 1.6808.6797.6786.6775.6764 1.6753.6742.6731.6720.6709 1.6698.6687.6676.6665.6654 1.6643.86603 588 573 559 544.86530 515 501 486 471.86457 442 427 413 398.86384 369 354 340 325.86310 295 281 266 251.86237 222 207 192 178.86163 148 133 119 104.86089 074 059 045 030.86015.86000.85985 970 956.85941 926 911 896 881.85866 851 836 821 806.85792 777 762 747 732.85717 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 I 0! - I Ir sin I tan | cot cos I 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.51504 529 554 579 604.51628 653 678 703 728.51753 778 803 828 852.51877 902 927 952.51977.52002 026 051 076 101.52126 151 175 200 225.52250 275 299 324 349.52374 399 423 448 473.52498 522 547 572 597.52621 646 671 696 720.52745 770 794 819 844.52869 893 918 943 967.52992.60086 126 165 205 245.60284 324 364 403 443.60483 522 562 602 642.60681 721 761 801 841.60881 921.60960.61000 040.61080 120 160 200 240.61280 320 360 400 440.61480 520 561 601 641.61681 721 761 801 842.61882 922.61962.62003 043.62083 124 164 204 245.62285 325 366 406 446.62487 1.6643.6632.6621.6610.6599 1.6588.6577.6566.6555.6545 1.6534.6523.6512.6501.6490 1.6479.6469.6458.6447.6436 1.6426.6415.6404.6393.6383 1.6372.6361.6351.6340.6329 1.6319.6308.6297.6287.6276 1.6265.6255.6244.6234.6223 1.6212.6202.6191.6181.6170 1.6160.6149.6139.6128.6118 1.6107.6097.6087.6076.6066 1.6055.6045.6034.6024.6014 1.6003.85717 702 687 672 657.85642 627 612 597 582.85567 551 536 521 506.85491 476 461 446 431.85416 401 385 370 355.85340 325 310 294 279.85264 249 234 218 203.85188 173 157 142 127.85112 096 081 066 051.85035 020.85005.84989 974.84959 943 928 913 897.84882 866 851 836 820.84805 I I I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 — 59 -38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0! - - I cos I cot tan I sin I I I - I cos cot I tan I sin I - I -. - 149~ 239~ 329~ 59~ 123 58~ 148~ 238~ 328~ 1220 212~ 302~ 32~ TABLE- IV 33~ 123~ 213~ 303~ I sin tan cot cos I sin I tan I.,.,,... 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.52992.62487.53017 527 041 568 066 608 091 649.53115.62689 140 730 164 770 189 811 214 852.53238.62892 263 933 288.62973 312.63014 337 055.53361.63095 386 136 411 177 435 217 460 258.53484.63299 509 340 534 380 558 421 583 462.53607.63503 632 544 656 584 681 625 705 666.53730.63707 754 748 779 789 804 830 828 871.53853.63912 877 953 902.63994 926.64035 951 076.53975.64117.54000 158 024 199 049 240 073 281.54097.64322 122 363 146 404 171 446 195 487.54220.64528 244 569 269 610 293 652 317 693.54342.64734 366 775 391 817 415 858 440 899.54464.64941 cos cot 1.6003.84805.5993 789.5983 774.5972 759.5962 743 1.5952.84728.5941 712.5931 697.5921 681.5911 666 1.5900.84650.5890 635.5880 619.5869 604.5859 588 1.5849.84573.5839 557.5829 542.5818 526.5808 511 1.5798.84495.5788 480.5778 464.5768 448.5757 433 1.5747.84417.5737 402.5727 386.5717 370.5707 355 1.5697.84339.5687 324.5677 308.5667 292.5657 277 1.5647.84261.5637 245.5627 230.5617 214.5607 198 1.5597.84182.5587 167.5577 151.5567 135.5557 120 1.5547.84104.5537 088.5527 072.5517 057.5507 041 1.5497.84025.5487.84009.5477.83994.5468 978.5458 962 1.5448.83946.5438 930.5428 915.5418 899.5408 883 1.5399.83867 tan sin 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.54464 488 513 537 561.54586 610 635 659 683.54708 732 756 781 805.54829 