/:'> s^f l/../\fiv£fi:^rr'i> . ' 6 >" ■ ':■ >* /»■ i ^ --*!•■ V>- ■ /'»V. WENTWORTH-SMITH MATHEMATICAL SERIES PLANE TRIGONOMETRY AND TABLES BY GEORGE WENTWORTH AND DAVID EUGENE SMITH GINN AND COMPANY BOSTON • NEW YORK • CHICAGO • LONDON ATLANTA • DALLAS • COLUMBUS • SAN FRANCISCO COPYRIGHT, 1914, BY GEORGE WENTWORTH AND DAVID EUGENE SMITH ALL RIGHTS RESERVED PRINTED IN THE UNITED STATES OF AMERICA 226.1 SCbensum GINN AND COMPANY • PRO- PRIETORS • BOSTON • U.SA. PKEFACE In preparing a work to replace the Wentworth Trigonometry, which has dominated the teaching of the subject in America for a whole generation, some words of explanation are necessary as to the desirability of the changes that have been made. Although the great truths of mathematics are permanent, educational policy changes from generation to generation, and the time has now arrived when some rearrangement of matter is necessary to meet the legitimate demands of the schools. The principal changes from the general plan of the standard texts in use in America relate to the sequence of material and to the number and nature of the practical applications. With respect to sequence the rule has been followed that the practical use of every new feature should be clearly set forth before the abstract theory is developed. For example, it will be noticed that the definite uses of each of the natural functions are given as soon as possible, that the need for logarithmic computation follows, that thereafter the secant and cosecant assume a minor place, and that a wide range of prac- tical applications of the right triangle awakens an early interest in the subject. The study of the functions of larger angles, and of the sum and difference of two angles, now becomes necessary to further progress in trigonometry, after which the oblique triangle is con- sidered, together with a large number of practical, nontechnical applications. The decimal division of the degree is explained and is used enough to show its value, but it is recognized that this topic has, as yet, only a subordinate place. It seems probable that the decimal frac- tion will in due time supplant the sexagesimal here as it has in other fields of science, and hence the student should be familiar with its advantages. Such topics as the radian, graphs of the various functions, the applications of trigonometry to higher algebra, and the theory of trigonometric equations properly find place at the end of the course in plane trigonometry. They are important, but their value is best appreciated after a good course in the practical uses of the subject. iii IV PEEFACE They may be considered briefly or at length as the circumstances may warrant. The authors have sought to give teachers and students all the material needed for a thorough study of plane trigonometry, with more problems than any one class will use, thus offering opportunity for a new selection of examples from year to year, and allowing for the omission of the more theoretical portions of Chapters IX-XII if desired. The tables have been arranged with great care, every practical device having been adopted to save eye strain, all tabular material being furnished that the student will need, and an opportunity being afforded to use angles divided either sexagesimally or decimally, as the occasion demands. It is hoped that the care that has been taken to arrange all matter in the order of difficulty and of actual need, to place the practical before the theoretical, to eliminate all that is not necessary to a clear understanding of the subject, and to present a page that is at the same time pleasing to the eye and inviting to the mind will com- mend itself to and will meet with the approval of the many friends of the series of which this work is a part. GEOKGE WENTWORTH DAVID EUGENE SMITH CONTENTS PLAJ^E TPIGOJ^OMETEY CHAPTER PAGE I. Trigonometric Functions of Acute Angles .... • 1 II. Use of the Table of Natural Functions 27 III. Logarithms 39 IV. The Right Triangle 63 V. Trigonometric Functions of any Angle 77 VI. Functions of the Sum or the Difference of Two Angles 97 VII. The Oblique Triangle 107 VIII. Miscellaneous Applications 133 IX. Plane Sailing 145 X. Graphs of Functions 151 XI. Trigonometric Identities and Equations 163 XII. Applications of Trigonometry to Algebra 173 The Most Important Formulas of Plane Trigonometry . . 185 V Digitized by the Internet Archive in 2016 with funding from Duke University Libraries https://archive.org/details/planetrigonometr01went PLAE^E TRIGONOMETRY CHAPTER I TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES 1. The Nature of Arithmetic. In aritlunetic we study computation, the working with numbers. We may have a formula expressed in algebraic symbols, such as a = Ih, but the actual computation involved in applying such a formula to a particular case is part of arithmetic. Arithmetic enters into all subsequent branches of mathematics. It plays such an important part in trigonometry that it becomes necessary to introduce another method of computation, the method which makes use of logarit hms . 2. The Nature of Algebra. In algebra w e generalize arithmetic . Thus, instead of saying that the area of a rectangle with base 4 in. and height 2 in. is 4 x 2 sq. in., we express a general law by saying that a = bh. Algebra, therefore, is a generalized arithm etic, and the equation is the chief object of attention. Algebra also enters into all subsequent branches of mathematics, and its relation to trigonometry will be found to be very close. 3. The Nature of Geometry. In geometry we study the forms and relations of figures, proving many propert ies and e ffecting numerous constructions concern ing them. Geometry, like algebra and arithmetic, enters into the work in trigonometry. Indeed, trigonometry may almost be said to unite arithmetic, algebra, and geometry in one subject. 4. The Nature of Trigonometry. We are now about to begin another branch of mathematics, one not chiefly relating to numbers although it uses numbers, not primarily devoted to equations although using equations, and not concerned principally with the study of geometric forms although freely drawing upon the facts of geometry. Trigonometry is concerne d chiefly with t he relation of certain lines in a triangle {trigon, "a triangle,” + metrein, "to measure”) and forms the basis of the mensuration used in surveying, engineering, mechanics, geodesy, and astronomy . 1 2 PLANE TKIGONOMETRY 5. How Angles are Measured. For ordinary purposes angles can be measured with a protractor to a degree of accuracy of about 30'. The student will find it advantageous to use the convenient protractor fur- nished with this hook and shown in the illustration below. For work out of doors it is customary to use a transit, an instru- ment by means of which angles can be measured to minutes. By turning the top of the transit to the right or left, horizontal angles can be measured on the horizontal plate. By turning the telescope up or down, ver- tical angles can be measured on the vertical circle seen in the illustration. For astronomical purposes, where great care is necessary in measuring angles, large circles are used. The degree of accuracy required in meas- uring an angle depends upon the nature of the problem. We shall now assume that we can measure angles in degrees, minutes, and seconds, or in degrees and decimal parts of a degree. Thus 15° 30' is the same as 15.5°, and 15° 30' 36" is the same as 15|° of 1°, or 15.51°. The ancient Greek astronomers had no good symbols for fractions. The best system they could devise for close approximations was the so-called sexagesimal one, in which there appear only the numerators of fractions whose denomi- nators are powers of 60. This system seems to have been first suggested by the Babylonians, but to have been developed by the Greeks. It is much inferior to the decimal system that was perfected about 1600, but the world still continues to use it for the measure of angles and time. 'Tlie decimal division of the angle is, however, gaining ground, and in due time will probably replace the more cumbersome one with which we are familiar. In this book we shall use both the ancient and modem systems, but with the chief attention to the former, since this is still the more common. FUNCTIONS OF ACUTE ANGLES 3 6. Functions of an Angle. In the annexed figure, if the line AR moves about the point A in the sense indicated by the arrow, from the position AX as an initial position, it generates the angle A. If from the points B,B\ B", . . ., on AR, we let fall the perpen- diculars BC, B'C, B”C”, . . ., on AX, we form a series of similar triangles ACB, AC'B', AC"B", and so on. The corresponding sides of these triangles are proportional. That is, BC B'C B"C" AB ~ AB' ~ AB" BC _ B^ _ B"C" AC ~ AC “ AC" AB _ A^ _ AB" AC ~ AC ~ AC" and similarly for the ratios AB AC’ Bc’ Ab’ each of which has a series of other ratios equal to it. AC AC For example, AB _ AB' _ AB" ^ ~ 'WC' ~ B"C" ' That is, these ratios remain unchanged so long as the angle remains unchanged, hut they change as the angle changes. Each of the above ratios is therefore 2 . function of the angle A. As already learned in algebra and geometry, a magnitude which depends upon another magnitude for its value is called a function o f the latter m ag- nitude. Thus a circle is a function of the radius, the area of a square is a function of the side, the surface of a sphere is a function of the diameter, and the volume of a pyramid is a function of the base and altitude. We indicate a function of x by such symbols as /(a:), F(x), f'(x), and {x), and we read these "/of x, /-major of x, /-prime of x, and phi of X ” respectively. For example, if we are repeatedly using some long expression like + 3 — 2 -f 7 a; — 4, we may speak of it briefly as f(x). If we are using some function of angle A, we may designate this as /(A). If we wish to speak of some other function of A, we may write it f(A), F{A), or {A). I n trigonomet ry we shall make much use of various functions of an angle , but we shall give to them special names and symbols. On this account the ordinary function symbols of algebra, mentioned above, will not be used frequently in trigonometry, but they will be used often enough to make it necessary that the student should understand their significance. 4 PLAIN’S TKIGONOMETEY 7. The Six Functions, Since with a given angle A we may take any one of the triangles described in § 6, we triangle ACB, lettered as here shown. It has long been the custom to letter in this way the hypotenuse, sides, and angles of the first triangle con- sidered in trigonometry, C being the right angle, and the hypotenuse and sides bearing the small letters corre- sponding to the opposite capitals. By the sides of the triangle is meant the sides a and b, c being called the hypotenuse. The sides a and b are also called the legs of the triangle, par- ticularly by early writers, since it was formerly the custom to represent the triangle as standing on the hypotenuse. shall consider the B - > — ) - j 7 5 and - have the following names ; c 0 a 0 a is called the sine of A, written sin A ; is called the cosine of A , written cos A ; is called the tangent of A, written tan A ; is called the cotangent of A , written cot A ; c . is called the secant of A, written sec A ; is called the cosecant of A, written esc A. The ratios -j c a c b c a b b a c b c a That is, a opposite side sin A = - — : ) c hypotenuse a _ opposite side b adjacent side ’ , b adiacent side cos A = - = — ; j c hypotenuse , , b adjacent side COtA= - = — r- rv-J a opposite side _ c _ hypotenuse b adjacent side’ , c hypotenuse CSC A = - = — — T7-- a opposite side These definitions must be thoroughly learned , since the y are the foundation upon which the whole science is built. The student should practice upon them, with the figure before him, until he can tell instantly what ratio is meant by sec A, cotA, sinA, and so on, in whatever order these functions are given. There are also two other fimctions, rarely used at present. These are the versed sine A = 1 — cos A, and the co versed sine A = 1 — sin A. These defini- tions need not be learned at this time, since they will be given again when the functions are met later in the work. FUNCTIONS OF ACUTE ANGLES o Exercise 1. The Six Functions 1. In the figure of § 7, sin 5 = -• Write the other five functions of the angle B. 2. Show that in the right triangle ACB (§ 7) the following relations exist : sin^ = cos 5, cos = sin£, tan^ = cot£, cot4 = tan£, sec 4 = cscB, esc A = seoB. State which of the following is the greater : 3. sin^ or tan^. 5. sec.I or tan^. 4. cos^ or cot^4. 6. csc^ or cot^. Find the values of the six functions of A, if a, b, c respectively have the following values : 7. 3, 4, 6. 9. 8, 15, 17. 11. 3.9, 8, 8.9. 8. 5, 12, 13. 10. 9, 40, 41. 12. 1.19, 1.20, 1.69. 13. What condition must be fulfilled by the lengths of the three lines a, h, c (fl) to make them the sides of a right triangle ? Show that this condition is fulfilled in Exs. 7-12. Find the values of the six functions of A, if a, b, c respectively have the following values: 14. 2n, — 1, + 1. 71 ^ — 1 + 1 15. 71, 16. 17. 2 77171, 771^ — 71^, 2 77171 — ) 771 + 71, 771 — 71 771 ^ + 71 ^. m? + 71 ^ m — n 18. As in Ex. 13, show that the condition for a right triangle is fulfilled in Exs. 14-17. Given (F -\-b^ = find the six functions of A when : 19. a = h. 20. a = 2b. 21. a = ^c. Given a^+b"^ = c^, fin^ the six functions of B when : 22. a = 24, b = 143. 24. a = 0.264, c = 0.265. 23. b = 9.5, c = 19.3. 25. b = 2 \/^, c — p -{■ (I- Given = F, find the six functions of A and also the six functions of B when : 26. a = b — ^2pq. 27. a - - + ^, c ^ + 1. 6 PLANE TPIGONOMETEY In the right triangle A CB, as shown in § 7: 28. Find the length of side a if sind = and c — 20.5. 29. Find the length of side b if cosd = 0.44, and c = 3.5. 30. Find the length of side a if tand = 3f, and h = 2^. 31. Find the length of side h if cotA = 4, and a = 1700. 32. Find the length of the hypotenuse if secA = 2, and J = 2000. 33. Find the length of the hypotenuse if cscA=6.4, and a = 35.6. Find the hypotenuse and other side of a right triangle, given : 34. 5 = 6, tanA = |. 36. 5 = 4, cscA = If. 35. a = 3.5, cos A = 0.5. 37. 5 = 2, sinA = 0.6. 38. The hypotenuse of a right triangle is 2.5 mi., sin A = 0.6, and cos A = 0.8. Compute the sides of the triangle. 39. Construct with a protractor the angles 20°, 40°, and 70°; determine their functions by measuring the necessary lines and compare the values obtained in this way with the more nearly correct values given in the following table : sin ^ ^ COS tan cot sec CSC 20° 0.342 0.940 0.364 2.747 1.064 2.924 40° 0.643 0.766 i 0.839 ; 1.192 1.305 1.556 70° 0.940 0.342 2.747 0.364 2.924 1.064 40. A = 20°, c = 1. 41. A =20°, c = 4. 42. A = 20°, c = 3.5. 43. A = 20°, c = 4.8. 44. A = 20°, c=7f. 50. A = 70°, c = 2. 51. A = 70°, a=2. 52. A = 70°, 5 = 2. 53. A = 70°, a = 25. 54. A = 70°, 5 =150. Find, by means of the above table, the sides and hypotenuse of a right triangle, given: 45. A = 40°, c=l. 46. A = 40°, c = 3. 47. A= 40°, c=7. 48. A = 40°, c=10.7. 49. A = 40°, c = 250. 55. By dividing the length of a vertical rod by the length of its horizontal shadow, the tangent of the angle of elevation of the sun at that time was found to be 0.82. How high is a tower, if the length of its horizontal shadow at the same time is 174.3 yd. ? 56. A pin is stuck upright on a table top and extends upward 1 in. above the surface. When its shadow is f in. long, what is the tangent of the angle of elevation of the sun ? How high is a tele- graph pole whose horizontal shadow at that instant is 21 ft. ? FUNCTIONS OF ACUTE ANGLES 7 8. Functions of Complementary Angles. In the annexed figure we see that B is the complement of A ; that is, B = 90° — A. Hence, sin ^ = - = cos B = cos (90° — A), cos A =- = sin B = sin (90° — A), tan^ “ ^ ~ ^ ~ cot A = - = tan B = tan (90° — A), Q sec ^ ^ = CSC A = CSC (90° — A), CSC A = - = sec B — sec (90° — A). a ' That is, each function of an acute angle is egual to the co-named function of the complementary angle. Co-sine means compZemeTit’s sine, and similarly for the other co-functions. It is therefore seen that sin 75° = cos (90° — 75°) = cos 15°, sec 82° 30' = CSC (90° — 82° 30') = CSC 7° 30', and so on. Therefore, any function of an angle between 45° and 90° may he found hy taking the co-named function of the complementary angle, which is between 0° and 45°. Hence, we need never have a direct table of functions beyond 45°. We shall presently see (§ 12) that this is of great advantage. B Exercise 2. Functions of Complementary Angles Express as functions of the complementary angle 1. sin 30°. 2. cos 20°. 3. tan 40°. 4. sec 25°. 5. sin 50°. 6. tan 60°. 7. sec 75°. 8. CSC 85°. 9. sin 60°. 10. cos 60°. 11. tan 45°. 12. sec 45°. 13. sin 75° 30'. 14. tan 82° 45'. 15. sec 68° 15'. 16. COS 88° 10'. Express as functions of an angle less than 45° : 17. sin 65°. 20. cos 52°. 23. sin 89°. 18. tan 80°. 21. cot 61°. 24. cos 86°. 19. sec 77°. 22. CSC 78°. 25. sec 88°. 26. sin 77^°. 27. cos 82^°. 28. tan 88.6°. Find A, given the following relations: 29. 90°-A = A. 31. 90°-A=2A. 30. cosA = sinA. 32. cosA = sin2A. 8 PLAICE TEIGONOMETBY 9. Functions of 45°. The functions of certain angles, among them 45°, are easily found. In the isosceles right triangle ACB we have A=i3 = 45°, and a = 6. Furthermore, since + = we have 2a^ = c^, a V2 = c, and a = \c V2. Hence, sin 45° = cos 45° = ^ ^ 1 ; c 2 ’ tan 45° = cot 45° = ^ = 1; sec 45° = CSC 45° = ^ = V2. a A b We have therefore found all six functions of 45°. For purposes of computa- tion these are commonly expressed as decimal fractions. Since V2 = 1.4142 +, we have the following values : sin 45° = 0.7071, cos 45° = 0.7071, tan 46° = 1, cot 46° = 1, sec 46° = 1.4142, esc 45° = 1.4142. 10. Functions of 30° and 60°. In the equilateral triangle AA'B here shown, BC is the perpendicular bisector of the base. Also, b = ^c, and a - - Vc^ — 5^ = | c* = i Hence, sec 30° = CSC 60° = - = — = ? V3 • CSC 30° = sec 60° = 7 = 2. b The sine and cosine of 30°, 46°, and 60° are easily remembered, thus : sin 30° = \ a/I, sin 46° = ^ a/2, sin 60° = ^ a/S ; cos 30° = ^ \/3, cos 45° = 4 a/^, cos 60° = i a/I. The functions of other angles are not so easily computed. The computation requires a study of series and is explained in more advanced works on mathe- matics. For the present we assume that the functions of all angles have been computed and are available, as is really the case. FUNCTIO^TS OF ACUTE ANGLES 9 Exercise 3. Functions of 30°, 45°, and 60° Given Vs - 1.7320, express as decimal fractions the following : 1. sin 30°. 4 . cot 30°. 7. sin 60°. 10. cot 60°. 2. cos 30°. 5. sec 30°. 8. cos 60°. 11. sec 60°. 3 . tan 30°. 6. esc 30°. 9. tan 60°. 12. esc 60°. Write the ratios of the following, simplifying the results : 13. sin 45° to sin 30°. 19. sin 30° to sin 60°. 14 . cos 45° to cos 30°. 20. cos 30° to cos 60°. 15 . tan 45° to tan 30°. 21. tan 30° to tan 60°. 16 . cot 45° to cot 30°. 22. cot 30° to cot 60°. 17. sec 45° to sec 30°. 23 . sec 30° to sec 60°. 18. CSC 45° to CSC 30°. 24 . CSC 30° to CSC 60°. ly' Express as functions of angles less than 46° : 25. sin 62° 17' 40". 29. sin 75.8°. 26. tan 75° 28' 35". 30 . cos 82.75°. 27. sec 87° 32' 51". 31. tan 68.82°. 28. cos 88° 0' 27". 32 . sec 85.95°. Find A, given the following relations: 33. 90° - 45° - ^A. 38. cos A = sin (45° — 34 . 90° - \A=A. 39. cot|^A = tanA. 35. 45° +A=90° —A. 40 . tan (45° + A) = cot A. 36 . 90° — 4.A=A. 41 . cos 4 A = sin A. 37 . 90°— A = nA. 42 . cot A = tanwA. 43. By wbiat must siu 45° be multiplied to equal tan 30° ? 44 . By ’what must sec 45° be multiplied to equal esc 30° ? 45 . By what must cos 45° be multiplied to equal tan 60° ? 46 . By what must esc 60° be divided to equal tan 45° ? 47 . By what must esc 30° be divided to equal tan 30° ? 48 . What is the ratio of sin 45° sec 45° to cos 60° ? 49 . What is the ratio of cos 45° esc 45° to cos 30° esc 30° ? 50. What is the ratio of sin 45° sin 30° to cos 45° cos 30° ? 51. What is the ratio of tan 30° cot 30° to tan 60° cot 60° ? 62. From the statement tan 30° = ^ Vs find cot 60°. 10 PLANE TRIGONOMETEY 11. Values of the Functions. The values of the functions have been computed and tables constructed giving these values. One of these tables is shown on page 11 and will suffice for the work required on the next few pages. This table gives the values of the functions to four decimal places for every degree from 0° to 90°. All such values are only approximate, the values of the functions being, in general, incommensurable with unity and not being ex- pressible by means of common fractions or by means of decimal fractions with a finite number of decimal places. 12 . Arrangement of the Table. As explained in § 8, cos 45° = sin 45°, cos 46° = sin 44°, cos 47° = sin 43°, and so on. Hence the column of sines from 0° to 45° is the same as the column of cosines from 45° to 90°. Therefore In finding the functions of angles from 0° to 45° read from the top down ; in finding the functions of angles from 45° to 90° read from the bottom up. Exercise 4. Use of the Table From the table on page 11 find the values of the following : 1. sin 5°. 9. COS 6°. 17. cot 5°. 25. sec 0°. 2. sin 14°. 10. sin 84°. 18. tan 85°. 26. CSC 90°, 3. sin 21°. 11. cos 14°. 19. cot 11°. 27. sec 15°. 4. sin 30°. 12. sin 76°. 20. tan 79°. 28. esc 75°. 6. cos 85°. 13. cos 24°. 21. tan 21°. 29. CSC 12°. 6. cos 76°. 14. sin 66°. 22. cot 69°. 30. sec 78°. 7. cos 69°. 15. cos 35°. 23. tan 45°. 31. CSC 44°. 8. cos 60°. 16. sin 55°. 24. cot 45°. 32. sec 46°. 33. Find the difference between 2 sin 9° and sin (2 x 9°). 34. Find the difference between 3 tan 5° and tan (3 x 5°). 36. Which is the larger, 2 sec 10° or sec (2 x 10°)? 36. Which is the larger, 2 cscl0° or esc (2 x 10°)? 37. Which is the larger, 2 cos 15° or cos (2 x 15°)? 38. Compare 3 sin 20° with sin (3 x 20°); with sin (2 x 20°). 39. Compare 3 tan 10° with tan (3 x 10°); with tan (2 x 10°). 40. Compare 3 cos 10° with cos (3 x 10°) ; with cos (2 x 10°). 41. Is sin (10° + 20°) equal to sin 10° + sin 20° ? 42. When the angle is increased from 0° to 90° which of the six functions are increased and which are decreased ? FUNCTIONS OF ACUTE ANGLES 11 Table of Trigonometric Functions for evert Degree FROM 0° TO 90° Angle sin cos tan cot sec CSC 0° .0000 1.0000 .0000 00 1.0000 00 90° 1° .0175 .9998 .0175 57.2900 1.0002 57.2987 89° 2° .0349 .9994 .0349 28.6363 1.0006 28.6537 88° 3° .0523 .9986 .0524 19.0811 1.0014 19.1073 87° 4° .0698 .9976 .0699 14.3007 1.0024 14.3356 86° 5° .0872 .9962 .0875 11.4301 1.0038 11.4737 85° 6^ .1045 .9945 .1051 9.5144 1.0055 9.5668 84° 7° .1219 .9925 .1228 8.1443 1.0075 8.2055 83° 8° .1392 .9903 .1405 7.1154 1.0098 7.1853 82° 90 .1564 .9877 .1584 6.3138 1.0125 6.3925 81° 0 © .1736 .9848 .1763 5.6713 1.0154 5.7588 80° 11° .1908 .9816 .1944 5.1446 1.0187 5.2408 79° 12° .2079 .9781 .2126 4.7046 1.0223 4.8097 78° 13° .2250 .9744 .2309 4.3315 1.0263 4.4454 77° 14° .2419 .9703 .2493 4.0108 1.0306 4.1336 76° 15° .2588 .9659 .2679 3.7321 1.0353 3.8637 75° 16° .2756 .9613 .2867 3.4874 1.0403 3.6280 74° 17° .2924 .9563 .3057 3.2709 1.0457 3.4203 73° 18° .3090 .9511 .3249 3.0777 1.0515 3.2361 72° 19° .3256 .9455 .3443 2.9042 1.0576 3.0716 71° 20° .3420 .9397 .3640 2.7475 1.0642 2.9238 70° 21° .3584 .9336 .3839 2.6051 1.0711 2.7904 69° 22° .3746 .9272 .4040 2.4751 1.0785 2.6695 68° 23° .3907 .9205 .4245 2.3559 1.0864 2.5593 67° 24° .4067 .9135 .4452 2.2460 1.0946 2.4586 66° 25° .4226 .9063 .4663 2.1445 1.1034 2.3662 65° 26° .4384 .8988 .4877 2.0503 1.1126 2.2812 64° 27° .4540 .8910 .5095 1.9626 1.1223' 2.2027 63° 28° .4695 .8829 .5317 1.8807 1.1326 2.1301 62° 29° .4848 .8746 .5543 1.8040 1.1434 2.0627 61° 30° .5000 .8660 .5774 1.7321 1.1547 2.0000 60° 31° .5150 .8572 .6009 1.6643 1.1666 1.9416 59° 32° .5299 .8480 .6249 1.6003 1.1792 1.8871 58° 33° .5446 .8387 .6494 1.5399 1.1924 1.8361 57° 34° .5592 .8290 .6745 1.4826 1.2062 1.7883 56° 35° .5736 .8192 .7002 1.4281 1.2208 1.7434 55° 36° .5878 .8090 .7265 1.3764 1.2361 1.7013 54° 37° .6018 .7986 .7536 1.3270 1.2521 1.6616 53° 38° .6157 .7880 .7813 ■ 1.2799 1.2690 1.6243 52° 39° .6293 .7771 .8098 1.2349 1.2868 1.5890 51° 40° .6428 .7660 .8391 1.1918 1.3054 1.5557 50° 41° .6561 .7547 .8693 1.1504 1.3250 1.5243 49° 42° .6691 .7431 .9004 1.1106 1.3456 1.4945 48° 43° .6820 .7314 .9325 1.0724 1.3673 1.4663 47° 44° .6947 .7193 .9657 1.0355 1.3902 1.4396 46° 45° .7071 .7071 1.0000 1.0000 1.4142 1.4142 45° COS sin cot tan CSC sec Angle 12 PLANE TKIGONOMETRY 13. Reciprocal Functions. Considering the definitions of the six functions, we see that, since sin A = - > c , c CSC A = - > a cos A - sec A = - ) 0 tanA - ^ > 0 cot A = - 1 a The sine is the reciprocal of the cosecant, the cosine is the reciprocal of the secant, and the tangent is the reciprocal of the cotangent. That is, ^ ^ sinA = -1 cosA = r? tanA= ^ cscA - - CSC A 1 sec A: sec A 1 cot A = cot A 1 sin A ' cos A tanA HLen.Qe _sinA esc A = 1, cosA secA = 1 . and t anA c ot A = J. For example, from the table on page 11 we find sin 27° esc 27° thus T sin 27° = 0.4540. CSC 27° = 2.2027. Therefore sin 27° csc27° = 0.4540 x 2.2027 = 1.00002580, or approximately 1. We have shown that sin A esc A = 1 exactly, but the'numbers given in the table are, as before stated, correct only to four decimal places. Exercise 5. Use of the Table Using the values .given in the table on page 11, show as above that the following are reciprocals : 1. sin 30°, CSC 30°. 4. sinl0°, cscl0°. 7. sin 75°, esc 75°. 2. sin 25°, CSC 25°. 5. tan 10°, cot 10°. 8. cos 75°, sec 75°. 3. cos 35°, sec 35°. 6. cosl0°, secl0°. 9. tan 75°, cot 75°. 10. From the table show that the ratio of sin 20° esc 20° to tan 50° cot 50° is 1. 11. Similarly, show that cos 40° sec 40° : tan 70° cot 70° =1. In the right triangle A CB, as shown in § 7 : 12. Find the length of side a. if A = 30°, and c = 75.2. 13. Find the length of side a if A = 45°, and c = 1.414. 14. Find the length of side h \i A = 30°, and c = 115.47. 15. Find the length of side a if A = 60°, and b — 34.64. 16. Find the length of side 5 if A = 60°, and c = 25.72. 17. Find the length of side a if A = 30°, and c = 45.28. FUNCTIONS OF ACUTE ANGLES 13 14. Other Relations of Functions. Since, from the figure in § 7, a* -f- z= c^, we have or - + - = 1 sin^A + cos*A = 1 . It is customary to write sin^A for (sinA^ and similarly for the other functions. This formula is one of the most important in trigonometry and should be memorized. From it we see that sin A - Vl — cos^A, cos A = Vl — sin^A. Furthermore, since tan A = sin A = -> and cos A = -? it follows he c , sinA tan A = cos A This is also an important formula to be memorized. From it we see that ta n A cos A = sinA, and, in general, that we can find any one of the functions, sine, cosine, or tangent, given the other two. Furthermore, from the same equation we see that . tt/ 1 = Hence we see that 1 + tan^A = secM. In a similar manner we may prove that 14 - — =:—; whence we have the formula i .p cotU = csc^A. These two formulas should be memorized. From these formulas the following relations can easily be deduced : sin X = cos X tana; = cos a;/cot x = tanx/sec x. cos X = cot X sin x = cot x/ese x = sin x/tanx. tanx = sin x sec x = sin x/cos x = sec x/ese x. cot X - - CSC X cos X = CSC x/sec x = cos x/sin x. sec X = tanx esc x = tan x/sin x = esc x/cot x. CSC X = sec X cot x = sec x/tanx = cot x/cos x. It is often convenient to recall thesg_relations, and this can be done by the aid of a simple mnemonic : .^''^tan x sin X sec x ' ,cosx esex; cotx In the above diagram, any function is equal to the product of the two adjacent functions, or to the quotient of either adjacent function divided try the one beyond it. 14 PLANE TRIGONOMETEY 15. Practical Use of the Sine. Since by definition we iiaYe - = sinA, c we see that a = c sin A. We might also derive the equation a c sin A But since — - — = esc A (§ 13), it is easier at present to use sin A c = a cscA, and this will be considered when we come to study the cosecant. 1. Given c = 38 and A = 40°, find a. As above, a = c sin A. From the table, sin 40° = 0.6428 and c = 38 51424 19 284 c sin A = 24.4264 But since the table on page 11 gives only the first four figures of sin 40'’, we can expect only the first four figures of the result to be correct. We therefore say that a = 24.43 — . If the third decimal place were less than 5, the value of a would be written 24.42 +. Some check should always be applied to the result. In this case we may proceed as follows : 24.4264 38 = 0.6428, which is sin 40°. 2. Given c = 10 and a = 6.293, find A. a Since we have c 6.293 10 = sin A, = 0.6293 = sinA. Looking in the table we see that 0.6293 = sin 39°; whence A - 39°. 3. Given a — 4.68^ and A = 22°, find c. As stated above, c may be found from the formula a = c sin A by using a and sin A, although we shall later use the cosecant for this purpose. Substituting the given values, we have 4.684 = sin 22°, or 4.6825 = 0.3746 c. Dividing by 0.3746, 12.5 = c. What check should be applied here and in Ex. 2 ? FUNCTIONS OF ACUTE ANGLES 15 Exercise 6. Use of the Sine Find a to four figures, given the following : 1. c = 10, 10’. 3. c = 58, A = 45°. 2. c = 15, A = 15°. 4. c = 75, A = 50°. Find A, given the following : b. c = 10, a = 2.079. 7. c = 2, a =1.2586. 6. c = 20, a = 6.840. 8. c = 50, a — 34.1. 9. A 50-foot ladder resting against the side of a house reaches a point 25 ft. from the ground. What angle does it make with the ground ? In all such cases the ground should he con.sidered level and the side of the building should be considered vertical unless the contrary is expressly stated. 10. From the top of a rock a cord is stretched to a point on the ground, making an angle of 40° with the horizontal plane. The cord is 84 ft. long. Assuming the cord to be straight, how high is the rock? 11. Find the side of a regular decagon in- scribed in a circle of radius 7 ft. What is the central angle? What is half of this angle ? Find BG and double it. By this plan we can find the perimeter of any inscribed regular polygon, given the radius of the circle. In this way we could approximate the value of tt. For example, we see that the semiperimeter of a polygon of 90 sides in a unit circle is 90 x sin 2°, or 90 x 0.0349, or 3.141. 12. The edge of the Great Pyramid is 609 ft. and makes an angle of 52° with the horizontal plane. What is the height of the pyramid ? 13. Wishing to measure BC, the length of a pond, a surveyor ran a line CA at right angles to BC. He measured AB and Z.A, finding that A5= 928 ft., and A = 29°. Find the length of BC. In practical surveying we would probably use an oblique triangle, although the work as given here is correct. The oblique triangle is considered later. 16 PLANE TRIGONOMETEY 16. Practical Use of the Cosine. Since by definition we have h - = eosA, c we see that h = c cos A. 1. Given c = 28 and A = 46°, find b. From the table, cos 46° = 0.6947 and e— 28 5 5576 13 894 19.4516 Hence, to four figures, h = 19.45. 2. Given c = 2 and b = 1.9022, find A. Since - = cosA, c we have 1.9022 ^ 2 = 0.9511 = cos A. From the table, 0.9511 = cos 18°. Hence A = 18°. What check should be applied here and in Ex. 1 ? Exercise 7. Use of the Cosine Find h to four figures, given the following : 1. c = 11, A = 10°. 6. c = 2.8, O * 00 II 2. c 14, A = 16°. 7. c = 9.7, A = 52°. 3. c = 28, A = 24°. 8. c = 11.2, A = 58°. 4. c = 41, A =3 39°. 9. c = 12.5, A = 67°. 5. c = 75, A = 42°. 10. c = 28.25; , A = 75°. Find A, given the following : 11. c = 10, 6 = 9.848. 16. c = 600, 6 = 205.2. 12. c = 20, 6 = 19.126. 17. c = 200, 6 = 117.56. 13. c = 40, 6 = 35.952. 18. c = 187, 6 = 93^. 14. c = 17.6, 6 = 8.8. 19. c = 300, 6 = 102|. 15. c = 500, 6 = 227. 20. c = 1000, 6 = 1044. 21. A flagstaff breaks off 22 ft. from the top and, the parts still holding together, the top of the staff reaches the earth 11 ft. from the foot. What angle does it make with the ground ? FUNCTIONS OF ACUTE ANGLES 17 22. Wishing to measure the length of a pond, a class constructed a right triangle as shown in the figure. If = 640 ft. and A --- 50°, required the distance AC. 23. In the same figure what is the length of AC when AB = 600 ft. and A = 40° ? 24. In the same figure, it AC = 731.4 ft. and AB= 1000 ft., what is the size of angle A ? 26. A regular hexagon is inscribed in a circle of radius 9 in. How far is it from the center to a side ? Having found this distance, the apothem, and knowing that a side of the regular hexagon equals the radius, we can find the area, as required in Ex. 26. 26. What is the area of a regular hexagon inscribed in a circle of radius 8 in. ? 27. A ship sails northeast 8 mi. It is then how many miles to the east of the starting point ? Northeast is 46° east of north. In all such cases in plane trigonometry the figure is supposed to be a plane. For long distances it would be necessary to consider a spherical triangle. A' 28. Some 16-foot roof timbers make an angle of 30° with the horizontal in an A-shaped roof, as shown in the figure. Find A A the span of the roof. 29. An equilateral triangle is inscribed in a circle of radius 12 in. How far is it from the center to a side ? 30. A crane AB, 30 ft. long, makes an angle of X degrees with the horizontal line AC. Find the distance A C when x = 20-, when a; = 45 ; when a; = 65 ; when a; = 0 ; when x = 90. 31. In Ex. 30. what angle does the crane make with the horizontal when AC = 15 ft. ? when ^ C = 30 ft. ? 32. The square AN, of which the side is 200 ft., is inscribed in the square CM. AC is 181.26 ft. Required the angles that the sides of the small square make with the large one. G 33. In Ex. 32 find the required angles when ^5 = 15 in. and BC = 1\ in.; when AB = 20 in. andRC = 10.3 in. 84. The edge of the Great Pyramid is 609 ft., and it makes an angle of 52° with the horizontal plane. What is the diagonal of the base ? 18 PLANE TKIGONOMETKY 17. Practical Use of the Tangent. Since by definition we have V = tanA, 0 we see that a = b tan A. Given h = 12 and A = 35°, find a. From the table, tan 35° = 0.7002 h= 12 1 4004 7 002 8.4024 Hence, to four figures, a = 8.402. The figures 1, 2, — ,9 are often spoken of as significant figures. In 8.402 the zero is, however, looked upon as a significant figure, but not in a case like 12,550. The first four significant figures in 0.6705067 are 6705. 18. Angles of Elevation and Depression. The angle of elevation, or the angle of depression, of an object is the angle which a line from the eye to the object makes with a horizontal line in the same vertical plane. Thus, if the observer is at 0, x is the angle of elevation of B, and y is the angle of depression of G. In measuring angles with a transit the height of the instru- ment must always be taken into account. In stating problems, however, it is not convenient to consider this every time, and hence the angle is supposed to be taken from the level on which the instrument stands, unless otherwise stated. 1. From a point 5 ft. above the ground and 150 ft. from the foot of a tree the angle of elevation of the top is observed to be 20°. How high is the tree ? W e have a = 6 tan A = 150 tan 20° = 150 X 0.3640 = 54.6. Hence the height of the tree is 54.6 ft. -f- 5 ft., or 59.6 ft. 2. From a point A on a cliff 60 ft. high, including the instrument, the angle of depression of a boat A on a lake is observed to be 25°. How far is the boat from C, the foot of the cliff ? We have Z AA C= 65°. Hence AC= 60 tan 65°. From the table, tan 65° = 2.1445. Hence AC = 60 x 2.1445 = 128.67. £ B FUNCTIONS OF ACUTE ANGLES 19 Exercise 8. Use of the Tangent Find a to four significant figures, given the following : 1. b = 37,A=^ 18°. 6. 5 = 4.8, A =51°. 2. 5 = 26, A = 23°. 7. 5 = 9.6, A =57°. 3. 5 = 48, A = 31°. 8. 5 = 23.4, A = 62°. 4. 5 = 62, A = 36°. 9. 5 = 28.7, A = 75°. 6. 5 = 98, A = 45°. 10. 5 = 39.7, A = 85°. Find A, given the following : 11. a = 6, 5 = 6. 14. a =13.772, 5 = 40. 12. a = 0.281, 5 = 2. 15. a = 2.424, 5 = 6. 13. a = 4.752, 5 = 30. 16. a = 20.503, 5 =10. 17. A man standing 120 ft. from the foot of a church spire finds that the angle of elevation of the top is 50°. If his eye is 5 ft. 8 in. from the ground, what is the height of the spire ? 18. When a flagstaff 55.43 ft. high casts a shadow 100 ft. long on a horizontal plane, what is the angle of elevation of the sun ? 19. A ship S is observed at the same instant from two lighthouses, L and L', 3 mi. apart. Z.L'LS is found to be 40° and Z.LL'S is found to he 90°. What is the distance of the ship from L' ? What is its distance from L ? 20. From the top of a rock which rises vertically, including the instrument, 134 ft. above a river bank the angle of depression of the opposite bank is found to be 40°. How wide is the river ? 21. An A-shaped roof has a span^ff'of 24 ft. The ridgepole A is 12 ft. above the horizontal line AA'. What angle does AR make with AA' ? with RA' ? with the perpendicular from R on AA'? 22. The foot of a ladder is 17 ft. 6 in. from a wall, and the ladder makes an angle of 42° with the horizontal when it leans against the wall. How far up the wall does it reach ? 23. A post subtends an angle of 7° from a point on the ground 50 ft. away. What is the height of the post ? 24. The diameter of a one-cent piece is f in. If the coin is held so that it subtends an angle of 40° at the eye, what is its distance from the eye ? 20 PLANE TRIGONOMETKY 19. Practical Use of the Cotangent. Since by definition we have we see that h - = cotA, a b = a cot A. For example, given a. = 71 and A = 28®, find b. From the table, cot 28° = 1.8807 and a= 71 1 8807 131 649 133.6297 Hence, to four significant figures, b = 133.5. What check should be applied in this case ? Exercise 9. Use of the Cotangent Find h to four significant figures, given the following : 1. a = 29, A = 48®. 2. a = 38, A = 72°. 3. a = 56, A = 19°. 4. a = 72, A = 40°. 6. a = 425, A = 38®. 6. a = 19^, A = 36°. 1. a = 24.8, A = 43°. 8. a = 256.8, A = 75®. Find A, given the following : 9. a=12,b = 72. 10. a = 60, =128.67. 11. How far from a tree 50 ft. high must a person lie in order to see the top at an angle of elevation of 60° ? 12. From the top of a tower 300 ft. high, in- cluding the instrument, a point on the ground is observed to have an angle of depression of 35°. How far is the point from the tower ? 300 13. From the extremity of the shadow cast by a church spire 150 ft. high the angle of elevation of the top is 53®. What is the length of the shadow ? 14. A tree known to be 50 ft. high, stand- ing on the bank of a stream, is observed from the opposite bank to have an angle of elevation of 20®. The angle is measured on a line 5 ft. above the foot of the tree. How wide is the stream ? FUNCTIONS OF ACUTE ANGLES 21 20. Practical Use of the Secant. Since by definition we have 7 = secA, 0 we see that c = h sec A. For example, given 5 = 15 and A = 30°, find c. From the table, sec 30° = 1.1547 and h = 15 5 7735 11 547 17.3205 Hence, to four significant figures, c = 17.32. Exercise 10. Use of the Secant Find c to four significant figures, given the following : 1. b = 36,A= 27°. 4. 6 = 22^, A = 48°. 2. 6 = 48, A = 39°. 5.6 = 33.4, A = 53°. 3. 6 = 74, A = 43°. 6.6 = 148.8, A = 64°. Find A, given the following : 7. 6 = 10, c = 131. 8. 6 = 17.8, 9. A ladder rests against tbe side of a build- ing, and makes an angle of 28° with tbe ground. Tbe foot of tbe ladder is 20 ft. from tbe building. How long is tbe ladder ? 10. From a point 50 ft. from a bouse a wire window so as to make an angle of 30° witb tbe tbe length of tbe wire, assuming it to be straight. 11. In measuring the distance A£ a surveyor ran tbe line AC, making an angle of 50° witb AB, and tbe line BC perpendicular to AC. He meas- ured AC and found that it was 880 ft. Eeqnired tbe distance AB. e = 35.6. horizontal. Find 12. From tbe extremity of tbe shadow cast by a tree tbe angle of elevation of tbe top is 47°. Tbe shadow is 62 ft. 6 in. long. How far is it from tbe top of tbe tree to tbe extremity of tbe shadow ? 13. Tbe span of this roof is 40 ft., and tbe roof timbers AB make an angle of 40° witb tbe hori- zontal. Find tbe length of AB. 22 PLAJSTE TRIGONOMETRY 21. Practical Use of the Cosecant. Since by definition we have — ~ GSC^, a we see that c = a csc^. For example, given a = 22 and A = 35°, find c. From the table. esc 35° = 1.7434 and a= 22 3 4868 34 868 38.3548 Hence, to four significant figures, c = 38.35. GUck. Since - = sin^, 22 38.35 = 0.5736 = sin 35°. c Exercise 11. Use of the Cosecant Find c to four significant figures, given the following : 1. a = 24, d = 29°. 4. a = 56^, A = 61°. 2. a = 36, ^ = 41°. 5. a = 75.8, A = 69°. 3. a = 56,A = 44°. 6. a = 146.9, A = 74°. Find A, given the following : 7. a = 10, c = 11.126. 9. a = 5^,c = 6.0687. 8. a = 13, c = 27.6913. 10. a = 75,c = 106.065. 11. Seen from a point on the ground the angle of elevation of an aeroplane is 64°. If the aeroplane is 1000 ft. above the ground, how far is it in a straight line from the observer ? 