854 878 902 927.54951 975.54999.55024 048.55072 097 121 145 169.55194 218 242 266 291.55315 339 363 388 412.55436 460 484 509 533.55557 581 605 630 654.55678 702 726 750 775.55799 823 847 871 895.55919.64941.64982.65024 065 106.65148 189 231 272 314.65355 397 438 480 521.65563 604 646 688 729.65771 813 854 896 938.65980.66021 063 105 147.66189 230 272 314 356.66398 440 482 524 566.66608 650 692 734 776.66818 860 902 944.66986.67028 071 113 155 197.67239 282 324 366 409.67451 cot j cos 1.5399.83867.5389 851.5379 835.5369 819.5359 804 1.5350.83788.5340 772.5330 756.5320 740.5311 724 1.5301.83708.5291 692.5282 676.5272 660.5262 645 1.5253.83629.5243 613.5233 597.5224 581.5214 565 1.5204.83549.5195 533.5185 517.5175 501.5166 485 1.5156.83469.5147 453.5137 437.5127 421.5118 405 1.5108.83389.5099 373.5089 356.5080 340.5070 324 1.5061.83308.5051 292.5042 276.5032 260.5023 244 1.5013.83228.5004 212.4994 195.4985 179.4975 163 1.4966.83147.4957 131.4947 115.4938 098.4928 082 1.4919.83066.4910 050.4900 034.4891 017.4882.83001 1.4872.82985.4863 969.4854 95-3.4844 936.4835 920 1.4826.82904 tan sin I 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2.1 0 I 0 I -I. I I =jI cos cot I I 147~ 237~ 3270 57~ 124 56~ 146~ 2360 326~ 1240 2140 304 0 T340 TABLE IV 350 1250 2150 3050 I sin [ tan I cot I cos I 0 2 3.4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.55919 943 968.55992.56016.56040 064 088 112 136.56160 184 208 232 256.56280 305 329 353 377.56401 425 449 473 497.56521 543 569 593 617.56641 665 689 713 736.56760 784 808 832 856.56880 904 928 952.56976.57000 024 047 071 095.57119 143 167 191 215.57238 262 286 310 334.57358.67451 493 536 578 620.67663 705 748 790 832.67875 917.67960.68002 045.68088 130 173 215 258.68301 343 386 429 471.68514 557 600 642 685.68728 771 814 857 900.68942.68985.69028 071 114.69157 200 243 286 329.69372 416 459 502 543.69588 631 673 718 761.69804 847 891 934.69977.70021 1.4826.4816.4807.4798.4788 1.4779.4770.4761.475 1.4742 1. 4733.4724.4715.4705.4696 1.4687.4678.4669.4659.4650 1.4641.4632.4623.4614.4605 1.4596.4586.4577.4568.4559 1.4550.4541.4532.4523.4514 1.4503.4496.4487.4478. 4469 1.4460.4451.4442.4433.4424 1.4415.4406.4397.4388. 4379 1.4370.4361.4352.4344.4335 1. 4326.4317.4308.4299.4290 1.4281.82904 887 871 853 839.82822 806 790 773 757.8274 1 724 708 692 675.82659 643 626 610 593.82577 561 544 528 511.82495 478 462 446 429.82413 396 380 363 347.82330 314 297 281 264.82248 231 214 198 181.82163 148 132 115 098.82082 065 048 032.82015.81999 982 965 949 932.81915 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 1 4 13 12 11 10 9 8 7 6 5 4 3 2 0 0 I I I sin I tan T cot I Cos I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 4 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60.57358 381 405 429 453.57477 501 524 548 572.57596 619 643 667 691.57715 738 762 786 810.57833 857 881 904 928.57952 976.57999.58023 047.58070 094 118 141 163.58189 212 236 260 283.58307 330 354 378 401.58423 449 472 496 519.58543 567 590 614 637.58661 684 708 731 753.58779.70021 064 107 151 194.70238 281 323 368 412 70455 499 542 586 629.70673 717 760 804 848.70891 935.70979.71023 066.71110 154 198 242 285.71329 373 417 461 505.71549 593 637 681 725.71769 813 857 901 946.71990.72034 078 122 167.72211 255 299 344 388.72432 477 521 565 610.72654 1.4281.4273.4264.4255.4246 1. 4237.4229.4220.4211.4202 1.4193.4185.4176. 4167.4158 1. 41 50.4141.4132.4124.4115 1.4106.4097.4089.4080.4071 1.4063.4054.4045.4037.4028 1.4019.4011.4002.3994.3985 1.3976.3968.3959.3951.3942 1.3934.3925.3916.3908.3899 1. 3891.3882.3874.3865.3857 1.3848.3840.3831.3823.3814 1.3806.3798.3789.3781.3772 1.3764.81915 899 882 865 848.81832 813 798 782 763.81748 731 714 698 681.81664 647 631 614 597.81580 563 546 530 513.81496 479 462 445 428.81412 395 378 361 344.81327 310 293 276 259.81242 225 208 191 174.81157 140 123 106 089.81072 055 038 021.81004 80987 970 953 936 919.80902 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 0 0 1 I I I cos cot I tan I sin i I I I Cos I i cot tan i sin 1450 235" 3250 550 125 540 1440 2340 3240 1260 216- 306o 360 TABLE IV 370 1270 2170 3070 Isin Itan I cot ICos il ~ sin T tan Icot I cosTI 0 2 3 4 5 6 7 8 9 10 I1I 1 2 1 3 14 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.58779 802 826 849 873.58896 920 943 967.58990.590 14 0-37 061 084 108.59131 154 178 201 225.59248 272 295 318 342.59365 389 412 436 459.59482 506 529 552 576.59599 622 646 669 693.59716 739 763 786 809.59832 856 879 902 926.59949 972.59995 60019 042.60065 089 112 135 158.60 182.72654 699 743 788 832.72877 921.72966.73010 055.73 100 144 189 234 278.73323 368 413 457 502.73547 592 637 681 726.7377 1 816 861 906 951.73996 74041 086 131 176.74221 267 312 357 402.74447 492 538 583 628.74674 719 764 810 855.74900 946.7499 1.75037 082.75128 1 73 219 264 310.75355 1. 3764.3755.3747.3739 3730 1.722.371 3.3705.3697 3688 1.3680.3672. 3663.3655.3647 1.3638.3630. 3622.3613.3605 1.3597.3588.3580.3572.3564 1. 3555.3547.3539.353 1.3522 1.3514.3506.3498.3490.348 1 1.3473.3465. 3457.3449.3440 1.3432.3424.3416.3408.3400 1.3392.3384.3375.3367.3359 1. 3351.3343.3335.3327.3319 1. 3311.3303.3295. 3287.3278 1. 3270.80902 885 867 850 833.80816 799 782 765 748.80730 713 696 679 662.80644 627 610 593 576.80558 541 524 507 489.80472 455 438 420 403.80386 368 351 334 316.8-0299 282 264 247 230.802 12 195 1 78 160 143.80 125 108 091 073 056.80038 021.80003.79986 968.7995 1 934 916 899 881.79864 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 0 2 3 4 5 6 7 8 9 10 11I 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 I40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.60182 205 228 251 274.60298 321 344 367 390.604 14 437 460 483 506.60529 553 576 599 622.60645 668 691 714 738.60761 784 807 830 853.60876 899 922 945 968.6099 1.61015 038 061 084.61107 130 153 176 199.61222 245 268 291 314.61337 360 383 406 429.61451 474 497 520 543.61566.75355 401 447 492 538.75584 629 675 721 767.75812 858 904 950.75996.76042 088 134 180 226.76272 318 364 410 456.76502 548 594 640 686.76733 779 825 871 918.76964.77010 057 103 149.77196 242 289 335 382.77428 475 521 568 615.7766 1 708 754 801 848.77895 941.77988.78035 082.78129 1.3270. 3262.3254.3246.3238 1.3230.3222.3214.3206. 3198 1. 3190.3182.3175.31 67.3159 1. 3151.3143.31 35.31 27.311 9 1. 3111.3103.3095.3087.3079 1. 3072.3064.3056.3048.3040 1. 3032. 3024.3017.3009.300 1 1.2993.2985. 2977.2970.2962 1. 2954.2946.2938.293 1.2923 1. 2915. 2907.2900.2892.2884 1.2876.2869.286 1. 2853.2846 1. 2838.2830. 2822.2815.2807 1. 