12. A ship sailing 47° east of north changes its latitude 28 mi. in 3 hr. What is its rate of sailing per hour ? 13. A ship sailing 63° east of south changes its latitude 45 mi. in 5 hr. What is its rate of sailing per hour ? 14. From the top of a lighthouse 100 ft., including the instrument, above the level of the sea a boat is observed under an angle of depres- sion of 22°. How far is the boat from the point of observation ? 15. Seen from a point on the ground the angle of elevation of the top of a telegraph pole 27 ft. high is 28°. How far is it from the point of observation to the top of the pole ? 16. What is the length of the hypotenuse of a right triangle of which one side is Ilf in. and the opposite angle 43° ? FUNCTIONS OF ACUTE ANGLES 23 / 22. Functions as Lines. The functions of an angle, being ratios, are numbers ; but we may represent them by lines if we first choose a unit of length, and then construct right tri- angles, such that the denominators of the ratios shall be equal to this unit. Thus in the annexed figure the radius is taken as 1, the circle then being spoken of as a unit circle. Then OA^OP=OB= 1. Drawing the four perpendiculars as shown, we have : MP siB.x = ~ = MP-, tana: = — = AT; OT sec a; = = 0T\ iJA. 021 GOSX = = 02I-, BS Gotx = — = BS-, OS CSC£C = — — = OS. OB In each case we have arranged the fraction so that the denominator is 1. 2IP A T For example, instead of taking for tan x we have taken the equal ratio , because OA = 1. OP OS Similarly, instead of taking for esc x we have taken the equal ratio , because OB = 1. This explains the use of the names tangent and secant, A T being a tangent to the circle, and OT being a secant. Formerly the functions were considered as lines instead of ratios and received their names at that time. The word sine is from the Latin sinus, a translation of an Arabic term for this function. We see from the figure that the sine of the complement of x is NP, which equals OM ; also that the tangent of the complement of X is BS, and that the secant of the complement of x is OS. Exercise 12. Functions as Lines 1. Kepresent by lines the functions of 45°. 2. Kepresent by lines the functions of an acute angle greater than 45°. Using the ahoj)e figure, deterr^ne which is the greater : 3. sina:ortan^. 5. see a: or tan x. 7. cos a; or cot a. 4. sin X or sec x. 6. esc x or cot a:. 8. cos x or esc x. 24 PLANE TRIGONOMETEY Construct the angle x, given the following : 9. tan a: = 3. 11. cosa; = ^. 13. sin a: = 2 coax. 10. csca: = 2. 12. sin a: = cos a;. 14. 4 sin a; = tan a:. 16. Show that the sine of an angle is equal to one half the chord of twice the angle in a unit circle. 16. Find x if sin x is equal to one half the side of a regular deca- gon inscribed in a unit circle. Given X and y^x + y being less than 90°, construct a line equal to : 17. sin (a: -h ?/) — sin a:. 20. cos a: — cos(a: -f ?/). 18. tan (a; + ^) — tan x. 21. cot x — cot(a: -f- y~). 19. sec {x H- ?/),— sec x. 22. esc x — esc (x -|- y). 23. tan(a: y)— sin (x y)-\- tan x — sin x. Given an angle x, construct an angle y such that : 24. sin ?/ = 2 sin ic. 28. tan ?/ = 3 tan a;. 25. cos cos a;. 29. secy = cscx. 26. siny = cosx. 30. sin tan x. 27. tan?/ = cot X. 31. siny = ftanx. 32. Show by construction that 2 sin^ > sin 2^, when A < 45°. 33. Show by construction that cos A< 2 cos 2 A, when A< 30°. 34. Given two angles A and B, A being less than 90°; show that sin (A + E) < sin A -f sin B. 35. Given sinx in a unit circle; find the length of a line in a circle of radius r corresponding in position to sinx. 36. In a right triangle, given the hypotenuse c, and sinA=??i; find the two sides, 37. In a right triangle, given the side b, and tan A = m; find the other side and the hypotenuse. Construct, or show that it is impossible to construct, the angle x, given the following : 38. sin X = ^. 41. cosx = 0. 44. tanx = J. 39. sin X = 1. 42. cos x = ^. 45. cot x = 4. 40. sin X = |. 43. cos x = \. 46. sec x = 4. 47. Using a protractor, draw the figure to show that sin 60° = cos(-^ af 60°), and sin 30° = cos (2 x 30°). FUNCTIONS OF ACUTE ANGLES 25 23. Changes in the Functions. If we suppose Z.AOP, or x, to in- crease gradually to 90°, the sine MP increases to M'P', M”P", and so on to OB. That is, the sine increases from 0 for the angle 0°, to 1 for the angle 90°. Hence 0 and 1 are called the limiting values of the sine. Similarly, AT and OT gradually in- crease in length, while OM, BS, and OS gradually decrease. That is, As an acute angle increases to 90°, its sine, tangent, and secant also increase, while its cosine, cotangent, and cosecant decrease. If we suppose x to decrease to 0°, OP coin- cides with OA and is parallel to BS. Therefore MP and AT vanish, OM becomes equal to OA, while BS and OS are each infinitely long and are represented in value by the symbol co. Similarly, we may consider the changes as x increases from 0° to 90°. ' Hence, as the angle x increases from 0° to 90°, we see that / I sinx increases from 0 to 1, . - cosx decreases from 1 to 0, tan X increases from 0 to oo, cot X decreases from oo to 0, secx increases from 1 to oo, CSC X decreases from oo to 1. ' We also see that sines and cosines are never greater than 1 ; secants and cosecants are never less than 1 ; tangents and cotangents may have any values from 0 to oo. In particular, for the angle 0°, we have the following values ; sin 0° = 0, tan 0° = 0, sec 0° = 1, cos 0° = 1, cot 0° = 00 , CSC 0° = 00 . For the angle 90° we have the following values : sin 90° = 1, tan 90° = oo, sec 90° = oo, cos 90° = 0, cot 90° = 0, CSC 90° = 1. By reference to the figure and the table it is apparent that the functions of 46° are never equal to half of the corresponding functions of 90°. Thus, sin 46° = 0.7071, tan 45° = 1, sec 45° = 1.4142, cos 46° = 0.7071, cot 46° = 1, esc 46° = 1.4142. PLAi^E TPIGONOMETEY H6 , Exercise 13. Functions as Lines / I. Draw a figure to show that sin 90° = 1. 2. What is the value of cos 90° ? Draw a figure to show this. 3. What is the value of sec 0° ? Draw a figure to show this. 4. What is the value of tan 90° ? Draw a figure to show this. 5. What is the value of cot 90° ? Draw a figure to show this. 6. As the angle increases, which increases the more rapidly, the sine or the tangent ? Show this by reference to the figure. 7. If you double an angle, does this double the sine ? Show this by reference to the figure. 8. If you bisect an angle, does this bisect the tangent ? Prove it. 9. State the angle for which these relations are true : sin X = cos X, tan x = cot x, sec x = esc x. Show this by reference to the figure. 10. If you know that sin 40° 15' = 0.6461, and cos 40° 15' = 0.7632, and that the difference between each of these and the sine and cosine of 40° 15' 30" is 0.0001, what is sin 40° 15' 30" ? cos 40° 15' 30" ? 11. If you know that tan 20° 12' is 0.3679, and that the difference between this and tan 20° 12' 15" is 0.0001, what is tan 20° 12' 15" ? 12. If you know that cot 20° 12' is 2.7179, and that the difference between this and cot 20° 12' 15" is 0.0006, what is cot 20° 12' 15" ? 13. If you know that tan 66.5° is 2.2998, and that the difference between this and tan 66.6° is 0.0111, what is tan 66.6°? 14. If you know that cos 57.4° is 0.5388, and that the difference between this and cos 57.5° is 0.0015, what is cos 57.5° ? Draw the angle x for which the functions have the following values and state (j>age 11') to the nearest degree the value of the angle : 16. sin X = 0.1. 16. sin X = 0.4. 17. sin X = 0.7. 18. cos X = 0.9. 19. cos X = 0.8. 20. cos X = 0.7. 21. tan a; = 0.1. 22. tana; = 0.23. 23. tan x = 0.4. 24. cot x = 4.0. 26. cot X = 2.9. 26. cot X = 0.9. 27. sec a: = 1.2. 28. secx = 1.3. 29. seca: = 1.7. 30. esex = 2.0. 31. esex = 3.6. 32. CSC X = 1.66. 33. Pind the value of sin x in the equation sin x ^ h 1.5 = 0. sinx Which root is admissible ? Why is the other root impossible ? CHAPTER II USE OF THE TABLE OF NATURAL FUNCTIONS 24. Sexagesimal and Decimal Fractions. The ancients, not having developed the idea of the decimal fraction and not having any con- venient notation for even the common fraction, used a system based upon sixtieths. Thus they had units, sixtieths, thirty-six hun- dredths, and so on, and they used this system in all kinds of theo- retical work requiring extensive fractions. For example, instead of 1-^^ they would use 1 28', meaning l|^ ; and instead of 1.51 they would use 1 30' 36", meaning l|-^ -1- -yff-Q. The symbols for de- grees, minutes, and seconds are modern. We to-day apply these sexagesimal (scale of sixty) fractions only to the measrrre of time, angles, and arcs. Thus 3 hr. 10 min. 15 sec. means (3 + -|^ + -g-H-o) hr., and 3° 10' 15" means (3 + + tIwo)° In medieval times the sexagesimal system was carried farther than this. For example, 3 10' 20" 30'" 45*^ was used for 3 -f — — -p —. Some ’ 60 602 003 ^ 604 writers used sexagesimal fractions in which the denominators extended to 60^2. Since about the year 1600 we have had decimal fractions with which to work, and these have gradually replaced sexagesimal frac- tions in most cases. At present there is a strong tendency towards using decimal instead of sexagesimal fractions in angle measure. On this account it is necessary to be familiar with tables which give the functions of angles not only to degrees and minutes, but also to degrees and hundredths, with provision for finding the functions also to seconds and to thousandths of a degree. Hence the tables which will be considered and the problems which will be proposed will in- volve both sexagesimal and decimal fractions, but with particular attention to the former because they are the ones still commonly used. The rise of the metric system in the nineteenth century gave an impetus to the movement to abandon the sexagesimal system. At the time the metric system was established in France, trigonometric tables were prepared on the decimal plan. It is only within recent years, however, that tables of this kind have begun to come into use. 27 28 PLANE TRIGONOMETEY 25. Sexagesimal Table. The following is a portion of a page from the Wentworth-Smith Trigonometric Tables : 41° 42° f sin cos tan cot f 0 6561 7547 8693 1.1504 60 1 6563 7545 8698 1.1497 59 2 6565 7543 8703 1.1490 58 3 6567 7541 8708 1.1483 57 4 6569 7539 8713 1.1477 56 5 6572 7538 8718 1.1470 55 t cos sin cot tan f 48° t sin cos tan cot f 0 6691 7431 9004 1.1106 60 1 6693 7430 9009 1.1100 59 2 6696 7428 9015 1.1093 58 3 6698 7426 9020 1.1087 57 4 6700 7424 9025 1.1080 56 5 6702 7422 9030 1.1074 55 t cos sin cot tan f 47° The functions of 41° and any number of minutes are found by reading down, under the abbreviations sin, cos, tan, cot. For example, sin 41° = 0.6561, cos 41° 2' = 0.7543, tan 41° 4' = 0.8713, cot 41° 5' = 1.1470, sin 42° = 0.6691, cos 42° = 0.7431, tan 42° 3' = 0.9020, cot 42° 5'= 1.1074. Decimal points are usually omitted in the tables when it is obvious where they should be placed. The secant and cosecant are seldom given in tables, being reciprocals of the cosine and sine. We shall presently see that we rarely need them. Since sin 41° 2' is the same as cos 48° 58' (§ 8), we may use the same table for 48° and any number of minutes by reading up, above the abbreviations cos, sin, cot, tan. For example, cos 48° 55' = 0.6572, sin 48° 56' = 0.7539, cot 48° 58' = 0.8703, tan 48° 59'= 1.1497, cos 47° 55' = 0.6702, sin 47° 56' =0.7424, cot 47° 57' = 0.9020, tan 47° 59'= 1.1100. Trigonometric tables are generally arranged with the degrees from 0" to 44° at the top, the minutes being at the left; and with the degrees from 45° to 89° at the bottom, the minutes being at the right. Therefore, in looking for functions of an angle from 0° to 44° 59', look at the top of the page for the degrees and in the left column for the minutes, reading the number below the proper abbreviation. For functions of an angle from 45° to 90° (89° 60'), look at the bot- tom of the page for the degrees and in the right-hand column for the minutes, reading the number above the proper abbreviation. NATUEAL FUNCTIONS 29 Exercise 14. Use of the Sexagesimal Table From the table on page 28 find the values of the following : 1. cos 41°. 2. tan 42°. 3. cos 41° 1'. 4. tan 42° 2'. 5. cos 41° 5'. 6. sin 48° 59'. 7. sin 47° 58'. 8. cos 48° 59'. 9. cos 47° 59'. 10. cos 48° 57'. 11. sin 42° 4'. 12. cos 47° 56'. 13. tan 41° 3'. 14. cot 48° 57'. 15. tan 48° 57'. In the right triangle A CB, in which C = 90° : 16. Given c = 27 and A = 41° 3', find a. 17. Given c = 48 and A - 42° 4', find a. 18. Given c = 61 and A = 41° 2', find h. 19. Given c = 12 and A = 42° 3', find b. 20. Given 5 = 24 and A — 41° 3', find a. 21. Given & = 28 and A = 42° 4', find a. 22. Given a = 42 and A = 41° 1', find b. 23. Given a = 60 and d — 42° 4', find b. 24. Given c = 86 and A — 48° 56', find a. 25. Given c = 92 and A = 48° 57', find a. 26. Given 6 = 45 and A = 47° 55', find a. 27. Given S = 85 and A = 47° 59', find a. 28. Given a. = 86 and A = 48° 56', find b. 29. Given a = 98 and A = 47° 58', find b. 30. Given 5 = 67 and s = 100, find A. 31. A hoisting crane has an arm 30 ft. long. When the arm makes an angle of 41° 3' with x, what is the length of v ? what is the length of a: ? 32. In Ex. 31 suppose the arm is raised until it makes an angle of 41° 5' with x, what are then the lengths of y and x ? 33. From a point 128 ft. from a building the angle of elevation of the top is observed, by aid of an instrument 5 ft. above the ground, to be 42° 4'. What is the height of the building ? 34. From the top of a building 62 ft. 6 in. high, including the instrument, the angle of depression of the foot of an electric-light pole is observed to be 41° 3'. How far is the pole from the building ? 30 PLAITE TRIGONOMETRY 26. Decimal Table. It would be possible to haye a decimal table of natural functions arranged as follows : 0 sin cos tan cot O 0.0 0000 1.0000 0000 00 90.0 0.1 0017 1.0000 0017 573.0 89.9 0.2 0035 1.0000 0035 286.5 89.8 0.3 0052 1.0000 0052 191.0 89.7 0.4 0070 1.0000 0070 143.2 89.6 0.5 0087 1.0000 0087 114.6 89.5 O cos sin cot tan O O sin cos tan cot e 4.0 0698 9976 0699 14.30 86.0 4.1 0715 9974 0717 13.95 85.9 4.2 0732 9973 0734 13.62 85.8 4.3 0750 9972 0752 13.30 85.7 4.4 0767 9971 0769 13.00 85.6 4.5 0785 9969 0787 12.71 85.5 O cos sin cot tan O Since, however, the decimal divisions of the angle have not yet become com- mon, it is not necessary to have a special table of this kind. It is quite con- venient to use the ordinary sexagesimal table for this purpose hy simply referring to the Table of Conversion of sexagesimals to decimals and vice versa. This table is given with the other Wentworth-Smith tables prepared for use with this book. Thus if we wish to find sin 27.75°, we see by the Table of Conversion that 0.75° = 45', so we simply look for sin 27° 45'. Tor example, using either the above table or, after conversion to sexagesimals, the common table, we see that : sin 0.4° = 0.0070, cos 4.1° = 0.9974, tan 0.5° = 0.0087, cot 4.3°= 13.30, sin 85.5°= 0.9969, cos 85.5° = 0.0785, tan 85.8°= 13.62, cot 85.9° = 0.0717. Exercise 15. Use of the Decimal Table From the above table find the values of the following : 1. sin 0.5°. 6. sin 4.1°. 11. sin 85.7°. 16. sin 89.5°. 2. tan 0.4°. 7. cos 4.3°. 12. sin 85.9°. 17. cos 85.9°. 3. sin 4°. 8. tan 4.4°. 13. cos 85.6°. 18. tan 89. 6°. 4. cos 4.2°. 9. cot 4.5°. 14. tan 85.9°. 19. cot 89.7°. 5. tan 4.5°. 10. cot 4.2°. 15. cot 85.6°. 20. cot 85.8°. 21. The hypotenuse of a right triangle is 12.7 in., and one acute angle is 85.5°. Eind the two perpendicular sides. 22. From a point on the top of a house the angle of depression of the foot of a tree is observed to be 4.4°. The house, including the instrument, is 30 ft. high. How far is the tree from the house ? 23. A rectangle has a base 9.5 in. long, and the diagonal makes an angle of 4.5° with the base. Find the height of the rectangle and the length of the diagonal. NATUEAL FUNCTIONS 31 27 . Interpolation. So long as we wish to find the functions of an acute angle expressed in degrees and minutes, or in degrees and tenths, the tables already explained are sufiicient. But when the angle is expressed in degrees, minutes, and seconds, or in degrees and hundredths, we see that the tables do not give the values of the functions directly. It is then necessary to resort to a process called interpolation. Briefly expressed, in the process of interpolation we assume that sin 42J° is found by adding to sin 42° half the difference between sin 42° and sin 43°. In general it is evident that this is not true. For example, in the annexed figure the line values of the functions of 30° and 60° are shown. It is clear that sin 30° is more than half sin 60°, that tan 30° is less than half tan 60°, and that sec 30° is more than half sec 60°. This is also seen from the table on page 11, where sin 30° = 0.5000, tan 30° = 0.5774, sec 30° = 1.1547, sin 60° = 0.8660, tan 60° = 1.7321, sec 60° = 2.0000. For angles in which the changes are very small, interpolation gives results which are correct to the number of decimal places given in the table. For example, from the table on page 11 we have sin 42° = 0.6691 sin 41° = 0.6561 Difference for 1°, or 60', = 0.0130 Difference for V = of 0.0130 = 0.0002. Adding this to sin 41°, we have sin 41° l' = 0.6563, a result given in the table on page 28. But if we wish to find tan 89.6° from tan 89.5° and tan 89.7°, we cannot use this method because here the changes are very great, as is always the case with the tangents and secants of angles near 90°, and with the cotangents and cosecants of angles near 0°. Thus, from the table on page 30, tan89.7° = 191.0 tan89.5°= 114.6 Difference for 0.2° = 76.4 Difference for 0.1° = 38.2 Adding this to tan 89.5°, tan 89.6° = 152.8, whereas the table shows the result to be 143.2. When cases arise in which interpolation cannot safely be used, we resort to the use of special tables that give the required values. These tables are explained later. Interpolation may safely be used in all examples given in the early part of the work. 32 PLANE TRIGONOMETEY 28. Interpolation applied. The following examples will illustrate the cases which arise in practical problems. The student should refer to the Wentworth-Smith Trigonometric Tables for the func- tions used in the problems. 1. Find sin 22° 10' 20". From the tables, sin 22° 11' = 0.3776 sin 22° 10' = 0.3773 Difference for 1', or 60", the tabular difference = 0.0003 Difference for 20" is of 0.0003, or 0.0001 Adding this to sin 22° 10', we have sin22f 10' 20" = 0.3774 2. Find cos 64° IT 30". From the tables, cos 64° 17' = 0.4339 cos 64° 18' = 0.4337 Tabular difference = 0.0002 Difference for 30" is of 0.0002, or 0.0001 Since the cosine decreases as the angle increases we must subtract 0.0001 from cos 64° 17', which gives us cos 64° 17' 30" = 0.4338 3. Find tan 37.54°. By the Table of Conversion, 0.54° = 32' 24". From the tables, tan 37° 33' tan 37° 32' Tabular difference Difference for 24" is f or 0.4, of 0.0004 Adding this to tan 37° 32', we have tan 37.54° = tan 37° 32' 24" 4. Given sin a: = 0.6456, find x. Looking in the tables for the sine that is a little less than 0.6456, and for the next larger sine, we have 0.6457 = sin 40° 13' 0.6455 = sin 40° 12' 0.0002 = tabular difference Therefore x lies between 40° 12' and 40° 13'. Furthermore, 0.6456 = sin x 0.6455 = sin 40° 12' 0.0001 = difference But 0.0001 is 4 of 0.0002, the tabular difference, so that x is halfway from 40° 12' to 40° 13'. Therefore we add 4 of 60", or 30", to 40° 12'. Hence x = 40° 12' 30". We interpolate in a similar manner when we use a decimal table. = 0.7687 = 0.7683 = 0.0004 = 0.0002 = 0.7685 NATUKAL FUNCTIONS 33 Exercise 16. Use of the Table Find the values of the following : 1. sin 27° 10' 30" 2. sin 42° 15' 30" 3. sin 56° 29' 40" 4. sin 65° 29' 40" 5. cos 36° 14' 30" 6. cos 43° 12' 20" 7. cos 64° 18' 45" 8. tan 28° 32' 20" 9. tan 32° 41' 30" 10. tan42° 38' 30" 11. tan 52° 10' 45". 12. tan 68° 12' 45". 13. tan 72° 15' 50". 14. tan 85° 17' 45". 15. tan 86° 15' 50". 16. cot5°27'30". 17. cot 6° 32' 45". 18. cot 7° 52' 50". ' 19. cot 8° 40' 10". 20. cot 9° 20' 10". Then find cos x. Then find cos x. Then find sin x. Then find cot x. Then find cot x. 21. Given since = 0.6391, find x. 22. Given since = 0.7691, find x. 23. Given cos ce = 0.3174, find x. 24. Given tan ce = 2.8649, find ce. 25. Given tan ce = 5.3977, find ce. First converting to sexagesimals, find the foUoiving : 26. sin 25.5°. 27. sin 25.55°. 28. sin 32.75°. 29. sin 41.65°. 30. sin 64.75°. 31. cos 78.52°. 32. tan 78.59°. 33. cos 81.43°. 34. tan 82.72°. 35. tan 84.68°. 36. cos 11.25° 37. cot 12.32° 38. cot 13.54° 39. cot 15.48° 40. cot 16.62° Find the value of x in each of the folloioing equations : 41. sin ce = 0.5225. 42. sin ce 0.5771. 43. since = 0.6601. 44. since = 0.7023. 45. cos ce = 0.7853. 46. cos x = 0.7716. 47. cos X = 0.9524. 48. cos ce = 0.7115. 49. tan ce = 2.6395. 50. tana: = 4.7625. 51. tan a: = 4.7608. 52. cot ce = 3.7983. 53. If since = 0.6431, what is the value of cosce ? 54. If cos a: = 0.7652, what is the value of sin ce ? 55. If tance = 0.6827, what is the value of since ? 56. If tance = 0.6537, what is the value of ce ? of cotcc ? 57. If cotcc = 1.6550, what is the value of ce ? of tance ? Verifj the second result by the relation tan ce = 1 /cot x. 34 PLANE TRIGONOMETRY 29. Application to the Right Triangle. In §§ 15-21 we learned how to use the several functions in finding various parts of a right triangle from other given parts, the angles being in exact degrees. In § § 25-28 we learned how to use the tables when the angles were not necessarily in exact degrees. We shall now review both of these phases of the work in connection with the solution of the right triangle. In order to solve a right triangle, that is, to find both of the acute angles, the hypotenuse, and both of the sides, two independent parts besides the right angle must be given. In speaking of the sides of a right triangle it should be repeated that we shall refer only to sides a and b, the sides which include the right angle, using the word hypotenuse to refer to c. It will he found that there is no confusion in thus referring to only two of the three sides by the special name sides. By independent parts is meant parts that do not depend one upon another. For example, the two acute angles are not independent parts, for each is equal to 90° minus the other. The two given parts may be : 1. An acute angle and the hypotenuse. That is, given A and c, or B and c. If A and c are given, we have to find a and h. The angle B is known from the relation B = 90° — A. If JB is given, we can find A from the equation A = 90° — B. 2. An acute angle and the opposite side. That is, given A and a, or B and b. If A and a are given, we have to find B, b, and c, and similarly for the other case. 3. An acute angle and the adjacent side. That is, given A and b, or B and a. If A and h are given, we have to find B, a, and c, and similarly for the other case. 4. The hypotenuse and a side. That is, given c and a, or c and b. If c and a are given, we have to find A, B, and &, and similarly for the other case. 5. The two sides. That is, given a and b, to find A, B, and c. Using side to include hypotenuse, we might combine the fourth and fifth of these cases in one. In each of these cases we shall consider right triangles which have their acute angles expressed in degrees and minutes, in de- grees, minutes, and seconds, or in degrees and decimal parts of a degree In this chapter the angles are given and required only to the nearest minute. NATURAL FUNCTIONS 35 30. Given an Acute Angle and the Hypotenuse. For example, given A = 43° 17', c = 26, find £, a, and h. 1. 90° -^ = 46° 43'. 2. - = sind ; . a — c sin^. c a 3. - = cos^ ; .'. b = c cos .4. c A b 0 sin^ = 0.6856 cosA= 0.7280 c= 26 c= 26 41136 4 3680 13 712 14 560 a = 17.8256 b = 18.9280 = 17.83 = 18.93 As usual, when a four-place table is employed, the result is given to four figures only. The check is left for the student. 31. Given an Acute Angle and the Opposite Side. For example, given A = 13° 58', a = 15.2, find B, b, and c. 1. 90° -A =76° 2'. 2. ~ = cotA; r. b = aootA. a B a . , a 3. - — smA ; c = . • c smA ^ b 0 a =15.2, cotA = 4.0207 a = 15.2, sin A = 0.2414 4.0207 62.97 = 15.2 2414)152000.00 80414 14484 20 1035 7160 40 207 4828 h = 61.11464 23320 = 61.11 21726 In dividing 15.2 by 0.2414, we adopt the modem plan of first multiplying each by 10,000. Only part of the actual division is shown. Instead of dividing a by sin^ to find c, we might multiply a by esc 4, as on page 22, except that tables do not generally give the cosecants. It will be seen in Chapter III that, by the aid of logarithms, we can divide by sin A as readily as multiply by esc J., and this is why the tables omit the cosecant. 36 PLANE TKIGONOMETEY 32. Given an Acute Angle and the Adjac^t Side. Eor example, given A = 27° 12', h = 31, find B, a, and c. 1. B= 90° - 62°48'. 2. - = tanA; .'.a — btdOiA. 3. - = cosd; .•. c = -• c cos A tanA= 0.5139 b= 31 5139 15 417 a = 15.9309 = 15.93 B b = 31, cosd = 0.8894 34.85 = c 8894)310000.00 26682 43180 35576 We might multiply 6 by sec A instead of dividing by cos A. The reason for not doing so is the same as that given in § 31 for not multiplying by cscA. 33. Given the Hypotenuse and a Side. For example, given a = 47, c = 63, find A, B, and b. 1. sin A = -• c 2. 90° - A. 3. b = y/c^ — = V(c 4- a) (c — a). In the case of V we can, of course, square c, square a, take the dif- ference of these squares, and then extract the square root. It is, however, easier to proceed by factoring — a? as shown. This will be even more apparent when we come, in Chapter III, to the short methods of computing by logarithms. B a = 47, c = 63 0.7460 63)47.0000 44 1 2 90 sin A = 0.7460 2 52 .-.A = 48° 15' 380 .-.B=41°45' 378 c -j- a = 110 c — a = 16 660 110 e^-a^ = 1760 .-. 6- = 1760 .-. b = V1760 = 41.95 NATURAL FUNCTIONS 87 34. Given the Two Sides. For example, given a = 40, b = 27, find Of course c can be found in other ways. For example, after finding tan A wc can find A, and hence can find sin A. Then, because sin A = a/c, we have c = a/sin A. When the numbers are small, however, it is easy to find c from the relation given above. a = 40, = 27 1 0 = 1.4815 tan A = 1.4815 .•.A = 55° 59' .-. R=34° V - 1600 729 = 2329 . c =V2329 = 48.26 35. Checks. As already stated, always apply some check to the results. For example, in § 34, we see at once that = 1600 and 6^ is less than 30^, or 900, so that is less than 2500, and c is less than 50. Hence the result as given, 48.26, is probably correct. We can also find B independently. For since tan B = —, a we see that tanS = = 0.6750, and therefore that B — 34° 1'. Exercise 17. The Right Triangle Solve the right triangle A CB, in which C = 90°, given : 1. a = 3, 5 = 4. 2. a = 7, c = 13. 3. a = 5.3, A = 12° 17'. 4. a =10.4, B= 43° 18'. 5. c = 26, A = 37° 42'. 6. c =140, R= 24° 12'. 7. h =19, c = 23. 8. 6 = 98, c= 135.2. 9. 6 = 42.4, A = 32° 14'. 10. 6 = 200, B= 46° 11'. 11. a = 95, b = 37. 12. a = 6, c = 103. 13. a = 3.12, 5=5° 8'. 14. a =17, c =18. 15. c = 57, A= 38° 29'. 16. a + c = 18, 5 = 12. 17. a + c = b = 30. 18. a + c = 45, b — 30. 38 PLANE TEIGONOMETEY Solve the right triangle ACB, in 19. a = 2.5, ^=35° 10' 30". 20. a = 5.7, A = 42° 12' 30". 21. a= 6.4, B= 29° 18' 30". 22. a =7.9, B= 36° 20' 30". 23. c = 6.8, A = 29° 42' 30". 24. c = 360, A = 34° 20' 30". 25. h = 250, A = 41° 10' 40". which C = 90°, given : 26. a = 48, A = 25.5°. 27. c = 25, A = 24.5°. 28. c = 40, A = 32.55°. 29. c = 80, A = 55.51°. 30. c = 75, A = 63.46°. 31. a = 45, .6= 50.59°. 32. h = 99,A = 68.25°. 33. Each equal side of an isosceles triangle is 16 in., and one of the equal angles is 24° 10'. What is the length of the base ? 34. Each equal side of an isosceles triangle is 25 in., and the ver- tical angle is 36° 40'. What is the altitude of the triangle ? 35. Each equal side of an isosceles triangle is 25 in., and one of the equal angles is 32° 20' 30". What is the length of the base ? 36. Each equal side of an isosceles triangle is 60 in., and the ver- tical angle is 50° 30' 30". What is the altitude of the triangle ? 37. Find the altitude of an equilateral triangle of which the side is 50 in. Show three methods of finding the altitude. 38. What is the side of an equilateral triangle of which the altitude is 52 in. ? 39. In planning a truss for a bridge it is necessary to have the upright BC = 12 ft., and the horizontal A C = 8 ft., as shown in the figure. What angle does AB make with AC? with BC ? 40. In Ex. 39 what are the angles if AB = 12 ft. and AC = 9ft. ? 41. In the figure of Ex. 39, what is the length of .BC if AA = 15 ft. and a: = 62° 10'? 42. Two angles of a triangle are 42° 17' and 47° 43' respectively, and the included side is 25 in. Find the other two sides. 43. A tangent AB, drawn from a point A to a circle, makes an angle of 51° 10' with a line from A through the center. If AB = 10 ft., what is the length of the radius ? 44. How far from the center of a circle of radius 12 in. will a tangent meet a diameter with which it makes an angle of 10° 20'? 45. Two circles of radii 10 in. and 14 in. are externally tangent. What angle does their line of centers make with their common exterior tangent ? a CHAPTER III LOGARITHMS 36. Importance of Logarithms. It has already been seen that the trigonometric functions are, in general, incommensurable with unity. Hence they contain decimal fractions of an infinite number of places. Even if we express these fractions only to four or five decimal places, the labor of multiplying and dividing by them is considerable. For this reason numerous devices have appeared for simplifying this work. Among these devices are various calculating machines, but none of these can easily be carried about and they are too expensive for general use. There is also the slide rule, an inexpensive instru- ment for approximate multiplication and division, hut for trigono- metric work this is not of particular value because the tables must be at hand even when the slide rule is used. The most practical device for the purpose was invented early in the seventeenth century and the credit is chiefly due to John Napier, a Scotchman, whose tables appeared in 1614. These tables, afterwards much improved by Henry Briggs, a contemporary of Napier, are known as tables of logarithms, and by their use the operation of multiplication is re- duced to that of addition ; that of division is reduced to subtraction ; raising to any power is reduced to one multiplication; and the extracting of any root is reduced to a single division. Eor the ordinary purposes of trigonometry the tables of functions used in Chapter II are fairly satisfactory, the time required for most of the operations not being unreasonable. But when a problem is met which requires a large amount of eomputation, the tables of natural functions, as they are called, to distinguish them from the tables of logarithmic functions, are not convenient. For example, we shall see that the product of 2.417, 3.426, 517.4, and 91.63 can he found from a table by adding four numbers which the table gives. In the case of x x we shall see that the result can be found 62.9 5.28 9283 from a table by adding six numbers. Taking a more difiBcult case, like that of J i v , we shall see that it \711 0.379 is necessary merely to take one third of the sum of four numbers, after whidi the table gives va the result. 39 40 PLANE TEIGONOMETEY 37. Logarithm. The exponent of the power to which a given m im- ber, called the base, must be raised in order to be equal to another given number is called the logarithm of this second given nnm hp.r. For example, since 10^ = 100, we have, to the base 10, 2 = the logarithm of 100. In the same way, since 10^ = 1000, we have, to the base 10, 3 = the logarithm of 1000. Similarly, 4 = the logarithm of 10,000, 5 = the logarithm of 100,000, and so on, whatever powers of 10 we take. In general, if = N, then, to the base &, z = the logarithm of N. 38. Symbolism. For "logarithm of iV” it is customary to write "logiV.” If we wish to specify log N to the base b, we write log^iV, reading this "logarithm of iV to the base b.” That is, as above, log 100 = 2, log 10,000 = 4, log 1000 = 3, log 100,000 = 5, and so on for the other powers of 10. 39. Base. Any positive number except unity may be taken as the base for a system of logarithms, but 10 is usually taken for purposes of practical calculation. Thus, since 2® =8, logjS = 3; since 31 = 81, logs 81 = 4: and since 5* = 625, logj 625 = 4. It is more convenient to take 10 as the base, however. For since 102 = 100 and 10^ = 1000, we can tell at once that the logarithm of any number between 100 and 1000 must lie between 2 and 3, and therefore must be 2 + some fraction. That is, by using 10 as the base we know immediately the integral part of the logarithm. When we write log 27, we mean log^j27 ; that is, the base 10 is to be imder- stood unless some other base is specified. Since log 10 - 1, because 10^ = 10, and log 1 = 0, because 10® =1, and log^= — 1, because 10-^=^, we see that the logarithm of the base is always 1, the logarithm of 1 is always zero, and the logarithm of a proper fraction is negative. That this is true for any base is apparent from the fact that 61 = 6, whence log6 6=1; 6® = 1, whence logjl =0; 6-" = —, whence log;,— =— n. 6n 6» LOGARITHMS 41 13. log.^343. 14. logg512. Exercise 18. Logarithms 1. Since 2® = 32, what is log^ 32 ? . 2. Since 4^ = 16, what is log^ 16 ? 3. Since 10^ = 10,000, what is log 10,000 ? Write the following logarithms : 4. logjlO. 8. logg243. 12. logg36. 5. loggOl. 9. loggT29. 6. log2l28. 10. log^256. 7. log2256. 11. loggl25. 15. logg6561. 20. Since 10“^ = ^, or 0.1, what is log 0.1 ? 21. What is log -j-^, or log 0.01 ? log 0.001 ? log 0.0001 ? 22. Between what consecutive integers is log 52 ? log 726 ? log 2400? log 24,000? log 175,000? log 175,000,000 ? 23. Between what consecutive negative integers is log 0.08 ? log 0.008 ? log 0.0008 ? log 0.1238 ? log 0.0123 ? log 0.002768 ? ^ 24. To the base 2, write the logarithms of 2, 4, 8, 64, 512, 1024, X ^ -J— _L 1 1 4>Te; 32> 64J 128J 256" 25. To the base 3, write the logarithms of 3, 81, 729, 2187, 6561, 1 1 _L 1 1 1 1 ^ 3) 2 7 j “ST ’ 2437 729’ 2187' 16. log 100. 17. log 1000. 18. log 100,000. 19. log 1,000,000. 26. To the base 10, write the logarithms of 1, 0.0001, 0.00001, 10 , 000 , 000 , 100 , 000 , 000 . Write the consecutive integers between which the logarithms of the following numbers lie : 27. 75. 31. 642. 35. 7346. 39. 243,481. 28. 75.9. 32. 642.75. 36. 7346.9. 40. 5,276,192. 29. 75.05. 33. 642.005. 37. 7346.09. ^41. 7,286,348.5. 30. 82.95. 34. 793.175. 38. 9182.735. 42. 19,423,076. Show that the following statements are true : ^'^3. log 24 + log 28 4 - log 2 l 6 + log^64 + log 22 + logg32 = 21. 44. log33 + loggO + logg81 + logg729 + logg27 + logg243 = 21, 45. log^ll + log^l21 + log^, 1331 + log^ 14,641 =. 10. 46. log 1 + log 10 + log 1000 + log 0.1 + log 0.001 = 0. 47. log 1 + log 100 + log 10,000 + log 0.01 + log 0.0001 = 0. 48. log 10,000 — log 1000 + log 100,000 — log 100 = 4. 42 PLANE TEIGONOMETRY 40. Logarithm of a Product. The logarithm of the product of two numbers is equal to the sum of the logarithms of the nurribers. Let A and B be the numbers, and x and y their logarithms. Then, taking 10 as the base and remembering that x = logA, and y = log B, we have ^ _ ^qx and B — 10*'. Therefore AB = 10^ + ’^, and therefore logAB— x + y = log A 4- logP. The proof is the same if any other base is taken. For example, if z = logi, A, we have A = ; and if y — log;, B, we have B = bv. Therefore AB = b^ + v, and log6 AB = X + y = logs A + logs B. The proposition is also true for the product of more than two numbers, the proof being evidently the same. Thus, log ABC — log A + logB + log C, and so on for any number of factors. 41. Logarithm of a Quotient. The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor. For if A = 1(F, and B = 10*', then — = 10*-!', and therefore log — = a; — y -- log A — logE. This proposition is true if any base h is taken. For, as in § 40, and therefore logs ■ X— y = logs A - logs B. It is therefore seen from §§ 40 and 41 that if we know the logarithms of all numbers we can find the logarithm of a product by addition and the logarithm of a quotient by subtraction. If we can then find the numbers of which these results are the logarithms, we shall have solved our problems in multiplication and division by merely adding and subtracting. LOGAEITHMS 43 42. Logarithm of a Power. The logarithm of a power of a number is equal to the logarithm of the number multiplied by the exponent. For if A = 1(F, raising to the ^th power, Hence log^^ - - px = ^logJ.. This is easily seen by taking special numbers. Thus if we take the base 2, we have the following relations : Since 2® = 32, then log 2 32 = 6 ; and since (2®)^ = 32^ = 1024, then logj 1024 = 2-5 = 2 logg 32. That is, logg 32^ = 21og2 32. 43. Logarithm of a Root. The logarithm of a root of a number is equal to the logarithm of the number divided by the index of the root. For if A = 10"^, 1 X taking the rth root, ^.’•=10''. _ log^ r The propositions of §§ 42 and 43 are true whatever base is taken, as may easily be seen by using the base h. From §§42 and 43 we see that the raising of a number to any power, integral or fractional, reduces to the operation of multiplying the logarithm by the ex- ponent (integral or fractional) and then finding the number of which the result is the logarithm. Therefore the operations of multiplying, dividing, raising to powers, and extracting roots will be greatly simplified if we can find the logarithms of num- bers, and this will next be considered. 44. Characteristic and Mantissa. Usually a logarithm consists of an integer plus a decimal fraction. The integral part of a logarithm is called the characteristic. The decimal part of a logarit hm is called the mantissa. Thus, if log 2353 = 3.37162, the characteristic is 3 and the mantissa 0.37162. This means that l03-8n62 = 2353, or that the 100,000th root of the 337,162d power of 10 is 2353, approximately. It must always be recognized that the mantissa is only an approximation, correct to as many decimal places as are given in the table, but not exact. Computations made with logarithms give results which, in general, are correct only to a certain number of figures, but results which are sufficiently near the correct result to answer the purposes of the problem. 44 PLANE TRIGONOMETRY 45. Finding the Characteristic. Since we know that 10=> = 1000 and 10^ == 10,000, therefore 3 = log 1000 and 4 = log 10,000. Hence the logarithm of a number between 1000 and 10,000 lies between 3 and 4, and so is 3 plus a fraction. Thus the characteristic of the logarithm of a number between 1000 and 10,000 is 3. Likewise, since 10-3 = 0.001 and lO-^^O.Ol, therefore — 3 = log 0.001 and — 2 = log 0.01. Hence the logarithm of a number between 0.001 and 0.01 lies between — 3 and — 2, and so is — 3 plus a fraction. Thus the char- acteristic of the logarithm of a number between 0.01 and 0.001 is — 3. Of course, instead of saying that log 1475 is 3 -|- a fraction, we might say that it is 4 — a fraction ; and instead of saying that log 0.007 is — 3 -t- a fraction, we might say that it is — 2 — a fraction. For convenience, however, the man- tissa of a logarithm is always taken as positive, but the characteristic may be either positive or negative. 46. Laws of the Characteristic. From the reasoning set forth in § 45 we deduce the following laws : 1. The characteristic of the logarithm of a number greater than 1 is positive and is one less than the number of integral places in the number. For example, log 75 = 1 + some mantissa, log 472.8 = 2 -1- some mantissa, and log 14,800.75 = 4-1- some mantissa. 2. The characteristic of the logarithm of a number between 0 and 1 is negative and is one greater than the number of zeros between the decimal point and the first significant figure in the number. For example, log 0.02 = — 2 -1- some mantissa, and log 0.00076 = — 4 -f some mantissa. The logarithm of a negative number is an imaginary number, and hence such logarithms are not used in computation. 