2799.79864 846 829 811 793.79776 758 741' 723 706.79688 671 653 635 618.79600 583 5 65 547 530.79512 494 477 459 441.79424 406' 388 371 353.79335 318 300 282 264.79247 229 211 193 1 76.79158 140 122 105 087.79069 051 033.79016.78998.78980 962 944 926 908.78891 873 85.5 837 819.7880 1 60 59 58 57 56 55 54 53. 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 2 1 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 I cos I cot I tan I sin I T o cot I tan I sin I ii 1430 2330 3230 530 165 ~ 12 30 32 126 520 1420 2320 3220 1280 2180 308~ 38~ TABLE IV 39~ 129~ 219~ 309~ r / I sin Itan cot | cos ' I sin 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.61566 589 612 635 658.61681 704 726 749 772.61795 818 841 864 887.61909 932 955.61978.62001.62024 046 069 092 115.62138 160 183 206 229.62251 274 297 320 342.62365 388 411 433 456.62479 502 524 547 570.62592 615 638 660 683.62706 728 751 774 796.62819 842 864 887 909.62932.78129 175 222 269 316.78363 410 457 504 551.78598 645 692 739 786.78834 881 928.78975.79022.79070 117 164 212 259.79306 354 401 449 496.79544 591 639 686 734.79781 829 877 924.79972.80020 067 115 163 211.80258 306 354 402 450.80498 546 594 642 690.80738 786 834 882 930.80978 1.2799.2792.2784.2776.2769 1.2761.2753.2746.2738.2731 1.2723.2715.2708.2700.2693 1.2685.2677.2670.2662.2655 1.2647.2640.2632.2624.2617 1.2609.2602.2594.2587.2579 1.2572.2564.2557.2549.2542 1.2534.2527.2519.2512.2504 1.2497.2489.2482.2475.2467 1.2460.2452.2445.2437.2430 1.2423.2415.2408.2401.2393 1.2386.2378.2371.2364.2356 1.2349.78801 783 765 747 729.78711 694 676 658 640.78622 604 586 568 550.78532 514 496 478 460.78442 424 405 387 369.78351 333 315 297 279.'78261 243 225 206 188.78170 152 134! 116 098.78079 061 043 025.78007.77988 970 952 934 916.77897 879 861 843 824.77806 788 769 751 733.77715 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.62932 955.62977.63000 022.63045 068 090 113 135.63158 180 203 225 248.63271 -293 316 338 361.63383 406 428 451 473.63496 518 540 563 585.63608 630 653 675 698.63720 742 765 787 810.63832 854 877 899 922.63944 966.63989.64011 033.64056 078 100 123 145.64167 190 212 234 256.64279 tan cot..80978 1.2349.81027.2342 075.2334 123.2327 171.2320.81220 1.2312 268.2305 316.2298 364.2290 413.2283.81461 1.2276 510.2268 558.2261 606.2254 655.2247.81703 1.2239 752.2232 800.2225 849.2218 898.2210.81946 1.2203.81995.2196.82044.2189 092.2181 141.2174.82190 1.2167 238.2160 287.2153 336.2145 385.2138.82434 1.2131 483.2124 531.2117 580.2109 629.2102.82678 1.2095 727.2088 776.2081 825.2074 874.2066.82923 1.2059.82972.2052.83022.2045 071.2038 120.2031.83169 1.2024 218.2017 268.2009 317.2002 366.1995.83415 1.1988 465.1981 514.1974 564.1967 613.1960.83662 1.1953 712.1946 761.1939 811.1932 860.1925.83910 1.1918.77715 696 678 660 641.77623 605 586 568 550.77531 513 494 476 458.77439 421 402 384 366.77347 329 310 292 273.77255 236 218 199 181.77162 144 125 107 088.77070 051 033.77014.76996.76977 959 940 921 903.76884 866 847 828 810.76791 772 754 735 717.76698 679 661 642 623.76604 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 I cos 5 4 3 2 1 0 I cos I cot I tan I sin I ' i cos! cot I tan I sin I 141~ 231~ 321~ 51~ 127 50~ 140~ 230~ 320~ 1300 2200 3100 400 TABLE IV 410 1310 2210 3110 f i 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 7 1 8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I I sin tan I cot I cos I I.64279 301 323 346 368.64390 412 435 457 479.64501 524 546 568 590.64612 635 657 679 701.64723 746 768 790 812.64834 856 878 901 923.64945 967.64989.65011 033.65055 077 100 122 144.65166 188 210 232 254.65276 298 320 342 364.65386 408 430 452 474.65496 518 540 562 584.65606.83910.83960.84009 059 108.84158 208 258 307 357.84407 457 507 556 606.84656 706 756 806 856.84906.84956.85006 057 107.85157 207 257 308 358.85408 458 509 559 609.85660 710 761 811 862.85912.85963.860 14 064 115.86166 216 267 318 368.864 19 470 521 572 623.86674 725 776 827 878 86929 1.1918.76604.1910 586.1903 567.1896 548.1889 530 1.1882. 76511.1875 492.1868 473.1861 455.1854 436 1.1847.76417.1840 398.1833 380.1826 361.1819 342 1.1812.76323.1806 304.1799 286.1792 267.1785 248 1.1778.76229 1771 210.1764 192.1757 173.1750 154 1.1743.76135.1736 116.1729 097.1722 078.1715 059 1.1708.76041.1702 022.1695.76003.1688.75984.1681 965 1.1674.75946.1667 927.1660 908.1653 889.1647 870 1.1640.75851.1633 832.1626 813.1619 794.1612 775 1.1606.75756.1599 738.1592 719.1585 700.1578 680 1.1571.75661.1565 642.1558 623.1551 604.1544 585 1.1538.75566.1531 547.1524 528.1517 509.1510 490 1.1504.75471 tan sin 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 1 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 0 0 1 2 3 4 5 6 7 8 9 i 10 11 12 13 1 4 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I j I sin tan cot F Cos.65606 628 650 672 694.65716 738 759 781 803.65825 847 869 891 913.65935 956.65978.66000 022.66044 066 088 109 131.66153 175 197 218 240.66262 284 306 327 349.66371 393 414 436 458.66480 501 523 545 566.66588 610 632 653 675.66697 718 740 762 783.66805 827 848 870 891.66913.86929.86980.87031 082 133.87184 236 287 338 389.87441 492 543 595 646.87698 749 801 852 904.87955.88007 059 110 162.882 14 265 317 369 421.88473 524 576 628 680.88732 784 836 888 940.88992.89045 097 149 201.89253 306 358 410 463.895 15 567 620 672 725.89777 830 883 935.89988.90040 1.1504.1497.1490.1483.1477 1.1470.1463.1456.1450.1443 1. 1436.1430.1423.1416.1410 1. 1403.1396.1389.1383.1376 1.1369.1363.1356.1349 1343 1.1336.1329.1323.1316.1310 1.1303.1296.1290.1283.1276 1.1270.1263.1257.1250.1243 1.1237.1230.1224 1217.1211 1.1204.1197.1191.1184.1178 1. 1171.1165.1158.1152.1145 1.1139.1132.1126.1119.1113 1.1106.75471 452 433 414 395.75375 356 337 318 299.75280 261 241 222 203.75184 165 146 126 107.75088 069 050 030.75011.74992 973 953 934 915.74896 876 857 838 818.74799 780 760 741 722.74703 683 664 644 625.74606 586 567 548 528.74509 489 470 451 431.744 12 392 373 353 334.74314 60 59 58 57 56 55 54. 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 110 9 8 7 6 5 4 3 2 0 0 1 cos I cot I I I I I Cos I cot I tan I i sin I, 1390 2290 3190 490 128 480 1380 2280 3180 1320 2220 3120, 420 TAB3LE IV I~ I, 430 1330 2230 3130 I I sin tan I cot Cos I O'.. tan I cot I Cos 0 2 3 4 5 6 7 8 9 10 11I 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 21 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60.66913 935 956 978.66999.6702 1 043 064 086 107.67 129 151 172 194 215.67237 258 280 301 323.67344 366 387 409 430.67452 473 495 516 538.67559 580 602 623 645.67666 688 709 730 752.67773 795 816 837 859.67880 901 923 944 965.67987.68008 029 051 072.68093 115. 