47. Negative Characteristic. If log 0.02 = — 2 -t- 0.30103, we cannot write it — 2.30103, because this would mean that both mantissa and characteristic are negative. Hence the form 2.30103 has been chosen, which means that only the characteristic 2 is negative. That is, 2.30103 =-2 -f 0.30103, and 5.48561 = - 5 -1- 0.48561. We may also write 2.30103 as 0.30103 — 2, or 8.30103 — 10, or in any similar manner which will show that the characteristic is negative. LOGAKITHMS 45 48. Mantissa independent of Decimal Point. It may be shown that 103-37107 ^ 2350 ; whence log 2350 = 3.37107. Dividing 2350 by 10, we have 103.87107-1 ^ 102-3713T = 235 ; whence log 235 = 2.37107. Dividing 2350 by 10^, or 10,000, we have 103.37107 - 4^ 101.37107 ^ 0.235 ; whence log 0.235 = T.37107. That is, the mantissas are the same for log 2350, log 235, log 0.235, and so on, wherever the decimal points are placed. The mantissa of the logarithm of a number is unchanged by any change in the position of the decimal point of the number. This is a fact of great importance, for if the table gives us the mantissa of log 235, we know that we may use the same mantissa for log 0.00235, log 2.35, log 23,500, log 235,000,000, and so on. Exercise 19. Logarithms Write the characteristics of the logarithms of the following : 1. 75. 6. 2578. 11. 0.8. 16. 0.0007. 2. 75.4. 7. 257.8. 12. 0.08. 17. 0.0077. 3. 754. 8. 25.78. 13. 0.88. 18. 0.00007, 4. 7.54. 9. 2.578. 14. 0.885. 19. 0.10007, 5, 7540. 10. 25,780. 15. 0.005. 20. 0.07007. Given 3.68681 as the logarithm of 3862, find the following : 21. log 38.62. 24. log 38,620. 27. log 0.3862. 22. log 3.862. 25. log 386,200. 28. log 0.03862. 23. log 386.2. 26. log 38,620,000. \/29. log 0.0003862. Given 1.67724 as the logarithm of 0.4756, find the following . 30. log 4756. 32. log 47,560. 34. log 0.04756. 31. log 4.756. 33. log 47,560,000. \y^5. log 0.00004756, Given 3.40603 as the logarithm of 2547, find the following : 36. log 2.547. 38. log 0.2547. 40. log 25,470. 37. log 25.47. 39. log 0.002547. ’'/41. log 25,470,000. Given 1.39794 as the logarithm of 25, find the following : 42. log 2 ^. 44. log 0.25. 46. log 25,000. 43. log:|^. 45. log 0.025. /47. iQg 25,000,000, 46 PLANE TKIGONOMETEY 49. Using the Table. The following is a portion of a page taken from the Wentworth-Smith Logarithmic and Trigonometric Tables : 250 — 300 N 0 1 2 3 4 5 6 7 8 9 250 39 794 39 811 39 829 39 846 39 863 39 881 39 898 39 915 39 933 39 950 251 39 967 39 985 40 002 40 019 40 037 40 054 40 071 40 088 40 106 40 123 252 40 140 40 157 40 175 40 192 40 209 40 226 40 243 40 261 40 278 40 295 253 40 312 40 329 40 346 40 364 40 381 40 398 40 415 40 432 40 449 40 466 254 40 483 40 500 40 518 40 535 40 552 40 569 40 586 40 603 40 620 40 637 255 40 654 40 671 40 688 40 705 40 722 40 739 40 756 40 773 40 790 40 807 Only the mantissas are given ; the characteristics are always to be determined by the laws stated in § 46. Always write the characteristic at once, before writing the mantissa. For example, looking to the right of 251 and under 0, and writing the proper characteristics, we have log251 = 2.39967, log25.1 = 1.39967, log 2510 = 3.39967, log 0.0251 = 2.39967. The first three significant figures of each number are given under N, and the fourth figure under the columns headed 0, 1, 2, . . . , 9. For example, log 252.1 = 2.40157, log 0.2547 = 1.40603, log 25.25 = 1.40226, log 2549 = 3.40637. Furthermore, log 251.1 = 2.39985 — , the minus sign being placed beneath the final 5 in the table to show that if only a four-place mantissa is being used it should be written 3998 instead of 3999. The logarithms of numbers of more than four figures are found by interpolation, as explained in § 27. For example, to find log 25,314 we have log 25,320 = 4.40346 log 25,310 = 4.40329 Tabular difierence = 0.00017 .f 0.000068 Difference to be added = 0.00007 Adding this to 4.40329, log 25314 = 4.40336 In general, the tabular difierence can be found so easily by inspection that it is unnecessary to multiply, as shown in this example. If any multiplication is necessary, it is an easy matter to turn to pages 46 and 47 of the tables, where will be found a table of proportional parts. On page 46, after the number 17 in the column of difierences (D), and under 4 (for 0.4), is found 6.8. In the same way we can find any decimal part of a difierence. LOGAEITHMS 47 Exercise 20. Using the Table Using the table, find the logarithms of the following : 1. 2. 9. 3485. 17. 0.7. 25. 12,340. 2. 20. 10. 4462. 18. 0.75. 26. 12,345. 3. 200. 11. 5581. 19. 0.756. 27. 12,347. 4. 0.002. 12. 7007. 20. 0.7567. 28. 123.47. 5. 2100. 13. 5285. 21. 0.0255. 29. 234.62. 6. 2150. 14. 68.48. 22. 0.0036. 30. 41.327. 7. 2156. 15. 7.926. 23. 0.0009. 31. 56.283. 8. 2.156. 16. 834.8. 24. 0.0178. 32. 0.41282. 33. In a certain computation it is necessary to find the sum of the logarithms of 45.6, 72.8, and 98.4. What is this sum ? 34. In a certain computation it is necessary to subtract the loga- rithm of 3.84 from the sum of the logarithms of 52.8 and 26.5. What is the resulting logarithm ? Perform the following operations : 35. log 275 + log 321 + log 4.26 + log 3.87 + log 46.4. 36. log 2643 + log 3462 -|- log 4926 + log 5376 -f log 2194. 37. log 51.82 + log 7.263 + log 5.826 log 218.7 + log 3275. __ 38. log 8263 + log 2179 + log 3972 — log 2163 — log 178. 39. log 37.42 + log 61.73 -f- log 5.823 — log 1.46 — log 27.83. 40. log 3.427 + log 38.46 + log 723.8 — log 2.73 — log 21.68. 41. In a certain operation it is necessary to find three times log 41.75. What is the resulting logarithm? 42. In a certain operation it is necessary to find one fifth of log 254.8. What is the resulting logarithm ? Perform the following operations : 43. 2 X log 3. 44. 3 X log 2. 45. 3 X log 25.6. 46. 5 X log 3.76. 47. 4 X log 21.42. 48. 5 X log 346.8. 49. 12 X log 42.86. 50. ^log2. 51. ^ log 2000. 52. -J- log 3460. 53. ^ log 24.76. 54. log 368.7. 55. flog 41.73. 66. flog 763.8. ^57. 0.3 log 431. 58. 0.7 log 43.19. 59. 0.9 log 4.007. 60. 1.4 log 5.108. 61. 2.3 log 7.411. 62. I log 16.05. 63. flog 23.43. 48 PLANE TRIGONOMETRY 50. Antilogarithm. The number corresponding to a given logarithm is called an antilogarithm. For " antilogarithm of N ” it is customary to -write " antilog N.” Thus if log 25.31 = 1.40329, antilog 1.40329 = 25.31. Similarly, -we see that antilog 5.40329 = 253,100, and antilog 2.40329 = 0.02531. 51. Finding the Antilogarithm. An antilogarithm is found from the tables by looking for the number corresponding to the given mantissa and placing the decimal point according to the character- istic. For example, consider the following portion of a table : 550 — 600 N 0 1 2 3 4 6 6 7 8 9 660 551 74 036 74 115 74 044 74123 74 052 74 060 74 068 74 131 74 139 74 147 74 076 74 155 74 084 74 092 74 099 74 107 74 162 74 170 74 178 74 186 If the mantissa is given in the table, we find the sequence of the digits of the antilogarithm in the column under N. If the mantissa is not given in the table, we interpolate. 1. Find the antilogarithm of 5.74139. We find 74139 in the table, opposite 551 and under 3. Hence the digits of the number are 5513. Since the characteristic is 5, there are six integral places, and hence the antilogarithm is 551,300. That is, log 551,300 = 5.74139, or antilog 5.74139 = 551,300. 2. Find the antilogarithm of 2.74166. We find 74170 in the table, opposite 551 and under 7. log 0.05517 = 2.74170 logO.05516 = 2.74162 . Tabular difference = 0.00008 Subtracting, -we see that, neglecting the decimal point, the tabular difference is 8, and the difference between log x and log 0.05516 is 4. Hence x is | of the way from 0.05516 to 0.05517. Hence x = 0.055165. 3. Find the antilogarithm of 7.74053. We find 74060 in the table, opposite 550 and under 3. log 55,030,000 = 7.74060 log 55,020,000 = 7.74052 Tabular difference = 0.00008 Seasoning as before, x is ^ of the way from 55,020,000 to 55,030,000. Hence, to five significant figures, x = 55,021,000. In general, the interpolation gives only one additional figure correct ; that is, with a table like the one above, the sixth figure will not be correct if found by interpolation. LOGARITHMS 4 ^ Exercise 21. Antilogarithms Find the antilogarithms of the following : 1. 0.47712. 9. 3.74076. 17. 0.23305. 25. 8.77425. 2. 3.47712. 10. 2.76305. 18. 1.43144. 26. 4.82966. /3. 3.47712. 11. 4.78497. 19. 2.56838. 27. 3.83547. 4. 2.48359. 12. T. 81954. 20. 1.58041. 28. 2.83604. 5. 4.56844. 13. 0.82575. 21. 3.63490. 29. 4.88960. 6. 1.66276. 14. 0.88081. 22. 4.63492. 30. 2.89523. 7. 2.66978. 15. 9.89237. 23. 0.63994. 31. 3.89858. 8. 5.74819. 16. 7.90282. 24. 2.69085. /32. 0.93223. ^ 33. If the logarithm of the product of two numbers is 2.94210, what is the product of the numbers ? ^ 34. If the logarithm of the quotient of two numbers is 0.30103, what is the quotient of the numbers ? 35. If we wish to multiply 2857 by 2875, what logarithms do we need ? What are these logarithms ? 36. If we know that the logarithm of a result which we are seek- ing is 3.47056, what is that result? ^ 37. If we know that log V0.000043641 is 3.81995, what is the value of Vo.000043641 ? 38. If we know that log '^0.076553 is 1.81400, what is the value of -^0.076553 ? 39. The logarithm of V8322 is 1.96012. Find V8322 to three decimal places. 40. The logarithm of the cube of 376 is 7.72557. Find the cube of 376 to five significant figures. 41. If we know that log 0.003278^ is 5.03122, what is the value ^ of 0.003278" ? 42. Find twice log 731, and find the antilogarithm of the result. 43. Find the antilogarithm of the sum of log 27.8 + log 34.6 + log 367.8. Find the antilogarithms of the following : 44. log 7 -j- log 2 — log 1.934. 47. 5 log 27.83. 45. log 63 -j- log 5.8 — log 3.415. 48. 2.8 log 5.683. 46. log 728 -1- log 96.8 — log 2.768. \) 49. f (log 2 -|- log 4.2). 60 PLANE TRIGONOMETEY 52. Multiplication by Logarithms. It has been shown (§ 40) that the logarithm of a product is equal to the sum of the logarithms of the numbers. This is of practical value in multiplication. Find the product of 6.15 x 27.05. From the tables, log 6.15 = 0.78888 log 27.06 = 1.43217 log a; =2.22105 Interpolating to find the value of x, we have log 166.4 = 2.22116 logx =2.22105 log 166.3 = 2.22089 log 166.3 = 2.22089 26 16 Annexing to 166.3 the fraction , we have X = 166.3|| = 166.36, the interpolation not being exact beyond one figure. If we perform the actual multiplication, we have 6.15 x 27.05 = 166.3676, or 166.36 to two decimal places. Exercise 22. Multiplication by Logarithms Using logarithms, find the following products : 1. 2 X 5. 11. 2 X 50. 21. 35.8 X 28.9. 2. 4 X 6. 12. 40 X 60. 22. 52.7 X 41.6. 3. 3 X 5. 13. 3 X 500. 23. 2.75 X 4.84. 4. 5 X 7. 14. 50 X 70. 24. 5.25 X 3.86. 6. 2 X 4. 15. 2 X 4000. 25. 14.26 X 42.35. 6. 3 X 7. 16. 30 X 700. 26. 43.28 X 29.64. 7. 2 X 6. 17. 200 X 60. 27. 529.6 X 348.7. 8. 3 X 6. 18. 30 X 600. 28. 240.8 X 46.09. 9. 7 X 8. 19. 7 >< c 80,000. 29. 34.81 X 46.25. 10. 2 X 9. 20. 200 X 900. 30. 5028 X 3.472. 31. Taking the circumference of a circle to be 3.14 times the diameter, find the circumference of a steel shaft of diameter 5.8 in. 32. Taking the ratio of the circumference to the diameter as given in Ex. 31, find the circumference of a water tank of diameter 36 ft. Using logarithms, find the following products : 33. 2 X 3 X 5 X 7. 36. 43.8 X 26.9 x 32.8. 34. 3 X 5 X 7 X 9. 37. 527.6 x 283.4 x 4.196. 36. 5 X 7 X 11 X 13. 38. 7.283 X 6.987 x 5.437. LOGAEITHMS 51 53. Negative Characteristic. Since the mantissa is always positive (§ 45), care has to be taken in adding or subtracting logarithms in which a negative characteristic may occur. In all such cases it is better to separate the characteristics from the mantissas, as shown in the following illustrations : 1. Add the logarithms 2.81764 and 1.41283. Separating the negative characteristic from its mantissa, we have 2.81764 = 0.81764 - 2 1.41283 = 1.41283 Adding, we have 2.23047 — 2 = 0.23047 2. Add the logarithms 4.21255 and 2.96245. Separating both negative characteristics from the mantissas, we have 4.21255 = 0.21255 - 4 2.96245 = 0.96245 - 2 Adding, we have 1. 17500 ^ 6 = 5.17500 Exercise 23. Negative Characteristics Add the following logarithms : 1. 2.41283 + 5.27681. 2. 2.41283 + 5.27681. 3. 2.41283 + 5.27681. 4. 0.38264 + 4.71233. 5. 0.57121 + 1.42879. 6. 2.63841 + 1.36158. 7. 2.41238 + 3.62701. 8. 5.58623 + 6.41387. 9. 6.41382 + 7.58617. flO. 4.22334 + 3.77666. Using logarithms, find the following products : 11. 256 X 4875. 12. 2.56 X 48.75. 13. 0.256 X 0.4875. 14. 0.0256 X 0.004875. 15. 0.1275 X 0.03428. 16. 0.2763 X 0.4134. 17. 0.00025 X 0.00125. 25. 26. 27. V 18. 0.725 X 0.3465. 19. 0.256 X 0.0875. 20. 0.037 X 0.00425. 21. 47.26 X 0.02755. 22. 296.8 X 0.1283 23. 45,650 X 0.0725. ^ 24. 127,400 X 0.00355. Given sin 25.75° = 0.4344, find 52.8 sin 25.75°. Given cos 37.25° = 0.7960, find 42.85 cos 37.25°. Given tan 30° 50' 30" = 0.5971, find 27.65 tan 30° 50' 30". 52 PLANE TPIGONOMETEY 54. Division by Logarithms. It has been shown (§ 41) logarithm of a quotient is equal to the logarithm of the minus the logarithm of the divisor. Care must be taken that the mantissa in subtraction become negative (§ 45). 1. Using logarithms, divide 17.28 by 1.44. Erom the tables, logl7. 28 = 1.23754 log 1.44 = 0.15836 1.07918 = log 12 Hence 17.28 1.44 = 12. 2. Using logarithms, divide 2603.5 by 0.015998. log 2603.5 = 3.41556 log 0.015998 = 2.20407 Arranging these in a form more convenient for subtracting, we have log 2603.5 = 3.415-56 log 0.015998 = 0.20407 - 2 3.21149 + 2 = 5.21149 = log 162,740 Hence 2603.6 -- 0.016998 = 162,740. 3. Using logarithms, divide 0.016502 by 127.41. log 0.016502 = 2.21753 = 8.21753 - 10 log 127.41 = 2.10520 = 2.10520 6.11233- 10 = 4.11233 = log 0.00012952 Hence 0.016502 --- 127.41 = 0.00012952. Here we increased 2.21753 by 10 and decreased the sum by 10. We might take any other number that would make the highest order of the minuend larger than the corresponding order of the subtrahend, but it is a convenient custom to take 10 or the smallest multiple of 10 that will serve the purpose. 4. Using logarithms, divide 0.000148 by 0.022922. log 0.000148 = 1.17026 = 16.17026 - 20 log 0.022922 = 2.36025 = 8.36025 - 10 7.81001 - 10 = 3.81001 = log 0.0064567 Hence 0.000148 h- 0.022922 = 0.0064567. 5. Using logarithms, divide 0.2548 by 0.05513. log 0.2548 = 1.40620 = 9. 40620 - 10 log 0.05613 = 174139 = 8.74139- 10 0.66481 = log 4.6218 Hence 0.2548 -f- 0.05513 = 4.6218. that the dividend does not LOGARITHMS 53 Exercise 24. Division by Logarithms Add the following logarithms : 1. 2.14755 + 3.82764. 2. 4.07256 + 1.58822. 3. 0.21783 + 1.46835. 4. 0.41722 + 3.28682. 5. 4.18755 + 2.81245. 6. 6.28742 + 3.41258. 7. 4.21722 + 4.78278. 8. 5.28720 + 3.71280. 9. rind the sum of 2.41280, 4.17623, 5.26453, 0.21020, 7.36423, 2.63577, 6.41323, and 3.28740. From the first of these logarithms subtract the second : 10. 0.21250, 2.21250. 11. 0.17286,3.27286. 12. 2.34222, A44222. 13. 3.14725,1.25625. 14. 4.17325, 2.17325. 15. 5.82340, 3.71120. 16. 3.14286,1.14000. ^ 17. 3.27283, 5.56111. Using logarithms. 18. 10 2. 19. 15^3. 20. 15^5. 21 . 12 - 3 . 22. 12 ^ 4. 23. 60 ^ 12. 24. 75 --25. 25. 125 ^25. divide as folloivs : 26. 25,284-- 301. 27. 51,742^631. 28. 47,348 ^ 623. 29. 19,224 H- 540. 30. 37,960 ^520. 31. 84,640 H- 920. 32. 65,100 ^ 620. 33. 45,990 H- 730. 34. 59.29 H- 0.77. 35. 2.451 -- 190. 36. 851.4 H- 0.66. 37. 0.98902 -- 99. 38. 0.41831 -- 5.9. 39. 0.08772 ^ 4.3. 40. 0.02275 -j- 0.35. Al. 0.02736 -- 0.057 Using logarithms, divide to four significant figures : 42. 15-^7. 45. 26.4-- 13.8. 48. 17.625 ^-3.4. 43. 7-J-15. 46. 4.21 -^- 3.75. 49. 43.826 ^0.72. 44. 0.7 -T- 150. 47. 63.25 ^4.92. 50. 5.483^8.4. Taking log 3.1416 as 0.49715 and interpolating for six figures on the same principle as for five, find the diameters of circles with circumferences as follows : 51. 62.832. 53. 2199.12. 55. 28,274.2. 57. 376,992. 52. 157.08. 54. 2513.28. 56. 34,557.6. 58. 0.031416. 59. By using logarithms find the product of 41.74 x 20.87, and the quotient of 41.74 -- 20.87. 54 PLANE TRIGONOMETRY 55. Cologarithm. The logarithm of the reciprocal of a number is called the cologarithm of the number. For "cologarithm of N” it is customary to write "colog A.” By definition colog x = log - = log 1 — logo; (§ 41). But log 1 = 0. Hence we have colog x—— log x. To avoid a negative mantissa (§ 45) it is customary to consider that colog a: = 10 — log x — 10, since 10 — log a: — 10 is the same as — log a:. For example, colog 2 = — log 2 = 10 — log 2 — 10 = 10 - 0.30103 - 10 = 9.69897 - 10 = 1.69897. 56. Use of the Cologarithm. Since to divide by a number is the same as to multiply by its reciprocal, instead of subtracting the logarithm of a divisor we may add its cologarithm. The cologarithm of a number is easily written by looking at the logarithm in the table. Thus, since log 20 = 1.30103, we find colog 20 by subtracting this from 10.00000 — 10. To do this we begin at the left and subtract the number represented by each figure from 9, except the right-hand significant figure, which we subtract from 10. In full form we have 10.00000 - 10 = 9. 9 9 9 9 10 - 10 log 20 = 1.30103 = 1. 3 0 1 0 3 colog 20 = 8. 6 9 8 9 7 - 10 = 2.69897 Similarly, we may find colog 0.03952 thus : 10.00000 - 10 = 9. 9 9 9 9 10 - 10 log 0.03952 = 159682 = 8. 5968 2 - 10 cologO.03952 = 1. 4 0 3 1 8 = 1.40318 Practically, of course, we would find log 0.03952 and subtract mentally. Exercise 25. Cologarithms Write the cologarithms of the following numbers: 1. 25. 6. 3751. 9. 0.5. 13 . 3.007. 2. 130. 6. 427.3. 10. 0.72. 14 . 62.09. 3. 27.4. 7. 51.61. 11. 0.083. 15 . 0.0006. 4. 5.83. 8. 7.213. 12. 0.00726. 16. O.OOOOT 17 . What number has 0 for its cologarithm ? 18 . What number has 1 for its cologarithm ? 19 . What number has oo for its cologarithm ? 20. Find the number whose cologarithm equals its logarithm. LOGARITHMS 55 57. Advantages of the Cologarithm. If, as is not infrequently the case in the computations of trigonometry and physics, we have the product of two or more numbers to be divided by the product of two or more different numbers, the cologarithm is of great advantage. Using logarithms and cologarithms, simplify the expression 17.28 X 6.25 X 16.9 1.44 X 0.25 X 1.3 This is so chosen that we can easily verify the answer by cancellation. By logarithms we have, log 17.28 = 1.23754 log 6.25 = 0.79588 log 16.9 = 1.22789 colog 1.44 = 9.84164 — 10 colog 0.25 = 0.60206 colog 1.3 = 9.88606 - 10 3.59107 = log 3900.1 In a long computation the fifth figure may be in error. Exercise 26. Use of Cologarithms Using cologarithms, find the value of the following to five figures •. 3x2 10. 172.8 X 1.44 19. 435 X 0.2751 4 xl.5 0.288 X 0.864 2.83 X 1.045 8x9 11. 57.5 X 0.64 20. 50.05 X 2.742 3x4 1.25 X 320 381.4 X 2.461 6 xl2 12. 1.28 X 13.41 21. 50730 X 2.875 3x8 1.49 X 6.4 34.48 X 1.462 4 X 24 13. 5.48 X 0.198 22. 3.427 X 0.7832 12 xl6 3.96 X 27.4 3.1416 X 0.0081 12 xl5 14. 1.176 X 10.22 23. 27.98 X 32.05 9 X 20 14.6 X 3.92 0.48 X 0.00062 12 X 28 15. 3 X 11 X 17 24. 2.1 X 0.3 X 0.11 8 X 21 7 xl3 17 X 0.05 3 X 22 16. 16 X 23 V 26. 1.1 X 3.003 18 X 33 3 X 7 X 41 0.2 X 0.07112 8 . 11x13 17x19’ 23 X 39 X 47 17 X 33 X 53* . 0.0347 X 0.117 • 3 X 11 X 170 15 xl7 0.2 X 0.3 11x13' 0.11 xl7i' 528.4 X 1.001 7.03 X 0.7281' 9. 27. 56 PLA^E TRIGONOMETRY 58. Raising to a Power. It has been shown (§ 42) that the logarithm of a power of a number is equal to the logarithm of the number multiplied by the exponent. 1. Find by logarithms the value of 11®. From the tables, log 11 =1.04139 Multiplying by 3, 3 log 118 ^ 3.12417 = log 1331.0 That is, 11® = 1331.0, to five figures. Of course we see that 11® = 1331 exactly, log 1331 being 3.12418. The last figure in log 11® as found in the above multh plication is therefore not exact, as is frequently the case in such computations. As usual, care must be taken when a negative characteristic appears. 2. Find by logarithms the value of 0.2413®. From the tables, log 0.2413 = 0.38256 — 1 Multiplying by 6, 6 log 0.2413® = 1.91280-6 - 4.91280 = log 0.00081808 Hence 0.2413® = 0.00081808, to five significant figures. As on page 18, we use the expression "significant figures” to indicate the figures after the zeros at the left, even though some of these figures are zero. Exercise 27. Raising to Powers By logarithms, find the value of each of the following to five significant figures: 1. 2®. 9. 1 “. 17. 25®. 25. 1.1®. 33. 12.55®. 2. 2®. 10. 7®. 18. 25h 26. 2.V. 34. 34.75®. 3. 2®. 11. 9’. 19. 125®. 27. 0 . 11 ®. 35. 1.275®. 4. 2“ 12. 8®. 20. 625®. 28. 0 . 211 . 36. 0.1254®. 6 . 3®. 13. IV. 21. 1750®. 29. 0.7®. 37. 0.4725®. 6. 3®. 14. 15®. 22. 2775®. 30. o o -0 38. 0.01234®. 7. 4®. 15. 1.5®. 23. 3146®. 31. 0.37h ^ 39. 0.00275®. 8. 5®. 16. 17h 24. 41351 32. 5.37®. V40. 0.000355®. 41. If log 7T = 0.49716, what is the value of tt® ? of tt® ? 42. U sing log TT as in Ex. 41, what is the value of wr when r = 7 ? of XT® when ?* = 7 ? of f Trr® when ?• = 9 ? LOGARITHMS 57 59. Fractional Exponent. It has been shown (§ 43) that the log- arithm of a root of a number is equal to the logarithm of the number divided by the index of the root. This law may, however, be com- bined with that of § 58, since means Va, and means The law of § 58 therefore applies to roots or to powers of roots, the exponent simply being considered fractional. 1. Find by logarithms the value of Vl, or 4^. From the tables, log 4 = 0.60206 Dividing by 2, 2 )0.60206 log Vi, or log 4^, = 0.30103 = log2 Hence Vi, or 4^, is 2. 2. Find by logarithms the value of 8^. From the tables, log 8 = 0.90309 Multiplying by logSt = 0.60206 = log 4 Therefore 8^ = 4. 3. Find by logarithms the value of 0.127^. From the tables, log 0.127 = 0.10380 — 1. Since we cannot divide — 1 by 6 and get an integral quotient for the new characteristic, we add 4 and subtract 4 and then have log 0.127 = 4.10380-5 Dividing by 5, log 0. 127^ = 0.82076 — 1 = log 0.66185 Hence 0.127^, or Vo.l27, is 0.66185. We might have written log 0.127 = 9.10380 — 10, 14.10380 — 15, and so on. Exercise 28. Extracting Roots By logarithms, find the value of each of the follovnng : 1. V2. 5. 2L 9. Vil. 13. 0.3^ iai. 127.8i 2. ^5. 6. 3L 10. 73. 14. 0.05L 18. 2.475i 3. 77. 7. 8i 11. 7^. 15. 0.0175^. 19. 5.135i 4. 7^. 8. 7L 12. TiM. 16. 0.0325^ V20. 0.00125f 21. If log 7T = 0.49715, what is the value of Vtt ? of ? 22. Using the value of log tt given in Ex. 21, what is the value of 7 T^ ? of 7T^ ? of ? of 7T“i ? of 7r“^ ? of ? 58 PLANE TRIGONOMETRY 60. Exponential Equation. An equation in which the unknown quantity appears in an exponent is called an exponential equation. Exponential equations may often be solved by the aid of loga- rithms. 1. Given 5"^ = 625, find by logarithms the value of x. Taking the logarithms of both sides, we have (§ 42) X log 5 =: log 625 _ log 625 Whence log 5 Check. 5* = 625. 2.79588 0.69897 In all such cases bear in mind that one logarithm must actually be divided by the other. If we wished to perform this division by means of logarithms, we should have to take the logarithm of 2.79588 and the logarithm of 0.69897, subtract the second logarithm from the first, and then find the antilogarithm. We may apply this principle to certain simultaneous equations. 2. Solve this pair of simultaneous equations 2* • 3*' = 72 (1) 4* . 27>' = 46,656 (2) Taking the logarithms of both sides, we have (§§ 40, 42) X log 2 -f- 2/ logs = log 72, (3) and X log 4 + y log 27 = log 46,656. (4) Then, since log 4 = log2^ = 2 log 2, and log 27 = log 3* = 3 log 3, we have 2 x log 2 4- 3 y log 3 = log 46,656. (5) Eliminating x by multiplying equation (3) by 2 and subtracting from equa- tion (5), we have _ log 46656 — 2 log 72 i^^3 _ 4.66890- 2 X 1.85733 “ 0.47712 _ 0.95424 _ ^ “ 0.47712 “ We may substitute this value of y in (1), divide by 3*, and then find x by taking the logarithms of both sides. It will be found that x = 3. We may check by substituting in (2). In the same way, equations involving three or more unknown quantities may be solved. Although the exponential equation is valuable in algebra, as in the solution of Exs. 22, 23, 25, and 26 of Exercise 29. we rarely have need of it in trigonometry. LOGARITHMS 69 Exercise 29. Exponential Equations By logarithms, solve the following exponential equations: V 1. 2^ = 8. 2. 3* = 81. 3. 5" = 625. 4. 4" - 256. 6. 1H = 1331. 6. 2"^ =19. 7. 3"^ ==75. 8 . 5 " = 1000 . 9. 4" = 2560. 10. 11^=1500. '■Al. 2-^ = f 12 . 2 -^ = 0 . 1 . 13. 0.3- * = 0.9. 14. 2*+^ = 3^-\ v'lS. 9"^+® = 53,143. Solve the following simultaneous equations : ^ 16. + = a^ a^~'> — c? 17. ^ mS- 18. 3"^ . 4*' =12 5" . 7*' = 35 19. 2* • 3*' = 36 4^ . 5^ = 400 20. 2"^ . 5" = 200 3^ . 3^ = 243 21. 2^ . 8*' = 256 8^ . 32*' = 65,536 Solve the following equations by logarithms : - 22. a=p(l+rf. ^25. a—p(l-\-rty. V 23. l=ar^~^. 26. s(r —I) — ar^ — a. 24. 2^+^^ — 8. ^' 27 . 3 =^- "'+1 = 27. Perform the following operations by logarithms : „ 2.47 X 84.96 34.8 X 96.55' / 5.75 X 3.428 V59.62 X 48.08/ ’ 29. i 42.4 X 0.075 3.64 X 0.009' J 31 ■ NV3. 07 X 0.00964Y 426 X 0.875 / 32. To what power must 7 be raised to equal 117,649 ? 33. To wbat power must a be raised to equal h ? 34. To wbat power must 5 be raised to equal n ? 35. Find tbe value of x when "v^ = 3 ; when ^f2 = 1.1 ; when ^ = 1.414 ; when = 1.73. 36. Find tbe value of x when "V^ = 3 ; when "Va = h ; when Va = a j when •^1331 = 11 ; when ■v'^20736 = 12. \5 37. Solve tbe equations Vi/ = a i+i/- , Sy = h 1 I 38. Wbat value of x satisfies tbe equation = Va? 60 PLANE TPIGONOMETPY 61. Logarithms of the Functions. Since computations involving trigonometric functions are often laborious, they are generally per- formed by the aid of logarithms. Eor this reason tables have been prepared giving the logarithms of the sine, cosine, tangent, and cotangent of the various angles from 0° to 90° at intervals of 1'. The functions of angles greater than 90° are defined and discussed later in this work when the need for them arises. Logarithms of the secant and cosecant are usually not given for the reason that the secant is the reciprocal of the cosine, and the cosecant is the reciprocal of the sine. Instead of multiplying by secx, for example, we may divide by cos X ; and when we are using logarithms one operation is as simple as the other, since multiplication requires the addition of a logarithm and division requires the addition of a cologarithm. In order to avoid negative characteristics the characteristic of every logarithm of a trigonometric function is printed 10 too large, and hence 10 must be subtracted from it. Practically this gives rise to no confusion, for we can always tell by a result if a logarithm is 10 too large, since it would give an antilogarithm with 10 integral places more than it should have. For example, if we are measuring the length of a lake in miles, and find 10.30103 as the logarithm of the result, we see that the characteristic must be much too large, since this would make the lake 20,000,000,000 mi. long. It would be possible to print 2.97496 for log sin 5° 25', instead of 8.97496, which is 10 too large. It would be more troublesome, however, for the eye to detect the negative sign than it would be to think of the characteristic as 10 too large. On pages 56-77 of the tables the characteristic remains the same throughout each column, and is therefore printed only at the top and bottom, except in the case of pages 58 and 77. Here the characteristic changes one unit at the places marked with the bars. By a little practice, such as is afforded on pages 61 and 62 of the text, the use of the tables will become clear. On account of the rapid change of the sine and tangent for very small angles log sin x is given for every second from 0" to 3' on page 49 of the tables, and log tan x has identically the same values to five decimal places. The same table, read uj)wards, gives the log cos X for every second from 89° 57' to 90°. Also log sin cr, log tan X, and log cos x are given, on pages 50-55 of the tables, for every 10" from 0" to 2°. Reading from the foot of the page, the cofunctions of the complementary angles are given. On pages 56-77 of the tables, log sinx, log cos a-, log tanrr, and log cot X are given for every minute from 1° to 89°. Interpolation in the usual manner (page 31) gives the logarithmic functions for every second from 1° to 89°. LOGAEITHMS 61 62. Use of the Tables. The tables are used in much the same way as the tables of natural functions. For example, log sin 5° 25' = 8.97496 — 10 Page 58 logtan40°55' = 9.93789-10 Page 75 log cos 52° 20' = 9.78609 — 10 Page 74 log cot 88° 59' = 8.24910 - 10 Page 56 logsin 0° 28' 40" = 7.92110 - 10 Page 51 logsin 0° 1' 52" = 6.73479 — 10 Page 49 Furthermore, if log cot x = 9.55910 — 10, then x = 70° 5'. Page 65 Interpolation is performed in the usual manner, whether the angles are expressed in the sexagesimal system or decimally. 1. Find log sin 19° 50' 30". From the tables, logsin 19° 50' = 9.53056 —10, and the tabular difference is 36. We must therefore add -|ff of 36 to the mantissa, in the proper place. We therefore add 0.00018, and have logsin 19° 50' 30" = 9.53074 — 10. 2. Find log tan 39.75°. From the tables, log tan 39.7° = 9.91919 — 10, and the tabular difference is 154. We therefore add 0.5 of 154 to the mantissa, in the proper place. Adding 0.00077, we have log tan 39.75° = 9.91996 — 10. Special directions in the case of very small angles are given on page 49 of the tables. It should be understood, however, that we rarely use angles involving seconds except in astronomy. If the function is decreasing, care must be taken to subtract instead of add in making an interpolation. 3. Find log cos 43° 45' 15". From the tables, log cos 43° 45' = 9.85876 — 10, and the tabular difference is 12. Taking ^ of 12, or l of 12, we have 0.00003 to be subtracted. Therefore log cos 4.3° 45' 15" = 9.85873 — 10. 4. Given log cotx = 0.19268, find x. From the tables, log cot 32° 41' = 10.19275 — 10 = 0.19275. The tabular difference is 28, and the difference between the logarithm 0.19275 and the given logarithm is 7, in each case hundred-thousandths. Hence there is an angular difference of of 1', or ^ of 1', or 15". Since the angle increases as the cotangent decreases, and 0.19268 is less than 10.19275 — 10, we have to add 15" to 32° 41', whence x = 32° 41' 15". 5. Given log tanx = 0.26629, find x. From the tables, log tan 61° 33' = 10.26614 — 10 = 0.26614. The tabular difference is 30, and the difference between the logarithm 0.26614 and the given logarithm is 15, in each case hundred-thousandths. Hence there is an angular difference of of 1', or 30". Slnce/(x) is increasing in this case, and x is also increasing, we add 30" to 61° S3'. Hence x = 61° 3.3' JO", 62 PLANE TRIGONOMETRY Exercise 30- Use of the Tables Find the value of each of the following : 1. log sin 27°. 16. log cos 42° 45". 31. log sin 0° 1 ' 7 '( f 2. log sin 69°. 17. log tan 26° 15". 32. log sin 1° 2 '5", 3. log cos 36°. 18. log cot 38° 30". 33. log tan 0° 2 '8" 4. log cos 48°. 19. log sin 21° 10' 4". 34. log tan 2° i "7' 1 6. log tan 75°. 20. log sin 68° 49' 56". 36. log cos 89° 50' 10" 6. log tan 12°. 21. log cos 15° 17' 3". 36. log cos 89° 10' 45". 7. log cot 15°. 22. log cos 74° 42' 57". 37. log cot 89° 15' 12" 8 log cot 78°. 23. log tan 17° 2' 10". 38. log cot 89° 25' 15" 9. log sin 9° 15'. 24. log tan 26° 3' 4". 39. log sin 1° 1 '1" 10. log cos 8° 27'. 26. log cot 48° 4' 5". 40. log cos 88° 58' 59". 11. log tan 7° 56'. 26. log cot 4° 10' 7". 41. log tan 2° 2 17' 2 !5". 12. log cot 82° 4'. 27. log sin 34° 30". 42. log cot 87° 32' 45". 13. log sin 4.5°. 28. log sin 27.45°. 43.. log sin 12° 12' 12". 14. log cos 7.25°. 29. log tan 56.35°. 44. log cos 77° 47' 48" 16. log tan 9.75°. \/30. log cos 48.26°. 45. log tan 68° 6' 43". Find the value of x, given the following logarithms, each of which is 10 too large : 46. log sin a: = 9.11570. 69. log sin X — 9.53871. 47. log sin cc = 9.72843. 60. log sin a: — 9.72868. 48. log sin X = 9.93053. 61. log sin X = 9.88150. 49. log sin X = 9.99866. 62. log sin a: = 9.89530. 60. log cos a: = 9.99866. 63. log cos X = 9.90151. 61. log cos a: = 9.93053. 64. log cos a: = 9.80070. 62. log cos a: = 9.71705. 65. log cos X = 9.99483. 63. log cos a: = 9.80320. 66. log tana: = 9.18854. 64. log tana; = 9.90889. 67. log tan X = 10.18750. 66. log tana: = 10.30587. 68. logtanx = 10.06725. 66. log tana: = 10.64011. 69. log cot X = 10.10134. 67. log cot a: = 9.28865. v^O. log cot X = 11.44442. 68. log cot X = 9.56107. 71. log cot X = 7.49849. CHAPTER IV THE RIGHT TRIANGLE 63. Given an Acute Angle and the Hypotenuse. In § 30 the solution of the right triangle was considered when an acute angle and the hypotenuse are given. We may now consider this case and the follow- ing cases with the aid of logarithms. For example, given A = 34° 28', c — 18.75, find B, a, and h. 1. .B = 90° - d = 55° 32'. 2. - = sin A c .-.a — c sin A. 3. - = cosd ; .‘.b = c cos A. c log a = logc + log sin A log c = 1.27300 log sin A = 9.7 527 6—10 loga = 1.02576 .-. a = 10.611 - 10.61 log b = log c -fi log cos A logc = 1.27300 log cos A = 9.91617 — 10 logJ = 1.18917 .-.b = 15.459 = 15.46 Check. 10.612 + 15.452 = 351 . 58 , and 18.752 351.55. This solution may be compared with the one on page 35. In this case there is a gain in using logarithms, since we avoid two multiplications by 18.75. The result is given to four figures (two decimal places) only, the length of c having been given to four figures (two decimal places) only, and this probably being all that is desired. In general, the result cannot be more nearly accurate than data derived from measurement. Consider also the case in which A = 72° 27' 42", c =147.35, to find B, a, and b as above. log a = log c -f log sin A log c = 2.16835 log sin A = 9.97933 ■10 log a = 2.14768 .'. a = 140.50 log b = logc + log cos A log c = 2.16835 log cosA= 9.47906 —10 log b = 1.64741 .-. b = 44.403 Check. What convenient check can be applied in this case ? 63 64 PLANE TRIGONOMETRY 64. Given an Acute Angle and the Opposite Side. Eor example, given A = 62° 10', a = 7S, find B, h, and c. log b — log a + log cotd log a = 1.89209 log eotd = 9.72262 - 10 log b = 1.61471 .-. b = 41.182 = 41.18 log c = log a -f- colog sin A log a = 1.89209 colog sin A = 0.05340 log c = 1.94549 .-. c = 88.204 = 88.20 GUeck. 88.202 _ 41182 _ 6083 + , and 782 = 0O84. This solution should be compared with the one given in § 31, page 3.5. It will be seen that this is much shorter, especially as to that part in which c is found. The difference is still more marked if we remember that only part of the long division is given in § 31. 65. Given an Acute Angle and the Adjacent Side. For example, given A = 50° 2', b — 88, find B, a, and c. log a - log b 4- log tan A log b = 1.94448 log tan A = 10.07670 — 10 loga= 2.02118 .-. a = 105.00 log c = log b 4- colog cos A log 5 = 1.94448 colog cos A = 0.19223 log c = 2.13671 .-.c = 137.00 Check. 1372 - 1062 = 7744 ^ and 882 = 7744 . This solution should he compared with the one given in § 32, page 36. Here again it will be seen that a noticeable gain is made by using logarithms, partic- ularly in finding the value of c THE EIGHT TEIAHGLE 65 66. Given the Hypotenuse and a Side. Eor example, given a — 47.55, We could, of course, find b from the equation 6 = V(c + a) (c — a), as in § 33, page 36. By taking b = a cot^, however, we save the trouble of first find- ing c - 1 - a and c — a. log sin A = log a -I- colog c log a - 1.67715 colog c = 8.23359 —10 log sin^ = 9.91074 —10 .-. A= 54° 31' .-. 35° 29' log d — log a -|- log cot A log a = 1.67715 log cot.4 = 9.85300 —10 log h = 1.53015 .-. h = 33.896 = 33.90 Check. 58.42 _ 33 .g 2 = 2261-1-, and 47.552 = 2261-1- . This solution should be compared with the one given in § 33, page 36. 67. Given the Two Sides. Eor example, given a = 40, b = 27, find B A, B, and c. , , a 1. tanG = -• b 2. 90° -A. a 3. - = sin A : c .\a — c sinG, and c = . ^ • sinA log tan A = log a -fi colog b log(t= 1.60206 colog 5= 8.56864—10 log tan A = 10.17070 —10 .-. A= 55° 59' .-. B= 34° 1' log c = log a 4- colog sin A log a = 1.60206 colog sin -4 = 0.08151 logc - 1.68357 .-. c = 48.258 = 48.26 Check. 272 + 402 = 2329, and 48.262 = 2329 + . This solution should be compared with the solution of the same problem given in § 34, page 37. There is not much gained in this particular example because the numbers are so small that the operations are easily performed. 66 PLANE TRIGONOMETRY 68. Area of a Right Triangle. The area of a triangle is equal to one half the product of the base by the altitude ; therefore, if a and h denote the two sides of a right triangle and 5' the area, then S=\ab. Hence the area may be found when a and b are known. Consider first the case in which an acute angle and the hypotenuse are given. For example, let A = 34° 28' and c = 18.75. Then, finding log a and log 6 as in § 63, we have log S = colog 2 + log a + log h colog 2 = 9.69897 - 10 log a = 1.02576 log^ = 1.18917 log S = 1.91390 .-. S = 82.016 = 82.02 Next consider the case in which the hypotenuse and a side are given. For example, let c = 58.4 and a = 47.55. Then, finding log b as in § 66, we have log S = colog 2 + log a + log b colog 2 = 9.69897 — 10 log a = 1.67715 log^> = 1.53015 log S = 2.90627 .-. S = 805.88 = 805.9 Finally, consider the case in which an acute angle and the opposite side are given. For example, let A =62° 10' and a = 78. Then, finding log i as in § 64, we have log 5 = colog 2 + log a + log b colog 2 = 9.69897-10 log a = 1.89209 log6 = 1.61471 log S = 3.20577 .-.5 = 1606.1 = 1606 We can easily verify this result, since, from § 64, a = 78 and 6 = 41.18 ; whence | o6 = 1606, to four significant figures. The case of an acute angle and the opposite side is treated in § 64 ; that of an acute angle and the adjacent side in § 66 ; and that of the two sides in § 67. THE RIGHT TRIANGLE 67 Exercise 31. The Right Triangle Using logarithms, solve the following right triangles, finding the sides and areas to four figures, and the angles to minutes : 1. a = 6, c = 12. 16. II So CO o CO II 2. J = 4, A = 60°. 17. a = 992, B= 76° 19' 3. a = 3, A = 30°. 18. a = 73, B= 68° 52' 4. a = 4, 1 4. 19. a = 2.189, B= 45° 25' 6. a = 2, r 2.89. 20. 5 = 4, A = 37° 56' 6. c = 627, i - : 23° 30'. 21. c - 8590, a = 4476. 7. c = 2280, A = 28° 5'. 22. c = 86.53, a = 71.78. 8. C : 72.15, A = 39° 34'. 23. c = 9.35, a = 8.49. 9. c = 1, A = 36°. 24. c = 2194, 5 = 1312.7. 10. c = 200, B = 21° 47'. 25. c - 30.69, 5 = 18.25. 11. c = 93.4, B = 76° 25'. 26. a = 38.31, 5 = 19.62. 12. a = 637, A = 4° 35'. 27. a - 1.229, 5 = 14.95. 13. a = 48.63, A = 36° 44'. 28. a - 415.3, 5 = 62.08. 14. a = 0.008, A = 86°. 29. a = 13.69, 5 = 16.92. 15. b = 50.94, B = 43° 48'. 30. c = 91.92, = 2.19. Compute the unknoum parts and also the area, having given : 31. a = 5, b = 6. 36. c = 68, A = 69° 54' 32. a = 0.615, c = 70. 37. c = 27, B= 44° 4'. 33. b=V2, c = V3. 38. a = 47, B= 48° 49' 34. a = l, A = 18° 14'. 39. 5 = 9, B= 34° 44' 35. 5 = 12, A = 29° 8'. 40. c = 8.462, B= 86° 4'. 41. Find the value of S in terms of c and A. 42. Find the value of S in terms of a and A. 43. Find the value of S in terms of b and A. 44. Find the value of S in terms of a and c. 46. Given S = 58 and a = 10, solve the right triangle. 46. Given S = 18 and b = 5, solve the right triangle. 47. Given 5 = 12 and A = 29°, solve the right triangle. 48. Given S = 98 and c = 22, solve the right triangle. 49. Find the two acute angles of a right triangle if the hypote- nuse is equal to three times one of the sides. 68 PLANE TEIGONOMETEY 50. The latitude of Washington is 38° 55' 15" N. Taking the radius of the earth as 4000 mi., what is the radius of the circle of latitude of Washington ? What is the circum- ference of this circle ? In all such examples the earth will he considered as a perfect sphere with the radius as above given, unless the contrary is stated. For more accurate data consult the Table of Constants. 61. What is the difference between the length of a degree of lati- tude and the length of a degree of longitude at Washington ? Use the data given in Ex. 50. 52. From the top of a mountain 1 mi. high, overlooking the sea, an observer looks toward the horizon. What is the angle of depres- sion of the line of sight ? In the figure the height of the mountain is necessarily exaggerated. The angle is so small that the result can he found by five-place tables only between two limits which differ by 3' 40". ^ 53. At a horizontal distance of 120 ft. from the foot of a steeple, the angle of elevation of the top is found to be 60° 30'. Find the height of the steeple above the instrument. 64. From the top of a rock which rises vertically 326 ft. out of the water, the angle of depression of a boat is found to be 24°. Find the distance of the boat from the base of the rock. 55. How far from the eye is a monument on a level plain if the height of the monument is 200 ft. and the angle of elevation of the top is 3° 30' ? 56. A distance AB of 96 ft. is measured along the bank of a river from a point A opposite a tree C on the other bank. The angle ABC is 21° 14'. Find the breadth of the river. 57. What is the angle of elevation of an inclined plane if it rises 1 ft. in a horizontal distance of 40 ft. ? 58. Find the angle of elevation of the sun when a tower 120 ft. high casts a horizontal shadow 70 ft. long. 59. How high is a tree which casts a horizontal shadow SO ft. in length when the angle of elevation of the sun is 50° ? SO. A rectangle 7.5 in. long has a diagonal 8.2 in. long. What angle does the diagonal make with the base ? THE EIGHT TRIANGLE 69 61 . A rectangle 85 in. long has an area of 49 J sq. in. Find the angle which the diagonal makes with the base. 62 . The length AB of a rectangular field ABCD is 80 rd. and the width is 60 rd. The field is divided into two equal parts by a straight fence PQ starting from a point P on AB which is 15 rd. from A. What angle does PQ make with AB? 63 . A ship is sailing due northeast at the rate of 10 mi. an hour. Find the rate at which she is moving due north, and also due east. 64 . If the foot of a ladder 22 ft. long is 11 ft. from a house, how far up the side of the house does the lad- der reach? 65 . In front of a window 20 ft. from the ground there is a flower bed 6 ft. wide and close to the house. How long is a ladder which will just reach from the outside edge of the bed to the window ? O'' 66. A ladder 40 ft. long can be so placed that it will reach a win- dow 33 ft. above the ground on one side of the street, and by tipj^ing it back without moving its foot it will reach a window 21 ft. above the ground on the other side. Find the width of the street. 67 . From the top of a hill the angles of depression of two suc- cessive milestones, on a straight, level road leading to the hill, are 5° and 15°. Find the height of the hill. 68. A stick 8 ft. long makes an angle of 45° with the floor of a room, the other end resting against the wall. How far is the foot of the stick from the wall ? 69 . A building stands on a horizontal plain. The angle of elevation at a certain point on the plain is 30°, and at a point 100 ft. nearer the building it is 45°. How high is the building ? ^ 70. From a certain point on the ground the angles of elevation of the top of the belfry of a church and of the top of the steeple are found to be 40° and 51° respectively. From a point 300 ft. fur- ther off, on a horizontal line, the angle of elevation of the top of the steeple is found to be 33° 45'. Find the height of the top of the steeple above the top of the belfry. B ^ 71. The angle of elevation of the top C of an inaccessible fort ^ observed from a point A is 12°. At a point B, 219 ft. from A and on a line AB perpendicular to AC, the angle is 61° 45'. Find the height of the fort. 70 PLANE TRIGONOMETPvY 69. The Isosceles Triangle. Since an isosceles triangle is divided by tlie perpendicular from the vertex to the base into two congruent right triangles, an isosceles triangle is determined by any two parts which determine one of these right triangles. In the examples which follow we shall represent the parts of the isosceles triangle ABC, among which the altitude CD is included, as follows : a = one of the equal sides, c = the base, h = the altitude, A = one of the equal angles, C = the angle at the vertex. For example, given a and c, find A, C, and h. 1. cos^ i£ — ^ . a 2 a 2. C + 2A = 180°; .-. C = 180° -2A = 2(90° - A). 3. h may be found by any one of the following equations : A- 1 = d-, whence /;. = V(a + c) (a — ^ c) ; or - = sinA, whence A = usinA; or — = tan A, whence Ti = \ ctanA. When c and h are known, the area can be found by the formula S = ^ch That is, S = • a sin A = ^ ac sin A, or S = I c • tan A = J tan A, or S = ^cV(a + i-c) (a — ^c). Consider also the case in which a and /i are given, to find A, (7, c, and S. 1. sin A = and hence A is known. a 2. C = 2(90° — A), as above, and. hence C is known. 3. ^ c = a. cos A, and hence c is known. 4. iS = -^ ch, and hence S is known. We can also find S from any of its other equivalents, such as those given above, or sin ^ G sinA, each of which is easily deduced. THE EIGHT TEIAIIGLE 71 Exercise 32. The Isosceles Triangle Solve the following isosceles triangles : 1. Given a and A, find C, c, and h. 2. Given a and C, find A, c, and h. 3. Given c and^, find C, a, and A. 4. Given c and C, find A, a, and h. 6. Given h and A, find C, a, and c. 6. Given h and C, find A, a, and c. 7. Given a and h, find A, C, and c. 8. Given c and h, find A, C, and a. 9. Given a = 14.3, c = 11, find A, C, and h. 10. Given a = 0.295, A = 68° 10', find c, h, and S. 11. Given c = 2.352, C = 69° 49', find a, h, and S. 12. Given h — 7.4847, ^ = 76° 14', find a, c, and S. 13. Given c = 147, S — 2572.5, find A, C, a, and h. 14. Given h = 16.8, S = 43.68, find A, C, a, and c. 15. Given a = 27.56, ^ = 75° 14', find c, h, and S. \ Given an isosceles triangle, AB C : 16. Find the value of S in terms of a and C. 17. Find the value of S in terms of a and A. 18. Find the value of S in terms of h and C. 19. A barn is 40 ft. by 80 ft., the pitch of the roof is 45°; find the length of the rafters and the area of the whole roof. 20. In a unit circle what is the length of the chord subtending the angle 45° at the center ? 21. The radius of a circle is 30 in., and the length of a chord is 44 in. ; find the angle subtended at the center. 22. Find the radius of a circle if a chord whose length is 5 in, subtends at the center an angle of 133°. 23. What is the angle at the center of a circle if the subtending chord is equal to § of the radius ? 24. Find the area of a circular sector if the radius of the circle is 12 in., and the angle of the sector is 30°. \\ 25. Find the tangent of the angle of the slope of an A-roof of a building which is 24 ft. 6 in. wide at the eaves, the ridgepole being 10 ft. 9 in. above the eaves. 72 PLAITE TEIGOXOMETRY 70. The Regular Polygon. We have already considered a few cases involving the regular polygon. It is evident from geometry that if the polygon shown below has n sides, the angle of the right triangle which has its vertex at the center is equal to J of 360°/n, or 180°/n. The triangle may evidently be solved if the radius of the circum- scribed circle (r), the radius of the inscribed circle (/i), or the side of the polygon (c) is given. In the exercises we shall let n = number of sides, c = length of one side, r = radius of circumscribed circle, h = radius of inscribed circle, p = the perimeter, S = the area. Then, by geometry, 5 = ^ hp. Exercise 33. The Regular Polygon Find the remaining parts of a regular polygon, given : 1.71=10,0=1. 3. 71 = 20, ?• = 20. 5. 77=11, S' = 20. 2. 77=18, 7’= 1. 4. 77 = 8, 77 =1. 6. 77 =7, S=T. 7. The side of a regular inscribed hexagon is 1 in.; find the side of a regular inscribed dodecagon. \J 8. Given n and c, and represent by b the side of the regular inscribed polygon having 2 n sides, find b in terms of n and c. 9. Compute the difference between the areas of a regular octagon and a regular nonagon if the perimeter of each is 16 in. 10. Compute the difference between the perimeters of a regular pentagon and a regular hexagon if the area of each is 12 sq. in. 11. Find the perimeter of a regular dodecagon circumscribed about a circle the circumference of which is 1 in. 12. AlHiat is the side of the regular inscribed polygon of 100 sides, the radius of the circle being unity ? What is the perimeter ? is. What is the perimeter of the regular inscribed polygon of 360 sides, the radius of the circle being unity ? ■ 14. The area of a regular polygon of twenty-five sides is 40 sq. in ; find the area of the ring included between the circumferences of the inscribed and circumscribed circles. THE EIGHT TEIANGLE 73 Exercise 34. Review Problems 1 . Prove that the area of the parallelogram here shown is equal to ab sin A. 2. Two sides of a parallelogram, are 5 in. and 6 in. respectively, and their included angle is 82° 45'. What is the area ? 3. Two sides of a parallelogram are 9 ft. and 12 ft. respectively, and their included angle is 74.5°.' the area ? 4. Each side of a rhombus is 7.35 in., and one angle is 42° 27'. What is the area ? 5. The area of a rhombus is 250 sq. in., and one of the angles is 37° 25'. What is the length of each side ? 6. A pole BD stands on the top of a mound BC. - Erom a point A the angles of elevation of the top and foot of the pole are 60° and 30° respectively. Prove that the height of the pole is twice the height of the mound. 7. A ladder 38 ft. long is resting against a wall. The foot of the ladder is 7 ft. 2 in. from the wall. What is the height of the top of the ladder above the ground ? 8. Erom a boat 1325 ft. from the base of a vertical cliff the angle of elevation of the top of the cliff is observed to be 14° 30'. Eind the height of the cliff. 9. On the top of a building 50 ft. high there is a flagstaff BD Erom a point A on the ground the angles of elevation of B and D are 30° and 45° respectively. Eind the length of the flagstaff and the distance AC of the observer from the building, as shown in the annexed figure. 50 50 I "u Since — = tan 30° and = tan 45°, x can evidently X X be eliminated. 10. A man whose eye is 5 ft. 8 in. above the ground stands midway between two telegraph poles which are 200 ft. apart. The elevation of the top of each pole is 48° 50'. What is the height of each ? 11. The captain of a ship observed a lighthouse directly to the east. After sailing north 2 mi. he observed it to lie 55° 30' east of south. How far was the ship from the lighthouse at the time of each observation ? 74 PLANE TKIGONOMETRY 12. A leveling instrument is placed at A on the slope MN, and the line M'N' is sighted to two upright rods. By measurement MM' is found to be 12.8 ft., NN' to be 3.4 ft., and M'N' to be 48.3 ft. Bequired the angle of the slope of MN and the distance MN. ^ 13. A wire stay is fastened to a telegraph pole 6.8 ft. from the ground and is stretched tightly so as to reach the ground 5.2 ft. from the foot of the pole. What angle does the wire stay make with the ground ? 14. The top of a conical tent is 8 ft. 7 in. above the ground, and the diameter of the base is 9 ft. 8 in. Find the inclination of the side of the tent to the horizontal. Check the result by drawing the figure to scale and measuring the angle with a protractor. 16. In this piece of iron construction work AC = 11 in. and AB makes an angle of 30° with BC. What is the length of ^C? 16. In Ex. 15 it is also known that BE and CD are each 9 in. long and make angles of 60° with BC produced. What is the length of ED ? 17. From the conditions given in Ex. 16, find the length of CF. 18. The base of a rectangle is 14| in. and the diag- onal is 19^ in. What angle does the diagonal make with the base ? Check the result by drawing the figure to scale and measuring the angle with a protractor. 19. In constructing the spire represented in the figure below it is planned to have AA = 42ft. and PM =92 ft. What angle of slope must the builders give to AP ? 20. In Ex. 19 find the length of AP and find the angle P. 21. In the figure of Ex. 19 the brace CD is put in 38 ft. above AB. What is its length ? 22. The spire of Ex. 19 rests on a tower. A man standing on the ground at a distance of 400 ft. from the base of the tower observes the angle of elevation of P to be 25° 38', the instrument being 5 ft. above the ground. What is the height of P above the ground ? 23. When the angle of elevation of the sun is 38.4°, what is the length of the shadow of a tower 175 ft. high ? 'U ^ u THE EIGHT TEIAHGLE 75 24. Two men, M and N, 3200 ft. apart, observe an aeroplane A at the same instant, and at a time when the plane MNA is vertical. The angle of elevation at M is 41° 27' and the angle at N is 61° 42'. Eequired AB, the height of the aeroplane. Show that h cot 41° 27' + h cot 61° 42' is known, whence h can be found. 25. A kite string 475 ft. long makes an angle of elevation of 49° 40'. Assuming the string to be straight, what is the altitude of the kite ? 26. A steel bridge has a truss ADEF in which it is given that AD = 20 ft., 6 ft. 8 in., and EA =12 ft., as ^ ^ shown in the figure. Eequired the angle of slope which AF makes with AD. A B D 27. Two tangents are drawn from a point P to a circle and contain an angle of 37.4°. The radius of the circle is 5 in. Eind the length of each tangent and the distance of P from the center. 28. Erom the top of a cliff 95 ft. high, the angles of depression of two boats at sea are observed, by the aid of an instrument 5 ft. above the gromid, to be 45° and 30° respectively. The boats are in a straight line with a point at the foot of the cliff directly beneath the observer. What is the distance between the boats ? 29. A carpenter’s square ^PCA/is held against the vertical stick BD resting on a sloping roof AD, as in the figure. It is found that AC =24 in. and CP = 11.5 in. Eind the angle of slope of the roof with the horizontal. ^ 30. In Ex. 29 find the length of AD. 31. A man 6 ft. tall stands 4 ft. 9 in. from a street lamp. If the length of his shadow is 19 it.,J how high is the light above the street ? ^ 32. The shadow of a city building is observed to be 100 ft. long, and at the same time the shadow of a lamp-post 9 ft. high is observed to be 5.2 ft. long. Eind the angle of elevation of the sun and the height of the building. 33. A man 5 ft. 10 in. tall walks along a straight line that passes at a distance of 2 ft. 9 in. from a street light. If the light is 9 ft. 6 in. above the ground, find the length of the man’s shadow when his distance from the point on his path that is nearest to the lamp is 3 ft. 8 in. PLANE TRIGONOMETKY 76 34 . A man on a bridge 35 ft. above a stream, using an instrument 5 ft. high, sees a rowboat at an angle of depression of 27° 30'. If the boat is approaching at the rate of 2| mi. an hour, in how many seconds will it reach the bridge ? 35 . A shaft 0, of diameter 4 in., makes 480 revolutions per minute. If the point P starts on the horizontal line OA . how far is it above OA after of a second? 36 . Assrmring the earth to be a sphere with radius 3957 mi., find the radius of the circle of latitude which passes through a place in latitude 47° 27' 10" N. 37 . When a hoisting crane AB, 28 ft. long, makes an angle of 23° with the horizontal A C, what is the length of A C ? Suppose that the angle CAB is doubled, what is then a— the length of A C ? | 38 . In Ex. 37 find the length of BC in I each of the two cases. 'o 39 . Wishing to measure the distance AB, a man swings a 100-foot tape line about B, describing an arc on the ground, and then does the same about A. The arcs intersect at C, and the angle ACB is found to be 32° 10'. What is the length of AP ? 40 . From the top of a mountain 15,250 ft. high, overlooking the sea to the south, over how many minutes of latitude can a person see if he looks southward ? Use the assumption stated in Ex. 36. 41 . The length of each blade of a pair of shears, from the screw to the point, is 5^- in. When the points of the open shears are 3|- in. apart, what angle do the blades make with each other ? 42 . Ill Ex. 41 how far apart are the points when the blades make an angle of 28° 45' with each other ? 43 . The wheel here represented has eight spokes, each being 19 in. long. How far is it from A to P ? from P to P ? ^ 44 . The angle of elevation of a balloon from a station directly south of it is 60°. From a second station lying 5280 ft. directly west of the first one the angle of elevation is 45°. The instrument being 5 ft. above the level of the ground, what is the height of the balloon ? CHAPTER V TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 71. Need for Oblique Angles. We have thus far considered only- right triangles, or triangles which can readily be cut into right tri- angles for purposes of solution. There are, however, oblique triangles which cannot conveniently be solved by merely separating them into right triangles. We are therefore led to consider the functions of oblique angles, and to enlarge our idea of angles so as to include angles greater than 180°, angles greater than 360°, and even negative angles and the angle 0°. 72. Positive and Negative Angles. We have learned in algebra that we may distinguish between two lines which extend in opposite direc- tions by calling one positive and the other negative. For example, in the annexed figure we consider OX T as positive and therefore OX' as negative. We also con- + sider OY as positive and hence OY' as negative. In gen- _ eral, horizontal lines extending to the right of a point Xr q 'X which we select as zero are considered positive, and those to the left negative. Vertical lines extending upward from zero are considered positive, and those extending down- x ward are considered negative. With respect to angles, an angle is considered positive if the rotat- ing line which describes it moves counterclockwise, that is, in the direction opposite to that taken by the hands of a clock. An angle is considered negative if the rotat- ing line moves clockwise, that is, in the same direction as that taken by the hands of a clock. Arcs which subtend positive angles are considered positive, and arcs which subtend negative angles are considered negative. Thus /.A OB and arc AB are considered positive; Z.AOB' and arc AB' are considered negative. For example, we may think of a pendulum as swinging through a positive angle when it swings to the right, and through a negative angle when it swings to the left. We may also think of an angle of elevation as positive and an angle of depression as negative, if it appears to be advantageous to do so in the solu- tion of a problem. 77 78 PLANE TEIGONOMETKY 73. Coordinates of a Point. In trigonometry, as in work with graphs in algebra, we locate a point in a plane by means of its distances from two perpendicular lines. These lines are lettered XX' and YY', and their point of intersection 0. The lines are called the axes and the point of intersection the origin. In some branches of mathematics it is more convenient to use oblique axes, but in trigonometry rectangular axes are used as here shown. The distance of any point P from the axis XX' , or the a;-axis, is called the ordinate of the point. Its distance from the axis YY', or the y-axis, is called the abscissa of the point. In the figure, y is the ordinate of P, and X is the abscissa of P. The point P is rep- resented by the symbol (x, y). In the figure the side of each small square may be taken to represent one unit, in which case P = (4, 3), because its abscissa is 4 and its ordinate 3. Following a helpful European custom, the points are indicated by small circles, so as to show more clearly when a line is drawn through them. The abscissa and ordinate of a point are together called the coordi- nates of the po int. 74. Signs of the Coordinates. Erom § 73 we see that odiscissas to the right of the y-axis are positive ; abscissas to the left of the y-axis are negative ; ordinates above the x-axis are positive ; ordinates below the x-axis are negative. A point on the line YY' has zero for its abscissa, and hence the abscis.sa may be considered as either positive or negative and may be indicated by ± 0. Simi- larly, a point on the line XX' has ± 0 for its ordinate. 75. The Four Quadrants. The axes divide the plane into four parts known as quadrants. Because angles are generally considered as generated by the rotating line moving counterclockwise, the four quadrants are named in a counterclockwise order. Quadrant XOY is spoken of as the first quadrant, YOX' as the second quadrant, X'OY' as the third quadrant, and Y'OX as the fourth quadrant. 76. Signs of the Coordinates in the Several Quadrants. Erom § 74 we have the following rule of signs : In quadrant I the abscissa is qoositive, the ordinate positive ; In quadrant II the abscissa is negative, the ordinate positive ; In quadrant III the abscissa is negative, the ordinate negative ; In quadrant IV the abscissa is positive, the ordinate negative. P(2 V) X' ■Y'- FUNCTIONS OF ANY ANGLE 79 77. Plotting a Point. Locating a point, having given its coordi- nates, is c,?il\edL plotting the point. L, 1 (- ) I X X 1 d n (- 3r 8) I rt F- 0 ■M P 0 0 (1 1) Xi Y jr 0 rr 2 ■/ 0 ■S < to; a ' Por example, in the first of these figures the point (—2, 4) is shown in quadrant II, the point (— 3, — 2) in quadrant III, and the point (1, — 1) in quadrant IV. In the second figure the point (— 2, 0) is shown on OX', between quad- rants II and III, and the point (1, 0) on OX, between quadrants I and IV. In the third figure the point (0, 1) is shown on OY, between quadrants I and II, and the point (0, — 3) on OY', between quadrants III and IV. In every case the origin 0 may be designated as the point (0,0). 78. Distance from the Origin. The coordinates of P being x and y, we may form a right triangle the hypotenuse of which is the distance of P from 0. Kepresenting OP by r, we have ?■ = V; Since this may be written r = ± ~Vx^ + y^, we see that r may be considered as either positive or negative. It is the custom, however, to consider the rotating line which forms the angle as positive. If r is produced through 0, the production is considered as negative. 1. What is the distance of the point (3, 4) from the origin ? r = V 3 F +42 = = 5 . 2. What is the distance of the point (— 3, — 2) from the origin ? r = V(- 3)2 - 1 - (- 2)2 = V9 -I- 4 = Vl3 = 3.61. 3. What is the distance of the point (5, — 5) from the origin ? r = V 52 + (- 5)2 = VEo = 7.07. 4. What is the distance of the point (— 2, 0) from the origin ? r = V(- 2)2 + 02 = \/i = 2, as is evident from the conditions of the problem. 80 PLANE TEIGONOMETRY Exercise 35. Distances from the Origin Using squared paper ^ or measuring with a ruler ^ plot the follow- ing points : 1. (2, 3). 8. (- -3, 2). 15. (3, -4). 22. (0, 0). 2. (3, 5). 9. (- -3, 4). 16. (4, -3). 23. (0, 24). 3. (4, 4). 10. (- -5, !)• 17. (5, -!)• 24. (0, - 34). 4. (2i, 3). 11. (- -4, 6). 18. (0, -)• 25. (44, 0). 5. (3i, 44). 12. (- -2). 19. (3, 0). 26. (54, 0). 6. (4i, 44). 13. (- -3, -5). 20. (0, -4). 27. (- 24, 0). 7. (5i, 34). 14. (- - 5, -3). 21. (- 2, 0). 28. o' CO Find the distance of each of the following points from the origin: 29. (6, 8). 32. (1^, 2). 35. (2, Vs). 38. (0, 7). 30. (9, 12). 33. (I, 1). 36. (- 3, 4). 39. (5, 0). 31. (5,12). 34. 37. (0,0). ^ 40. (-12,-9). 41. Find the distance from (3, 2) to (— 2, 3). 42. Find the distance from (— 3, — 2) to (2, — 3). 43. Find the distance from (4, 1) to (— 4, —1). V;44. Find the distance from (0, 3) to (— 3, 0). 45. A point moves to the right 7 in., up 4 in., to the right 10 in., and up 18f in. How far is it then from the starting point ? 46. A point moves to the right 9 in., up 5 in., to the left 4 in., and up 3 in. How far is it then from the starting point ? 47. Find the distance from (— 4^3) to (^, — 4 Vs). 48. A triangle is formed by joining the points (1, 0), (— 4, 4 and (— — 4'^)- Find the perimeter of the triangle. Draw the figure to scale. 49. Find the area of the triangle in Ex. 48. U 50. A hexagon is formed by joining in order the points (1, 0), (i,iV3), (-^,^V3), (-1,0), (-^,-^V3), (i, -^V3), and (1, 0). Is the figure a regular hexagon ? Prove it. 51. A polygon is formed by joining in order the points (1, 0), (iV2, iV2), (0, 1), (-^V2,iV2), (-1, 0), (-^V2, -i^), (0, — 1), (■4- V2, — 4- V2), and (1, 0). Draw the figure, state the kind of polygon, and find its area. FUNCTIONS OF AKY A^sGLE 81 79. Angles of any Magnitude. In the following figures, if the rotat- ing line OP revolves about 0 from the position OX, in a counterclock- wise direction, until it again coincides with OX, it will generate all angles in every quadrant from 0° to 360°. The line OX is called the initial side of the angle, and the line OP the ter- minal side of the angle. An angle is said to be an angle of that quadrant in which its terminal side lies. Angles between 0° and 90° are angles of quadrant I. Angles between 90° and 180° are angles of quadrant II. Angles between 180° and 270° are angles of quadrant III. Angles between 270° and 360° are angles of quadrant IV. The rotating line may also pass through 360°, forming angles from 360° to 720°. It may then make another revolution, forming angles greater than 720°, and so on in- definitely. For example, in using a screwdriver we turn through angles of 360°, 720°, 1080°, and so on, depending upon the number of revolutions. In the same way, the minute hand of a clock turns through 8640° in a day, and the drive wheel of an engine may turn through thousands of degrees in an hour. We might, if necessary, speak of an angle of 400° as an angle of quadrant I, because its terminal side is in that quadrant, but we have no occasion to do so in practical cases. As stated in § 72, if the line OP is rotated clockwise, it generates negative angles. In this way we may form angles of — 40° or — 140°, as here shown, and the rotation may continue until we have angles of — 360°, — 720°, — 1080°, — 1440°, and so on indefinitely. We shall have but little need for the negative angle in the practical work of trigonometry, but we shall make ex- tensive use of angles between 0° and 180°, and some use of those between 180° and 360°. / H i \ I ^ P<(ni "ivy 82 PLANE TPIGONOMETKY 80. Functions of Any Angle. Since we have now seen that we may have angles of any magnitude, it is necessary to consider their func- tions. Although we must define these functions anew, it will be seen that the definitions hold for the acute angles which we have already considered. In whatever quadrant the angle is, we designate it by A. We take a point P, or (x, y), on the rotating line, and let OP = r. Then the angle XOP, read counterclockwise, is the angle A. We then define the functions as follows : . , y ordinate sin A = - = -r— , r distance , X abscissa cosA= - = , r distance , , y ordinate tanA= -= ^ , X abscissa CSC A = sec A - cot A 1 distance sin A y ordinate ’ 1 distance cos A X abscissa 1 _ abscissa tan A y ordinate It will be seen that these definitions are practically the same as those already learned for angles in quadrant I. Their application to the other quadrants is apparent. The general definitions might have been given at first, but this plan offers difficulties for a beginner which make it undesirable. By counting the squares on squared paper and thus getting the lengths of certain lines, the approximate values of the functions of any given angle may be found, but the exercise has no practical significance. The values of the functions are determined by series, these being explained in works on the calculus. FUNCTIONS OF ANY ANGLE 83 81. Angles determined by Functions. Given any function of an angle, it is possible to construct the angle or angles which satisfy the value of the function. 1. Given sinA = f, construct the angle A. If we take a line parallel to X'X and 3 units above it, and then rotate a line OP, 5 units long, about 0 until P rests upon this parallel, we shall have In other words, we have constructed two angles, each of which has 3- for its sine. Furthermore, we could construct an infinite number of such angles, for we see that 360° + A terminates in OP and has the same sine that A has, and that tue same may be said of 360° + A', 720° + A, 720° + A', 1080° + A, and so on. In general, therefore, the angle n x 360° + A has the same functions as A, ?i being any integer. Hence if we know the value of any particular function of an angle, the angle cannot be uniquely determined ; that is, there is more than one angle which satisfies the condition. In general, as we see, an infinite number of angles will satisfy the given condition, although this gives no trouble because only two of these angles can be less than 360°. 2. Given tanA= construct the angle A. If we take an abscissa 4 and an ordinate 3, as in quadrant I of the figure, we locate the point (3, 4). Then angle XOP has for its tangent But it is evident that we may also locate the point (— 3, — 4) in quadrant III, and thus find an angle between 180° and 270° whose tangent is 82. Functions found from Other Functions. Given any function of an angle, it is possible not only to construct the angle but also to find the other functions. For in Ex. 1 above, after constructing angles A and A', we see that sin A = 5 cos A = - or ■, 5 5 tan A = - or , 4 - 4 cscA = ^ 5 5 sec A = - or , 4 - 4 . 4 - 4 cot A = - or 3 3 That is, if sin A = |^, then cos A = ± tan A = ± f, esc A = sec A = ± and cot A = ± |. 84 PLANE TRIGONOMETRY Exercise 36. Construction of Angles and Functions Using the protractor, construct the following angles: 1. 30°. 4. 150°. 7. 270°. 10. 405°. 13. -45°. 2. 60°. 6. 180°. 8. 300°. 11. 450°. 14. - 90°. 3. 80°. 6. 200°. 9. 360°. 12. 720°. 15. - 180°. State the quadrants in which the terminal sides of the following angles lie : 16. 45°. 19. 150°. 22. 390°. 25. 660°. 28. 930°. 17. 75°. 20. 210°. 23. 495°. 26. 765°. 29. 990°. 18. 120°. 21. 315°. 24. 570°. 27. 820°. 30. 1080°. Construct two angles A, given the following . \/ 31. sin A = 36. sin A = — |. 41. sin A = — 1. 32. cos A= v/ 37. cosA = — !■. 42. cos A = — 1. 33. tan A \/38. tanA = — 43. tanA = — 1. 34. cot A = ^. 39. cot A = — 1/44. cot A = — 1. 35. sec A= 2. 40. sec A = — 1. 46. sec A = — 2. Given the following functions of angle A, construct the other functions : 46. sin A = 2 T- 61. sin A = - -t- 56. sin A = — 47. cos A - 3 52. cos A-- -1. 57. cos A = — 48. tanA = 4 X- 63. tan A = - _ 3 S- 58. tan A——h. 49. cot A = 3 64. sec A = - - 2. 59. cot A = — ^. 60. CSC A = 3. 65. CSC A = - -1. 60. sec A = — 24. 61. If tan4 = V2, show that cotT is half as large. What are the values of sinA, cosA, secA, and cscA ? 62. The product 2 sin 45° cos 45° is equal to the sine of what angle? 63. The product 2 sin 30° cos 30° is equal to the sine of what angle ? 64. To the diagonal A C of a square A BCD a perpendicular AM is drawn. Find the values of the six functions of angle BAM. 65. In the figure of Ex. 64, suppose AM rotates further, until it is in line with BA. What are then the six functions of angle BAM? FUNCTIONS OF ANY ANGLE 85 83. Line Values of the Functions. As in the case of acute angles (§ 22) we may represent the trigonometric functions of any angle by means of lines in a circle of radius unity. Thus in each of the following figures sin X = MP, cos X = OM, tan X — AT, cot X - BS, seex = OT, CSC X = OS. By examining the figures we see that In quadrant I all the functions are positive ; In quadrant II the sine and cosecant only are positive ; In quadrant III the tangent and cotangent only are positive ; In quadrant IV the cosine and secant only are positive. It will be seen as we proceed that the laws and relations which have been found for the functions of acute angles hold for the func- tions of angles greater than 90°. For example, it is apparent from the above figures that, in every quadrant, MP" + = 1, and hence that sinM + cosM = 1, as shown in § 14. It is also evident that AT _ ^ 1 ~ om’ sin.4 and hence that tan^ = -• cosJ. Other similar relations are easily proved by reference to the figures. 86 PLAXJirTRIGONOMETRY 84. Variations in the Functions. A study of the line values of the functions shows how they change as the angle increases from 0° to 360°. 1. The Sine. In the first quadrant the sine MP is positive, and increases from 0 to 1 ; in the second it remains positive, and decreases from 1 to 0 ; In the third it is negative, and increases in absolute value from 0 to 1 ; in the fourth it is negative, and decreases in absolute value from 1 to 0. The absolute value of the sine varies, therefore, from 0 to 1, and its total range of values is from + 1 to — 1. In the third quadrant the sine decreases from 0 to — 1, but the absolute value (the value without reference to its sign) increases from 0 to 1, and similarly for other cases on this page in which the absolute value is mentioned. 2. The Cosine. In the first quadrant the cosine OM is positive, and decreases from 1 to 0 ; in the second it becomes negative, and increases in absolute value from 0 to 1 ; in the third it is negative, and decreases in absolute value from 1 to 0 ; in the fourth it is positive, and increases from 0 to 1. The absolute value of the cosine varies, therefore, from 0 to 1. 3. The Tangent. In the first quadrant the tangent A T is positive, and increases from 0 to oo ; in the second it becomes negative, and decreases in absolute value from oo to 0 ; in the third it is positive, and increases from 0 to oo ; in the fourth it is negative, and decreases in absolute value from oo to 0. 4. The Cotangent. In the first quadrant the cotangent BS is posi- tive, and decreases from oo to 0; in the second it is negative, and increases in absolute value from 0 to oo ; in the third and fourth quad- rants it has the same sign, and undergoes the same changes as in the first and second quadrants respectively. The tangent and cotangent may therefore have any values whatever, positive or negative. 5. The Secant. In the first quadrant the secant Or is positive, and increases from 1 to oo ; in the second it is negative, and decreases in absolute value from oo to 1 ; in the third it is negative, and increases in absolute value from 1 to oo ; in the fourth it is positive, and decreases from 00 to 1. 6. The Cosecant. In the first quadrant the cosecant OS is positive, and decreases from oo to 1 ; in the second it is positive, and increases from 1 to 00 ; in the third it is negative, and decreases in absolute value from oo to 1; in the fourth it is negative, and increases in absolute value from 1 to oo. FUNCTIONS OF ANY ANGLE 87 It is evident, therefore, that the sine can never be greater than 1 nor less than — 1, and that it has these limiting values at 90° and 270° respectively. We may also say that its absolute value can never be greater than 1, and that it has its limiting value 0 at 0° and 180°, and its limiting absolute value 1 at 90° and 270°. If we have an equation in which the value of the sine is found to he greater than 1 or less than — 1, we know either that the equation is wrong or that an error has been made in the solution. Of course the values of the functions of 360° are the same as those of 0°, since the moving radius has returned to its original position and the initial and terminal sides of the angle coincide. In the same way, the absolute value of the cosine cannot be greater than 1, and it has its limiting value 0 at 90° and 270°, and its limit- ing absolute value 1 at 0° and 180°. Similarly we can find the limiting values of all the other functions. For convenience we speak of oo as a limiting value, although the function increases without limit, the meaning of the expression in this case being clear. Summarizing these results, we have the following table : Function 0° 90° 180° 270° 360° Sine y 0 + 1 ±0 -1 y 0 Cosine y 1 ±0 -1 y 0 + 1 Tangent y 0 ± CO y 0 ± oo y 0 Cotangent y CO ±0 y CO ±0 y 00 Secant y 1 ± CO -1 y 00 y 1 Cosecant y CO y 1 ± 00 - 1 y CO \ Sines and cosines vary in value from +1 to — 1 ; tangents and cotangents, from -h 00 to — 00 ; secants and cosecants, from -f oo to -f 1, and from — 1 to — oo . In the table given above the double sign ± or y is placed before 0 and oo . From the preceding investigation it appears that the functions always change sign in passing through 0 or through oo ; and the sign ± or y prefixed to 0 or oo simply shows the direction from which the value is reached. For example, at 0° the sine is passing from — (in quadrant IV) to y (in quadrant I). At 90° the tangent is passing from y (in quadrant I) to — (in quadrant II). 85. Functions of Angles Greater than 360°. The functions of 360° -t- x are the same in sign and in absolute value as tdiose of x. If n is a pc^itive integer. The funotions af (n x 360° -(- x) are the sarm as those of x. For example, the functions of 2200°, or 6 x 360° y 40°, are the same in sign and in absolute value as the functions of 40°. 88 PLANE TEIGONOMETKY Exercise 37. Variations in the Functions Represent the following functions hy lines in a unit circle : 1. sin 135°. 2. cos 120°. 3. tan 150°. 4. cot 135°. 5. sec 120°. 6. CSC 150°. 7. sin 210°. 8. cos 225°. 9. tan 240°. 10. cot 210°. 11. sec 225°. 12. CSC 240°. 13. sin 300°. 14. cos 315°. 15. tan 330°. 16. cot 300°. 17. sec 315°. 18. CSC 330°. 19. sin 270°. 20. cos 180°. 21. tan 180°. 22. cot 270°. 23. sec 180°. 24. CSC 270°. 25. Prepare a table showing the signs of all the functions in each of the four quadrants. 26. Prepare a table showing which functions always have the minus sign in each of the four quadrants. Represent the following functions by lines in a unit circle : 27. sin 390°. 30. cos 390°. 33. sin 460°. 36. tan 475°. 28. tan 405°. 31. cot 405°. 34. sin 570°. 37. sec 705°. 29. sec 420°. 32. esc 420°. 35. sin 720°. 38. esc 810°. Show by lines in a unit circle that 39. sin 150° = sin 30°. 40. cos 150° = — cos 30°. 41. sin 210° = - sin 30°. 42. COS 210° = — cos 30°. 43. sin 330° = — sin 30°. 44. cos 330°= cos 30°. 45. tan 120° = — tan 60°. 46. cot 120° = — cot 60°. 47. tan 240° = tan 60°. 48. cot 240° = cot 60°. 49. tan 300° = — tan 60°. y^50. cot 300° = — cot 60°. 51. Write the signs of the functions of the following angles: 340°, 239°, 145°, 400°, 700°, 1200°, 3800°. 52. How many values less than 360° can the angle x have if sin a; = + f , and in what quadrants do the angles lie ? Draw a figure. 53. How many values less than 720° can the angle x have if cos a: = + f , and in what quadrants do the angles lie ? Draw a figure. 54. If we take into account only angles less than 180°, how many values can x have if sin a; = f ? if cos x = \l if cos x = — %? if tan a; = § ? if cot a: = — 7 ? 55. Within what limits between 0° and 360° must the angle x lie if cos X =— I ? if cot X = 4 ? if sec x = 80 ? if esc x = — 3 ? FUNCTIONS OF ANY ANGLE 89 66. Why may cot 360° be considered as either + oo or — oo ? 57. Find the valnes of sin 460°, tan 540°, cos 630°, cot 720°, sin810° CSC 900°, cos 1800°, sin 3600°. 58. What functions of an angle of a triangle may be negative ? In what cases are they negative ? \X 59. In what quadrant does an angle lie if sine and cosine are both negative ? if cosine and tangent are both negative ? 60. Between 0° and 3600° how many angles are there whose sines have the absolute value f ? Of these sines how many are positive ? Compute the values of the following expressions : '\ 61. a sin 0° + ^ cos 90° — c tan 180°. 62. a cos 90° — h tan 180° + c cot 90°. 63. a sin 90° — h cos 360° -\-(a — b) cos 180°. V 64. (a^ — 6^) cos 360° — 4 sin 270° + sin 360°. 65. cos 180°+ (a" + 1“^) sin 180° + (a" + tan 135°. 66. (a^ + 2a6 + 5^)sin90° +(a^ — 2ab-\- 6^)cos 180°— 4aS tan225°. 67. (a — b c — c^sin 210° —(a — b + c — d')GOS, 180° + a tan 360°. State the sign of each of the six functions of the following angles : 68. 75°. 70. 155°. 72. 275°. 74. 355°. 69. 125°. 71. 185°. 73. 325°. 75. - 66°. Find the four smallest angles that satisfy the following conditions : 76. sind = ^. 78. sin.4 = -|V3. 80. tan^ = -jV3. 77. cos^=|-V3. 79. cosd.= -^. v^81. tan^=V3. Find two angles less than 360° that satisfy the following conditions : 82. sinJ. = -4- 84. sind = — -^V^. 86. tan.4 = — 1. 83. cos^ = — I-. 85. cosd = — ^ V2. 87. cotd = — 1. iff A, B, and C are the angles of any triangle AB C*, prove that : 88. cos -l-d. = sin-^(5 + C). 90. cos = sin + C). 89. sin-|-C= cos-^(d +5). 91. sin-|-d.= cos-^(iJ + C). As angle A increases from 0° to 360°, trace the changes in sign and magnitude of the following : 92. sin d, cos d-. 94. sin^ — cos^. 96. tand + cotA. 93. sinA + cosA. 95. sinA-^cosA. 90 PLANE TPIGONOMETPY 86. Reduction of Functions to the First Quadrant. In tlie annexed figure BB' is perpendicular to the horizontal diameter AA \ and the diameters PR and QS are so drawn as to B make Z.AOP— Z.SOA. It therefore fol- lows from geometry that A MOP, MOS, NOQ, and NOR are congruent. Considering, therefore, only the absolute values of the functions, we have sin A OP = sin AOQ = sin A OR, = sin A OS, cos AOP = cos A OQ = cos = cos^O.?, and so on for the other functions. Hence, For every acute angle there is an angle in each of the h igher quadrants whose functions, in absolute value, are equal to those of this acute angle. If we let AAOP = x and Z.POB = y, noticing that Z.AOP — ZQOA'= ZA'OR = ZSOA = X, and Z.POB ^ FBOQ ^ Z.ROB’ = Z.B'OS = y, and prefixing the proper signs to the functions (§ 83), we have : Angle in Quadrant II ^ sin (180° — x)= sin x ^ cos (180° — x) = — cos X V tan (180° — x) = — tan x cot (180° — a;) = — cot x sin (90° r y}= cos y cos (90° -f ?