136 157 179.68200.90040 093 146 199 251.90304 357 410 463 516.90569 621 674 727 781.90834 887 940.90993.9 1046.91099 153 206 259 31 3.91366 419 473 526 580.9 1633 687 740 794 847.91901.91955.92008 062 116.92 170 224 277 331 385.92439 493 547 601 655.92709 763 817 872 926.92980.93034 088 143 197.93252 1. 1106.1100.1093.1087.1080 1. 1074.1067.1061.1054.1048 1.1041.1035.1028.1022.1016 1.1009.1003.0996.0990.0983 1.0977.0971.0964.0958.095 1 1.0945.0939.0932.0926.0919 1.0913.0907.0900.0894.0888 1.0881. 0875.0869.0862.0856 1. 0850.0843.0837.0831.0824 1.0818.0812.0805.0799.0793 1.0786.0780.0774.0768.0761 1. 0755.0749.0742.0736.0730 1.0724.743 14 295 276 256 237.74217 198 1 78 159 139.74120 100 080 061 041.74022.74002.73983 963 944.73924 904 885 865 846.73826 806 787 767 747.73728 708 688 669 649.73629 610 590 570 551.7353 1 511 491 472 452.73432 413 393 373 353.73333 314 294 274 254.73234 215 195 175 *155.73 135 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 0 2 3 4 5 6 7 8 9 10 11I 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 20 2 1 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.68200 221 242 264 285.68306 327 349 370 391.68412 434 455 476 497.685 18 539 561 582 603.68624 645 666 688 709.68730 751 772 793 814.68835 857 -878 899 920.6894 1 962.68983.69004 025.69046 067 088 109 130.69151 1 72 193 214 235.69256 277 298 319 340.6936 1 382 403 424 445.69466.93252 306 360 415 469.93524 578 633 688 742.93797 852 906.93961.94016.9407 1 1 25 180 235 290.94345 400 455 510 565.94620 676 731 786 841.94896.94952.95007 062 118.95 173 229 284 340 395.95451 506 562 61 8 673.95729 785 841 897.95952.96008 064 120 1 76 232.96288 344 400 457 513.96569 1.0724.0717.0711.0705.0699 1.0692.0686.0680.0674.0668 1. 0661.0655.0649.0643.0637 1.0630.0624.0618.0612.0606 1.0599.0593.0587.058 1. 0575 1.0569.0562.0556.0550.0544 1.0538.0532.0526.0519.0513 1. 0507.050 1.0495.0489.0483 1. 0477. 0470.0464.0458 1,0452 1.0446.0440.0434.0428. 0422 1.0416.0410.0404.0398.0392 1.0385.0379.0373.0367.0361 1.0355.731 35 116 096 076 056.73036.73016.72996 976 957.72937 917 897 877 857.72837 817 797 777 757.72737 71 7 697 677 657.72637 617 597 577 557.72537 517 497 477 457.72437 417 397 377 357.72337 317 297 277 257.72236 216 196 176 156.721 36 116 095 075 055.72035.72015.71995 974 954.7 1934 60 59 58 57 56 55 54 53 52 5 1 50 49 48 47 46 45 44 43 42 4 1 40 39 38 37 36 35 34 33 32 3 1 30 29 28 27 26 25 24 23 22 21 20 1 9 1 8 1 7 1 6 15 1 4 1 3 1 2 1 1 10 9 8 7 6 5 4 3 2 0 I I I cos - cot _ I.tan I sin I I I Cos Icot L tan Ila in I' 1370 2270 3j7o 470 1 94 ~ 1 6 20 3 6 129 0 46, 1360 2260 3160 TABLE IV 44~ 134~ 224~ 314~ ' I sin I tan cot | cos | 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60.69466 487 508 529 549.69570 591 612 633 654.69675 696 717 737 758.69779 800 821 842 862.69883 904 925 946 966.69987.70008 029 049 070.70091 112 132 153 174.70195 215 236 257 277.70298 319 339 360 381.70401 422 443 463 484.70505 525 546 567 587.70608 628 649 670 690.70711.96569 625 681 738 794.96850 907.96963.97020 076.97133 189 246 302 359.97416 472 529 586 643.97700 756 813 870 927.97984.98041 098 155 213.98270 327 384 441 499.98556 613 671 728 786.98843 901.98958.99016 073.99131 189 247 304 362.99420 478 536 594 652.99710 768 826 884.99942 1.0000 1.0355.0349.0343.0337.0331 1.-0325.0319.0313.0307.0301 1.0295.0289.0283.0277.0271 1.0265.0259.0253.0247.0241 1.0235.0230.0224.0218.0212 1.0206.0200.0194.0188.0182 1.0176.0170.0164.0158.0152 1.0147.0141.0135.0129.0123 1.0117.0111.0105.0099.0094 1.0088.0082.0076.0070.0064 1.0058.0052.0047.0041.0035 1.0029.0023.0017.0012.0006 1.0000.71934 914 894 873 853.71833 813 792 772 752.71732 711 691 671 650.71630 610 590 569 549.71529 508 488 468 447.71427 407 386 366 345.71325 305 284 264 243.71223 203 182 162 141.71121 100 080 059 039.71019.70998 978 957 937.70916 896 875 855 834.70813 793 772 752 731.70711 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 - I cos j cot tan sin | I0 1350 2 3 130 45~ 135~ 225~ 315~ TABLE V. RADIAN MEASURE, 0~ TO 180~, RADIUS = 1. Degrees I Minutes I Seconds --- - - -- 4t-. —Y 00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 -51 52 53 54 55 56 57 58 59 60 0.00000 00 0.01745 33 0.03490 66 0.05235 99 0.06981 32 0.08726 65 0.10471 98 0.12217 30 0.13962 63 0.15707 96 0.17453 29 0.19198 62 0.20943 95 0.22689 28 0.24434 61 0.26179 94 0.27925 27 0.29670 60 0.31415 93 0.33161 26 0.34906 59 0.36651 91 0.38397 24 0.40142 57 0.41887 90 0.43633 23 0.45378 56 0.47123 89 0.48869 22 0.50614 55 0.52359 88 0.54105 21 0.55850 54 0.57595 87 0.59341 19 0.61086 52 0.62831 85 0.64577 18 0.66322 51 0.68067 84 0.69813 17 0.71558 50 0.73303 83 0.75049 16 0.76794 49 0.78539 82 0.80285 15 0.82030 47 0.83775 80 0.85521 13 0.87266 46 0.89011 79 0.90757 12 0.92502 45 0.94247 78 0.95993 11 0.97738 44 0.99483 77 1.01229 10 1.02974 43 1.04719 76 600 61 62 63 64 65 66 -67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 10Q 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 1.04719 76 1.06465 08 1.08210 41 1.09955 74 1.11701 07 1.13446 40 1.15191 73 1.16937 06 1.18682 39 1.20427 72 1.22173 05 1.23918 38 1.25663 71 1.27409 04 1.29154 36 1.30899 69 1.32645 02 1.34390 35 1.36135 68 1.37881 01 1.39626 34 1.41371 67 1.43117 00 1.44862 33 1.46607 66 1.48352 99 1.50098 32 1.51843 64 1..53588 97 1.55334 30 1.57079 63 1.58824 96 1.60570 29 1.62315 62 1.64060 95 1.65806 28 1.67551 61 1.69296 94 1.71042 27 1.72787 60 1.74532 93 1.76278 25 1.78023 58 1.79768 91 1.81514 24 1.83259 57 1.85004 90 1.86750 23 1.88495 56 1.90240 89 1.94986 22 1.93731 55 1.95476 88 1.97222 21 1.98967 53 2.00712 86 2.02458 19 2.04203 52 2.05948 85 2.07694 18 2.09439 51!! -. I I I L20~ 121 122 123 124 L25 126 127 128 129 130 131 132 133 134 L35 136 137 138 139 140 141 142 143 144 L45 146 147 148 149 L50 151 152 153 154 L55 156 157 158 159 160 161 162 163 164 165 166 167 168 169 L70 171 172 173 174 L75 176 177 178 179 L80 2.09439 51 2.11184 84 2.12930 17 2.14675 50 2.16420 83 2.18166 16 2.19911 49 2.21656 82 2.23402 14 2.25147 47 2.26892 80 2.28638 13 2.30383 46 2.32128 79 2.33874 12 2.35619 45 2.37364 78 2.39110 11 2.40855 44 2.42600 77 2.44346 10 2.46091 42 2.47836 75 2.49582 08 2.51327 41 2.53072 74 2.54818 07 2.56563 40 2.58308 73 2.60054 06 2.61799 39 2.63544 72 2.65290 05 2.67035 38 2.68780 70 2.70526 03 2.72271 36 2.74016 69 2.75762 02 2.77507 35 2.79252 68 2.80998 01 2.82743 34 2.84488 67 2.86234 00 2.87979 33 2.89724 66 2.91469 99 2.93215 31 2.94960 64 2.96705 97 2.98451 30 3.00196 63 3.01941 96 3.03687 29 3.05432 62 3.07177 95 3.08923 28 3.10668 61 3.12413 94 3.14159 27 0' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0.00000 00 0.00029 09 0.00058 18 0.00087 27 0.00116 36 0.00145 44 0.00174 53 0.00203 62 0.00232 71 0.00261 80 0.00290 89 0.00319 98 0.00349 07 0.00378 15 0.00407 24 0.00436 33 