/) = — sin y tan (90° -f y) = — cot y cot (90° -p y) = — tan y Angle in sin (180° -p x) — — sin x cos (180° -\-F) —— cos X tan (180° -p a;) = tan x cot (180° -p a:) = cot x Angle in Quadrant III sin (270° — y) —— cos y cos (270° — y) = — sin y tan (27 0° — y) = cot y cot (270° — y) = tany Quadrant IV sin (360° — x) =— sin x sin (270° + y) = — cos y cos (360° — x) = cos X cos (270° -P y) = sin y tan (360° — x) = — tan x tan (270° -f y) - — cot y cot (360° — x) = — cot X cot (270° -P y) = — tan y For example, sin 127° = sin (180° — 53°) = sin 53° = cos 37°, sin 210° = sin (180° -p 30°) = — sin 30° = — cos 60°, sin 350° = sin (360° — 10°) = — sin 10° = — cos 80°. and FUNCTIONS OF ANY ANGLE 91 It appears from the results set forth on page 90 that the functions of any angle, however great, can he reduced to the functions of an angle in the first quadrant. Por example, suppose that we have a polygon with a reentrant angle of 247° 30', and we wish to find the tangent of this angle. We may proceed by finding tan (180° + x) or by finding tan (270° — x). We then have tan 247° 30' = tan (180° + 67° 30') = tan 67° 30', and tan 247° 30' = tan (270° - 22° 30') = cot 22° 30'. That these two results are equal is apparent, for tan 67° 30' = cot (90° - 67° 30') = cot 22° 30'. It also appears that, for angles less than 180°, a given value of a sine or cosecant determines two swpjplementary angles, one acute, the other obtuse ; a given value of any other function determines only one angle, this angle being acute if the value is positive and obtuse if the value is negative. For example, if we know that sin x = ^, we cannot tell whether x = 30° or 160°, since the sine of each of these angles is But if we know that tanx = 1, we know that x = 45°. Similarly, if we know that cot x = — 1, we know that x = 135°, there being no other angle less than 180° whose cotangent is — 1. Since sec x is the reciprocal of cos x and esc x is the reciprocal of sin x, and since by the aid of logarithms we can divide by cos x or sin x as easily as we can multiply by sec x or esc x, we shall hereafter pay but little attention to the secant and cosecant. Since the invention of logarithms these functions have been of little practical importance in the work of ordinary mensuration. Exercise 38. Reduction to the First Quadrant Express the following as functions of angles less than 90° : 1. sin 170°. 11. sin 275°. 21. sin 148° 10' 2. cos 160°. 12. sin 345°. 22. cos 192° 20' r 3. tan 148°. 13. tan 282°. 23. tan 265° 30 t 4. cot 156°. 14. tan 325°. 24. cot 287° 40' sin 180°. ^15. cos 290°. {^26. sin 187° 10' 3". 6. tan 180°. 16. cos 350°. 26. cos 274° 5' ; 14". 7. sin 200°. 17. cot 295°.^ 27. tan 322° 8' 15". 8. cos 225°. 18. cot 347°. 28. cot 375° 10' 3". 9. tan 258°. 19. sin 360°. 29. sin 147.75°. ^ 10. cot 262°. ^20. cos 360°. 1/30. cos 232.25°. 92 PLANE TKIGONOMETPY 87. Functions of Angles Differing by 90°. It was shown in the case of acute angles that the function of any angle is equal to the co-func- tion of its complement (§ 8). That is, tan 28° = cot (90° — 28°) = cot 62°, sinx =: cos (90° — x), and so on. It will now be shown for all angles that if two angles differ hy 90°, the func- tions of either are equal in absolute value to the co-functions of the other. In the annexed figure the diameters PR and Q.S are perpendicular to each other, and from P, Q, R, and S perpendiculars are drawn to AA\ Then from the congruent triangles OMP, QHO, OKR, and SNO we see that and OM= QH= OK = SN, MP = OH = KR - - ON. Hence, considering the proper signs (§ 83), sin^OQ = cosAOP, cos^ OQ = — sinAOP, sinfi OP = cosAOC^, cos A OP =— sin A OQ, sin A 08’ = cos A OP, cos A 08 = — sin A OP. In all these equations, if x denotes the angle on the right-hand side, the angle on the left-hand side is 90° -f x. Therefore, if x is an angle in any one of the four quadrants, sin (90° -|- x) = cos x, cos (90° -|- x) = — sin x ; and hence tan (90° -|- x) = — cot x, cot (90° -}- x) = — tan x. It is therefore seen that the algebraic sign of the function of the resulting angle is the same as that found in the similar case in § 86. 88. Functions of a Negative Angle. If the angle x is generated by the radius moving clockwise from the initial position OA to the terminal position OS, it will be negative (§ 72), and its terminal side will be identical with that for the b angle 360° — x. Therefore the functions of the angle — x are the same as those of the angle 360° — x ; or sin (— x) — sin x, cos (— x) = cos X, tan (— x) = — tan x, cot (— x) = — cot X. FUNCTIONS OF ANY ANGLE 93 Exercise 39. Reduction of Functions Express the following as functions of angles less than 45° : 1. sin 100°. 2. sin 120°. 3. sin 110°. 4. sin 130°. 5. cos 95°. 6. cos 97°. 7. cos 111°. 8. cos 127°. 9. tan 91°. 10. tan 99°. 11. tan 119°. 12. tan 129°. Express the following as functions of positive angles : 17. sin (—3°). 18. sin (—9°). 19. sin(— 86°). 20. cos (—75°). 21. cos(— 87°). 22. cos (—95°). 23. tan (—100°). 24. tan (—150°). 13. cot 94° 1’. 14. cot 97° 2'. 15. cot 98° 3'. y 16. cot 99° 9'. 9 25. tan(— 200°). 26. cot (—1.5°). 27. cot(-7.8°). 1 / 28 . cot (— 9.1°). Find the following hy aid of the tables : 29. sin 178° 30'. 37. log sin 127.5°. 30. cos 236° 45', 38. log cos 226.4°. 31. tan 322° 18' 39. log tan 327.8°. 32. cot 423° 15'. 40. log cot 343.3°. 33. sin (-7° 29 ' 30"). 41. log sin 236° 13 '5". 34. cos (- 29° 42' 19"). 42. log cos 327° 5' 11". 36. tan (-172° 16' 14"). 43. log tan (— 125° 27'). 36. cot (— 262° 17' 15"). 44. log cot (— 236‘ ' 15'). 45. Show that ■ the angles 42°, 138°, - 318 °, 402°, and - 222° have the same sine. 46. Find four angles between 0° and 720° which satisfy the equa- tion sin x=— ^ V2. 47. Draw a circle with unit radius, and represent by lines the sine, cosine, tangent, and cotangent of — 325°. 48. Show by drawing a figure that sin 195° = cos (— 105°), and that cos 300° = sin (— 210°). ^49. Show by drawing a figure that cos 320° = — cos (—140°), and that sin 320° = — sin 40°. 50. Show by drawing a figure that sin 765° = J V2, and that tan 1395° = -!. 51. In the triangle ABC show that cosd = — cos (A -|- C), and that cos B — — cos (A -f C). 94 PLANE TRIGONOMETRY 89. Relations of the Functions. Certain relations between tbe func- tions have already been proved to exist in the case of acute angles (§§ 13, 14), and since the relations of the functions of any angle to the functions of an acute angle have also been considered (§§ 80, 85, 86, 88), it is evident that the laws are true for any angle. These laws are so important that they will now be summarized, and others of a similar kind will be added. These laws should be memorized. They will be needed frequently in the subsequent work. The proof of each should be given, as required in § 14. The ± sign is placed before the square root sign, since we have now learned the nooning of negative functions. To find the sine we have : 1 sm X — CSC X To find the cosine we have : 1 cos X = sec a: sin X = ± Vl— cos^x cos X = ± V 1 — sin^x To find the tangent we have : tan X = — ; — cot X , sinx tan X = ± —i=== V 1 — sin^x tan X = ± Vsec^x — 1 To find the cotangent we have : cot X - tan X cot X — ± cos X VT cos“x cot X = ± Vcsc^x — 1 To find the secant we have : 1 sec X = cos X To find the cosecant we have : 1 CSCX = sin X , sinx tanx = cos X , Vl — cos^x tanx = ± I- COSX iCi_ I tan X = sin x sec x f .. ^ cos X cot X = Sin X , Vl — siVx cot X = + ^ sm X cot X = cos X CSC X sec x = ± Vl -f- taVx CSC X — ± Vl-|- cot^x FUNCTIONS OF ANY ANGLE 95 Exercise 40. Relations, of the Functions 1. Prove each, of the formulas given in § 89. Prove the following relations : 2. sin a; ± 3. cos X = ± tana: Vl + tan^a: cot a: Vl+ cot^a; 6. Find sin x in terms of cot x. 7. Find cos x in terms of tan x. Prove the following relations : 10. tan X cos x = sin x. 11. cos^a; = cot^a; — cot^a: cos^a:. 12. tw?x — sin^a: + sin^x tan^x. 13. cos^x + 2sin^x =1 + sin^x. 4. tan X = ± 5. cot X = ± V csc^x — 1 1 Vsec^x —1 8. Find sec x in terms of sin x. 9. Find esc x in terms of cos x. 14. cot^x = cos^x + cos^x coUx. 15. cot^x sec^x = 1 + cot"x. 16. csc^x — cot^x = 1. 17. sec^x + csc^x = sec^x csc^x. y 18. Show that the sum of the tangent and cotangent of an angle is equal to the product of the secant and cosecant of the angle. Recalling the values given on page 8, find the value of x when : 19. 2cosx = secx, 25. tanx = 2sinx. 20. 4sinx = cscx. 26. sec x = V2 tan x. 21. sin^x = 3 eos^x. ■ '27. sin^x — cos x = 22. 2 sin^x + cos^x = f. 28. tan"x — secx = 1. 23. 3 tan^x — sec'^x = 1. 29. tan^x + csc'^x = 3. 24. tan x + cot x = 2. M 30. sin x + VS cos x = 2. '^31. Given (sin x + cos x)^ — 1 = (sin x — cos x)‘^ + 1, find x. 32. Given 2 sin x = cos x, find sin x and cos x. 33. Given 4 sin x = tan x, find sin x and tan x. 34. Given 5 sin x = tan x, find cos x and sec x. 35. Given 4cotx tanx, find the other functions. 36. Given sin x = 4 cos x, find sin x and cos x. 37. If sin X : cos x = 9 : 40, find sinx and cos x. sinx 38. From the formula tanx = ± under which tan x = sin x. Vl find the condition sm'x 96 PLANE TEIGONOMETPY Solve the following equations ; that is, find the value of x when : 44. 2 cos X + sec a: = 3. 45. cos^a: — sin^a: = sin x. 46. 2 sin a: + cot a: = 1 + 2 cos x. 47. sin^a; + tan^x = 3 cos^x. 48. tan X + 2 cot x = f esc x. 39. cos X = seex. 40. cos X = tan x. 41. cos X = sin X. 42. tan X = cot X. 43. sec X = esex. Prove the following relations : 49. sin^ +COS (l-|-tanyl)cos4. 51. cos x ; cot x = Vl — cos'^ x. 50. cotx = Vl + cot'^ 52. taVx cos X ■ cos^x Find the values of the other functions of A when: - 1 . 53. sin ^ = f. 54. cos ^ = f . 55. tanyl = 1.6. 56. cot = 0.75. 57. sec^ = 1.5. 58. sin^ = 59. sinyl = 0.8. 60. cos^ = fy. 61. cos .4 =0.28. 62. tan = f. 63. cot^ = l. 64. cot^l=0.5. 65. sec A=2. 66. CSC A = V2. 67. sin .4 - m. 68. Given sin ^ = 2 m : (1 + find the value of tand. 69. Given cos A = 2 mn : (rrd‘ + V), find the value of sec.4. 7 0. Given sin 0° = 0, find the other functions of 0°. 71. Given sin 90° = 1, find the other functions of 90°. 72. Given tan 90° = oo, find the other functions of 90°. 73. Given cot 22° 30' = V2 + 1, find the other functions of 22° 30'. 74. Write taV^ + cot^d so as to contain only cosd. In the triangle ABC, prove the following relations : 75. sin d = sin(£ + C). 83. sin d = — eos(|d+.^54-.lC). 76. cos d = — cos (A + C). 84. cosd = — cos(2 d + .B + C). 77. tand =— tan(5+ C). 85. cosd = sin(fd + 1-5 + 78. cotd = — cot(5 + C). 86. sin (-Id 4-5) = cos (-15 — -I C). 79. sin d =— sin (2d 4-54- C). 87. sin(l-C— Id) =— cos (-1-54-5). 80. sin 5 = — sin (d 4-254- C). 88. cos5 = — cos(d 4- 2 5 4- C). 81. cos 5 = — cos (d 4-54-25). 89. tand = tan(2 d 4- 5 4- 5). 82. cot 5 = cot (d 4-2 5 4- 5). 90. cot d = tan (|-54-f 5 4- 1-d). In the quadrilateral ABCD, prove the following relations : 91. — sin d = sin (5 4- 5 4- 5). 93. — tand = tan (5 4- 5 4- 5). 92. cos d= cos (5 4- 5 4- 5). 94. — cotd = cot(5 4- 5 -f -D)- CHAPTER VI FUNCTIONS OF THE SUM OR THE DIFFERENCE OF TWO ANGLES 90. Formula for sin(jf + y). In this figure there are shown two acute angles, x and y, with Z.AOC acute and equal to x-\-y, two perpendiculars are let fall from C, and two from D, as shown. Then by geometry the triangles CGD and EOD are similar and hence Z.GCD - Z.EOD = X. Considering the radius as unity, OD = cos y and CD - sin y. Hence we have sin (x A-y)= CF = DE + CG. ^ . DE , But sin X = j whence DE = sin x- OD OD = sin X cos y ; CG , and cos x = > whence CG= cos x ■ CD = cos X sin y. Hence sin (x-\-y) = sin jt cos y + cos x sin y. This is one of the most important formulas and should he memorized. For example, sin (30° + 60°) = sin 30° cos 60° + cos 30° sin 60° Vs Vs which we have already found to he sin 90°. 1 1 _ 1 3_ 2 ’ 91. Formula for cos (x-\- y). Using the above figure we see that cos (x y)= OF = OE — DG. OE But cos X — , whence OE = cos x ■ OD = cos x cos y ; DG and sin x = , whence DG = sin x- CD = sin x sin y. Hence cos (jr + y) = cos x cos y — sin x sin y. This important formula should he memorized. For example, cos (45° + 45°) = cos 45° cos 45° — sin 45° sin 45° _ 1 ^ V2 V2 V 2 V 2 2 2 which we have already found to he cos 90°. 97 98 PLANE TKIGONOMETRY 92. The Proofs continued. In the proofs given on page 97, x, y, and X + y were assumed to be acute angles. If, however, x and y are acute but x + y is obtuse, as shown in this figure, the proofs remain, word for word, the same as before, the only differ- ence being that the sign of OF will be nega- tive, as Ziff is now greater than OE. This, ^ however, does not affect the proof. The above formulas, therefore, hold true for aU acute angles x and y. Furthermore, if these formulas hold true for any two acute angles X and y, they hold true when one of the angles is increased by 90°. Thus, if for x we write x' = 90° -f- x, then, by § 87, sin (x' + y) = sin (90° -j- x + y) = cos (x + y) = cos X cos y — sin x sin y. But by § 87, cos x = sin (90° + x) = sin x\ and sin x = — cos (90° + x) = — cos x'. Hence, by substituting these values, sin (x' -t- y) = sin x' cos y -f cos x' sin y. That is, § 90 holds true if either angle is repeatedly increased by 90°. It is therefore true for all angles. Similarly, by § 87, cos (x' y) = cos (90° + X -(- y) = — sin(x + y) = — sin X cos y — cos x sin y = cos x' cos y — sin x' sin y, by substituting cos x' for — sin x and sin x' for cos x as above. That is, § 91 also holds true if either angle is repeatedly increased by 90°. It is therefore true for all angles. Exercise 41. Sines and Cosines Given sin 30° = cos 60° = ^, cos 30° = sin 60° = and sin 43° — cos 45° = j v^, find the values of the following : 1. sin 15°. 2. cos 15°. 3. sin 75°. 4. cos 75°. 6. sin 90°. 6. cos 90°. 7. sin 105°. V '8. cos 105°. 9. sin 120°. 10. cos 120°. 11. sin 135°. 12. COS 135°. 13. sin 150'. 14. COS 150°. 15. sin 105°. 16. cos 166°. SUM OR UIRFERENCE OF TWO ANGLES 99 93. Formula for tan (jr + y). Since tan A = tan (x y) = sin A therefore cos A sin (x + y) _ sin x cos y + cos x sin y cos (x + y) cos X cos y — sin x sin y whatever the size of the angles x and y 92). Dividing each term of the numerator and denominator of the last of these fractions by cos x cos y, we have tan {x y) - sin X , sin y 1 cos X cos y 1 - sina; siny cos X cos y But since we have sin a; , ^ sin?/ = tan X, and = tan y, Gosx cosy , . tan X 4- tan y '' 1 — tan X tan y This important formula should be memorized. 94. Formula for cot ('jr+y'). Since cotA= — -j therefore ^ ^ sm A , , ^ cos (x 4- ?/) cos X cos y — sin x sin y cot (x + 'll) = - 7—7 : — - > sin(a: + y) sin x cos y + cos x sin y whatever the size of the angles x and y (§ 92). Dividing each term of the numerator and denominator of the last of these fractions by sin x sin y, and then remembering that cos X ^ ^ cos y , , — = cot a: and ^ = cot y, we have sin X sin y cot(jf+y) = cot Jf cot y — 1 cot y + cot a: This important formula should be memorized. Exercise 42. Tangents and Cotangents Given tan 30° = cot 60° = | cot 30° = tan 60° = tan 45° = cot 45° = i, find the values of the following : 1. tan 15°. y 2. cot 15°. 3. tan 75°. 4. cot 75°. 5. tan 90°. 6. cot 90°. 7. tan 105°. 8. cot 105°. 9. tan 120°. 10. cot 120°. 11. tan 135°. 12. cot 135°. 13 . tan 150°. 14 . cot 150°. ^15. tanl65°i 16 . cot 165° 100 PLANE TKIGONOMETEY 95 . Formula for sin (x — y). In this figure there are shown two acute angles, A OB = x and COB = y, with /LAOC equal tox — y, two per- pendiculars are let fall from C, and two from D. The perpendiculars from D are BE and BG, BG being drawn to FC produced. Then, considering the radius as unity, we have sin (x — y) = CF= BE - CG. But DE = sin x • OD = sin x cos y, and GC = cos x • CD = cos x sin y. Hence, by substituting these values of DE and GC, sin (x—y) = sin xcosy — cos x sin y. This is one of the most important formulas and should be memorized. 96 . Formula for cos (x—y). Using the above figure we see that cos (x — y)= OF — OE -f- DG. But OE — cos X • OD = cos x cos y, ^ and DG = sin x • CD = sin x sin y. Hence it follows that cos (x — y)z= cos jr cos -p sin x sin y. This important formula should be memorized. The proof in §§ 95 and 96 refers only to acute angles, but the formulas are entirely general if due regard is paid to the algebraic signs. The general proof may follow the method of § 92, or it may be based upon it; the latter plan is followed in § 97. 97 . The Proofs continued. Since cc = (x — y) -p y, we see that sin X = sin {(x — y) + y} = sin (x — y) cos y -p cos (x — y) sin y, cos X = cos {(x — y) + y} = cos (x — y) cos y — sin (x — y) sin y. Multiplying the first equation by cos y, and the second by sin y, sin X cos y = sin (x — y) cos^y -p cos (x — y) sin y cos y, cos X sin y = — sin (x — y) sin^y + cos (x — y) sin y cos y. Hence sin x cos y — cos x sin y = sin (x — y) (sin^y -p cos'^y). But by § 14 sin^y -p cos^y = 1. Therefore sin (x — y) = sin x cos y — cos x sin y. Similarly, cos (x — y) — cos x cos y + sin x sin y. Therefore the formulas of §§95 and 96 are tmiversally true. SUM on DIFFERENCE OF TWO ANGLES 101 sin A 98. Formula for tan (x — y). Since tan /I = ' - ^ cos A i/ tai.(»=-y)=Sl&^ we have cos {x — y) _ sin X cos y — cos x sin y cos X cos y + sin x sin y Dividing numerator and denominator by cos x cos y, as in § 93, we obtain sin x sin y cos X cos y tan (x — y~) = 1 + sin X sm y cos X cos y m, a. • a. / X tan j: — tan 1 / That IS, tan (x—y) = . ’ •' l+tanxtanz/ This important formula should be memorized. 99. Formula for cot (x— jr). we niav show that cot (x — y) = Following the plan suggested in § 98, cos (x- — ij) sin (x — y) cos X cos y -f- sin x sin y sin X cos y — cos x sin y cos X sin X cos y sin 7/ cosy sin y cos X sin X That is. cot {x — y) = cot xcotyA - 1 cot y —cot X / This important formula should be memorized. 100. Summary of the Addition Formulas. The formulas of §§ 90-99 may be combined as follows : sin {x ±_y) — sin a; cos y ±_ cos a; sin y, cos (a; ± y) = cos x cos y zf sin x sin y, tan(a: ± y) = tan X ± tan y 1 ^ tan X tan y cot (x ±y)^ cot X cot y ^ 1 cot y ± cot X When the signs ± and occur in the same formula we should be careful to take the — of T with the + of ± . That is, the upper signs are to be taken together, and the lower signs are to be taken together. 102 PLANE TRIGONOMETRY Exercise 43. The Addition Formulas G-iven sinx = j, cosx = j^, siny = cosy = find the value of: 1. sm(£c + y). 3. cos(cc + y). 5. tan(x + ?/). 2. sin {x — y). 4. cos {x — y). 6. tan (a; — y). By letting x = 90° in the formulas, find the following : 7. sin(90°-?/). 8. cos(90°-?/). 9. tan(90°-y). Similarly, hy substituting in the formulas, find the following , 10. sin(90'’ + ?/). 11. sin (180° — y) 12. sin(180° + ?/) 13. sin (270° — y) 14. sin(270° + y) 15. sin(360° — ?/) 16. sin(360° + y) 24. sin (— y). 25. sin (45° — y). 26. cos (45° — y). 27. tan(45° — y), 28. cot(30° + y). 29. cot (60° — y). 30. cot (90° — y). 17. cos {x — 90°). 18. cos (x — 180°). 19. cos (x — 270°). 20. tan (x — 90°). 21. tan (a: — 180°). 22. cot (x — 90°). 23. cot (x — 180°). 31. If tan X = 0.5 and tan y = 0.25, find tan (x + y) and tan (x — y) 32. If tan a: = 1 and tan y = ^ Vs, find tan (x + y) and tan(a; — y). 33. If tan x — ^ and tan y = -^, find tan (x + y) and tan (x — y), and find the number of degrees in a: + y. 34. If tana: = 2 and tant/ = what is the nature of the angle X y? Consider the same question when tan x — d and tan y = \, and when tan x = a and tan y = 1/a. 35. Prove that the sum of tan (x — 45°) and cot (x + 45°) is zero. 36. Prove that the sum of cot (x — 45°) and tan (x + 45°) is zero. 37. If sin X = 0.2 Vs and sin y — 0.1 VTo, prove that x-\-y = 45° May a: + y have other values ? If so, state two of these values. 38. Prove that if an angle x is decreased by 45° the cotangent of the resulting angle is equal to — a; -f- 1 _ 39. Prove that if an angle x is increased by 45° the cotangent of the resulting angle is equal to — r’ cot a: + 1 ^ and tan y = ^ > prove that tan (a; + y) = 1. 40. If tana: = 1 a l+2a 41. If a righr angle is divided into any three angles x, y, 1 — tan y tan g prove that tan x = tan y + tan z SUM OE DIFFERENCE OF TWO ANGLES 103 101. Functions of Twice an Angle. By substituting in the formulas for the functions of x + y we obtain the following important for- mulas for the functions of twice an angle : sin 2 jr = 2 sin x cos x, cos 2 jr = cos^ X — sin*^ x, tan 2 x = 2 tan jc 1 — tan^ X ’ cot 2 jr = cot^ Jf — 1 2 cot Jf Letting 2x = y we have the following useful formulas : sin y = 2 sin ^ y cos y, cos y = cos '^ — sin^ ^ y, tany = 2 tan ^ y 1 — tan^ i y coty = cot^ \y — 1 2 cot ^ 1 / Exercise 44. Functions of Twice an Angle As suggested above, deduce the formulas for the following : 1. sin 2 a;. 2. cos 2 a:. 3. tan 2 a;.- 4. cot 2 a:. Find sin2x, given the following values of sin x and cosx: 5. sin a; = ^ V2, cos a; = V2. 6. sin a: = cos a: = -I" Find cos 2 x, given the following values of sin x and cos x : 7. sin x = \ Vs, cos x = ^. 8. sin a; = f , cos a; = Find tan 2 x, given the following values of tan x : 9. tan X = 0.3673. 10. tan x = 0.2701. Find cot 2 x, given the following values of cot x and tan x : 11. cota; = 0.3673. 12. tan a; = 0.2701. Find sin 2 x, given the following values of sin x : 13. sina: = .j^. 14. sina: = ^. 16. As suggested in § 101, find sin 3 a: in terms of sin a:. 16. As suggested in § 101, find cos 3a; in terms of cosx. 104 PLANE TRIGONOMETRY 102. Functions of Half an Angle. If we substitute ^ z for x in the formulas cos^ x + sin"* x = 1 (§ 14) and cos^ x — sin^ x = cos 2 x (§ 101), so as to find the functions of half an angle, we have cos^ ^ z + sin^ ^z =1, and cos^ i — sin^ i s = cos z. Subtracting, 2 sin’^ ^z =1— cos z ; whence sill 2 zt N 2 In the above proof, if we add instead of subtract we have 2 cos^ ^z =1+ cos z ; whence cos |z = ±^ + cosz c- s. 1 sin 4 2 ! j cos-4^; , Since tan — > and cot -^z = — — — > we have, by dividing, COS ^ sm ^ tan-z = ±^ 1 — cos z 1 + cos z and cot cos z cos z These four formulas are important and should be memorized. From the formula for tan 4 s can be derived a formula which is occasionally used in dealing with very small angles. In the triangle ACB we have — tan-^ A — cos A + CCS A = ± N 1-^ c 1+5 c = ± N c — b c + 6 Exercise 45. Functions of Half an Angle Given sin 30° = find the values of the following : 1 . sin 15°. 2 . cos 15°. 3 . tanl5°. 4 . cotl5°. 6 . cot7-|°. Given tan 45° = 1, find the values of the following : 6 . sin22.5°. 7 . cos 22.5°. 8 . tan22.5°. 9 . cot22.5°. 10 . cotlli° 11. Given sinx = 0.2, find sin-|-x and cos4x. 12. Given cosx = 0.7, find sin 4x, cos -^x, tan ^x, and cot^x. SUM OR DIFFERENCE OF TWO ANGLES 105 103. Sums and Differences of Functions. Since we liave (§§ 92, 97) sin (cc + 1 /) = sin x cos y + cos x sin y, and sin (x — y')= sin x cos y — cos x sin y, we find, by addition and subtraction, that sin (x -\- y)-\- sin (x — y) — 2 sin x cos y, and sin (x + y) — sin (x — y)= 2 cos x sin y. Similarly, by using the formulas for cos (x ± y), we obtain cos (x + y) + cos (x — y)z= 2 cos x cos y, and cos (x y')— cos (x — y) — — 2 sin x sin y. By letting x y = A, and x — y — B, we have x = ^(A + B), and y = ^(A — -B), whence sinA + sinR= 2 sini(A + R) cos |(A — 5), sin A — sin R= 2 cos i(A + R) sin ^(A — R), cosA + cosR= 2 cos |(A + R) cos i(A — R), and cos A — cos R = — 2 sin \(A + R) sin \(A — R). By division we obtain sin A + sinR . , . „ tan A (A + R) cot (A — R) : sin A — sin R ^ J tv y? 1 and since cot ^{A — B) — we have tan -^(A — R) sin A -f sin R tan|(A4-R) sin A — sin R tan | (A — R) This is one of the most important formulas in the solution of oblique triangles. Exercise 46. Formulas Prove the following formulas: 1. sin 2 X 2. cos 2 X 2 tan a; 1 + tan^cc 1 — tan^a; 1 + tan^a; 3. tan^a: = 4. cot^a: = sin a; 1 + cos X sin a: 1 — cos X If A, B, C are the angles of a triangle, prove that : 5. sin A + sinR + sin C = 4 cos A cos ^ R cos ^ C. 6. cos A + cosR + cosC = 1 + 4sin4A sin-|-R sin -I- (7. 7. tan A + tanR + tan C = tan A tanR tan C. 106 PLANE TPIGONOMETKY 8. Given tan = 1, find cos x. 9. Given cot ^x = VS, find sin x. -r, i. H oo sin 33° + sin 3° 10. Prove that tan 18 = cos 33° + cos 3° .^Lll. Prove that sin-^x ± cos-|-x = Vl ± sinx, .r, , , . tan X + tan y ^ 12. Prove that — ; ; — = + tanx tan w. cot X ± cot y 13. Prove that tan (45° — x) = 1 — tan X 1 + tan X .^14. In the triangle ABC prove that cot ^ A + cot j^B + cot Y G = cot cot j-B cot ^ C. Change to a form involving products instead of sums, and hence more convenient for computation by logarithms : 15. cot X + tan x. 20. 1 + tan x tan y. 16. cotx — tanx. 21. 1 — tanx tany. 17. cotx + tany. 22. cot x cot 2 / + 1. ^18. cotx — tan y. 23. cotxcoty — 1. 19. 1 — COS 2 X 1 + cos 2 X ^ 24 . tan X 4- tan y cot X + cot y / 26. Prove that tan x + tan y = ^ • cos X cos y -1 , , , sin (x — y) 26. Prove that cot y — cotx = — ^ sin X sin y 27. Given tan (x + y) = 3, and tan x — 2, find tan y. 28. Prove that (sin x + cos xf = 1 + sin 2 x. 29. Prove that (sin x — cos x)'^ = 1 — sin 2 x. 30. Prove that tan x + cot x = 2 esc 2 x. 31. Prove that cot x — tan x = 2 cos 2 x esc 2 x. 32. Prove that 2 sin^(45° — x) = 1 — sin 2 x. 33. Prove that cos 45° + cos 75° = cos 15°. 34. Prove that 1 + tan x tan 2 x = tan 2 x cot x — 1. Prove the folloiving formulas : ^35. (cos X + cos yf + (sin x + sin yf =2 + 2 cos (x — y). 36. (sin X + cos yf + (sin y + cos x)^= 2 + 2 sin (x + y). 37. sin (x + ?/) + cos (x — y) = (sin x + cos x) (sin y + cos y). 38. sin (x + y) cos y — cos (x + y) sin y — sin x. CHAPTER Vn THE OBLIQUE TRIANGLE 104. Geometric Properties of the Triangle. In solving an oblique triangle certain geometric properties are involved in addition to those already mentioned in the preceding chapters, and these should be recalled to mind before undertaking further work with trigono- metric functions. These properties are as follows : The angles opposite the equal sides of an isosceles triangle are equal. If two angles of a triangle are equal, the sides opposite the equal angles are equal. If two angles of a triangle are unequal, the greater side is opposite the greater angle. If two sides of a triangle are unequal, the greater angle is opposite the greater side. A triangle is determined, that is, it is completely fixed in form and size, if the following pa7'ts are given: 1. Two sides and the included angle. 2. Tivo angles and the included side. 3. Two angles and the side opposite one of them. 4. Two sides and the angle opposite one of them. 5. Three sides. The fourth case, however, will be recalled as the ambiguous case, since the triangle is not in general completely determined. If we have given and sides a and h in this figure, either of the triangles ABC &ndAB'C will satisfy the conditions. If a is equal to the perpendicular from C on AB, how- ever, the points B and B' will coincide, and hence the two triangles become congruent and the triangle is completely determined. The five cases relating to the determining of a triangle may be summarized as follows : A triangle is determined when three independent parts are given. This excludes the case of three angles, because they are not independent. That is, A = 180° — {B + C), and therefore A depends upon B and C. 107 108 PLANE TPIGONOMETEY 105. Law of Sines. In the triangle ABC, using either of the figures as here shown, we have the following relations. In the first figure, and in the second figure. ^ • r> - = smS, a - = sin(180° a B) = sin B. Therefore, whether h lies within or without the triangle, we obtain, by division, the following relation : a sinA b sinE In the same way, by drawing perpendiculars from the vertices A and B to the opposite sides, we may obtain the following relations : b sin£ c and sinC sin A sin C This relation between the sides and the sines of the opposite angles is called the Law of Sines and may be expressed as follows : The sides of a triangle are proportional to the sines of the opposit-e angles. If we multiply - = by 6, and divide by sin A, we have b sinJ? a _ b sin A sin 5 Similarly, we may obtain the following : a _ 6 _ c sin A sinB sin C and this is frequently given as the Law of Sines. It is also apparent that a sin B = b sin A, a sin C = c sin A, and b sin C = c sin B, three relations which are still another form of the Law of Sines. THE OBLIQUE TKIANGLE 109 106. The Law of Sines extended. There is an interesting extension of the Law of Sines with respect to the diameter of the circle circum- scribed about a triangle. Circumscribe a circle about the triangle ABC and draw the radii OB, OC, as shown in the figure. Let J? denote the radius. Draw OM perpendicular to BC. Since the angle BOC is a central angle intercepting the same arc as the angle A, the angle BOC = 2 A-, hence the angle BOM = A\ then Therefore In like manner, and Therefore BM — R smBOM = R sin^l. a = 2 R sin4. b = 2 R sin B, c = 2 R sin C. a b c 2R = sinG sin 5 sinC A That is, The ratio of any side of a triangle to the sine of the oppo- site angle is numerically equal to the diameter of the circumscribed circle. Exercise 47. Law of Sines y 1. Consider the formula % — when B = 90° ; when A = 90° : 1 „ 1 , 5 sin 5 ’ when A — B-, when a = b. ^ 2. Prove by the Law of Sines that the bisector of an angle of a triangle divides the opposite side into parts proportional to the adjacent sides. 3. Prove Ex. 2 for the bisector of an exterior angle of a triangle. //d. The triangle ABC has A = 78°, B = 72°, and c = 4 in. Find the diameter of the circumscribed circle. ^ 5. The triangle ABC has A = 76° 37', B = 81° 46', and c = 368.4 ft. Find the diameter of the circumscribed circle. 6. What is the diameter of the circle circumscribed about an equi- lateral triangle of side 7.4 in. ? What is the diameter of the circle inscribed in the same triangle ? l/ 7. What is the diameter of the circle circumscribed about an isos- celes triangle of base 4.8 in. and vertical angle 10° ? \js. What is the diameter of the circle circumscribed about an isos- celes triangle whose vertical angle is 18° and the sum of the two equal sides 18 in. ? 110 PLAJSTE TEIGONOMETRY 107 . Applications of the Law of Sines. If we have given any side of a triangle, and any two of the angles, we are able to solve the tri- angle by means of the Law of Sines. Thus, if we have given a, A, and B, in this triangle, we can find the remaining parts as follows : 1 . 2 . 3. C = 180° - (A + B). c sin C a sin C a - — .■.c = — — — -xsinC. a sinA sinA sinA For example, given a = 24.31, A = 45° 18', and B = 22° 11', solve the triangle. The work may be arranged as follows : a = 24.31 A = 45° 18' B= 22° 11' A + E = 67° 29' .*. C = 112°31' log a = 1.38578 colog sin A = 0.14825 log sinE = 9.57700 logi = 1.11103 .-. b = 12.913 = 1.38578 = 0.14825 log sin C = 9.96556 log c = 1.49959 .-. c = 31.593 When — 10 is omitted after a logarithm or cologarithm to which it belongs, it must still be remembered that the logarithm or cologarithm is 10 too large. The length of a having'been given only to four significant figures, the values of h and c are to be depended upon only to the same number of significant figures in practical measurement. In the above example a is given to only four significant figures, and hence we say that 6 = 12.91, and c = 31.59. Exercise 48. Law of Sines Solve the triangle ABC, given the following parts : . 1. a = 500, A = 10° 12', B = 46° 36'. 2. a = 795, A = 79° 59', S = 44° 41'. 3. a = 804, A = 99° 55', B = 45° 1'. 4. a = 820, A = 12° 49', B = 141° 59'. 6. c = 1005, A = 78° 19', B = 54° 27'. 6. 5 = 13.57, B = 13° 57', C = 57° 13'. 7. a = 6412, A = 70° 55', C = 52° 9'. Y8. b = 999, A = 37° 58', C = 65° 2'. THE OBLIQUE TRIANGLE 111 Solve Exs. 9 -14 without using logarithms : 9. Given i = 7.071, A = 30°, and C = 105°, find a and c. 10. Given c = 9.562, A = 45°, and B = 60°, find a and b. i/ll. The base of a triangle is 600 ft. and the angles at the base are 30° and 120°. Find the other sides and the altitude. ^l2. Two angles of a triangle are 20° and 40°. Find the ratio of and the side oppo- / site the smallest angle is 3. Find the other sides. 14. Given one side of a triangle 27 in., and the adjacent angles each equal to 30°, find the radius of the circumscribed circle. 15. The angles B and C of a triangle ABC are 50° 30' and 122° 9’ respectively, and RC is 9 mi. Find AB and AC. 16. In a parallelogram, given a diagonal d and the angles x and y which this diagonal makes with the sides, find the sides. Compute the results when d — 11.2, x = 19° 1', and y = 42° 54'. 17. A lighthouse was observed from a ship to bear N. 34° E.; after the ship sailed due south 3 mi. the lighthouse bore N. 23° E. Find the distance from the lighthouse to the ship in each position. The phrase to hear N. 34° E. means that the line of sight to the lighthouse is in the northeast quarter of the horizon and makes, with a line due north, an angle of 34°. / 18. A headland was observed from a ship to bear directly east ; ; after the ship had sailed 5 mi. N. 31° E. the headland bore S. 42° E. FindAhe distance from the headland to the ship in each position. 19. In a trapezoid, given the parallel sides a and b, and the angles X and y at the ends of one of the parallel sides, find the nonparallel sides. Compute the results when a = 15, b — 7, x = 70°,y = 40°. 20. Two observers 5 mi. apart on a plain, and facing each other, find that the angles of elevation of a balloon in the same vertical I plane with themselves are 55° and 58° respectively. Find the dis- tance from the balloon to each observer, and also the height of the balloon above the plain. 21. A balloon is directly above a straight road 7^ mi. long, joining two towns. The balloonist observes that the first town makes an angle of 42° and the second town an angle of 38° with the perpen- dicular. Find the distance from the balloon to each townj and also the height of the balloon above the plain. / ^ . the opposite sides, v/ 131 'i'he^ angles of a triangle are as 5:10: 21, 112 PLANE TRIGONOMETRY 108 . The Ambiguous Case. As mentioned in § 104, if two sides of a triangle and the angle opposite one of them are given, the solu- tion will lead, in general, to two triangles. Thus, if we have the two sides a and b and the angle A given, we proceed to solve the triangle as follows : C = 180° - (d + B) ; hence we can find C if we can find B. „ c sin C I urthermore, - = ? a sin A , a sin C whence c = — : — — ; sm A hence we can find c if we can find C, and we can also find c if we can find B. But to find B we have sin B _h sin A a , . Zisind whence sin B = — • a Therefore we do not find B directly, but only sin B. But when an angle is determined by its sine, it admits of two values which are supplements of each other (§ 86) ; hence either of the two values of B may be taken unless one of them is excluded by the conditions of the problem. In general, therefore, either of the triangles ABC and AB'C fulfills the given conditions. Exercise 49. The Ambiguous Case In the triangle AB C given a, b, and A, prove that : 1. If a > h, then d > R, A is acute, and there is one and only one triangle which will satisfy the given conditions. 2. If a = h, both A and B are acute, and there is one and only one triangle which will satisfy the given conditions, and this triangle is isosceles. 3. If a < h, then d must be acute to have the triangle possible, and there are in general two triangles which satisfy the given conditions. 4. If a = 6 sind, the required triangle is a right triangle. 6. If a. 0, then sin 5 > 1, and hence the triangle is impossible. If log sin 5 < 0, there is one solution when a>b ; there are two solutions when a 90°. Since a 90°, 5 must also be greater than 90°. But a triangle cannot have two obtuse angles. Therefore the triangle is impossible. 2. Given a = 36, b 80, and A = 30°, find the remaining parts. Here we have hsinH = 80 x | = 40 ; so that a < hsinJ. and the triangle is impossible. Draw the figure to illustrate this fact. 3. Given a = 25, b = 50, and A = 30°, find the remaining parts. Here we have b sin ^ = 60 x ^ = 25 ; but a is also equal to 25. Hence 5 must be a right angle. ABC is therefore a right triangle and there is only one solution. 4. Given a = 30, b = 30, and A = 60°, find the remaining parts. Here we have a = b, and A an acute angle. Hence there is one solution and only one. It is evident, also, that the triangle is not only isosceles but equilateral. 6. Given a = 3.4, b = 3.4, and A = 45°, find the remaining parts. Here we have a = b, and A an acute angle. Hence there is one solution and only one. It is evident, also, that the triangle is not only isosceles but right. 114 PLANE TKIGONOMETRY 6. Given a = 72,630, h = 117,480, and A = 80° 0' 50", find B, C, and c. log h = 5.06997 Here log sin 5 > 0. log sin A = 9.99337 Therefore sin iJ>l, which is impossible. colog a = 5.13888 log sin B = 0.20222 Therefore there is no solution. 7. Given a = 13.2, h = 15.7, and A = 57° 13' 15", find B, C, and c. logs = 1.19590 log sin A = 9.92467 colog a = 8.87943 log sin B = 0.00000 J5 = 90° C = 32° 46' 45" c = h cos A log h = 1.19590 log cos A = 9.73352 logc = 0.92942 c = 8.5 Therefore there is one solution. Since B = 90°, the triangle is a right triangle. 8. Given a = 767, S = 242, and A = 36° 53' 2", find B, C, and c. logs = 2.38382 log sin A = 9.77830 colog a ■ 7.11520 log sin B = 9.27732 .-.5= 10° 54' 58" .-. C=132°12'0" log a = 2.88480 log sin C = 9.86970 colog sin A = 0.22170 logc = 2.97620 c = 946.68 = 946.7 Here a > S, and log sin .B < 0. Therefore there is one solution. 9. Given a = 177.01, S = 216.45, and A = 35° 36' 20", find the other parts. logs = 2.33536 log sin A = 9.76507 colog a = 7.75200 log sin B = 9.85243 A = 45° 23' 28" or 134° 36' 32" e=99°0'12"or 9° 47' 8" log a = 2.24800 log sin C = 9.99462 colog sin A = 0.23493 logc = 2.47755 2.24800 9.23035 0.23493 1.71328 c = 300.29 or 51.675 = 300.29 or 51.68 Here a < b, and log sin B < 0. Therefore there are two solutions. THE OBLIQUE TEIANGLE 115 Exercise 50. The Oblique Triangle Find the number of solutions, given the following : 1. a = 80, b = 100, A = 30°. 2. a = 50, b = 100, A = 30°. 3. a = 40, b = 100, A = 30°. 4. (z = 100, b = 100, A = 30°. 5. a = 13.4, b = 11.46, A = 77° 20'. 6. a = 70, b = 75. A = 60°. 7. a = 134.16, b = 84.54, B = 52° 9'. 8. a = 200, b = 100, A = 30°. the triangles, given the following : 9. a = 840, b = 485, A = 21° 31'. 10. a = 9.399, b = 9.197, A = 120° 35'. 11. a = 91.06, b = 77.04, A = 51° 9'. 12. a = 55.55, b = 66.66, B = 77° 44'. 13. a = 309, b = 360, A = 21° 14'. 14. a = 34, b = 22, B = 30° 20'. 15. b = 19, c = 18. C=15° 49'. 16. a = 8.716, b = 9.787, A = 38° 14' 12' 17. a = 4.4, b = 5.21, A =57° 37' 17 18. Given a = 75, b = 29, and B = 16° 15', find the difference be- tween the areas of the two triangles which meet these conditions. 19. In a parallelogram, given the side a, a diagonal d, and the angle A made by the two diagonals, find the other diagonal. As a special case consider the parallelogram in which a = 35, d = 63, and A = 21° 36'. 20. In a parallelogram ABCD, given AD = 3 in., BD = 2.5 in., and A = 47° 20', find AB. 21. In a quadrilateral ABCD, given AC = 4 in., /LB AC = 35°, AB = 75° 20', Ad = 38° 30', and ABAD = 70° 40', find the length of each of the four sides. 22. In a pentagon ABCDE, given AA = 110° 50', AB = 106° 30', ZE = 104°10', ABAC = 3i)°, ADAE = 34.° 5&, AADC = 52° 30', and AC = 6 in., find the sides and the remaining angles of the pentagon. 116 PLANE TPIGONOMETEY 111. Law of Cosines. This law gives the value of one side of a triangle in terms of the other two sides and the angle included between them. In either figure, + BD-. In the first figure, B' —c—AD. In the second figure, BD =AD — c. In either case, BD^ =AD^ — 2 c x AD + Therefore, in all cases, -\- AD^ — 2 c x AD. Now K^-\-AD^ — V^, and AD = b cos^l. Therefore a* = 6^ + — 2 &c cos A. In like manner it may be proved that b“^ — + a? — 2 ca cosB, and (^ = + b^ — 2 ab cos C. The three formulas have precisely the same form, and the Law of Cosines may be stated as follows : The square on any side of a triangle is equal to the sum of the squares on the other two sides diminished hy twice their product into the cosine of the included angle. It will be seen that if A = 90°, we have = 6- + — 2 6c cos 90° = 62 + c2. In other words we have the Pythagorean Theorem as a special case. Hence this is sometimes called the Generalized Pythagorean Theorem. It will also be seen that the law includes two other familiar propositions of geometry, one of which is the following : In an obtuse triangle the square on the side opposite the obtuse angle is equivalent to the sum of the squares on the other two sides increased by twice the product of one of those sides by the projection of the other upon that side. This and the analogous proposition are given as exercises on page 117. THE OBLIQUE TRIANGLE 117 Exercise 51. Law of Cosines 1. Using the figures on page 116, prove that, whether the angle B is acute or obtuse, c = a cos B -\-h cos^l. 2. What are the two symmetrical formulas obtained by changing the letters in Ex. 1 ? What does the formula in Ex. 1 become when R = 90° ? 3. Show that the sum of the squares on the sides of a triangle is equal to 2(ab cos C + be cos J + ca cosR). 4. Consider the Law of Cosines in the case of the triangle a = 5, b = 12, c = 6. 5. Given a = 5, b = 5, and C = 60°, find c. 6. Given a ~ 10, b — 10, and C = 45°, find c. 7. Given a = S, b = 5, and C — 60°, find c. 8. From the formula = b^ + — 2 be cos A deduce a formula for cosd. From this result find the value of A when b^ z= a^. 9. Prove that if right. cos A cos B a the triangle is either isosceles or ^ cos A , cos B cos C 10. Prove that 1 ; 1 = — ^ -- — a b c 2 abc 11. Prove that — cos A + t cos B + — cos C = abc 2 abc 12. From the Law of Cosines prove that the square on the side op- posite an acute angle of a triangle is equal to the sum of the squares on the other two sides minus twice the product of either side and the projection of the other side upon it. 13. As in Ex. 12, consider the geometric proposition relating to the square on the side opposite an obtuse angle. 14. In the parallelogram ABCD, given AB = 4 in., AD = 5 in., and A = 38° 40', find the two diagonals. 15. In the parallelogram ABCD, given AB = 7 in., AC = 10 in., and /.BAC = 36° 7', find the side BC and the diagonal BD. 16. In the quadrilateral ABCD, given AC = 3.6 in., AZ) = 4 in., BC = 2.4 in., Z.ACB = 29° 40', and Z. CAD = 71° 20', find the other two sides and all four angles of the quadrilateral. 17. In the pentagon ABCDE, given AB= 3.4 in., AC = 4.1 in., AZ> = 3.9in., AE=2.2in., ZBAC = ?,^°V, Z CAD = 41° 22', and ZDAE = 32° 5', find the perimeter of the pentagon. 118 PLANE TRIGONOMETRY iin -r , ^ 0(- ® sin 4 112. Law of Tangents. Since t = ^ > ^ b sinR follows by the theory of proportion that by the Law of Sines, it a — b _ sin 4 — sinR a b sinA + sinR This is easily seen without resorting to the theory of proportion. For, since a sin J5 = 6 sin A (§ 105), we have a sinil — 6 sinA = 6 sinA— a sinB Adding, a sin A — 6 sin i? = a sin A—h sin B a sinA + asinB — hsin A — 6sinJ5 = a sinA — a sin5 + 6 sin A — 6 sin 5, or (a — h) (sin A + sin B) = {a + 6) (sin A — sin B ) ; , , .... a — b sin A — sin B whence, by division, a + 6 sin A + sin B But by § 103, Therefore sinA — sinR _ tan 1 (A — B) sin A + sin R tan ^ (A + R) a — b tan |(A — R) a-\- b tan |(A + R) By merely changing the letters. a — c _ tan ^ (A — C) a + c tan ^ (A + O) ’ and ^^ tan^(R-C )^ b + c tan^(R+C) Hence the Law of Tangents : The difference between two sides of a triangle is to their sum as the tangent of half the difference between the ojojoosite angles is to the tangent of half their sum. In the case of a triangle, if we know the two sides a and b and the included angle C, we have our choice of two methods of solving. From the Law of Cosines we can find c, and then, from the Law of Sines, we can find A and R. Or we can find A B by taking C from 180°, and then, since we also know a-\-b and a — b, we can find A — B. From A + R and A — R we can find A and R. This second method is usually the simpler one. If 6 > a, then R > A . The formula is still true, but to avoid negative numbers the formula in this case should be written b — a _ tan ^ (R — A) 6 + a tan ^ (R + A) THE OBLIQUE TRIANGLE 119 Exercise 52. Law of Tangents Find the form to which a^h tan 1(A — B) tan 1 (A + B) reduces when : 1. C= 90°. 3. A=B^C. 2. a = b. A. A -B = 90°, and B = C, Prove the following formulas ; b - c 7. 8 . 9. 10 . = tan^(R-C)cot^(R + C'). b — c ■ cot ^A. b + c tan \(B~C) , ^ ^ ^ b c a -\-b _ cot ^{A~B) a — b cot -^ ( A + A) sin A 4- sinR _ tan ^ (A + B) sin A — sin R tan-|-(A— R) sinR + sin C _ 2 sin (R + C) cos ^(B — C) sin R — sin C 2 cos ^ (R + C) sin (R — C) sin A + sinR sin A — sinR = tan ^ (A + R) cot ^ (A — R). 11. To what does the formula in Ex. 8 reduce when A = R ? 12. To what does the formula in Ex. 9 reduce when B — C = 60° ? 13. To what does the formula in Ex. 10 reduce when the triangle is equilateral ? 14. To what does the Law of Tangents, in the form stated at the top of this page, reduce in the case of an isosceles triangle in which a = b? What does this prove with respect to the angles opposite the equal sides ? 15. By the help of the Law of Tangents prove that an equilateral triangle is also equiangular. 16. By the help of the Law of Tangents prove that an equiangular triangle is also equilateral. . 17. Given any three sides and any three angles of a quadrilateral, show how the fourth side and the fourth angle can be found. Show also that it is not necessary to have so many parts given, and find the smallest number of parts that will solve the quadrilateral. 18. What sides, what diagonals, and what angles of a pentagon is it necessary to know in order, by the aid of the Law of Tangents alone, to solve the pentagon ? 120 PLANE TPIGONOMETKY 113. Applications to Triangles. The Law of Cosines and the Law of Tangents are frequently used in the solution of triangles. This is particularly the case when we have given two sides, a and h, and the included angle C. There are two convenient ways of finding the angles A and B, the first being by the Law of Tangents. This law may be written tan ^(A—B)= X tan J (A + A). Since -^(A + 5) = ^(180°— C), the value of ^(A + A) is known, so that this equation enables us to find the value of ^(A—B). We then have ^(^a+B) + i(A—B) = A, and ^(A+B) — ^(A—B)=B. The second method of finding A and B is as follows : In the above figure let BD be perpendicular to A C. BD BD Then Now and tanA = - — = AD AC— DC BD = a sin C, DC = a cos C. a sin C tanA b — a cos C Since A and C are now known, B can be found. This is not so convenient as the first method, because it is not so well adapted to work with logarithms. The side c may now be found by the Law of Sines, thus : a sin C b sin C c = — : — — j or c = — ^ — — • sin A sin A Instead of finding A and B first, and from these values finding c, we may first find c and then find A and B. To find c first we may write the Law of Cosines (§ 111) as follows : c = ^ o? b‘‘‘ — 2 oib cos C. Having thus found c, and already knowing a, 6, and C, we have a sin C . b sin C sin A = > sin5 = c c In general this is not so convenient as the first method given above, because the formula for c is not so well adapted to work with logarithms. THE OBLIQUE TKIAHGLE 121 114. Illustrative Problems. 1. Given C = 63° 35' 30", a =748, and b = 375, find A, B, and c. We see that a. + 5 = 1123, a~h— 373, and A -\-B = 180° — C = 116° 24' 30". Hence ^(A +B)=5S° 12' 15". log (a — b')— 2.57171 colog (a + Z») = 6.94962 log tan ^ (G -}- i?) z= 0.207 66 log tan ^ (A —B)= 9.72899 .-. ^(A-B)=28° 10' 54" log 5 = 2.57403 log sinC = 9.95214 colog sin B = 0.30073 log c = 2.82690 .-. c = 671.27 After finding —B) we combine this with j-(A +B) and find A = 86° 23' 9" and B = 30° 1' 21". In the above example, in finding the side c we use the angle B rather than the angle A, because A is near 90°. The use of the sine of an angle near 90° should be avoided, because it varies so slowly that we cannot determine the angle accurately when the sine is given. 2. Given a. = 4, c = 6, and B = 60°, find the third side b. Here the Law of Cosines may be used to advantage, because the numbers are so small as to make the computation easy. We have b = Va^ + — 2 ac cos B = Vl6 + 36 — 24 = ; log 28 = 1.44716, log = 0.72358, = 5.2915 ; that is, to three significant figures, b = 5.292. Exercise 53. Solving Triangles Solve these triangles, given the folloiving parts : 1. II b = 83.39, C = 72° 15'. 2. b = 872.5, c = 632.7, A = 80°. 3. a =17, II C = 59° 17'. 4. II c = V3, A = 35° 53'. 6. a = 0.917, b = 0.312, C = 33° 7' 9". 6. a =13.715, c =11.214, A =15° 22' 36". 7. b = 3000.9, c= 1587.2, A = 86° 4' 4". 8. a - 4527, b - 3465, C = 66° 6' 27". 9. a = 55.14, b = 33.09, C = 30° 24'. 10. a = 47.99, b = 33.14, C=175° 19' 10". 11. a = 210, b =105, C = 36° 52' 12". 12. a = 100, b = 900, C = 65°. 122 PLANE TKIGONOMETEY Solve these triangles, given the following parts : 13. a = 409, &=169, C= 117.7°. 14. a = 6.25, 5 = 5.05, C= 105.77°. 15. a = 3718, 5 =1507, C = 95.86°. 16. a = 46.07, h = 22.29, C = 66.36°. 17 . h = 445, c = 624, A = 10.88°. 18. b = 15.7, c = 43.6, A = 57.22°. 19. If two sides of a triangle are each, equal to 6, and the in- cluded angle is 60°, find the third side by two different methods. 20. If two sides of a triangle are each equal to 6, and the in- cluded angle is 120°, find the third side by three different methods. 21. Apply the first method given on page 120"to the case in which a is equal to b ; that is, the case in which the triangle is isosceles. 22. If two sides of a triangle are 10 and 11, and the included angle is 50°, find the third side. 23. If two sides of a triangle are 43.301 and 25, and the included angle is 30°, find the third side. r 24. In order to find the distance between two objects, A and B, separated by a swamp, a station C was chosen, and the distances CA = 3825 yd., CB = 3475.6 yd., together with the.^ngle ACB = 62° 31', were measured. Eind distance from to P. \/ 25. Two inaccessible objects, A and B, are each viewed from two stations, C and B, on the same side of AB and 562 yd. apart. The angle ACB is 62° 12', PCZ» 41° 8', DP 60° 49', and ^BC 34° 51'. Eequired the distance AB. 26. In order to find the distance between two objects, A and B, separated by a pond, a station C was chosen, and it was found that CA = 426 yd., CB = 322.4 yd., and ACB = 68° 42'. Eequired the distance from A to B. i 27. Two trains start at the same time from the same station and move along straight tracks that form an angle of 30°, one train at the rate of 30 mi. an hour, the other at the rate of 40 rui. an hour. How far apart are the trains at the end of half an hour ? 28. In a parallelogram, given the two diagonals 5 and 6 and the angle which they form 49° 18', find the sides. THE OBLIQUE TKIAHGLE 123 115. Given the Three Sides. Given the three sides of a triangle, it is possible to find the angles by the Law of Cosines. Thus, from a? z= — 2 be cos 4, ~ a? we have cosyI = 2 be This formula is not, however, adapted to work with logarithms. In order to remedy this difficulty we shall now proceed to change its form. Let s equal the semiperimeter of the triangle ; that is. let Then and Hence a-\-b-\-e — 2 s. b e — a = 2 s — 2 a = 2 (s — a), e a — b = 2(s — b), a b — e = 2 (s — e'). 1 - cos .4 = 1 — b‘^ + e^ a 2be-b^-(^ + a^ a 2 be 2 be ^ - -(b — eY _ (a b — e){a — b e) 2 be 2be _ 2(s-b){s-e) be In the same way the value of 1 + cos .4 is 1 + b"^ + by A and the area by 5, we have 6’ = ^ ch. But h — a sin B. Therefore S = ^ac sin B. Also S = ^ab sin C, and S = ^hc sin A. Exercise 56. Area of a Triangle Find the areas of the triangles in which it is given that ; 1. 1-7 II c = 32, B = 40°. 2. a = 35, c = 43. B = 37°. 3. a = 4.8, c — 0.3, B = 39° 27'. 4. a = 9.8, c = 7.6, B = 48.5°. 6. a = 17.3, b = 19.4, C = 56.25°. 6. a = 48.35, b = 64.32, C = 62° 37'. 7. h = 127.8, c = 168.5, A = 72° 43'. 8. b = 423.9, c = 417.8, A = 68° 27'. 9. b = 32.78, c = 29.62, A = 57° 32' 20". 10. b = 1487, c = 1634, A = 61° 30' 30". 11. Prove that the area of a parallelogram is equal to the product of the base, the diagonal, and the sine of the angle included by them. 12. Eind the area of the quadrilateral ABCD, given AB = 3 in., JC = 4.2 in., .4L»=3.8 in., Z£dD = 88° 10', ZA.4C = 36° 20'. 13. In a quadrilateral ABCD, BC = 5.1 in., ^1C = 4.8 in., CD = 3.7 in., Z. A CB = 123° 4:2', and ZDCA =117° 2&. Draw the figure approximately and find the area. 14. In the pentagon ABODE, AB = 3.1 in., .4C = 4.2 in., AD = 3.7 in., 2.9 in., Z.4 =132° 18', Z5.4C = 38° 16', and ZDAE = 53° 9'. Find the area of the pentagon. THE OBLIQUE TRIANGLE 129 Case 2. Given two angles and any side. If two angles are known the third can be found, so we may consider that all three angles are given. G it follows that And since we have a sin C c = —. — — • sind S = ^ac sinR (Case 1), = asinC . „ sin 2? sin (7 — — smA = — ^ — Sind 2 Sind Since all three angles are known we may use this formula; or, since sin (R -f- C) = sin (180° — d) = sind, we may write it as follows : sin B sin C 2 sin (B + C) Exercise 57. Area of a Triangle Find the areas of the triangles in which it is given that : 1. a=n, B = 48°, C = 52°. 2. a - 182, B = 63.5°, C = 78.4°. 3. a = 298, B = 78.8°, C = 95.5°. 4. a = 19.8, 5 = 39° 20', 88° 40'. 5. a = 2487, 87° 28', 69° 32'. 6. b - 483.7, d = 84° 32', C = 78° 49'. 7. h = 527.4, d = 73° 42', 63° 37'. 8 c = 296.3, A = 58° 35', B - - 42° 36'. 9. c = 17.48, d = 36° 27' 30", B = 73° 50'. 10. c = 96.37, d = 42° 23' 35", B = 69° 52' 50". 11. In a parallelogram ABCD the diagonal AC makes with the sides the angles 27° 10' and 32° 43' respectively. AB is 2.8 in. long. What is the area of the parallelogram ? . , 130 PLANE TRIGONOMETRY Case 3. Given the three sides. Since, by § 101, sinR = 2 sin cos and, by § 115, and sin ^ (s — g) (s — c) ac cos \B = N by substituting these values for sin-|-R and cos in the above equation, we have 2 , sinR = — Vs(s - a){s~b){s - c). By putting this value for sin R in the formula of Case 1, we have the following important formula for the area of a triangle : S = ^ s(s — d)(s — ft) (s — c). This is known as Heron’s Formula for the area of a triangle, having been given in the works of this Greek writer. It is often given in geometry, but the proof by trigonometry is much simpler. A special case of finding the area of a triangle when the three sides are given is that in which the radius of the circumscribed circle or the radius of the inscribed circle is also given. If R denotes the radius of the circumscribed circle, we have. If r denotes the radius of the inscribed circle, we may divide the triangle into three triangles by lines from the center of this circle to the vertices ; then the altitude of each of the three triangles is equal to r. Therefore , S = ir(a + 6 + c) = rs. By putting in this formula the value of S from Heron’s Formula, we have (s — a) (s — b) (s — c) From this formula, 7-, as given in § 116, is seen to be equal to the radius of the inscribed circle. THE OBLIQUE TRIANGLE 131 Exercise 58. Area of a Triangle Find the areas of the triangles in which it is given that : 1. a = 3, 11 c = 5. 4. a - 1.8, CO II II 2. a = 15, b = 20, c = 25. 5. a = 5.3, b = 4.8, c = 4.6. 3. a = 10, b = 10, O 1— ( II 6. a = 7.1, b = 5.3, c = 6.4. 7. There is a triangular piece of land with sides 48.5 rd., 52.3 rd., and 61.4 rd. Find the area in square rods ; in acres. Find the areas of the triangles in which it is given that : 8. a = 2.4, b - 3.2, c = 4, R = 2. 9. a = 2.7, CO II c = 4.5, R = 2.25. 10. a — 3.9, b = 5.2, c = 6.5, R = 3.25. 11. a =12, II c=12, R = 6.928. 12. Given a = 60, A = 40° 35' 12", area =12, find the radius of the inscribed circle. Find the areas of the triangles in which it is given that : 13. a - 40, b = 13, c = 37. 14. a = 408, II c = 401. 15. a = 624, b = 205, c = 445. 16. II c = 5, O * O II 17. a = l, e = 3, A = 60°. 18. b - 21.66, c = 36.94, A = 66° 4' 19". 19. a = 215.9, c = 307.7, A = 25° 9' 31". 20. b = 149, A = 70° 42' 30", B = 39° 18' 28". 21. a = 4474.5, b = 2164.5, C = 116° 30' 20". 22. a = 510, c = 173, B = 162° 30' 28". 23. If a is the side of an equilateral triangle, show that the area is 1 a^ Vs. 24. Two sides of a triangle are 12.38 ch. and 6.78 eh., and the included angle is 46° 24'. Find the area. 25. Two sides of a triangle are 18.37 ch. and 13.44 ch., and they form a right angle. Find the area. 26. Two angles of a triangle are 76° 54' and 57° 33' 12", and the included side is 9 ch. Find the area. 27. The three sides of a triangle are 49 eh., 50.25 ch., and 25.69 ch. Find the area. 132 PLANE TEIGONOMETKY 28. The three sides of a triangle are 10.64 ch., 12.28 ch., and 9 ch. Find the area. 29. The sides of a triangular field, of which the area is 14 A., are proportional to 3, 5, 7. Find the sides. 30. Two sides of a triangle are 19.74 ch. and 17.34 ch. The first bears N. 82° 30' W. ; the second S. 24° 15' E. Find the area. 31. The base of an isosceles triangle is 20, and its area is 100 Vs ; find its angles. 32. Two sides and the included angle of a triangle are 2416 ft., 1712 ft., and 30°; and two sides and the included angle of another triangle are 1948 ft., 2848 ft., and 150°. Find the sum of their areas. 33. Two adjacent sides of a rectangle are 52.25 ch. and 38.24 ch. Find the area. 34. Two adjacent sides of a parallelogram are 59.8 ch. and 37.05 ch., and the included angle is 72° 10'. Find the area. 35. Two adjacent sides of a parallelogram are 15.36 ch. and 11.46 ch., and the included angle is 47° 30'. Find the area. 36. Show that the area of a quadrilateral is equal to one half the product of its diagonals into the sine of the included angle. 37. The diagonals of a quadrilateral are 34 ft. and 56 ft., inter- secting at an angle of 67°. Find the area. 38. The diagonals of a quadrilateral are 75 ft. and 49 ft., inter- secting at an angle of 42°. Find the area. 39. In the quadrilateral A A CD we have A A, 17.22 ch.; AD, 7.45 ch.; CD, 14.10 ch.; DC, 5.25 ch. ; and the diagonal AC, 15.04 ch. Find the area. 40. Show that the area of a regular polygon of n sides, of which . . na^ 180° one side is a, is — — cot 4 n 41. One side of a regular pentagon is 25. Find the area. 42. One side of a regular hexagon is 32. Find the area. 43. One side of a regular decagon is 46. Find the area. 44. If r is the radius of a circle, show that the area of the regular 180° circumscribed polygon of n sides is ni^ tan ^ > and the area of the regular inscribed polygon is ^ sin ~~~ • 45. Obtain a formula for the area of a parallelogram in terms of two adjacent sides and the included angle. CHAPTER VIII MISCELLANEOUS APPLICATIONS 119. Applications of the Right Triangle. Although the formulas for oblique triangles apply with equal force to right triangles, yet the formulas developed for the latter in Chapter IV are somewhat simpler and should be used when possible. It will be remembered that these formulas depend merely on the definitions of the functions. Exercise 59. Right Triangles 1. If the sun’s altitude is 30°, find the length of the longest shadow which can be cast on a horizontal plane by a stick 10 ft. in length. 2. A flagstaff 90 ft. high, on a horizontal plane, casts a shadow of 117 ft. Pind the altitude of the sun. 3. If the sun’s altitude is 60°, what angle must a stick make with the horizon in order that its shadow in a horizontal plane may be the longest possible ? 4. A tower 93.97 ft. high is situated on the bank of a river. The angle of depression of an object on the opposite bank is 25° 12'. Find the breadth of the river. 6. The angle of elevation of the top of a tower is 48° 19', and the distance of the base from the point of obser- vation is 95 ft. Find the height of the tower and the distance of its top from the point of observation. 6. From a tower 58 ft. high the angles of depression of two objects situated in the same horizontal line with the base of the tower, and on the same side, are 30° 13' and 45° 46'. Find the distance between ^ ^ these two objects. 7. From one edge of a ditch 36 ft. wide the angle of elevation of the top of a wall on the opposite edge is 62° 39'. Find the length of a ladder that wiU just reach from the point of observation to the top of the wall. 133 134 PLANE TRIGONOMETEY 8. The top of a flagstaff has been partly broken off and touches the ground at a distance of 15 ft. from the foot of the staff. If the length of the broken part is 39 ft., find the length of the whole staff. 9. From a balloon which is directly above one town the angle of depression of another town is observed to be 10° 14'. The towns being 8 mi. apart, find the height of the balloon. 10. A ladder 40 ft. long reaches a , window 33 ft. high, on one side of a street. Being turned over upon its foot, the ladder reaches another window 21 ft. high, on the opposite side of the street. Find the width of the street. 11. From a mountain 1000 ft. high the angle of depression of a ship is 27° 35' 11". Find the distance of the ship from the summit of the mountain. 12. From the top of a mountain 3 mi. high the angle of depres- sion of the most distant object which is visible on the earth’s sur- face is found to be 2° 13' 50". Find the diameter of the earth. 13. A lighthouse 54 ft. high is situated on a rock. The angle of elevation of the top of the lighthouse, as observed from a ship, is 4° 52', and the angle of elevation of the top of the rock is 4° 2'. Find the height of the rock and its distance from the ship. 14. The latitude of Cambridge, Massachusetts, is 42° 22' 49". What is the length of the radius of that parallel of latitude ? 15. At what latitude is the circumference of the parallel of lati- tude equal to half the equator ? 16. In a circle with a radius of 6.7 is inscribed a regular polygon of thirteen sides. Find the length of one of its sides. 17. A regular heptagon, one side of which is 5.73, is inscribed in a circle. Find the radius of the circle. 18. When the moon is setting at any place, the angle at the mooir subtended by the earth’s radius passing through that place is 57' 3". If the earth’s radius is 3956.2 mi., what is the moon’s distance from the earth’s center ? 19. A man in a balloon observes the angle of depression of an object on the ground, bearing south, to be 35° 30'; the balloon drifts 2 \ mi. east at the same height, when the angle of depression of the same object is 23° 14'. Find the height of the balloon. 20. The angle at the earth’s center subtended by the sun’s radius is 16' 2", and the sun’s distance is 92,400,000 mi. Find the sun’s diameter in miles. MISCELLANEOUS APPLICATIONS 135 21. A man standing south of a tower and on the same horizontal plane observes its angle of elevation to be 54° 16'; he goes east 100 yd. and then finds its angle of elevation is 50° 8'. Find the height of the tower. 22. A regular pyramid, with a square base, has a lateral edge 150 ft. long, and the side of the base is 200 ft. Find the inclination of the face of the pyramid to the base. 23. The height of a house subtends a right angle at a window on the other side of the street, and the angle of elevation of the top of the house from the same point is 60°. The street is 30 ft. wide. How high is the house ? 24. The perpendicular from the vertex of the right angle of a right triangle divides the hypotenuse into two segments 364.3 ft. and 492.8 ft. in length respectively. Find the acute angles of the triangle. 26. The bisector of the right angle of a right triangle divides the hypotenuse into two segments 431.9 ft. and 523.8 ft. in length respectively. Find the acute angles of the triangle. 26. Find the number of degrees, minutes, and seconds in an arc of a circle, knowing that the chord which subtends it is 238.25 ft., and that the radius is 196.27 ft. 27. Calculate to the nearest hundredth of an inch the chord which subtends an arc of 37° 43' in a circle having a radius of 542.35 in. 28. Calculate to the nearest hundredth of an inch the chord which subtends an arc of 14° in a circle having a radius of 47 5.23 in. 29. In an isosceles triangle ABC the base AB is 1235 in., and Z.4 = ZS = 64° 22'. Find the radius of the inscribed circle. 30. Find the number of degrees, minutes, and seconds in an arc of a circle, knowing that the chord which subtends it is two thirds of the diameter. 31. Find the number of degrees, minutes, and seconds in an arc of a circle, knowing that the chord which subtends it is three fourths of the diameter. 32. The radius of a circle being 2548.36 in., and the length of a chord BC being 3609.02 in., find the angle BAC made by two tangents drawn at B and C respectively. 33. Find the ratio of a chord to the diameter, knowing that the chord subtends an arc 27° 48'. If the diameter is 8 in., how long is the chord ? If the chord is 8 in., how long is the diameter ? 136 PLANE TEIGONOMETRY 34 . Find the length of the diameter of a regular pentagon of which the side is 1 in., and the length of the side of a regular pentagon of which the diameter is 1 in. 36 . Two circles of radii a and h are externally tangent. The com- mon tangents AP, BP, and the line of centers CC’P are drawn as shown in the figure. Find sin APC. 36 . In Ex. 35 find /.APC, know- ing that a — 2>b. 37 . In ZA= 68°26'27", /B = 75° 8' 23", and the altitude h, from C, is 148.17 in. Eequired the lengths of the three sides. 38. Two axes, OX and OY, form a right angle at 0, the center of a circle of radius 1091 ft. Through P, a point on OX 1997 ft. from 0, a tangent is drawn, meeting OF at C. Ee- quired OC and the angle CPO. 39. Find the sine of the angle formed by the intersection of the diagonals of a cube. 40. The angle of elevation of the top of a tower observed at a place A, south of it, is 30° ; and at a place B, west of A , and at a distance of a from it, the angle of elevation is 18°. Show that the height of the tower is ~~r =^ , the tangent of 18° being ~ j: '^2 + 2V6 ViFTTv! 41. Standing directly in front of one corner of a flat-roofed house, which is 150 ft. in length, I observe that the horizontal angle which the length subtends has for its cosine V^, and that the vertical angle 3 subtended by its height has for its sine n— • What is the height of the house ? 42. At a distance a from the foot of a tower, the angle of eleva- tion A of the top of the tower is the complement of the angle of elevation of a flagstaff on top of it. Show that the length of the staff is 2 a cot 2 A. 43. A rectangular solid is 4 in. long, 3 in. wide, and 2 in. high. Calculate the tangent of the angle formed by the intersection of any two of the diagonals. 44. Calculate the tangent as in Ex. 43, the solid being I units long, w units wide, and A units higln MISCELLANEOUS APPLICATIONS 137 120 . Applications of the Oblique Triangle. As stated in § 119, when conditions permit of using a right triangle in making a trigono- metric observation it is better to do so. Often, however, it is impos- sible or inconvenient to use the right triangle, as in the case of an observation on an inclined plane, and in such cases resort to the oblique triangle is necessary. Exercise 60. Oblique Triangles 1. Show how to determine the height of an inaccessible object situated on a horizontal plane by observing its angles of elevation at two points in the same line with its base and measuring the distance between these two points. 2. Show how to determine the height of an inaccessible object standing on an inclined plane. 3. Show how to determine the distance between two inaccessible objects by observing angles at the ends of a line of known length. 4. The angle of elevation of the top of an inaccessible tower stand- ing on a horizontal plain is 63° 26' ; at a point 500 ft. farther from the base of the tower the angle of elevation of the top is 32° 14'. Find the height of the tower. 5. A tower stands on the bank of a river. From the opposite bank the angle of elevation of the top of the tower is 60° 13', and from a point 40 ft. further off the angle of elevation is 50° 19'. Find the width of the river. 6. At the distance of 40 ft. from the foot of a vertical tower on an inclined plane, the tower subtends an angle of 41° 19'; at a point 60 ft. farther away the angle subtended by the tower is 23° 45'. Find the height of the tower. 7. A building makes an angle of 113° 12' with the inclined plane on which it stands ; at a distance of 89 ft. from its base, measured down the plane, the angle subtended by the building is 23° 27'. Find the height of the building. 8. A person goes 70 yd. up a slope of 1 in 3^ from the bank of a river and observes the angle of depression of an object on the oppo- site bank to be 2i°. Find the width of the river. 9. A tree stands on a declivity inclined 15° to the horizon. A man ascends the declivity 80 ft. from the foot of the tree and finds the angle then subtended by the tree to be 30°. Find the height of the tree. 138 PLANE TRIGONOMETRY 10. The angle subtended by a tree on an inclined plane is, at a certain point, 42° 17', and 325 ft. further down it is 21° 47'. The inclination of the plane is 8° 53'. Find the height of the tree. 11. From a point B at the foot of a mountain, the angle of elevation of the top A is 60°. After ascending the mountain one mile, at an inclination of 30° to the horizon, and reaching a point C, an observer finds that the angle A CB is 135°. Find the number of feet in the height of the mountain. 12. The length of a lake subtends, at a certain point, an angle of 46° 24', and the distances from this point to the two ends of the lake are 346 ft. and 290 ft. Find the length of the lake. 13. Along the bank of a river is drawn a base line of 500 ft. The angular distance of one end of this line from an object on the oppo- site side of the river, as observed from the other end of the line, is 53°; that of the second extremity from the same object, observed at the first, is 79° 12'. Find the width of the river. 14. Two observers, stationed on opposite sides of a cloud, observe its angles of elevation to be 44° 56' and 36° 4'. Their distance from each other is 7 00 ft. What is the height of the cloud ? 15. From the top of a house 42 ft. high the angle of elevation of the top of a pole is 14° 13'; at the bottom of the house it is 23° 19'. Find the height of the pole. 16. From a window on a level with the bottom of a steeple the angle of elevation of the top of the steeple is 40°, and from a second window 18 ft. higher the angle of elevation is 37° 30'. Find the height of the steeple. 17. The sides of a triangle are 17, 21, 28. Prove that the length of a line bisecting the longest side and drawn from the opposite angle is 13. 18. The sum of the sides of a triangle is 100. The angle at is double that at B, and the angle at B is double that at C. Determine the sides. 19. A ship sailing north sees two lighthouses 8 mi. apart in a line (iue west ; after an hour’s sailing, one lighthouse bears S.W., and the other S. 22° 30' W. (22° 30' west of south). Find the ship’s rate. 20. A ship, 10 mi. S.W. of a harbor, sees another ship sail from the harbor in a direction S. 80° E., at a rate of 9 mi. an hour. In what direction and at what rate must the first ship sail in order to catch up with the second ship in 1^ hr. ? MISCELLAITEOUS APPLICATIONS 139 21. Two ships are a mile apart. The angular distance of the first ship from a lighthouse on shore, as observed from the second ship, is 35° 14' 10" ; the angular distance of the second ship from the light- house, observed from the first ship, is 42° 11' 53". Find the distance in feet from each ship to the lighthouse. 22. A lighthouse bears N. 11° 15' E., as seen from a ship. The ship sails northwest 30 mi., and then the lighthouse bears east. How far is the lighthouse from the second point of observation ? 23. Two rocks are seen in the same straight line with a ship, bearing N. 15° E. After the ship has sailed N.W. 5 mi., the first rock bears E., and the second N.E. Find the distance between the rocks. 24. On the side OX of a given angle XOY a point A is taken such that OA = d. Deduce a formula for the length AS of a line from A to 0 F that makes a given angle a with OX. From this formula, a; is a minimum when what sine is the maximum? Under those circumstances what is the sum of 0 and a ? Then what is the size of ZS? State the conclusion as to the size of Zu in order that x shall be the minimum. 25. Three points. A, B, and C, form the vertices of an equilateral triangle, AB being 500 ft. Each of the two sides AB and AC is seen from a point P under an angle of 120° ; that is, Z A PB = 120°=Z CPA. Find the length of AP. [/26. A lighthouse facing south sends out its rays extending in a quadrant from S.E. to S.W. A steamer sailing due east first sees the light when 6 mi. away from the lighthouse and continues to see it for 45 min. At what rate is the ship sailing ? 27. If two forces, represented in intensity by the lengths a and b, pull from P in the directions PC and PA, respectively, and if Z.APC is known, the resultant force is represented in intensity and direction by f, the diagonal of the parallelogram ABCP. Show how to find/ and A APB, given a, b, and AAPC. 'y28. Two forces, one of 410 lb. and the other of 320 lb., make an angle of 51° 37'. Find the intensity and the direction of their resultant. 29. An unknown force combined with one of 128 lb. produces a resultant of 200 lb., and this resultant makes an angle of 18° 24' with the known force. Find the intensity and direction of the unknown force. 140 PLANE TRIGONOMETEY 30. Wishing to determine the distance between a church A and a tower B, on the opposite side of a river, a man measured a line CD along the river ((7 being nearly opposite A), and observed the angles ACB, 58° 20'; ACD, 95° 20'; ADB, 53° 30'; BDC, 98° 45'. CD is 600 ft. What is the distance required ? 31. Wishing to find the height of a summit A, a man measured a horizontal base line CD, 440 yd. At C the angle of elevation of A is 37° 18', and the horizontal angle between D and the summit of the mountain is 76° 18'; at D the horizontal angle between C and the summit is 67° 14'. Eind the height. 32. A balloon is observed from two stations 3000 ft. apart. At the first station the horizontal angle of the balloon and the other station is 75° 25', and the angle of elevation of the balloon is 18°. The hori- zontal angle of the first station and the balloon, measured at the second station, is 64° 30'. Find the height of the balloon. 33. At two stations the height of a kite subtends the same angle A. The angle which the line joining one station and the kite subtends at the other station is B ; and the distance between the two stations is a. Show that the height of the kite is ^ a sin A sec B. 34. Two towers on a horizontal plain are 120 ft. apart. A person standing successively at their bases observes that the angle of eleva- tion of one is double that of the other ; but when he is halfway be- tween the towers, the angles of elevation are complementary. Prove that the heights of the towers are 90 ft. and 40 ft. 35. To find the distance of an inaccessible point C from either of two points A and B, having no instruments to measure angles. Prolong CA to a, and CB to b, and draw AB, Ah, and Ba. Measure AB, 500 ft.; aA, 100 ft.; aB, 560 ft.; bB, 100 ft.; and A 5, 550 ft. Compute the distances A C and B C. 36. To compute the horizontal distance between two inaccessible points A and B when no point can be found whence both can be seen. Take two points C and D, distant 200 yd., so that A can be seen from C, and B from D. From C measure CF, 200 yd. to F, whence A can be seen ; and from D measure DE, 200 yd. to E, whence B can be seen. Measure AEC, 83°; ACD, 53° 30'; ACF, 54° 31'; BDE, 54° 30' ; BDC, 156° 25' ; DEB, 88° 30'. Compute the distance AB. 37. A column in the north temperate zone is S. 67° 30' E. of an observer, and at noon the extremity of its shadow is northeast of him. The shadow is 80 ft. in length, and the elevation of the column at the observer’s station is 45°. Find the height of the column. MISCELL AJJ^EOUS APPLICATIONS 141 121. Areas. In finding the areas of rectilinear figures the effort is made to divide any given figure into rectangles, parallelograms, triangles, or trapezoids, unless it already has one of these forms. For example, the dotted lines show how the above figures may be divided for the purpose of computing the areas. A regular polygon would be conveniently divided into congruent isosceles triangles by the radii of the circumscribed circle. Exercise 61. Miscellaneous Applications 1. In the trapezoid A jBCZ) it is known that Z A =90°, Z5 = 32°25', AB = 324.35 ft., and CD = 208.15 ft. Find the area. 2. Find the area of a regular pentagon of which each side is 4 in. 3. Find the area of a regrdar hexagon of which each side is 4 iu 4. The area of a regular polygon inscribed in a circle is to that of the circumscribed regular polygon of the same number of sides as 3 to 4. Find the number of sides. 5. The area of a regular polygon inscribed in a circle is the geometric mean between the areas of the inscribed and circumscribed regular polygons of half the number of sides. 6. Find the ratio of a square inscribed in a circle to a square cir- cumscribed about the same circle. Find the ratio of the perimeters. 7. The square circumscribed about a circle is four thirds the in- scribed regular dodecagon. 8. In finding the area of a field ABODE a surveyor measured the lengths of the sides and the angle which each side makes with the meridian (north and south) line through its extremities. AD happened to be a meridian line. Show how he could compute the area. 9. Two sides of a triangle are 3 and 12, and the included angle is 30°. Find the hypotenuse of the isosceles right triangle of equal area. 10. In the quadrilateral A B CD we'have given A B, BC,/-A,Z.B, and Z C. Show how to find the area of the quadrilateral. 11. In Ex. 10, suppose A5 = 175 ft., AC = 198 ft., ZA= 95°, Z A = 92° 15', and Z C = 96° 45'. What is the area ? 142 PLANE TRIGONOMETRY 122. Surveyor’s Measures. In measuring city lots surveyors com- monly use feet and square feet, with decimal parts of these units. In measuring larger pieces of land the following measures are used : 16^ feet (ft.) = 1 rod (rd.) 66 feet = 4 rods = 1 chain (ch.) 100 links (li.) = 1 chain 10 square chains (sq. ch.) = 160 square rods (sq. rd.)= 1 acre (A.) We may write either 7 ch. 42 li. or 7.42 ch. for 7 chains and 42 links. The decimal fraction is rapidly replacing the old plan, in which the word livk was used. Similarly, the parts of an acre are now written in the decimal form instead of, as formerly, in square chains or square rods. Areas are computed as if the land were flat, or projected on a horizontal plane, no allowance being made for inequalities of surface. 123. Area of a Field. The areas of fields are found in various ways, depending upon the shape. In general, however, the work is reduced to the finding of the areas of triangles or trapezoids. For example, in the case here shown we may draw a north and south line E'A' and then And the sum of the areas of the trapezoids ABB'A', BCC'B', CBJD'C', and DEE'D' . From this we may subtract the sum of the trapezoids A GG'A', GFF'G' a,nd FEE'F' . The result will be the area of the field. Instead of running the imaginary line E'A' outside the field, it would be quite as convenient to let it pass through F, A, E, or C. The method of computing the area is substantially the same in both cases. For details concerning surveying, beyond what is here given and is included in Exercise 60, the student is referred to works upon the subject. Exercise 62. Area of a Field 1. Find the number of acres in a triangular field of which the sides are 14 ch., 16 ch., and 20 ch. 2. Find the number of acres in a triangular field having two sides 16 ch. and 30 ch., and the included angle 64° 15'. 3. Find the number of acres in a triangular field having two angles 68.4° and 47.2°, and the included side 20 ch. 4. Required the area of the field described in § 123, knowing that AA' = 8 ch., BB’ = 12 ch., CC = 13 ch., DD' = 12 ch., EE' = 8 ch., FF' = 1 ch., GG' = 2 ch., A'G’ = 6 ch., G'B' = 1.5 ch., B'F' = 2.3 ch., F'C = 3 ch., CD' = 4 ch., D'E' = 2.9 ch. MISCELLANEOUS APPLICATIONS 143 5. In a quadrangialar field ABCD, AB runs N. 27° E. 12.5 ch., BC runs N. 30° W. 10 cli., CD runs S. 37° W. 15 ch., and DA runs S. 47° E. 11.5 eh. Find the area. That AB is N. 27°E. means that it makes an angle of 27° east of the line running north through A. 6. In a triangular field ABC, AB runs N. 10° E. 30 ch., BC runs S. 30° W. 20 ch., and CA runs S. 22° E. 13 ch. Find the area. 7. In a field ABCD, AB runs E. 10 ch., BC runs N. 12 ch., CD runs S. 68° 12' W. 10.77 ch., and DA runs S. 8 ch. Find the area. 8. A field is in the form of a right triangle of which the sides are 15 ch., 20 ch., and 25 ch. From the vertex of the right angle a line is run to the hypotenuse, making an angle of 30° with the side that is 15 ch. long. Find the area of each of the triangles into which the field is divided. Using a protractor, draw to scale the fields referred to in the following examples, and find the areas : 9. AB, N. 72° E. 18 ch., BC, N. 10° E. 12.5 ch.. 10. AB, N. 45° E. 10 ch., BC, S. 75° E. 11.55 ch., 11. AB, N. 5°30' W. 6.08 ch., BC, S. 82° 30' W. 6.51 ch.^ 12. AB, N. 6° 15' W. 6.31 ch., BC, S. 81° 50' W. 4.06 ch.. CD, N. 68° W. 21 ch., DA, S. 12° E. 26.3 ch. CD, S. 15° W. 18.21 ch., DA, N.45° W. 19.11 ch. CD, S. 3° E. 5.33 ch., DA, E. 6.72 ch. CD, S. 5° E. 5.86 ch., DA, N. 88° 30' E, 4.12 ch. 13. A farm is bounded and described as follows : Beginning at the southwest corner of lot No. 13, thence N. 1^° E. 132 rods and 23 links to a stake in the west boundary line of said lot; thence S. 89° E. 32 rods and 15-^ links to a stake ; thence N. 1^° E. 29 rods and 15 links to a stake in the north boundary line of said lot ; thence S. 89° E. 61 rods and 18 links to a stake ; thence S. 32J° W. 54 rods to a stake ; thence S. 35^° E. 22 rods and 4 links to a stake ; thence S. 48° E. 33 rods and 2 links to a stake ; thence S. 7 ^° W. 7 6 rods and 20 links to a stake in the south boundary line of said lot ; thence N. 89° W. 96 rods and 10 links to the place of beginning. Containing 85.65 acres, more or less. Verify the area given and plot the farm. TMs is a common way of describing a farm in a deed or a mortgage. 144 PLANE TRIGONOMETRY 124. The Circle. It is learned in geometry that c — 2 irr, and a — where c = circumference, r = radius, a — area, and tt = 3.14159+ = 3.1416— = about 3i. Eor practical purposes ^ may be taken. Furthermore, if we have a sector with angle n degrees, 7t the area of the sector is evidently of odU From these formulas we can, by the help of the formulas relating to triangles, solve numerous prob- lems relating to the circle. Exercise 63. The Circle 1. A sector of a circle of radius 8 in. has an angle of 62.5°. A chord joining the extremities of the radii forming the sector cuts off a segment. What is the area of this segment ? 