0.00465 42 0.00494 51 0.00523 60 0.00552 69 0.00581 78 0.00610 87 0.00639 95 0.00669 04 0.00698 13 0.00727 22 0.00756 31 0.00785 40 0.00814 49 0.00843 58 0.00872 66 0.00901 75 0.00930 84 0.00959 93 0.00989 02 0.01018 11 0.01047 20 T'h.01076 29 0.01105 38 0.01134 46 0.01163 55 0.01192 64 0.01221 73 0.01250 82 0.01579 91 0.01 09 00 0.01338 09 0.01367 17 0.01396 26 0.01425 35 0.01454 44 0.01483 53 0.01512 62 0.01541 71 0.01570 80 0.01599 89 0.01628 97 0.01658 06 0.01687 15 0.01716 24 0.01745 33 0" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0.00000 00 0.00000 48 0.00000 97 0.00001 45 0.00001 94 0.00002 42 0.00002 91 0.00003 39 0.00003 88 0.00004 36 0.00004 85 0.00005 33 0.00005 82 0.00006 30 0.00006 79 0.00007 27, 0,00007 76 0.00008 24 0.00008 73 0.00009 21 0.00009 70 0.00010 18 0.00010 67 0.00011 15 0.00011 64 0.00012 12 0.00012 61 0.00013 09 0.00013 57 0.00014 06 0.00014 54 0.00015 03 0.00015 51 0.00016 00 0.00016 48 0.00016 97 0.00017 45 0.00017 94 0.00018 42 0.00018 91, 0.00019 39 0.00019 88 0.00020 36 0.00020 85 0.00021 33 0.00021 82 0.00022 30 0.00022 79 0.00023 27 0.00023 76 0.00024 24 0.00024 73 0.00025 21 0.00025 70 0.00026 18 0.00026 66 0.00027 15 0.00027 63 0.00028 12 0.00028 60 0.00029 09 Degrees I Minutes I Seconds 131 TABLE VI. CONSTANTS AND THEIR LOGARITHMS. Circumference of a circle in degrees......... =360 Circumference of a circle in minutes..........= 21,600 Circumference of a circle in seconds.........= 1,296,000 Number of radians in one degree............=0.017 4533 Number of radians in one minute........... =0.000 2909 Number of radians in one second........... =0.000 0048 Number of degrees in one radian............= 57.2957795 Number of minutes in one radian............ =3437.7468 Number of seconds in one radian............ =206,264.806 7r=3.141 592653 589793....................................... Also: 27r= 6.283 1853 47r =12.566 3706 - r= 1.570 7963 2 4 4-= 4.188 7902 1-r= 0.785 3982 r-= 0.523 5988 6 - = 0.318 3099 7r 7-2 = 9.869 6044 Logarithm 0.798 1799 1.099 2099 0.196 1199 0.622 0886 9.895 0899- 10 9.718 9986-10 9.502 8501-10 0.994 2997 /7 = 1.772 4539 1 -= 0.564 1896 M =0.434 2945 1 =2.302 5851 e =2.718 2818 Logarithm 2.556 3025 4.334 4538 6.112 6050 8.241 8774-10 6.463 7261-10 4.685 5749-10 1.758 1226 3.536 2739 5.314 4251 0.497 1499 0.248 5749 9.751 4251-10 9.637 7843-10 0.362 2157 0.434 2945 9.565 7055-10 0.150 5150 0.238 5606 0.369 4850 e =0.367 8794 /2 =1.414 2136 =/3 = 1.732 0508 V/5 =2.236 0680 1 mile per hour = 1.466 667 feet per second. 1 foot per second = 0.681 818 miles per hour. 1 cu. ft. of water weighs 62.5 lb. = 1000 oz. (approximate). 1 gal. of water weighs 83 lb. (approximate). 1 gal. = 231 cu. in. (by law of Congress). 1 bu. = 2150.42 cu.in. (by law of Congress). 1 bu. = 1.2446 cu. ft. = cu. ft. (approximate). 1 cu. ft. = 7- gal. (approximate). 1 bbl. = 4.211 cu. ft. (approximate). 1 meter = 39.37 inches (by law of Congress). 1 ft. = 30.4801 cm. 1 kg. = 2.20462 lb. 1 gram = 15.432 grains. 1 lb. (avoirdupois) = 453.592 4277 grams = 0.45359 kg. 1 lb. (avoirdupois) = 7000 grains (by law of Congress). 1 lb. (apothecaries) = 5760 grains (by law of Congress). 1 liter = 1.05668 qt. (liquid) = 0.90808 qt. (dry). 1 qt. (liquid) = 946.358 cc. = 0.946 358 liters, or cu. dm. 1 qt. (dry) = 1101.228 cc. = 1.101 228 liters, or cu. dm. 132