2. A sector of a circle of diameter 9.2 in. has an angle of 29° 42'. A chord joining the extremities of the radii forming the sgctor cuts off a segment. What is the area of the remainder of the circle ? 3. In a circle of radius 3.5 in., what is the area included between two parallel chords of 6 in. and 5 in. respectively ? (Give two answers.) 4. A regular hexagon is inscribed in a circle of radius 4 in. What is the area of that part of the circle not covered by the hexagon ? 5. In a circle of radius 10 in. a regular five-pointed star is inscribed. What is the area of the star ? What is the area of that part of the circle not covered by the star ? 6. In a circle of diameter 7.2 in. a regular five- pointed star is inscribed. The points are joined, thus forming a regular pentagon. There is also a regular pentagon formed in the center by the crossing of the lines of the star. The small pentagon is what fractional part of the large one ? 7. A circular hole is cut in a regular hexagonal plate of side 8 in. The radius of the circle is 4 in. What is the area of the rest of the plate ? 8. A regular hexagon is formed by joining the mid-points of the sides of a regular hexagon. Find the ratio of the smaller hexagon to the larger. CHAPTER IX PLANE SAILING 125. Plane Sailing. A simple and interesting application of plane trigonometry is found in that branch, of navigation in which the surface of the earth is considered a plane. This can be the case only when the distance is so small that the curvature of the earth may be neglected. This chapter may be omitted if further applications of a practical nature are not needed. 126. Latitude and Departure. The difference of latitude between two places is the arc of a meridian between the parallels of latitude which pass through those places. Thus the latitude of Cape Cod is 42° 2' 21" N. and the latitude of Cape Hat- teras is 35° 15' 14" N. The difference of latitude is 6° 47' 7". The departure between two meridians is the length of the arc of a parallel of latitude cut off by those meridians, measured in geographic miles. The geographic mile, or knot, is the length of 1' of the equator. Taking the equator to he 131,385,456 ft., of of this length is 6082.66 ft., and this is generally taken as the standard in the United States. The British Admiralty knot is a little shorter, being 6080 ft. The term "mile” in this chapter refers to the geographic mile, and there are 60 mi. in one degree of a great circle. Calling the course the angle between the track of the ship and the meridian line, as in the case of E". 20° E., it will be evident by drawing a figure that the difference in latitude, expressed in distance, equals the distance sailed multiplied by the cosine of the course. That is diff. of latitude = distance x cos C. In the same way we can find the departure. This is evidently given by the equation departure = distance x sin C. For example, if a ship has sailed E". 30° E. 10 mi., the difference in latitude, expressed in miles, is 10 cos 30° = 10 X 0.8660 = 8.66, and the departure is 10 sin 30° = 10 x 0.5 = 5- 145 146 PLANE TRIGONOMETRY 127. The Compass. The mariner divides the circle into 32 equal parts called points. There are therefore 8 points in a right angle, and a point is 11° 15'. To sail two points east of north means, therefore, to sail 22° 30' east of north, or north- northeast (N.N.E.) as shown on the compass. Northeast (N.E.) is 45° east of north. One point east of north is called north by east (N. by E.) and one point east of south is called south by east (S. by E.). The other terms used, and their significance in angular measure, will best be understood from the illustration and the following table : North Points 0-1 0-^ 0-i 1 o / // 2 48 45 5 37 30 8 26 15 11 15 0 Points 0-i 0-1 0-1 1 South N. by E. N. by W. S. by E. S. by W. N.N.E. N.N.W. 1-1 1-1 1-1 2 14 3 45 16 52 30 19 41 15 22 30 0 1-i i-i 2 S.S.E. S.S.W. N.E. by N. N.W. by N. t 3-i 2-1 3 25 18 45 28 7 30 30 56 15 33 45 0 2-i 2-i 2-i 3 S.E. by S. S.W. by S. N.E. N.W. 3-1 3-3 3-i 4 36 33 45 39 22 30 42 11 15 ' 45 0 0 3-i 3-^ 3-i 4 S.E. S.W. N.E. by E. N.W. by W. 4-1 4-i 4-1 5 47 48 45 50 37 30 53 26 15 56 15 0 4-i 4-i 4-1 5 S.E. by E. S.W. by W. E.N.E. W.N.W. 5-1 5-1 5-1 6 59 3 45 61 52 30 64 41 15 67 30 0 5-i 5-i 5-i 6 E.S.E. W.S.W. E. by N. W. by N. 6-1 6-1 6-1 7 70 18 45 73 7 30 75 56 15 78 45 0 6-i 6-i 6-i 7 E. by S. W. by S. E. W. 7-1 7-1 7-1 8 81 33 45 84 22 30 87 11 15 90 0 0 7-i 7-i 7-i 8 E. W. The compass varies in different parts of the earth ; hence, in sailing, the compass course is not the same as the true course. The true course is the com- pass course, with allowances for variation of the needle in different parts of the earth, for deviation caused by the iron in the ship, and for leeway, the angle which the ship makes with her track. PLANE SAILING 147 Exercise 64. Plane Sailing 1. A ship sails from latitude 40° N. on a course N.E. 26 mi. Find the difference of latitude and the departure. 2. A ship sails from latitude 35° N. on a course S.W. 53 mi. Find the difference of latitude and the departure. 3. A ship sails from a point on the equator on a course N.E. by N. 62 mi. Find the difference of latitude and the departure. 4. A ship sails from latitude 43° 45' S. on a course N. by E. 38 mi. Find the difference of latitude and the departure. 5. A ship sails from latitude 1° 45' N. on a course S.E. by E. 25 mi. Find the difference of latitude and the departure. 6. A ship sails from latitude 13° 17' S. on a course N.E. by E. | E., until the departure is 42 mi. Find the difference of latitude and the latitude reached. 7. A ship sails from latitude 40° 20' N. on a N.N.E. course for 92 mi. Find the departure. 8. If a steamer sails S.W. by W. 20 mi. what is the departure and the difference of latitude ? 9. If a sailboat sails N. 25° W. until the departure is 25 mi., what distance does it sail ? 10. A ship sails from latitude 37° 40' N. on a N.E. by E. course for 122 mi. Find the departure. 11. A yacht sails 6^ points west of north, the distance being 12 mi. What is the departure ? 12. A steamer sails S.W. by W. 28 mi. It then sails N.W. 30 mi. How far is it then to the west of its starting point ? 13. A ship sails on a course between S. and E. 24 mi., leaving latitude 2° 52' S. and reaching latitude 2° 58' S. Find the course and the departure. 14. A ship sails from latitude 32° 18' N., on a course between N. and W., a distance of 34 mi. and a departure of 10 mi. Find the course and the latitude reached. 15. A ship sails on a course between S. and E., making a differ- ence of latitude 13 mi. and a departure of 20 mi. Find the distance and the course. 16. A ship sails on a course between N. and W., making a differ- ence of latitude 17 mi. and a departure of 22 mi. Find the distance and the course. 148 PLANE TRIGONOMETRY 128 . Parallel Sailing. Sailing dne east or due west, remaining on the same parallel of latitude, is called parallel sailing. 129 . Finding Difference in Longitude. In parallel sailing the dis- tance sailed is, by definition (§ 126), the departure. Erom the departure the difference in longitude is found as follows ; Let be the departure. Then in rt. A OAD - lat. DA Hence - = sin (90° — lat.) = cos lat (lA ^ ' The triangles DAB and OEQ, are similar, the arcs being (§ 125) considered straight lines. Therefore Hence Therefore ^ _ dR DA _ AB oe~eq’ oa~eq' AB coslat = — . AB EQ = - AB X sec lat. cos iat. That is, Diff. long. = depart, x sec lat. That is, the number of minutes in the difference in longitude is the product of the number of miles in the departure by the secant of the latitude, the nautical, or geographic, mile being a minute of longitude on the equator. Exercise 65. Parallel Sailing 1. A ship in latitude 42° 16’ N., longitude 72° 16' W., sails due east a distance of 149 mi. What is the position of the point reached ? 2. A ship in latitude 44° 49' S., longitude 119° 42' E., sails due west until it reaches longitude 117° 16' E. Find the distance made. 3. A ship in latitude 60° 15' N., longitude 60° 15' W., sails due west a distance of 60 mb What is the position of the point reached ? PLANE SAILING 149 130. Middle Latitude Sailing. Since a ship rarely sails for any length, of time due east or due west, the difference in longitude can- not ordinarily be found as in parallel sailing (§ § 128, 129). Therefore, in plane sailing the departure between two places is measured gen- erally on that parallel of latitude which lies midway between the parallels of the two places. This is called the method of middle latitude sailing. Hence, in middle latitude sailing, Diff. long. = depart, x sec mid. lat. This assumption produces no great error, except in very high latitudes or excessive runs. Exercise 66. Middle Latitude Sailing 1. A ship leaves latitude 31° 14' N., longitude 42° 19' W., and sails E.N.E. 32 mi. Find the position reached. 2. Leaving latitude 49° 57' N., longitude 15° 16' W., a ship sails between S. and W. till the departm'e is 38 mi. and the latitude is 49° 38' N. Find the course, distance, and longitude reached. 3. Leaving latitude 42° 30' N., longitude 58° 51' W., a ship sails S.E. by S. 48 mi. Find the position reached. 4. Leaving latitude 49° 57' N., longitude 30° W., a ship sails S. 39° W. and reaches latitude 49° 44' N. Find the distance and the longitude reached. 5. Leaving latitude 37° N., longitude 32° 16' TV., a ship sails be- tween N. and W. 45 mi. and reaches latitude 37° 10' N. Find the course and the longitude reached. 6. A ship sails from latitude 40° 28' N., longitude 74° W., on an E.S.E. course, 62 mi. Find the latitude and longitude reached. 7. A ship sails from latitude 42° 20' N., longitude 71° 4' W., on a N.N.E. course, 30 mi. Find the latitude and longitude reached. 150 PLANE TEIGONOMETRY 131. Traverse Sailing. In case a ship sails from one point to an- other on two or more different courses, the departure and difference of longitude are found by reckon- ing each course separately and com- bining the results. For example, two such courses are shown in the figure. This is called the method of traverse sailing. No new principles are involved in traverse sailing, as will be seen in solv- ing Ex. 1, given below. Exercise 67. Traverse Sailing 1. Leaving latitude 37° 16' S., longitude 18° 42' W., a ship sails N.E. 104 mi., then N.N.W. 60 mi., then W. by S. 216 mi. Find the position reached, and its bearing and distance from the point left. For the first course we have difierence of latitude 73.5 N., departure 73.5 E.; for the second course, difference of latitude 55.4 N., departure 23 W.; for the third course, difierence of latitude 42.1 S., departure 211.8 W. On the whole, then, the ship has made 128.9 mi. of north latitude and 42.1 mi. of south latitude. The place reached is therefore on a parallel of latitude 86.8 mi. to the north of the parallel left ; that is, in latitude 35° 49.2' S. ^ In the same way the departure is found to be 161.3 mi. W., and the middle latitude is 36° 32.6'. With these data we find the difierence of longitude to be 201', or 3° 21' W. Hence the longitude reached is 22° 3' W. With the difierence of latitude 86.8 mi. and the departure 161.3 mi., we find the course to be N. 61° 43' W. and the distance 183.2 mi. The ship has reached the same point that it would have reached if it had sailed directly on a course N. 61° 43' W. for a distance of 183.2 mi. 2. A skip leaves Cape Cod (42° 2' N., 70° 3' W.) and sails S.E. by S. 114 mi., then N. by E. 94 mi., then W.N.AY. 42 mi. Find its position and the total distance. 3. A ship leaves Cape of Good Hope (34° 22' S., 18° 30' E.) and sails N.W. 126 mi., then N. by E. 84 mi., then W.S.W. 21 T mi. Find its position and the total distance. 4. A ship in latitude 40° N. and longitude 67° 4' W. sails N.W. 60 mi., then N. by W. 52 mi., then W.S.W. 83 mi. Find its position. 6. A ship sailed S.S.W. 48 mi., then S.W. by S. 36 mi., and then '. N.E. 24 mi. Find the difference in latitude and the departure. 6. A ship sailed S. E. 18 mi., S.W. ^ S. 37 mi., and then S.S.W I W. 56 mi. Find the difference in latitude and the departure. CHAPTER X GRAPHS OF FUNCTIONS 132. Circular Measure. Besides the methods of measuring angles which have been discussed already and are generally used in practical work, there is another method that is frequently employed in the theoretical treatment of the subject. This takes for the unit the angle subtended by an arc which is equal in length to the radius, and is known as circular measure. 133. Radian. An angle subtended by an arc equal in length to the radius of the circle is called a radian. The term "radian” is applied to both the angle and arc. In the annexed figure we may think of a radius bent around the arc AB so as to coincide with it. Then | AAOB is a radian. 134. Relation of the Radian to Degree Measure. The number of radians in 360° is equal to the number of times the length of the radius is contained in the length of the circumference. It is proved in geometry that this number is 2 7T for all circles, tt being equal to 3.1416, nearly. Therefore the radian is the same angle in all circles. The circumference of a circle is 2 tt times the radius. Hence 2 tt radians = 360°, and tt radians = 180°. 1 80° Therefore 1 radian = = 57.29578° = 57° 17' 45", and 1 degree = radian = 0.017453 radian. loO 135. Number of Radians in an Angle. Prom the definition of radian we see that the number of radians in an angle is equal to the length of the subtending arc divided by the length of the radius. Thus, if an arc is 6 in. long and the radius of the circle is 4 in., the number of radians in the angle subtended by the arc is 6 in. ^ 4 in., or 1^. This may be reduced to degrees thus : 1 J X 57.29578° = 85.94367°, or, for practical purposes, 11 x 57.3° = 85.9° = 85° 54'. 151 152 PLANE TKIGONOMETKY 136. Reduction of Radians and Degrees. Erom the values found in § 134 the following methods of reduction are evident : To reduce radians to degrees, multiply 57° 17' 45" , or 57.29578°, by the number of radians. To reduce degrees to radians, multiply 0.017453 by the number of degrees. These rules need not be learned, since we do not often have to make these reductions. It is essential, however, to know clearly the significance of radian measure, since we shall often use it hereafter. In solving the following problems the rules may be consulted as necessary. In particular the student should learn the following : 360° = 2 7T radians, 60° = ^ w radians, 180° = 7T radians, 30° = ^ tt radians, 90° = ^7T radians, 15° = w radians, 45° = 5 7t radians, 22.5° = ^ tt radians. The word radians is usually understood without being written. Thus sin 1-n means the sine of 2 tt radians, or sin 360° ; and tan tt means the tangent of ^ TT radians, or 45°. Also, sin 2 means the sine of 2 radians, or sin 114.59156°. Exercise 68. Radians Express the following in radians : 1. 270°. 3. 56.25°. 5. 196.5°. 7. 200°. < 2. 11.25°. > 4. 7.5°. 1 6. 1440°. *~8. 3000°. Express the following in degree measure : 9. 1\TT. 11. l^TT. •^3. A^tt. 2 4 15. 6 77. 10. l^TT. 12. l\ir. ^^^4. Stt. 16. 10 77. State the quadrant in which the following angles lie 17. |7T. 19. 1|7T. 21. 2.5 TT. 23. 1. ^-18. JTT. 20. IjTT. ^ 22 . —3.4 77. <^4. — 2. Express the following in degrees and also in radians : 25. f of four right angles. 27. f of two right angles. 26. I of four right angles. 28. | of one right angle. ^..29. What decimal part of a radian is 1°? 1'? 30. How many minutes in a radian ? How many seconds ? 31. Express in radians the angle of an equilateral triangle. ^32. Over what part of a radian does the minute hand of a clock move in 15 min. ? GEAPHS OF FUNCTIOi^lS 153 137. Functions of Small Angles. Let A OP be any acute angle, and let X be its circular measure. Describe a circle of unit radius about 0 as center and take Z.AOP' -=—/-AOP. Draw the tangents to the circle at P and P', meeting OA in T. Then we see that chord PP' < arc PP' > cos X. Therefore the value of between cos x and 1. If, now, the angle x is constantly diminished, cos x approaches the value 1. sm X Accordingly, the limit of ■ ^ > as x approaches 0, is 1. Hence when x denotes the circular measure of an angle near 0° we may use X instead of sin x and instead of tan x. For example, required to find the sine and cosine of V. If x is the circular measure of 1', 2 7T 3.14159 + 360 X 60 “ 10800 0.00029088 +, the next figure in x being 8. Nowsinx > Obut Vl- (0.0003)2 > 0.9999999. Hence cos 1' = 0.9999999 +. But, as above, sin x~> x cos x. .-. sinl' > 0.000290888 x 0.9999999 > 0.000290888 (1 - 0.0000001) > 0.000290888 - 0.0000000000290888 > 0.000290887. Hence sin V lies between 0.000290887 and 0.000290889 ; that is. to eight places of decimals, sinl'= 0.00029088 +, the next figure being 7 or 8. 154 PLANE TRiaONOMETRY Y P' r X' 0 A Y 138. Angles having the Same Sine. If we let Z.XOP = x, in this figure, and let P' be symmetric to P with respect to the axis YY’, we shall have ZA0P' = 180° — a:, ovir — x. And since - = sin x = sin (tt — x) we see that x and IT — X have the same sine. Furthermore, sin x = sin (360° x), and sin (180° — a;) = sin (360° + 180° — x). That is, we may increase any angle by 360° without changing the sine. Hence we have sina: = sin(7i. • 360° + x), and sin (180° — x) = sin(?i • 360° + 180° — x). Using circular measure we may write these results as follows : sin X = sin (2 kir + a;), and sin (tt — a-) = sin (2 ^ + 1 tt — a;). These may be simplified still more, thus : ' sin X = sin [nw + (— l)"a;] where n is any integer, positive or negative. Thus if n = 0 we have sin a: = sin (0 • tt + (— l)“x) = sin z ; if n = 1 we have sin X = sin (tt — x) ; if n = 2 we have sin x = sin (2 tt + x) ; and so on. Since the sine is the reciprocal of the cosecant, it is evident that x and mr + (— l)"x have the same cosecant. To find four angles whose sine is 0. 2-588, we see by the tables that sin 1 5°= 0.2588. Hence we have sin 15° = sin [mr + (— 1)" • 15°] = sin (tt — 1-5°) = sin (2 tt -f 15°) = sin(37T — 15°); and so on. Exercise 69. Sines and Small Angles 1. Find four angles whose sine is 0.2756. 2. Find six angles whose sine is 0.5000. 3. Find eight angles having the same sine as I ir. 4. Find four angles having the same cosecant as | tt. 6. Find four angles having the same cosecgint as 0.1 tt. Given Tt = 3.141592653589, compute to eleven decimal places : 6. cos V. 7 . sin 1'. 8. tan 1'. 9. sin 2'. 10. From the results of Exs. 6 and 7, and by the aid of the formula sin 2x = 2 sin x cos x, compute sin 2', carrying the multiplication to six decimal places. Compare the result with that of Ex. 9. 11. Compute sinl° to four decimal places. o X X“ 12. From the formula cos x = 1 — 2 sin^ > show that cos x > 1 — -w • GRAPHS OF FUNCTIONS 165 139. Angles having the Same Cosine. If we let Z.XOP = x, in this figure, and let P' be symmetric to P with respect to the axis XX', we shall have Z.XOP' — 360° — x, or — x, depending on whether we think of it as a positive or a negative angle. In either case . . b , its cosme is - > the same as cos x. r In either case cos x = cos {n • 360° — x'). In general, cos x = cos (2 wtt ± x), where n is any integer, positive or negative. Thus if ji = 0, we have cosx = cos(± x); if ti = 1, we have cosz = cos( 27 t± x); if n = 2, we have cos x = cos (4 tt ± x) ; and so on. Since the cosine is the reciprocal of the secant, it is evident that x and 2n7r ±x have the same secant. 140. Angles having the Same Tangent. Since we have tan x = -> — a ° and tan (180° x) = — - > we see that tan x = tan (180° + x). In general we may say that tan x = tan (2 kir + x) = tan (2 k^r + tt + a:). This may be written more simply thus : tan x = tan (wtt + x~), where x is any integer, positive or negative. Thus if we have tan 2(P given, we know that nir + 20° has the same tangent. Writing both in degree measure, we may say that n • 180° + 20° has the same tangent. If n = 1, we have 200° ; if n = 2, we have 380° ; if n = 3, we have 560° ; and so on. Furthermore, if n = — 1, we have —160°; and so on. Since the cotangent is the reciprocal of the tangent, it is evident that x and KTT + X have the same cotangent. Exercise 70. Angles having the Same Functions 1. Find two positive angles that have \ as their cosine. 2. Find two negative angles that have ^ as their cosine. 3. Find four angles whose cosine is the same as the cosine of 25°- 4. Find four angles that have 2 as their secant. 5. Find two positive angles that have 1 as their tangent. 6. Find two negative angles that have 1 as their tangent. 7. Find four angles that have V§ as their tangent. 8. Find four angles that have Vs as their cotangent. 9. Find four angles that have 0.5000 as their tangent. 10. Find four negative angles whose cotangent is 0.6000. 156 PLANE TKIGONOMETRY 141. Inverse Trigonometric Functions. If y = 5iux, then x is the angle whose sine is y. This is expressed by the symbols x = sin~^ y, or X = arc sin y. In American and English books the symbol sin-i y is generally used ; on the continent of Europe the symbol arc sin y is the one that is met. The symbol sin~^ y is read ” the inverse sine of y,” ” the antisine of y,” or " the angle whose sine is y.” The symbol arc sin y is read " the arc whose sine is y,” or " the angle whose sine is y.” The symbols cos-i a;, tan-i z, cot-^ x, and so on are similarly used. The symbol sin-^y must not be confused with (sin The former means the angle whose sine is y ; the latter means the reciprocal of siny. We have seen (§ 138) that sin~^ 0.5000 may be 30°, 150°, 390°, 510°, and so on. In other words, there are many values for sin~^ x ; that is. Inverse trigonometric functions are many-valued. 142. Principal Value of an Inverse Function. The smallest positive value of an inverse function is called its 'principal value. For example, the principal value of sin~i 0.6000 is 30°; the principal value of cos-10.5000 is 60°; the principal value of tan-i(— 1) is 135°; and so on. 11. Find the value of the sine of the angle whose cosine is that is, the value of sin(cos"^^). lind the values of the following : Exercise 71. Inverse Functions 12 . sin(cos“^ Vs). 13 . sin(tan-^l). 14 . cos(cot~^l). Prove the following formtdas ; GRAPHS CP PUHCTIOHS 157 Find four values of each of the following : 19. tan“^ 0.5774. 21. sin“^ 0.9613. 20. cot“^ 0.6249. 22. sin“^ 0.3256. 25. Solve the equation y — sin“^^. 26. Pind the value of sin(tan~^-|- + tan“^-^). 27. If sin“^a: = 2 cos“^x, find the value of x. Prove the folloiving formidas : 28. cos (sin~^ x) — Vl — P. 29. cos (2 sin“^ cc) = 1 — 2 a;^. 30. sin(sin~^a:) = a:. 31. sin (sin'^a: + sin"^?/) = x Vl — y^ + y Vl — 32. tan~^ 2 + tan“^ 4 = ^ tt. 33. 2 tan“^x = tan“^[2 a; : (1 — x^)]. 34. 2 sin“^x = sin~^(2 X Vl — x^). 35. 2 cos“^x = cos“^(2 x^ — 1). 36. 3 tan~^x = tan“^[(3 x — x^) : (1 — 3 x^)]. 37. sin~’- Vx : y = tan“^ Vx : (yy — x). 38. sin“^ V(x — ?/) ; (x — z) = tan~' V(x — y~) : (yy — z) 39. sin"^x = sec“^(l : Vl — x^). 40. 2 sec~^x = tan~^ [2 Vx^ — 1 ; (2 — 3p)\. 41. tan~^ \ + tan“^ i = i 42. tan“^^ + tan~^ J = tan~^ |. 43. sin“^f + sin"^^ = sin“^|-|. 44. sin“^-^ VS2 4- sin-^^ ViT = J tt. 45. sec-^ f + sec-i || = 75° 45'. 46. tan~^(2 + Vs) — tan“^(2 — Vs)=; sec~^2. 47. tan“^ J + tan~^ i + tan"^ | + tan~^ i = i 48. sin“^x 4- sin“^ Vl — x^ = ^ tt. 49. sin~^0.5 + sin”^ ^ V3 = sin“^l. 50. tan~^ ^ = tan~^ i + tan~^ |. 51. tan~^0.5 + tan“^0.2 + tan~^0.125 = \ ’tt. 52. tan~^l + tan“^2 + tan"^3 = tt. 53. tan~^ f + tan~^ + tan“^ — \ir. 54. cos“^^ VlO + sin"^ i VS = \ 'ir. 23. eot~^ 0.2756. 24. eos“^ 0.9455. 158 PLANE TRIGONOMETRY 143. Graph of a Function. As in algebra, so in trigonometry, it is possible to represent a function graphically. Before taking up the subject of graphs in trigonometry a few of the simpler cases from algebra will be considered. Suppose, for example, we have the expression 3x + 2. Since the value of this expression depends upon the value of x, it is called a function of x. This fact is indicated by the equation fix) = 3 X + 2, read " function x = 3 x + 2.” But since /(x) is not so easil}* written as a single letter, it is customary to replace it by some such letter as y, writing the equation y = 3 X + 2. If X = 0, we see that y = 2 ; if x = 1, then ?/ = 5 ; and so on. We may form a table of such values, thus : X y X y 0 2 0 2 1 6 - 1 - 1 2 8 - 2 -4 3 11 -3 — I We may then plot the points (0, 2), (1, 5), (2, 8), • • •, 1, — 1)? (— 2, — 4), • • ., as in § 77, and connect them. Then we have the graph of the function 3 x + 2. The graph shows that the function, y or f(x), changes in value much more rapidly than the variable, x. It also shows that the function does not become negative at the same time that the variable does, its value being 2 when x = 0, and ^ when x = — J. This kind of function in which x is of the first degree only is called a linear function because its graph is a straight line. Exercise 72 . Graphs Plot the graphs of the following functions : 1. 2x. 2. \x. 3. — X. 4. X 4- 1. 6. X — 1. 6. 2 X + 1. 7. 3 — X. 8. 4 — lx. 9. — 2 — X. 10. 2 X + 3. 11. 2x — 3. 12. 3 — 2x. 13. 0.5 X + 1.5. 14. 1.4 X — 2.3. 15. - if X — 2^. 16 . -259-X + 3I. GRAPHS OF FUHCTIOHS 159 144. Graph of a Quadratic Function. We shall now consider func- tions of the second degree in the variable. Such a function is called a quadratic fimction. Taking the function x — 2, we write y X — 2. Preparing a table of values, as on page 158, we have X y X y 0 -2 0 -2 1 0 - 1 -2 2 4 -2 0 3 10 -3 4 4 18 -4 10 In order to see where the function lies between y = — 2 and y = — 2, we let X = — We find that when x = — 1, y =— 2t. Similarly if we give to x other values between 0 and — 1, we shall find that y in every case lies between 0 and — 2. Plotting the points and drawing through them a smooth curve, we have the graph as here shown. This curve is 2 ,parabola. All graphs of functions of the form y = ox^ + 6x + c are parabolas. Graphs of functions of the form x^ + 2/^ = or j/ = ± Vr^ — x^, are circles with their center at O. Graphs of functions of the form a^x* + h^y- — c^ are ellipses, these becoming circles if a = 6. Graphs of functions of the form a^x^ — b-y"^ = are hyperbolas. There are more general equations of all these conic sections, but these suffice for our present purposes. The graph of every quadratic function in x and y is always a conic section. Exercise 73. Graphs of Quadratic Functions Plot the graphs of the following functions : 1. x^. 5. x^ — 1. 9. 2cc^-f-3a:. 2. 2x^. 6. + a: -f- 1. 10. 3a;^ — 4x. 3. ix\ 7. x^ — x + 1. 11. ± VT x . 4. + 1. 8. + a: — 1. 12. ± V9 — 4 ; 13. ± V4 — 3 14. ± Vs — 2 X*. 15. ± V4 + 3x^. 16. ± Vs -h 2 a^. 160 PLANE TEIGONOMETRA 145. Graph of the Sine. Since sin a; is a function of x, we can plot the graph of sin x. We may represent x, the arc (or angle), in de- grees or in radians on the x-axis. Representing it in degrees, as more familiar, we may prepare a table of values as follows : X — 0° 16° 30° 45° 60° 75° 90° 105° 120° 135° 150° 165° 180° . . . y = 0 .26 .5 .7 .87 .97 1 .97 .87 .7 .5 .26 0 ... If we represent each unit on the y-axis by and each unit on the x-axis by 30°, the graph is as follows : The graph shows very clearly that the sine of an angle x is positive between the values x = 0° and x = 180°, and also between the values x = — 360° and X = — 180° ; that it is negative between the values x = — 180° and x = 0°, and also between the values x = 180° and x = 360°. It also shows that the sine changes from positive to negative as the angle increases and passes through — 180° and 180°, and that the sine changes from negative to positive as the angle increases and passes through the values — 360°, 0°, and 360°. These facts have been found analytically (§84), but they are seen more clearly by studying the graph. If we use radian measure for the arc (angle), and represent each unit on the a:-axis by 0.1 vr, the graph is as follows : The nature of the curves is the same, the only difference being that we have used different units of measure on the x-axis, thus elongating the curve in the second figure. 146. Periodicity of Functions. This curve shows graphically what we have already found, that periodically the sine comes back to any given value. Thus sin x = 1 when x = — 270°, 90°, 450°, • • •, returning to this value for increase of the angle by every 360°, or 2 :r radians. The period of the sine is therefore said to be 360° or 2 tt. GRAPHS OF FUNCTIONS 161 Exercise 74. Graphs of Trigonometric Functions 1. Verify the following plot of the graph of cos x : rP X 1 1 f 1 VI \ 1 r2 IQ WftV 9f * 1 3^60 ■ 1 i < fin M7 lb f 1 / 1 l\ ill) ■*111 2. What is the period of cos x ? 3. Verify the following plot of the graph of tan x : i-i ' 1 ’ 1 1 ! i i - ■90-^ 0 90^ * •>? * 1 1 ' /I 7 i 1 il I ;T 4. What is the period of tan x ? 6. Verify the following plot of the graph of cot x : 1 ■ . ' 1 ^ ' 1 ■ ; 1 TW 1 V V V V -360 Sii * 80* *27( pla, S60 ■ : ' ' ' ~r 1 1 6. What is the period of cot x ? 7. Verify the following plot of the graph of sec os: 8. What is the period of sec x ? 9. Plot the graph of esc x, and state the period. Also state at what values of x the sign of csccc changes. 10. Plot the graphs of sin x and cos x on the same paper. What does the figure tell as to the mutual relation of these functions ? 162 PLANE TRIGOXOMETKY Exercise 75. Miscellaneous Exercise Find the areas of the triangles in which : 1. a = 25, = 25, c = 25. 3. a =74, h = 75, c = 92. 2. a = 25, 6 = 33J, c = 41f . 4. a = 2^,h = 3^, c = 4i. 5. Consider the area of a triangle with sides 17.2, 26.4, 43.6. 6. Consider the area of a triangle with sides 26.3, 42.4, 73.9. 7. Two inaccessible points A and B are visible from D, but no other point can be found from which both points are visible. Take some point C from which both A and D can be seen and measure CD, 200 ft. ; angle ADC, 89°; and angle A CD, 50° 30'. Then take some point E from which both D and B are visible, and measure DE, 200 ft.; angle BDE, 54° 30'; and angle BED, 88° 30'. At D measure angle ADB, 72° 30'. Compute the distance AB. 8. Show by aid of the table of natural sines that sin x and x agree to four places of decimals for all angles less than 4° 40'. 9. If the values of log x and log sin x agree to five decimal places, find from the table the greatest value x can have. 10. Find four angles whose cosine is the same as the cosine of 175°. 11. Find four angles whose cosine is the same as the cosine of 200°. 12. How many radians in the angle subtended by an arc 7.2 in. long, the radius being 3.6 in. ? How many degrees ? 13. How many radians in the angle subtended by an arc 1.62 in. long, the radius being 4.86 in. ? How many degrees ? Draw the following angles : 14. — 7T. 16. —^7T. 18. 2.7 7T. 20. 3 7T — 9. 16. —2. 17. — 19. 2 7t — 6. 21. 4— 7T. 22. Find four angles whose tangent is • 23. Find four angles whose cotangent is • Vs 24. Plot the graphs of sin x and esc x on the same paper. What does the figure tell as to the mutual relation of these functions ? 26. Plot the graphs of cos x and sec x on the same paper. What does the figure tell as to the mutual relation of these functions ? 26. Plot the graphs of tan x and cot x on the same paper. What does the figure tell as to the mutual relation of these functions ? CHAPTER XI TRIGONOMETRIC IDENTITIES AND EQUATIONS 147. Equation and Identity. An expression of equality which, is true for one or more values of the unknown quantity is called an equation. An expression of equality which is true for all values of the literal quantities is called an identity. For example, in algebra we may have the equation 4 a: — 3 = 7, which is true only if x = 2.5. Or we may have the identity (a + 5)2 = a2 + 2ab + b'^, which is true whatever values we may give to a and b. Thus sin x = 1 is a trigonometric equation. It is true for x = 90° or ^ tt, X = 450° or 2^7 t, X = 810° or 41 tt, and so on, with a period of 360° or 2 7 t. In general, therefore, the equation sin x = 1 is true for x = (2 n + ^) tt. It is this general value that is required in solving a general trigonometric equation. On the other hand, the equation sin^x = 1 — cos^x is true for all values of x. It is therefore an identity. The symbol s is often used instead of = to indicate identity, but the sign of equality is very commonly employed unless special emphasis is to be laid upon the fact that the relation is an identity instead of an ordinary equation. ^ 148. How to prove an Identity. A convenient method of proving a trigonometric identity is to substitute the proper ratios for the functions themselves. a c Thus to prove that sin x = 1 : esc x we have only to substitute - for sin x and - a c c a for CSC X. We then see that Similarly, to prove that tan x = sin x sec x, we may substitute - for tanx, - for sinx, and - for seex. We then have be b a _a c b c b We can often prove a trigonometric identity oy reference to formulas already proved. This was done in proving the identity sin2x = 2slnxcosx (§ 101), and in tanx + tanw proving tan(x + y) = (§ 93). 1 — tan X tan y In some cases it may be better to draw a figure and use a geometric proof, as was done in § 90. 163 164 PLANE TRIGONOMETRY Exercise 76. Identities Prove the following identities : 2 tan ^ X tan X = 2. sin a: = 3. sin 2 X = 1 — tan^ X 2 tan ^ X 1 + tan^ \ X 2 tan X 6. tan 3 X = 3 tan X — tan*x 1 + tan^x 8 . 1—3 tan^x tan 2 X + tan x _ sin 3 x tan 2 X — tan x sin x 3 cos X 4- cos 3 X 3 sin X — sin 3 x = cot®x. 4. 2sinx + sin2x = 6. sin 3 X = 2 sin®x 1 — cos X sin^ 2 X — sin^ x sin 3 X + sin 5 x 9. ^ r — = cot X . sin X 10 . 11. sin X + sin 3 x + sin 5 x = 12. tan 2 X + sec 2 x = 13. tanx + tany 14. tan (x + y) = cos 3 X — cos 5 X sin 3 X 4- sin 5 x sin X 4- sin 3 x sin^ 3 X = 2 cos 2 X. sinx cos X + sin X cos X — sin X _ sin (x + y) cos X cos y sin 2x4- sin 2 y 16. cos 2x4- cos 2 y sin X + cos y _ tan fi (^ + y) + sin X — cos y tan [-^ (x — y) — 45°] 16. sin 2 X 4- sin 4 x = 2 sin 3 x cos x. 17. sin 4 X = 4 sin X cos x — 8 sin^x cos x. 18. sin 4 X = 8 cos®x sin x — 4 cos x sin x. 19. cos 4 X = 1 — 8 cos^x + 8 cos*x =1 — 8 sin^x + 8 sin'x. 20. cos 2 X + cos 4 X = 2 cos 3 X cos x. 21. sin 3 X — sin X = 2 cos 2 X sin x. 22. sin^x sin 3 X 4- cos^x cos 3 x = cos® 2 x 23. cos*x — siiPx = cos 2 x. 24. cos^x + siiPx = l — i sin^ 2 x. 25. cos®x — sin®x = (1 — siiRx cos®x) cos 2 x. 26. cos®x 4- sin®x = 1 — 3 ,sin®x cos®x. 27. CSC X — 2 cot 2 X cos X = 2 sin X. IDENTITIES AND EQUATIONS 165 Prove the following identities : 28. (sin 2x — sin 2 y) tan (x y) = 2 (sin^x — sin^i/). 29. sin 3 a; = 4 sin x sin (60° + x') sin (60° — x'). 30. sin 4 £c = 2 sin x cos 3 a: + sin 2 x. 3 1. sin X + sin (a: — f tt) + sin (fir — x)= 0. 32. cos X sin (y — s) + cos y sin (z — x')-\- cos z sin (x — y~)= 0. 33. cos (x + y) sin y — cos (x + z) sin z = sin (x + y) cos y — sin (x + z) cos z. 34. cos (a; + y + s) + cos (x y — z)-\- cos (x — y z~) + cos ( 2 / + « — a;) = 4 cos x cos y cos z. 35. sin (x + y) cos (x — y)-\- sin (7/ + z) cos {y — z) + sia (z + a;) cos (z — x)= sin 2 x -f- sin 2y sin 2 z. 36. sin (x + ?/) + cos (x — y) = 2 sin (x + ^ tt) sin (y + i tt). 37. sin (x + y)— cos (x — y) = — 2 sin (x — ^ tt) sin {y — ^ tt). 38. cos (x + y) cos (x — y)= cos^ x — sin^ y. 39. sin(x + ?/)sin(x — y)= sin^x — sin^y. 40. sin X + 2 sin 3 x + sin 5 x = 4 cos^x sin 3 x. If A, B, C are the angles of a triangle, prove that : 41. sin 2 A 4- sin 2B + sin 2 C = 4 sin4 sin A sin C. 42. cos 2 A + cos 2B + cos 2 C = — 1 — 4 cos4 cos£ cos C. 43. sin ZA + sin 3A + sin 3 C = — 4 cos |4 cos cos | C. 44. cos^4 + cos^A + cos^ (7 = 1 — 2 cos A cos A cos C. If A + B + C = 90°, prove that : 45. tanA tanA + tan5 tan C + tan C tan A = 1. 46. sin^A + sin^A + sin^ C = 1 — 2 sin A sinA sin C. 47. sin 2A + sin 2B + sin 2(7 = 4 cos A cos5 cosC. 48. Prove that cot~^ 3 + csc“^ Vs = J tt. 49. Prove that x + tan~^ (cot 2 x) = tan~^ (cot x). Prove the following statements : sin 76° + sin 15° ^ 50. . r— ^ = tan60°. sin 75 — sin 16 51. sin 60° + sin 120° = 2 sin 90° cos 30°. 62. cos 20° + cos 100° + cos 140° = 0. 53. cos 36° + sin 36° = V2 cos 9°. 54. tan 11° 15' + 2 tan 22° 30' + 4 tan 45° = cot 11° 15'. 166 PLANE TRIGONOMETRY 149 . How to solve a Trigonometric Equation. To solve a trigonometric equation is to find for the unknown quantity the general value which satisfies the equation. Practically it suffices to find the values between 0° and 360°, since we can then apply our knowledge of the periodicity of the various functions to give us the other values if we need them. There is no general method applicable to all cases, but the follow- ing suggestions will prove of value : 1. If functions of the sum or difference of two angles are involved, reduce such functions to functions of a single angle. Thus, instead of leaving sin (x y) in an equation, substitute for sin (x -t- y) its equal sin x cos y + cosx siny. Similarly, replace cos2x by cos^x — sin^x, and replace the functions of ^x by the functions of x. 2. If several functions are involved, reduce them to the same function. This is not always convenient, but it is frequently possible to reduce the equation so as to involve only sines and cosines, or tangents and cotangents, after which the solution can be seen. 3. If possible, employ the method of factoring in solving the final equation. 4. Check the results by substituting in the given equation. For example, solve the equation cos x = sin 2 x. By § 101, sin 2 X = 2 sin x cos x. .-. cosx = 2 sinx cosx. (1 — 2 sin x) cos x = 0. .-. cosx = 0, orl — 2sinx = 0. .•. X = 90° or 270°, 30° or 160°, or these values increased by 2n?r. Each of these values satisfies the given equation. Exercise 77. Trigonometric Equations Solve the following equations: 1. sin X = 2 sin (■!■ tt -|- x). 2. sin 2 X = 2 cos x. 3. cos 2 X = 2 sin x. 4. sin X + cos X = 1. 6. sin X + cos 2 X = 4 sin^x. 6. 4 cos 2 X -h 3 cos x = 1. iCh cos 0 -f cos 26 = 0. 11. cot 4 6 -f CSC 6 = 2. 12. cot X tan 2 X = 3. 7. sinx = cos 2x. 8. tan X tan 2x = 2. 9. secx = 4 cscx. IDENTITIES AND EQUATIONS 167 Solve the following equations : u 13. sin X + sin 2 x = sin 3 x. 33. sin X sec 2 x = 1. 14. sin 2 X = 3 sin^x — cos^x. 34. sin^x + sin 2 x = 1. 15. cot 6 = ^ tan 6. 35. cos X sin 2 x esc x = 1. u 16. 2 sin 6 = cos 6. 36. cot X tan 2 X = see 2 x. 17. 2 sin^x + 5 sin x = 3. 37. sin 2 X = cos 4 x. 18. tan X sec x = V2. 38. sin 2 « cot z — sin^ s = •§■• it' 19. cos X — cos 2 X = 1. 39. tan^x = sin 2 x. 20. cos 3 X + 8 cos®x = 0. 40. sec 2 X + 1 = 2 cos x. 21. tan X + cot X = tan 2 x. 41. tan 2 X + tan 3 x = 0. 22. tan X + sec x = a. 42. CSC X = cot X + Vs. 23. cos 2 X = a (1 — cos x). 43. tan X tan 3 x = — f . 24. sin“^^x = 30°. 44. cos 5 X + cos 3 X + cos x = 0 25. tan“^x + 2 cot~^x = 135°. 45. sin^ X — cos^ X = k. 26. sec X — cot X = CSC x — tan x. 46. sin X + 2 cos x = 1. 27. tan 2 X tan x = 1. 47. sin 4 X — cos 3 x = sin 2 x. 28. tan'^x + cot^x = 48. sin X + cos X = sec x. 29. sin X + sin 2 x = 1 — cos 2 x. 49. 2 cos X cos 3 X + 1 = 0. 30. 4 cos 2 X + 6 sin x = 5. 50. cos 3 X — 2 cos 2 X + cos x = 0. 31. sin 4 X — sin 2 x = sin x. 51. sin (x — 30°) = Vs sin x. 32. 2 sin^x + sin^ 2x = 2. 52. sin~^x + 2 eos“^x = f tt. 53. sin“^x + 3 COS' ■^x = 210°. 64. tana: = cos 2 X. 1 + tan X 55. tan ir x) tan tt — x) = 4. 56. Vl -|- sinx — VI — sin X = 2 cos x. 57. sin (45° + x) + cos (45° — x) = 1. 58. (1 — tan x) cos 2 x = a (1 + tan x). 59. sin®x + cos®x = — ^ 60. sec (x + 120°) + sec (x — 120°) = 2 cos x. 61. sin^x cos^x — cos^x — sin^x + 1 = 0. 62. sin X + sin 2 x + sin 3 x = 0. 63. sin 6 + 2 sin 2 6 + 3 sin 36 = 0. 64. sin 3 X = cos 2 x — 1. 66. sin (x + 12°) + sin (x — 8°) = sin 20°. 168 PLANE TRIGONOMETRY Solve the following equations : 66 . tan (60° + a:) tan (60° — x)=~ 2. 67. tan x + tan 2 a: = 0. 68 . sin (x + 120 °) + sin (x + 60°) = |. 69. sin (x + 30°) sin (x — 30°) = J. 70. sin 2 0 = cos 3 0. 71. sin*x + cos^x = |. 72. sin^a; — cos^a: = 7 3. tan (x + 30°) = 2 cos x. 74. sec X = 2 tan x + 75. sin 11a; sin 4x + sin 5 a: sin 2x = 0. 76. cos X + cos 3 a: + cos 6 a; + cos 7 a: = 0. 77. sin (x + 12°) cos (a; — 12°) = cos 33° sin 57° 78. sin'^a: + sin~^ ^ x =120°. 79. tan~^a: + tan~^ 2x = tan~^ 3 V3. 80. tan“^(a: + 1 ) + tan~^ (a: — 1 ) = tan" ^ 2 a;. 81. (3 — 4 cos^a:) sin 2 a: = 0. 82. cos 2 0 sec 0 + sec 0 + 1 = 0. 83. sin X cos 2 x tan x cot 2 x sec x esc 2x =1. 84. tan (0 -f 45°) = 8 tan 0. 85. tan(0 + 45°) tan 0 = 2. 86 . sin X + sin 3 a: = cos x — cos 3 x. 87. sin \ X (cos 2 x — 2) (1 — tan^x) = 0. 88 . tan X + tan 2 x = tan 3 x. 89. cot X — tan X = sin x + cos x. Prove the following identities : ... , , V / • s secx esex 90. (1 4- cotx 4- tanx) (sinx — cosx )= — 5 ^ ^ csc'^x sec'^x 91. 2 CSC 2x cotx = 1 4- cot^x. 92. sin a + sin b + sin (a -\-h')= 1 cos ^ a cos ^ b sin ^ (a + i). 93. tan (45° + x) — tan (45° — x) = 2 tan 2 x. 94. cot^x — cos^x = cot**x cos^x. 95. tan^x — sin^x = tan^x sin^x. 96. cot^x + cot^x = csc^x — CSC'^X. 97. cos^x 4- sin^x cos^y = cos^y 4- sin^^y cos®x. IDENTITIES AND EQUATIONS 169 150. Simultaneous Equations. Simultaneous trigonometric equations are solved by the same principles as simultaneous algebraic equations. 1. Eequired to solve for x and y the system X sin a -1- y sin b = m (1) X cos a + y cos b = n (2) From (1), X sin a cos a + y sin 6 cos a = m cos a. (3) jj'rom (2), X sin a cos a-\- y cos 6 sin a = n sin a. (4) From (3) and (4), y sin b cos a — y cos 6 sin a = m cos a — n sin a. or y sin {p — a) = m cos a — n sin a ; whence 771 COS a — n sin a y — , sin (6 — a) Similarly, n sin b — m cos b X = • sin (6 — a) 2. Eequired to solve for x and y the system sin X + sin y = a (1) cos X + cos y = b (2) By § 103, 2 sin \(x + y) cos \ {x — y) = a, (3) and 2 cos ^ (x + 2 /) cos \ {x — y) = b. Dividing, tan ^(x + y) = ^. (4) sin h{x + y) = — Va^ + Substituting the value of sin J (x + j/) in (3), cos ^{x — y)= J Va^ + (5) From (4), X + 2/ = 2 tan- 1 - . b (6) From (5), X — y = 2 cos- 1 J Va^ + 6*. (7) From (6) and (7), x = tan- ^ ^ + cos-i ^ Va^ + and y — tan- ^ ^ ~ cos- ^ ^ Va^ + 6^. 3. Eequired to solve for x and y the system y sin X — a (1) y cos X = b (2) Dividing, tan X = - . h * 1 ® .-. X = tan-i-. Adding the squares of (1) and (2), y‘‘ (sin^x + cos^x) = + b^. Therefore + 6^, and y = ± Va^ + 6^. 170 PLANE TPIGONOMETPY 4, Eequired to solve for x and y tlie system y sin (a: + — x 1 — sin<^ --- 0.5 2. Solve for 6 and x : 1 — sin 6 = x 1 + sin 6 = a 3. Solve for X and /x : sin X = V2 sin ju, tan X = Vs tan /a 4. Solve for 0 and : sin 0 + cos (f) = a sin + cos 6 = b 5. Solve for 0 and : a siid0 — b sm*cj> = a a cos^ 6 — b cos^ = b Since sin^^ + cos*0 = 1 , we have -{■ = 1, the eliminant. 2. Find the eliminant, with respect to X, of sec X = m tan X — n Since sec^X — tan^X = 1, we have m?‘ — rfi = 1 , the eliminant. 3. Find the eliminant, with respect to ft., of m sin ft. + cos ju. = 1 n sin ft. — cos ft. = 1 Writing the equations m sin /* = 1 — cos ft, n sin = 1 + cos /x, and multiplying, we have mn sin^yu = 1 — cos^/x = sin^yix. Hence mn = 1 is the eliminant. Exercise 80. Elimination Find the eliminant with respect to a, 6, X, fi, or of the folloiv- ing equations : 1. sin 1 = a cos (j) — 1 = b 2. tan A — 0 . = 0 cot X — b = 0 3. sin a m = n cos a p = q 4. a + sec = b p cot = q 5. c sin 2=l 6. X = r(0 — sin 0) y = r(l — cos 6) 9 = versine- 1 y/r. 7. sin + cos 2 + tan 2 = k sin 2 (f> — tan 2 4> = I 12. p = a cos 0 cos q = b cos 6 sin ^ Vsin / Similarly, = — -y/q cot J 4>. For example, if -f 1.1102 x — 3.3594 = 0 we have whence log tan 0 = 0.51876, = 73° 9' 2.6". log tan \ = 9.87041 —10. and Therefore and and Hence log Vq = logV3.3594 = 0.26313. logXj = 0.13354, Xj = 1.360. X2=- 2.470. 173 Similarly, 174 PLANE TKIGONOMETEY 155. De Moivre’s Theorem. Expressions of the form cos X + i sin X, where i = V— 1, play an important part in modern analysis. Since (cos x i sin x) (cos y i sin y) = cos X cos y — sin a; sin y + i (cos cc sin y + sin x cos y') = cos (x + y) + i sin (x + y), we have (cos x + i sin x^ = cos 2x + i sin 2 x ; and again, (cos x-\- i sin (cos x i sin (cos x i sin x) = (cos 2 X + i sin 2 x) (cos x + i sin x) = cos 3 X + i sin 3 x. Similarly, (cos x i sin x)"= cos nx i sin nx. To find the nth power of cos x-\-i sin x, n being a positive integer, we have only to multiply the angle x hy n in the expression. This is known as De Moivre’s Theorem, from the discoverer (c. 1725). 156. De Moivre’s Theorem extended. Again, if n is a positive integer as before. i X .. x\ cos - + t sin - I = cos X + i sin x. , . . .-r X . . X .'. (cos X 1 sin x) = cos - + i sin - • ^ ' n n However, x may be increased by any integral multiple of 2 tt with- out changing the value of cos x -f i sin x. Therefore the following n expressions are the «th roots of cos x i sin x : X...X x-|-27r,..x-}-27r cos — \- 1 sin - 5 cos h t sin > n n n n x + Itt . . x-flvT cos h % sin > • ■ • j cos -f(?i — 1)2 7 t , . . x-f(?i — 1)2 7 t -f i sin Hence, if is a positive integer. (cos X -t- i sin x)" X + 2 kir = cos , . . x + 2Jc7r n 1 o -.N -f tsm (* = 0,1, 2, . • •,re— 1). Similarly, it may be shown that — Tft (cos x-\-i sin xfi = cos— (x -f 2 *7t) -|- i sin— (x -f 2 kif). (* = 0, 1, 2, • • •, w — 1, m and n being integers, positive or negative.) APPLICATIONS TO ALGEBEA 175 157. The Roots of Unity. If we have the binomial equation x” — 1= 0, we see that x’‘ — 1, and X = the nth root of 1, of which the simplest positive root is "VI or 1. Since the equation is of the nth degree, there are n roots. In other words, 1 has n nth roots. These are easily found by De Moivre’s Theorem. There are no other roots than those in § 156. For, letting k = n, n + 1, and so on, we have X + n(2tr) . . X + n(2Tr) cos t sm — — - n n ( X \ /x \ X X - + 2 7T I + i sin ( — + 2 7T ) = cos - + i sin — , n / \n / n n and x + (n+l) 27 T . . ®+(n+l) 27 r cos h i sin h = cos I ’x + 2 7T n X + 2 7T + 2 7T + ^ sin ■^ + i sin X + 27t 'x + 2 7T + 2 n n and so on, all of which we found when & = 0, 1 , 2, • • • , n — 1 . Por example, required to find the three cube roots of 1. If COS 0 + i sin 0=1, the given number, then 0 = 0, 2 7T, 4 7T, • • • . Also (cos 0 + i sin 0 )’^ = 1 ^ = the three cube roots of 1. k(2Tr) + Also (COS0 + ^ sin0)7 = cos — ^ ^ + i sm — ^ ^ - , where A: = 0, 1, • • • , 6, and 0 = ir, 3 tt, • • • . That is, in this case I ^ 1 • • (2fc + l)7T . . (2fc + l)7T (cos0 + I sin0)t = cos^^ — h i sm ^ — - — Hence the seven 7th roots of 1 are cos^ + i siuy = cos 25° 42' 51^" + i sin 25* 42' 51^", 3 7T 3 7T cos h i sin — = cos 77° 8' 34t" + i sin 77° 8' 342", 7 7 ^ 57t . . 5tt . . 9ir . . 9 tt cos h i sm — , cos 7T + ^ sm tt, cos — • + i sm — , 7 7 7 7 Htt . . IItt 13-7r . . 13 7T cos H i sm , cos h i sm 7 7 7 7 All these values may be found from the tables. For example, cos 25° 42' 51f" + i sin 25° 42' 51^" = 0.9010 + 0.4339 V^. and Exercise 81. Roots of Unity 1. Find by De Moivre’s Theorem the two square roots of 1. 2. Find by De Moivre’s Theorem the four 4th roots of 1. 3. Find three of the nine 9th roots of 1. 4. Find the five 5th roots of 1. 5. Find the six 6th roots of + 1 and of — 1. 6. Find the four 4th roots of — 1. 7. Show that the sum of the three cube roots of 1 is zero. 8. Show that the sum of the five 5th roots of 1 is zero. 9. From Exs. 7 and 8 infer the law as to the sum of the nth roots of 1 and prove this law. 10. From Ex. 9 infer the law as to the sum of the nth roots of k and prove this law. 11. Show that any power of any one of the three cube roots of 1 is one of these three roots. 12. Investigate the law implied in the statement of Ex. 11 for the four 4th. roots and the five 5th roots of 1. APPLICATIONS TO ALGEBEA 177 158. Roots of Numbers. We have seen that the three cube roots oflare ^ gog 120° + i sin 120° = - ^ ^ V^, = cos 240° + i sin 240° = — ^V— 3, and = cos 360° + i sin 360° = cos 0° + i sin 0° = 1. Furthermore, x^ is the square of x^, because (cos 120° 4- i sin 120°)^ = cos (2 • 120°) + i sin (2 • 120°), by De Moivre’s Theorem. We may therefore represent the three cube roots by to, a?, and either to^ or 1. In the same way we may represent all n of the Tith roots of 1 by If we have to extract the three cube roots of 8 we can see at once that they are j o 2 2j 2 CO5 and 2 1. If n is not greater than 1 the series is not convergent ; that is, the sum approaches no definite limit. The further discussion of convergency belongs to the domain of algebra. When a: = 1 we have i 1-i 1 + ^ 1 = 1 + 1 + ■ 2! + 3! + ( 2 ) But Hence, from (1) and (2), 1 - - 1 + 1 + -^ + 2 ! = 1 + a: + ■ 3! -) A + (*-J) 2! 3! (3) If we take n infinitely large, (3) becomes (l + 1 + ...)= 1 + a; +fj + ... ; that is, e* = 1 + a; + — 4- ^ 4- • • •. In particular, if a; = 1 we have e = 1 4- 1 4- 4- 4- • • •. (4) We therefore see that we can compute the value of e by simply adding 1, 1, ^ of 1, of of 1, and so on, indefinitely, and that to compute the value to only a few decimal places is a very simple matter. We have merely to proceed as here shown. Here we take 1, 1, of 1, of ^ of 1, i of ^ of ^ of 1, and so on, and add them. The result given is correct to five decimal places. The result to ten decimal places is 2.7182818284. 1.000000 2 1.000000 3 0.500000 4 0.166667 5 0.041667 6 0.008333 7 0.001388 8 0.000198 9 0.000025 0.000003 e = 2.71828. 180 PLANE TEIGONOMETKY 162. Expansion of sin x, cos x, and tan x. Denote one radian by 1, and let cos l-\- i sin 1 — k. Then cos x i sin x = (cos 1 + i sin ly = and, putting — x for x, cos (— x)-{- i sin(— x) — cos x — i sinx = That is, cos x i sin x — k^, and cos x — i sin x = kr"^. By taking the sum and difference of these two equations, and dividing the sum by 2 and the difference by 2 i, we have cos x. — \. (7c* + 7:“*), and sin x = ^. (k^ — k~^). Jd % But /fc* = (e’og*)* = and kr^ = (log a;^(log kY and and •. = 1 + X log k + g— “clog*: _ 2! 3! + cosx = I ( 4 . + ^- .) = 1 + + 1 sin X = - -< X log k + x®(log7;)® x®(log7:)® ^ Dividing the last equation by x, we have smx = - log 7: + • (log ky ^ x^ (log ky 3! 5! + But remembering that x represents radians, it is evident that the smaller x is, the nearer sin x comes to equaling x ; that is, the more nearly the sine equals the arc. Sin X Therefore the smaller x becomes, the nearer comes to 1, and the nearer the second member of the equation comes to t log k. We therefore say that, as x approaches the limit 0, the limits of these two members are equal, and 1 = T log 7: ; whence and log k = i, k = e*'. APPLICATIONS TO ALGEBEA 181 Therefore, we have 1 » I -Xi\ 1 ^ , cosa;=-(e« + e “) = 1 “ ^ + 4] “ ^ + 1 . a:*' X" X' , smx = -(e“-6-“) = x + „5 ^ ^ 3! ' 5! 7! From the last two series we obtain, by division. tan X = since cos X 2x® . 17 cc’ "^+3+15+315 By the aid of these series, which rapidly converge, the trigonometric func- tions of any angle are readily calculated. In the computation it must be remembered that x is the circular measure of the given angle. Thus to compute cosl, that is, the cosine of 1 radian or cos 57.29578°, or approximately cos 57.3°, we have , , 1111 cos 1 = 1 1 — 1 ••• 2 ! 4 ! 6 ! 8 ! = 1 - 0.5 -f- 0.04167 - 0.00139 + 0.00002 - • • • = 0.5403 = cos 57° 18'. 163. Euler’s Formula. An important formula discovered in the eighteenth century by the Swiss mathematician Euler will now be considered. We have, as in § 162, /y >3 /yt? sinx = x-:^ + --- + ..; and /y »2 /y »4 . *K/ *Ay cosa; = l-- + --- + .... ix* ix} By multiplying by i in the formula for sin x, we have i sin x — ix Adding, /y»2 'i "3 ( • . , tK/ l/vU vU fjUL/ COS X + ^ sm X = 1 + ^x — — — — + — -f- — — . . .. By substituting ix for x in the formula for we see that i^x? iV tV i®x® 0” = l + “+ 2 T + 3 T+ 4! /y»2 ^ 'j ■y*^ j , iL tfiAy tAy t/tAy In other words, e“ = cos x + i sin x. 182 PLANE TEIGONOMETRY 164. Deductions from Euler’s Formula. Euler’s formula is one of the most important formulas in all mathematics. From it several important deductions will now be made. Since e“ = cos x + i sin x, in which x may have any values, we may let a: = tt. We then have e‘^ : cos 7T + i sin tt = — 1 + 0, or e'” = — 1. In this formula we have combined four of the most in teres ting numbers of mathematics, e(the natural base), f (the imaginary unit, V— l), 7r(the ratio of the circumference to the diameter), and — 1 (the negative unit). Furthermore, we see that a real number (e) may be afiected by an imaginary exponent (itr) and yet have the power real (— 1). Taking the square root of each side of the equation e’"' = — 1, we have = V— 1 = i. Taking the logarithm of each side of the equation e'”’ = — 1, ITT = log (-1). Hence we see that — 1 has a logarithm, but that it is an imaginary number and is, therefore, not suitable for purposes of calculation. Since cos + i sin (f> = cos (2 kir + (f>)+ i sin (2 + ), we see that e**”’, which is equal to cos + i sin , may be written + or we may write gi _ g(2t7r + (i)i _ ^ i sin (j) = cos (2 fcTT + <^) + i sin (2 Jctt + )+ i sin (2 kTr + <^)]. If ^ = 0, 2 kiri - log 1. If fc = 0, this reduces to 0 = log 1. If fc = I we have 2 rri = log 1 ; if k = 2, we have 4-rri = log 1, and so on. In other words, log I is multiple-valued, but only one of these values is real. If <^ = 7T, (2 kir + 7r)i = (2 k + l)7ri = log (— 1). Hence the logarithms of negative numbers are always imaginary ; for if A: = 0 we have ttI = log (— 1) ; if A: = 1 we have Sm — log (— 1) ; and so on. If we wish to consider the logarithm of some number A, we have = N (cos 2 A'tt -b i sin 2 Att). Hence log A -f 2 kiri = log N -t- log (cos 2 Att -f- i sin 2 krr) = log A -f- log 1 = log N. That is, log A = log A -|- 2 kiri. Hence the logarithm of a number is the logarithm given by the tables, -J- 2 km. If A = 0 we have the usual logarithm, but for other values of k we have imaginaries. APPLICATIONS TO ALGEBEA 188 Exercise 83. Properties of Logarithms Prove the following properties of logarithms as given in § 159, using h as the base : 1. Properties 1 and 2. 3. Property 4. 5. Property 6. 2. Property 3. 4. Property 5. 6. Property 7. Find the value of each of the following : 7. 5! 8. 7! Simplify the following : 12 . 10! 3! ■ 13. 10! 8 ! ' 9. 6! 10. 8! 11. 10! 7! 15! 20! 14. 15. — — • 16. — 5! 14! 17! / 1 1 17. Pind to five decimal places the value of (1+1+ ^ ^ + ' ' ' ) • /II 18. Find to five decimal places the value ^ ^ ^ + ' ' ’ ) • By the use of the series for cos x find the following : 20. cos 21. cos 2. 22. cos 0. By the use of the series for sin x find the following : 23. sinl. 24. sin 25. sin 2. 26. sinO. By the use of the series for tan x find the following : 27. tan 0. 28. tan 1. 29. tan J. 30. tan 2. Prove the following statements: 7T 31. e'^”^ - 1. 32. = i\ 33. s’" = 34. d = V— 1. Given log = 0.6931, find two logarithms (to the base e) of: 36. 2. 36. 4. 37. V§. 38. — 2. Given log ^5 = 1.609, find three logarithms (to the base e) of: 39. 5. 40. 25. 41. 125. 42. - 5. Given logfiO = 2.302585, find two logarithms (to the base e) of: 43. 100. 44. — 10. 45. 1000. 46. VTO. 47. From the series of § 162 show that sin(— <^) = — sin - 48. Prove that the ratio of the circumference of a circle to the diameter equals — 2 log (i*) = — 2 i log i. 184 PLAISTE TEIGONOMETEY Exercise 84. Review Problems 1. The angle of elevation of the top of a vertical cliff at a point 575 ft. from the foot is 32° 15'. Find the height of the cliff. 2. An aeroplane is above a straight road on which are two observers 1640 ft. apart. At a given signal the observers take the angles of ele- vation of the aeroplane, finding them to be 58° and 63° respectively. Find the height of the aeroplane and its distance from each observer. 3. Prove that (Vcscx + cotx — Vcsca: — cotx)^ = 2 (esc a; — 1). 4. Given sina: = 2 rn/(m? + 1) and sin y = 2 n/(r^ 1), find the value of tan(£c -F y). 5. Find the least value of cos'^a: + sec^a;. 6. Prove that 1 — sin^a:/sin^y = cos^a;(l — tan^a:/tan^y). 7. Prove this formula, due to Euler ; tan~^^ -f tan~^^ — 8. Prove that cot — cot x = esc x. 9. Prove that (sin x + i cos a;)" = cos n(^7r — x) + i sin nQvr — x). 10. Show that log i J 'rri and that log (— i) = — |. tti. 11. Through the excenters of a triangle ABC lines are drawn parallel to the three sides, thus forming another triangle A'B'C’. Prove that the perimeter of AA'B'C' is 4 r cot cot cot -^C, where r is the radius of the circumcircle. 12. Given two sides and the included angle of a triangle, find the perpendicular drawn to the third side from the opposite vertex. 13. To find the height of a mountain a north-and-south base line is taken 1000 yd. long. From one end of this line the summit bears N. 80° E., and has an angle of elevation of 13° 14' ; from the other end it bears N. 43° 30' E. Find the height of the mountain. 14. The angle of elevation of a wireless telegraph tower is observed from a point on the horizontal plain on which it stands. At a point a feet nearer, the angle of elevation is the complement of the former. At a point b feet nearer still, the angle of elevation is double the first. Show that the height of the tower is [(a + by — ^ Prove the following formulas : 15. 2cos*a: = cos 2a;-t-l. 17. 8 cos^jc = cos 4x -f 4cos 2 a; + 3. 16. 2 sin^a; = — cos 2 a; + 1. 18. 4 cos® a: = cos 3a; + 3 cos a:. 19. 4 sin®x = — sin 3 a; + 3 sin x. 20. 8 sin^a; = cos 4 a; — 4 cos 2 x + 3. FOKMULAS 185 THE MOST IMPOETANT FOEMULAS OF PLANE TEIGONOMETEY Eight Triangles (§§ 15-21) 1. y = r sin . 6. r = y esc (f>. Eelations of Functions (§§ 13, 14, 89) 7. sin ^ = 8. cos fj) = 9. tan = CSC 4 > 1 sec — 13. sec 1 -1. CSC 4 > = tan (ji 1 COS 1 sin (j> 17. sin = 18. tan ^ = cos <}> COt , „ , , cos d> 19. cot — — ^ sin 10. sin^csct^=l. 15. tan c/> cot ^ = 1. 20. 1 +tan^<^ = sec^<^. 11. cos^sec<^=l. 16. sin^(^ + cos^^=l. 21. 1 + cot^<^ = csc^^. Functions ofx ±y (§§ 90-100) 22. sin (x + y)= sin x cos y + cos x sin y. 23. sin (x — y)= sin x cos y — cos x sin y. 24. cos (x -\-y)= cos x cos y — sin x sin y. 25. cos {x — y)= cos x cos y + sin x sin y. , tan a; 4- tan?/ x coticcot?/ — 1 ^ ± — tan X tan y \ y _j_ ^ ^ . . tan cc — tan?/ cot x cot ?/ 4- 1 27. tan(a; — w)=- 29. cot(a: — ?/) = — : ^ ' l + tana:tan^ ' cot?/ — cota; Functions of Twice an Angle (§ 101) 30. sin 2 = 2 sin (f> cos (j>. 32. cos 2 (f> = cos^ = 2 tan 33. cot 2 = cot^ <^ — 1 1 — tan^ Functions of Half an Angle (§ 102) 36. tan ^ = ± cos 4> 34. sin^^ =±^ h + cos d> 36. cosi=±^- 1 — cos 37. cot^^ 1 + cos ^ cos cos 186 PLANE TRIGONOMETRY Functions involving Half Angles (§ 101) c. ■ X X 38. sin X = 2 sm - cos - • ^ Jj 40. COS X = cos^- — Jj ■ 2^ sin^-- 39. tan X — 2 tan^ 41. cot X = 2cot^ Sums and Differences of Functions (§ 103) 42. sinal + sin A = 2 sin ^(^4 + A) cos — B). 43. sin^I — sin 5 = 2 cos ^(A + A) sin \(A — B). 44. cosal + cos A = 2 cos ^(A + 'A)cos ^(^1 — A). 45. cosal — cos A — — 2 sin + A) sin ^(^4 — A), sin A + sin A _ tan ^ (^4 + A) 46. sina4 — sin A tan ^ (^4 — A) Laws of Sines, Cosines, and Tangents (§§ 105, 111, 112) a _ sinyl b sin A a h c 47. Law of sines. ^ 48. Law of cosines, ^ 49. Law of tangents. a + & + c 50. ^ — = s. sin^l sin A sinC = b^ — 2 be cos A . a — b tan 1 (.4 — A) ^ 7 = ■; , 7 » if a > 0 ; a b tan (^.4 -|- A) b — a tanl(A— -4) .. = TV 7 ) if G < 0. b + a tan -^A + .4 ) Formulas in Terms of Sides (§§ 115, 116) 53 -a){s-b)(s-c) ^ s 51. 62. cos ^A = j(s — b)(s — c) 54. tan LG = ^ • ^ > s(s — a) j s(s — a) be 55. tan ^A -- Areas of Triangles (§ 118) 56. Area of triangle *4 AC = ^ ac sin A = ^ ?■(« + 5 + c) = ra = , abc sin A sin C Vs(s _ a) (s - 5) (s - c) = — = 4 A 2 sin (A + C) INDEX PAGE Abscissa 78 Addition formulas .... 97, 101 Algebra, applications to . . . . 173 Ambiguous case 112 Angle, functions of an .... 3, 4 of depression 18 of elevation 18 negative 77, 92 positive 77 Angles, difference of 100 differing by 90° 92 greater than 360° 87 having the same functions 154, 155 how measured 2 sum of 97 Autilogarithm 48 Areas ...... 66, 128, 141, 142 Base '40 Briggs 39 Changes in the functions ... 25 Characteristic 43 negative 44, 61 Circle 144 Circular measure 151 Cologarithm 54 Compass 146 Complementary angles .... 7 Conversion table 30 Coordinates 78 Cosecant 4, 22 Cosine 4, 16, 116, 180 Cosines, law of 116 Cotangent 4, 20 Course 145 Coversed sine 171 . . . . 30 . ... 174 . ... 145 187 PAGE Depression, angle of 18 Difference of two angles .... 100 of two functions 105 Division by logarithms ... 42, 52 Elevation, angle of 18 Eliminant 171 Equation .... 163, 166, 169, 173 Euler 181 Euler’s Formula 181 Expansion in series 180 Exponential equation 58 series 179 Formulas, important 185 Fractional exponent 57 Functions as lines 23 changes in the 25 graphs of 158 inverse 156. line values of 85 logarithms of 60 of a negative angle .... 92 of an angle 3, 10 of any angle 82 of half an angle . . . 104, 123 of small angles 153 of the difference of two angles 100 of the sum of two angles . . 97 of 30°, 4.5°, 60° 8 of twice an angle . . . . . 103 reciprocal 12 relations of 12, 13 variations in 86 Graphs of functions 158 Half angles 104, 123 Decimal table . . . De Moivre’s Theorem Departure .... Identity . . Interpolation . . 163 31, 32, 48 188 INDEX PAGE Inverse functions 156 Isosceles triangle 70 Latitude 145 Laws of the characteristic ... 44 of cosines 116 of sines 108 of tangents 118 Logarithm 40 Logarithms 39 of functions 60 properties of 178 systems of 178 use of tables of .... 46, 61 Mantissa 43 Middle latitude sailing .... 149 Multiplication by logarithms . 42, 50 Napier 39 Negative angle 77, 92 characteristic 44, 51 lines 77 Oblique angles 77 triangle 107 Ordinate 78 Origin 78 Parallel sailing 148 Plane sailing . 145 trigonometry 1 Polygon, regular 72 Positive angle 77 Power, logarithm of .... 43, 56 Practical use of the cosecant . . 22 of the cosine 16 of the cotangent 20 of the secant 21 of the sine 14 of the tangent 18 PAGE Quadrant 78 Radian 151 Reciprocal functions 12 Reduction of functions to first quadrant 90 Regular polygon 72 Relations of the functions . 12, 13, 94 Right triangle 34, 63, 133 Root, logarithm of 43, 57 Roots of numbers 177 of unity 175 Secant 4, 21 Series, exponential 179 Sexagesimal table 28 Signs of functions 86 Simultaneous equations .... 169 Sine 4, 14, 108, 180 Sines, law of 108 Sum of two angles 97 of two functions 105 Surveyor’s measures 142 Symbols 3, 4, 40, 171 Tables explained 10, 28, 30, 46, 48, 61 Tangent 4, 18 Tangents, law of 118 Traverse sailing 150 Trigonometric equation .... 163 identity 163 Trigonometry, nature of ... . 1 plane 1 Unity, roots of 175 Variations in the functions . . 86 Versed sine 171 AJN^SWERS ANSWEES PLANE TRIGONOMETRY Exercise 1. Page 5 1. cos5=-; tan5=-; cotB=~\ seoB^-; csc-B=-- c a b a b 4. cotJ.. 5. sec^. 6. cscA. I ; cos-4 = i ; tan A = | ; cot^ = | ; sec f ; csc^ = ■j®g- ; cos-4 = i| ; tan-4 = ; cot-4 = JJ. ; sec-4 = 1 1 ; csc^ = ; cos-4 = jj', tan-4 = Jg ; cot-4 = Jg®- ; sec-4 = ; csc-4 = ; ccs-4 = I ° ; tan A = cot A = ; sec -4 = ; esc -4 = 4; cos-4 = M; tan-4 = j cot-4 = |g; sec-4 = ; csc-4 = ; cos-4 l|a ; tanJ. = 119 ; cot^ = ifg ; sec4 = {W> 15. 16. ir. 19. 20 . 21 . 22 . 23. 24. tan-1, sin -4 = sin -4 = sin -4 = sin -4 = sin-1 = sin-1 (;■' -1 = -f. &2 sin-1 = sec -4 : sin-4 : sec -4 : sin -4 : sec -4 : sin -4 = sec-1 = sin -4 = sin -4 = csc-4 = sin -4 : csc-1 : sinl? CSC B sinB csc£ sinB cscB 3 „ ■gs 1 1 ^ T6 9 1B9 119’ = C2. 2n + l’ + 1 n^ — 1 2n Tl^ + 1 ’ M- + 1 — 1 2mn ; cos-1 = csc-1 = cos-1 = csc-1 -- vP- — 1 vP + 1 vP + 1 2n vP - 1 vP + 1 4- 1 tan-1 2n — 1 ; cot -4 2n tan -4 = 2n n'^ — 1 ; cot-1 = — 1 2n — 1 2 n w? + V? w? + m? —V?’ 2mn w? + ’ + V? - ; cos-1 = csc-4 = cos-1 = ; csc-1 = m2 - m2 mP' + rP m2 + «2 2 mn m2 — n2 m2 + n2 mP + n2 : tan-4 : tan -4 = 2mn 2mn n ,2 — 7)2 ' cot A = n% — 2 mn cot -4 = 2 mn ^2 _ jj 2 2 mn 1 V2 = cos-4 ; tan^ = 1 = cot-4 ; sec-4 = Vi = csc-4. §■ V5; cos-4 = 4 Vs ; tan-4 = 2 ; cot-4 = ^ ; sec-4 = Vs ; iVs. ; cos-4 = ^ Vs ; tan-4 = |V5; cot-4 = -^VS; sec -4 = 3- VS; cos B = yW ; tan B = 1/^1 ; cot B = ; sec B = lj\5. ; cosB= jfl; tanB = ^^^s-, cotB = J^e, • 3 - 3 - J- -143. ■ T4 5 > - 1 4 S ■ T4^ secB= III; = cosB=||-4; tanB=j2^\; cotB = 2^^--, seoB = ||f; :-W- 1 2 PLANE TRIGONOMETRY 26. sinR = seoR = 26. sin A cos A = tan A = 27. sin A Co:-, si = cosR = ^; P+Q P+q tanR = 2 p-q ’ cotB = p + q . p -q' Vp2 + ( + 9 P + g Vp2 + \/2pq -v^2 ^ p p + 1 1 Vp + 1 cscR = + g 2pg • = cos jB ; cot A = Vp^ + : = tan B ; = sin B ; .sec A = ^ = cscB : = cotB; CSC A = Vp2 + : = secR. = cosR : = sinR ; Vp cot A = — — = tan R ; ;A - Vp + 1 = cscR; tan A = Vp = cotR ; CSC A = + p „ — = — — sec R. 28. 12.3. 37. 2.5; 1.5. 47. a = 4.501 ; h = 5.362. 29. 1.64. 38. 1.5mi. ; 2 mi. 48. a = 6.8801 ; 6 = 8.1962. 30. 9. 40. a = 0.342 ; 6 = 0.94. 49. a = 160.75; 6 = 191.-5. 31. 6800. 41. a = 1.368; h = 3.76. 50. a = 1.88 ; 6 = 0.684. 32. 4000. 42. a = 1.197 ; h = 3.29. 51. c = 2.128; 6 = 0.728. 33. 227.84. 43. a = 1.6416 ; 6 = 4.512. 52. c = 5.848 ; a = 5.494. 34. 3V13; 9. 44. a = 2.565 ; 6 = 7.05. 53. c =26.6; 6= 9.1. 35. 45. a = 0.643 ; h = 0.766. 54. a = 412.05; c =438.6. 36. 5; 3. 46. a = 1.929; h = 2.298. 55. 142.926 yd. 56. 11 ; 24 ft. Exercise 2. Page 7 1. cos 60°. 5. COS 40°. 9. cos 30°. 13. cos 14° 30'. 17. cos 2-5°. 21. tan 29°, 2. sin 70°. 6. cot 30°. 10. sin 30°. 14. cot 7° 15'. 18. cot 10°. 22. sec 12°. 3. cot 50°. 7. CSC 15°. 11. cot 45°. 15. CSC 21° 45'. 19. CSC 13°. 23. cos 1°. 4. CSC 65°. 8. sec 5°. 12. CSC 45°. 16. sin 1° 50'. 20. sin 38°. 24. sin 4°. 25. CSC 2°. 27. sin7-t° 29. 45°. 31. 30°. 26. cos 12^°. 28. cot 1.4°. CO p 32. 30°. Exercise 3. Page 9 1. 0.5. 5. 1.1547. 9. 1.7320. 13. V2. 17. 21. 2. 0.8660. 6. 2. 10. 0.5773. 14. We. 18. ;^V2. 22. 3. 3. 0.5773. 7. 0.8660. 11. 2. 15. V3. 19. IV3. 23. 4. 1.7320. 8. 0.5. 12. 1.1547. 16. IV3. 20. A 3. 24. 25. cos 27° 42' '20". 27. CSC 2° 27' 9". 29. COS 14.2°. 31. cot 21.18°. 26. cot 14° 31' 25'' '. 28. sin 1° 59' 33". 30. sin 7.25°. 32. CSC 4.05°. 33. 90°. 37. 90° 40. 22° 30'. 43. jVe. 44. V2. 47. 2a'3. 51. 1. 2. 68. iVs. 34. 60°. n+1 41.18°. 48. 35. 36. 22° 30'. 18°. 38. 39. 90°. 42 90° 60°.-^l ■ ■a + 1 45. Ve. 46. fVi. 49. 50. iVs. ^V3. ANSWEJIS 3 Exercise 4. Page 10 1. 0.0872. 7. 0.3584. 13. 0.9136. 19. 5.1446. 25. 1.0000. 31. 1.4396. 2. 0.2419. 8. 0.5000. 14. 0.9135. 20. 5.1446. 26. 1.0000. 32. 1.4396. 3. 0.3684. 9. 0.9945. 15. 0.8192. 21. 0.3839. 27. 1.0353. 33. 0.0038. 4. 0.5000. 10. 0.9945. 16. 0.8192. 22. 0.3839. 28. 1.0353. 34. 0.0054. 5. 0.0872. 11. 0.9703. 17. 11.4301. 23. 1.0000. 29. 4.8097. 35. 2 sec 10°. 6. 0.2419. 12. 0.9703. 18. 11.4301. 24. 1.0000. 30. 4.8097. 36. 2 CSC 10°. 37. 2 cos 15°. 38. 3 sin 20° > sin (3 x 20°) and > sin (2 x 20°). 39. 3 tan 10° < tan (3 x 10°) and > tan (2 x 10°). 40. 3 cos 10° > cos (3 x 10°) and > cos (2 x 10°). 41. No. 42. The sin, tan, sec increase and the cos, cot, esc decrease. Exercise 5. Page 12 12. 37.6. 13. 1. 14. 100. 15. 60. 16. 12.86. 17. 22.64. Exercise 6. Page 15 1. 1.736. 4. 57.45, 7. 39° 10. 54 ft. 13. 449.9 ft. 2. 3.882. 5. 12°. 8. 43° 11. 4.326 ft. 3. 41.01. 6. 20°. 9. 30° 12. 479.9 ft. Exercise 7. Page 16 1. 10.83. 8. 5.935. 15. 63°. 22. 411.4 ft. 29. 6 in. 2. 13.46. 9. 4.884. 16. 70°. 23. 383 ft. 30. 28.19ft.; 21.21 ft.; 3. 25.58. 10. 7.311. 17. 54°. 24. 43°. 12.68 ft.; 30 ft.; 0ft 4. 31.86. 11. 10°. 18. 60°. 25. 7.794 in. 31. 60°; 0°. 5. 55.73. 12. 17°. 19. 70°. 26. 166.272 sq. in. 32. 2.5°; 65°. 6. 1.873. 13. 26°. 20. 84°. 27. 5.657. 33. 30° and 60°; 7. 5.972. 14. 60°. 21. 60°. 28. 27.71 ft. 31° and 59°. 34. 749. £ Ift. Exercise 8 . Page 19 1. 12.02. 6. 5.928. 11. 45°. 16. 64°. 20. 159.7 ft. 2. 11.04. 7, , 14.78. 12. 8°. 17. 148 ft. 8 in. 21. 45°; 90°; 45°. 3. 28.84. 8. 44.01. 13. 9°. 18. 29°. 22. 15.76 ft. 4. 45.04. 9. 107.1. 14. 19°. 19. 2.517 mi; 23. 6.14 ft. 5. 98. 10. 453.8. 15. 22°. 3.916 mi. 24. 1.03 in. Exercise 9. Page 20 1. 26.11. 4. 85.81. 7. 26.60. 10. 26' ) 13. 113 ft. 2. 12.36. 5. 544.0. 8. 68.80. 11. 28. 87 ft. 14. 123.6 ft 3. 162.6. 6. 26.84. 9. 45°. 12. 428.4 ft. Exercise 10. Page 21 1. 40.40. 4. 33.63. 7. 41°. 10. 57. .74 ft. 13. 26.11ft 2. 61.77. 5. 65.50. 8. 60°. 11. 1369 ft. 3. 101.2. 6. 339.4. 9. 22.66 ft. 12. 91, ,64 ft. 4 PLANE TRIGONOMETRY Exercise 11. Page 22 1. 49,60. 3. 80.62. 5. 81.19. 2. 54.87. 4. 64.60. 6. 152.8. 13. 19.82 mi. 14. 267.0 ft. 7. 64°. 9. 65°. 11. 1113 ft. 8. 28°. 10. 45°. 12. 13.69 mi 15. 57.51ft. 16. 17.23 in. Exercise 12. Page 23 3. tana;. 4. secx. 5. secx. 6. csc x. 7. cotx. 8. cscx. 16.18°. 35. rsinx. 36. a = cm; b = c Vl — m^. 37. a = bm; c = b + 1. Exercise 13. Page 26 2. 0. 8. No. 13. 2.3109. 19. 37°. 25. 19°. 31. 16^ 3. 1. 9. 45°. 14. 0.5373. 20. 46°. 26. 48°. 32. 31^. 4. 00. 10. 0.6462; 15. 6°. 21. 6°. 27. 34°. 33. 1 5* 5. 0. 0.7631. 16. 24°. 22. 13°. 28. 40°. 6. The tangent. 11. 0.3680. 17. 44°. 23. 22°. 29. 54°. 7. No. 12. 2.7173. 18. 26°. 24. 14°. 30. 30°. Exercise 14. Page 29 1 . 0.7647. 7. 0.7428. 13. 0.8708. 19. 53.47. 25. 69.38. 31. 19.70 ft.; 2. 0.9004. 8. 0.6563. 14. 0.8708. 20. 20.90. 26. 49.83. 22.62 ft. 3. 0.7545. 9. 0.6693. 15. 1.1483. 21. 25.27. 27. 94.35. 32. 19.72 ft.; 4. 0.9015. 10. 0.6567. 16. 17.73. 22. 48.29. 28. 74.93. 22.61 ft. 5. 0.7538. 11. 0.6700. 17. 32.16. 23. 66.48. 29. 88.35, 33. 120.5 ft. 6. 0.7546. 12. 0.6700. 18. 46.01. 24. 64.84. 30. 47° 56'. 34. 71.77 ft. Exercise 15. Page 30 1. 0.0087. 6. 0.0715. 11. 0.9972. 16. 1.0000. 21. 12.66 in. ; 2. 0.0070. 7. 0.9972. 12. 0.9974. 17. 0.0715. 0.9970 in. 3. 0.0698. 8. 0.0769. 13. 0.0767. 18. 143.2. 22. 390 ft. 4. 0.9973. 9. 12.71. 14. 13.95. 19. 0.0052. 23. 0.7477 in. 5. 0.0787. 10. 13.62. 15. 0.0769, 20. 0.0734. 9.530 in. Exercise 16. Page 33 1. 0.4567. 14. 12.1524. 24. 70° 45' 30"; 35. 10.7389. 48. 44° 38' 30" 2. 0.6725. 15. 15.3140. 0.3490. 36. 0.9808. 49. 69° 15'. 3. 0.8338. 16. 10.4652. 25. 79° 30' 15"; 37. 4.5787. 50. 78° 8' 30". 4. 0.9099. 17. 8.7149. 0.1852. 38. 4.1525. 51. 78° 8' 15". 5. 0.8065. 18. 7.2246. 26. 0.4305. 39. 3.6108. 52. 14° 45'. 6. 0.7289. 19. 6.6585. 27. 0.4313. 40. 3.3502. 53. 0.7658. 7. 0.4335. 20. 6.0826. 28. 0.5410. 41. 31° 30'. 54. 0.6438. 8. 0.5438. 21. 39° 43' 30"; ; 29. 0.6646. 42. 35° 15'. 55. 0.5639. 9. 0.6418. 0.7691. 30. 0.9045. 43. 41° 18' 30". 56. 33° 10' 15". 10. 0.9209. 22. 50° 16' SO"; 31. 0.1990. 44. 44° 36' 30" 1.5298. 11. 1.2882. 0.6391. 32. 4.9550. 45. 38° 15'. 57. 31° 8' 30"; 12. 2.5018. 23. 71° 29' 40"; ; 33. 0.1490. 46. 39° 30'. 0.6042. 13. 3.1266. 0.9483. 34. 7.8279. 47. 17° 46'. ANSWERS 5 Exercise 17. Page 37 1. A = 36°62', R=53°8', c = 5. 2. A = 32° 35', R= 67° 25', 6 = 10.95. 3.5 = 77° 43', b = 24.34, c = 24.93. 4. A = 46° 42', 6 = 9.801, c = 14.29. 5. 5 = 52° 18', a = 15.90, 6 = 20.57. 6. A = 65° 48', a = 127.7, 6 = 57.39. 7. A = 34° 18', 5= 5-5° 42', a = 12.96. 15. 5 = 51° 31', a 16. A = 22° 37', B 17. A = 53° 8', B 18. A = 22° 37', B 8. A = 43° 33', 5=46°27',a = 93.14. 9. 5 = 67° 46', a = 26. 73, c = 50.12. 10. A = 43° 49', a = 191.9, c = 277.2. 11. A = 68° 43', 5=21°17', c = 102.0. 12. A = 3° 20', B = 86° 40', 6 = 102.8. 13. A = 84° 52', 6 = 0.2802, c = 3.133. 14. A = 70° 48', 5=19° 12', 6 = 5.916. 35.47, 6 =44.62. 67° 23', a = 5, c = 13. 36° 52', a = 40, c = 50. 67° 23', a = 12.5, c = 32.5. 19. 5 = 54° 49' 30", 6 = 3.547, c= 4.340. 21. A = 60° 41' 30", 6=3.593, c = 7.339. 20. 5 = 47°47'30", 6 = 6.284, c=8.485. 22. A = 63° 39' 30", 6=5.812, c=9.808. 23. 5 = 60° 17' 30", a = 3.370, 6 = 5.906. 24. 5 = 55° 39' 30", a = 203.08, 6 = 297.25. 25. 5 = 48° 49' 20", a = 218.68, c = 332.14. 26. 5 = 64.5°, 6 = 100.6, c = 111.5. 27. 5 = 65.5°, a = 10.37, 6 = 22.76. 28. 5 = 57.45°, a = 21.52, 6 = 33.72. 29. 5 = 34.49°, a = 65.94, 6 = 45.30. 30. 5 = 26.54°, a = 67.10, 6 = 33.51. 31. A = 39.41°, 6 = 54.77, c = 70.88. 32. B = 21.75°, a = 225.6, c = 242.8. 33. 29.20 in. 34. 23.73 in. 35. 42.25 in. 36. 64.26 in. 37. 43.30 in. 38. 60.05 in. 39. 66° 18' 36", 33° 41' 24". 40. A = 41° 24' 30", B = 48° 35' 30". 41. 13.26 ft. 42. 16.82 in.; 18.50 in. 43. 12.42 ft. 44. 66.89 in. 45. 9° 35' 40". Exercise 18. Page 41 1. 5. 3. 4. 5. 6. 7. 8. 9. 6. 11. 3. 13. 3. 15. 4. 17. 3. 19. 6. 2. 2. 4. 4. 6. 7. 8. 5. 10. 4. 12. 2. 14. 3. 16. 2. 18. 6. 20. -1. 21. — 2; -3; - 4. 24. 1; 2; 3 ; 6 ; 9; 10: ; — 2; -4; 22. 1 and 2 ; 2 and 3 ; 3 and 4 ; -5; - 6; - -7 ; - 8 4 and 5 ; 5 and 6 ; 8; and 9. 25. 1 ; 4; 6 ; 7; 8 ; - 1 ) 2; -3; 23. — 2 and — 1; 3 and — 2 ; -4; - 5 ; - - 6; - 7 — 4 and — 3; - 1 and 0 ; 26. 0; -4; C 7 ; 8. — 2 and 1; - 3 and — 2. 27. 1 and 2. 31. 2 and 3. 35. 3 and 4. 39. 5 and 6. 28. 1 and 2. 32. 2 and 3. 36. 3 and 4. 40. 6 and 7. 29. 1 and 2. 33. 2 and 3. 37. 3 and 4. 41. 6 and 7. 30. 1 and 2. 34. 2 and 3. 38. 3 and 4. 42. 7 and 8. Exercise 19. Page 45 1. 1. 6. 3. 11. - 1. 16. - 4. 21. 1.58681. 2. 1. 7. 2. 12. - 2. 17. ■ - 3. 22. 0.58681. 3. 2. 8. 1. 13. - 1. 18. • - 5. 23. 2.58681. 4. 0. 9. 0. 14. - 1. 19. - 1. 24. 4.58681. 5. 3. 10. 4. 15. - ■ 3. 20. - 2. 25. 5.58681. 6 PLANE TRIGONOMETRY 26. 7.68681. 32. 4.67724. 38. T.40603. 44. 1.39794, 27. T.58681. 33. 7.67724. 39. 3.40603. 46. 2.39794. 28. 2.58681. 34. 167724. 40. 4.40603. 46. 4.39794. 29. 4.68681. 35. 6.67724. 41. 7.40603. 47. 7.39794. 30. 3.67724. 36. 0.40603. 42. 0.39794. 31. 0.67724. 37. 1.40603. 43. 1.39794. Exercise 20. Page 47 1. 0.30103. 14. 1.83556. 27. 4.09157. 40. 3.20732. 53. 0.464.58. 2. 1.30103. 15. 0.89905. 28. 2.09157. 41. 4.86198. 54. 0.64167. 3. 2.30103. 16. 2.92158. 29. 2.37037. 42. 0.48124. 55. 1.08030. 4. 3.30103. 17. T.84510. 30. 1.61624. 43. 0.95424. 56. 2.16224. 5. 3.32222. 18. 1.87506. 31. 1.75037. 44. 0.90309. 57. 0.79034. 6. 3.33244. 19. 1.87852 . 32. 1.61576. 45. 4.22472. 58. 1.14477. 7. 3.33365. 20. T.87892. 33. 5.51409. 46. 2.87595. 59. 0.54254. 8. 0.33365. 21. 2.40654. 34. 2.56155. 47. 5.32328. 60. 0.99155. 9. 3.54220. 22. 3.55630. 35. 7.82948. 48. 12.70040. 61. 2.00072. 10. 3.64953. 23. 4.95424. 36. 17.72.562. 49. 19.58460. 62. 0.75343. 11. 3.74671. 24. 2.25042. 37. 9.19605. 50. 0.15052. 63. 1.19855. 12. 3.84663. 25. 4.09132. 38. 5.26893. 51. 1.65052. 13. 3.72304. 26. 4.09150. 39. 2.51989. 62. 1.17969. Exercise 21. Page 49 1. 3. 14. 7.6. 27. 6846.5. 39. 91.226. 2. 3000. 15. 7,805,000,000. 28. 685.5.5. 40. 53,159,000. 3. 0.003. 16. 79,950,000. 29. 77,553. 41. 0.000010745. 4. 304.5. 17. 1.7102. 30. 785.65. 42. 5.72784; 5. 37,020. 18. 27.005. 31. 7917.3. 534,360. 6. 46. 19. 370.15. 32. 8.5552. 43. 353,780. 7. 467.6. 20. 0.38065. 33. 875.18. 44. 7.2388. 8. 0.000056. 21. 0.0043142. 34. 2. 45. 107. 9. 6505. 22. 43,144. 35. 3.45591 ; 46. 25,459. 10. 0.06796. 23. 4.3646. 3.45864. 47. 16,693,000. 11. 0.0006095. 24. 0.049074. 36. 2955. 48. 129.66. 12. 0.66. 25. 594,640,000. 37. 0.0066062. 49. 4.9341. 13. 6.696. 26. 0.00067656. 38. 0.65163. Exercise 22. Page 50 1. 10. 9. 66. 17. 12,000. 25. 603.9. 33. 210. 2. 24. 10. 18. 18. 18,000. 26. 1282.8. 34. 945. 3. 15. 11. 100. 19. 660,000. 27. 184,670. 35. 5005. 4. 35. 12. 2400. 20. 180,000. 28. 11,099. 36. 38,645. 5. 8. 13. 1500. 21. 1034.6. 29. 1609.9. 37. 627,400 6. 21. 14. 3500. 22. 2192.3. 30. 17,458. 38. 276.67, 7. 12. 15. 8000. 23. 13.31. 31. 18.212 in. 8. 18. 16. 21,000. 24. 20.265. 32. 113.04 ft. ANSWERS 7 Exercise 23. Page 51 1. 7.68964. 7. 4.03939. 13. 0.1248. 19. 0.02240. 25. 22.936, &. 3.68964. 8. 2.00010. 14. 0.0001248. 20. 0.00015725. 26. 34.108, 3. 7.68964. 9. 1.99999. 15. 0.0043707. 21. 1.3020. 27. 16.51. 4. 3.09497. 10. 0.00000. 16. 0.11422. 22. 38.079. 5. 0.00000. 11. 1,248,000. 17. 0.0000003125. 23. 3309.6. 6. 1.99999. 12. 124.8. 18. 0.25121. 24. 452.27. Exercise 24. Page 53 1. 1.97519. 13. 3.89100. 25. 5. 37. 0.00999. 49. 60.87. 2. 3.66078. 14. 2.00000. 26. 84. 38. 0.0709. 50. 0.6527. 3. 1.68618. 15. 2.11220. 27. 82.002. 39. 0.0204. 51. 20. 4. 3.70404. 16. 2.00286. 28. 76. 40. 0.065. 52. 50. 5. 5.00000. 17. 1.71172. 29. 35.6. 41. 0.48001. 53. 700. 6. 9.70000. 18. 5. 30. 73.002. 42. 2.143. 54. 800. 7. 7.00000. 19. 5. 31. 92. 43. 0.4667. 55. 9000. 8. 7.00000. 20. 3. 32. 105. 44. 0.004667. 56. 11,000. 9. 3.76439. 21. 4. 33. 63. 45. 1.913. 57. 120,000. 10. 2.00000. 22. 3. 34. 77. 46. 1.123. 58. 0.01. 11. 2.90000. 23. 5. 35. 0.0129. 47. 12.86. 59. 871.1 ; 2, 12. 6.90000. 24. 3. 36. 1290. 48. 5.184. Exercise 25. Page 54 1. 2.60206. 5. 4.42585. 9. 0.30103. 13. 1.52187. 17. 1. 2. 3.88606. 6. 3.36927. 10. 0.14267. 14. 2.20698. 18. 0.1, 3. 2.56225. 7. 2.28727. 11. 1.08092. 15. 3.22185. 19. 0. 4. T.23433. 8. 1.14188. 12. 2.13906. 16. 4.15490. 20. 1. Exercise 26. Page 55 J. 1. 8. 0.44272. 15. 6.1649. 22. 105.47. 2. 6. 9. 1.7833. 16. 0.42742. 23. 3,013,400. 3. 3. 10. 1000. 17. 1.4179. 24. 0.081528. 4. 0.5. 11. 0.092. 18. 0.031169. 25. 232.24. 5. 1. 12. 1.8. 19. 40.464. 26. 0.0000007237. 6. 2. 13. 0.01. 20. 0.14621. 27. 103.33. 7. 0.11111. 14. 0.21. 21. 2893.2. Exercise 27. Page 56 1. 4. 6. 728.98. 11. 4,782,800. 16. 83,522. 2. 8. 7. 64. 12. 16,777,000. 17. 15,625. 3. 32. 8. 125. 13. 19,486,000. 18. 6,103,600,000. 4. 1024. 9. 1. 14. 11,391,000. 19. 15,625. 5. 80.998. 10. 40,355,000. 15. 11.391. 20. 244,140,000. 8 PLANE TRIGONOMETRY 21. 16,413,000,000,000,000. 29. 0.0.5765. 37. 0.023551. 22. 7,700,500. 30. 0.00000011765. 38. 0.0001.5228. 23. 31,137,000,000. 31. 0.018741. 39. 0.000007.5624. 24. 292,360,000,000,000. 32. 154.85. 40. 0.00000012603. 25. 2.1435. 33. 157.5. 41. 9.8696; 31.006. 26. 180.11. 34. 41,961. 42. 21.991 ; 153.94; 27. 0.000000000001. 35. 2.0727. 3053.6. 28. 0.00000002048. 36. 0.-0019720. Exercise 28. Page 57 1. 1.4142. 7. 5.6569. 13. 0.54773. 19. 3.9095. 2. 1.71. 8. 3.0403. 14. 0.3684. 20. 0.0028827. 3. 1.3205. 9. 3.3166. 15. 0.067405. 21. 1.7725; 1.4645. 4. 1.2394. 10. 1.4422. 16. 0.064491. 22. 1.3313; 2.1450; 5. 1.1487. 11. 2.802. 17. 20.729. 5-5684; 0.42378; 6. 2.2795. 12. 1.2023. 18. 1.9733. 0.40020; 0.79537. Exercise 29. Page 59 1. X = 3. 6. X = 4.2479. 11. X = 3. 16. X = 3, y = 1. 2. X = 4. 7. X = 3.9300. 12. X = 3.3219. 17. X = 5, 2/ = 1. 3. X = 4. 8. X = 4.2920. 13. X = - 0.087515. 18. x = l,y= 1. 4. X = 4. 9. X = 5.6610. 14. X = 4.4190. 19. X = 2, y = 2. 5. X = 3. 10. X = 3.0499. 15. X = - 0.047954. 20. x = 3, y = 2. 21. X = 2, ?/ = 2. 27. X = 2, - 1. 35. 2; 7.2730; 22. X: 23. X: 24. X = 25. X = 26. X = log a — logp log(l + r) log r + log I — log a log r :1, -3. log a — log p log(l + r£) log [s (r — 1) + a] — log a 28. 0.062457. 29. 3.1389. 30. 0.036161. 31. 0.03475. 6 . log& logo logw log 5 2.0009; 2.0043. 36. 1; log a loi 32. 33. 34. 37. 38. X = lb’ log 6 1;3;4 log a — log 6 - 1 . Exercise 30. Page 62 1. 9.65705 - 10. 13. 8.89464- 10. 25. 9.95340 - 10. 37. 8.11503- 10. 2. 9.97015- 10. 14. 9.99651 - 10. 26. 11.13737- 10. 38. 8.00469 - 10. 3. 9.90796 - 10. 15. 9.23510 - 10. 27. 9.74766 - 10. 39. 8.24915 - 10. 4. 9.82551 - 10. 16. 9.87099 - 10. 28. 9.66368 - 10. 40. 8.24915- 10. 5. 10.57195 - 10. 17. 9.68826 - 10. 29. 10.17675- 10. 41. 8.63254- 10. 6. 9.32747- 10. 18. 10.10706 - - 10. 30. 9.82332 - 10. 42. 8.63205 - 10. 7. 10.57195- 10. 19. 9.55763 - 10. 31. 6.51165-10. 43. 9.32507- 10. 8. 9.32747 - 10. 20. 9.96966 - 10. 32. 8.25667- 10. 44. 9.32507 - 10. 9. 9.20613- 10. 21. 9.98436 - 10. 33. 6.79257 - 10. 45. 10.39604 - -10 10. 9.99526 - 10. 22. 9.42095 - 10. 34. 8.56813 - 10. 46. .7° 30'. 11. 9.14412-10. 23. 9.48632 - 10. 35. 7.45643 - 10. 47. 32° 21'. 12. 9.14412-10. 24. 9.68916- 10. 36. 8.15611 - 10. 48. 58° 27. ANSWERS 9 49. 86° 30'. 55. 63° 41' 23". 61. 49° 34 ' 12". 67. 57° 4f". SO. 4°3(K. 56. 77° 6'. 62. 61° 47 ' 36". 68. 49° 25' 7". 51. 31° 33'. 57. 79°. 63. CO O 00 48". 69. 38° 22' 30' 52. 58° 35'., 58. 70°. 64. 50° 48 1' 15". 70. 2° 3' 30". 63. 50° 32'. 59. 20° 13' 30". 65. O 00 30". 71. 89° 49' 10' 54. 39° 2'. 60. O CO 22' 15". 66. 0 00 CO o Exercise 31. Page 67 1. A = 30°, B = 60°, 6 = 10.39, S= 31.18. 2. B= 30°, a = 6.928, c = 8, 5 = 13.86. 3.5=60°, 6 = 5.196, c = 6, 5= 7.794. 4. A = 45°, 5 = 45°, c = 5.657, 5 = 8. 5. A = 43° 47', 5 = 46° 13', 6 = 2.086, 5 = 2.086. 6. 5 = 66° 30'. a = 250, 6 = 575, 5 = 71,880. 7. 5 = 61° 55', a = 1073, 6 = 2012, 5=1,079,500. 8. 5 = 50° 26', a = 45.96, 6 = 55.62, 5 = 1278. 9. 5 = 54°, a = 0.5878, 6 = 0.8090, 5 = 0.2378. 10. A = 68° 13', a = 185.7, 6 = 74.22, 5 = 6892. 11. A = 13° 35', a = 21.94, 6 = 90.79, 5 = 995.8. 12. 5= 85° 25', 6 = 7946, c = 7972, 5 = 2,531,000. 13. 5= 53° 16', 6 = 65.03, c = 81.14, 5= 1578. 14. 5 = 4°, 6 = 0.0005594, c = 0.00802, 5 = 0.000002238. 15. A = 46° 12', a = 53.12, c = 73.60, 5 = 1353. 16. A = 86° 22', a = 31.50, c = 31.56, 5= 31.50. 17. A = 13° 41', 6 = 4075, c = 4194, 5 = 2,021,000. 18. A = 21° 8', 6 = 188.9, c = 202.5, 5 = 6893. 19. A = 44° 35', 6 = 2.221, c = 3.119, 5= 2.431. 20. 5 =52° 4', a = 3.118, c = 5.071, 5= 6.235. 21. A = 31° 24', 5 = 58° 36', 6 = 7333, 5 = 16,410,000. 22. A = 56° 3', 5 = 33° 57', 6 = 48.32, 5 = 1734. 23. A = 65° 14', 5 = 24° 46', 6 = 3.917, 5=16.63. 24. A = 53° 15', 5 = 36° 45', a = 1758, 5 = 1,154,000. 25. A = 53° 31', 5 = 36° 29', a = 24.68, 5= 225.2. 26. A = 63°, 5 = 27°, c = 43, 5 = 373.9. 27. A = 4° 42', 5 = 85° 18', c = 15. 5= 9.187. 28. A = 81° 30', 5=8° 30', c = 419.9, 5 = 12,890. 29. A = 38° 59', 5 = 51° 1', c = 21.76, 5=115.8. 30. A = 1° 22', 5 = 88° 38', b = 91.89, 5 = 100.6. 31. A = 39° 48', 5 = 50° 12', c = 7.811, 5= 16. 32. A = 30' 12", 5 = 89° 29' 48", 1 6 = 70, 5= 21.53. 33. A = 43° 20', 5 = 46° 40', a - 1.189, 5 = 0.7488. 34. 5= 71° 46', 6 = 21.25, c = 22.37, 5 = 74.37. 35. 5 = 60° 52', a = 6.688, c = 13.74, 5=40.13. 36. 5 = 20° 6', a = 63.86, b = 23.37, 5= 746.15. 37. A = 45° 56', a = 19.40, b = 18.78, 5=182.16. 38. A = 41° 11', 6 = 53.72, c = 71.38, 5 = 1262.4. 39. A = 55° 16', a = 12.98, c = 15.80, 5 = 58.42. 40. A = 3° 56', a = 0.5805, b = 8.442, 5 = 2.450. 10 PLANE TRIGONOMETRY 41. 5 = 1C2 sin-i4 L cosA. 43. S= 1 b^ tan A. 42. 5: = ia^ cot A. 44. S=i a Vc^ — 1 2^. 45. A = 40° 45' 48", B = 49° 14' 12", b = :11.6, C = 15.315 46. A = 55° 13' 20", B = 34° '46' 40", a = : 7.2, C = 8.766. 47. B = 61° > a = 3.647, 6 = ■ 6.58, c = 7.523. 48. A = 27° 2' 30 ", B = 62° 57' 30", a = : 10.002, b = 19.595 49. 19° 28' 17" ; 70° 31' 43 51. 15 1.498 mi. 50. 3112 mi. ; 19,553 mi. 52. Between 1° 15' 30" and 1°19' 10 ", 53. 212.1 ft. 58. 59° 44' 35". 63. 7.071 mi. ; 67. 685.9 ft. 54. 732.2 ft. 59. 95.34 ft. 7.071 mi. 68. 5.657 ft 55. 3270 ft. 60. 23° 50' 40". 64. 19.05 ft. 69. 136.6 ft. 56. 37.3 ft. 61. 36° 1' 42". 65. 20.88 ft. 70. 140 ft. 57. 1° 25' 56' 62. 69° 26' 38". 66. 56.65 ft. 71. 84.74 ft. Exercise 32 . Page 71 1. . C = 2(90' °-A), c = 2 a cos A, A = a sin A 2, , A = 90° - -iC, c = 2a cos A, A = a sin A 3. C = 2(90° — A), a = — ^ , ^ = asinA. 2 cos A 4. A = 90°— iC, a = — - — , A r= a sin A. ^ 2 cos A 6. C = 2(90°- A), a=z— c = 2acosA. sin A 6. A = 90° — ,T (7, a — — ~ — , c = 2 a cos A . . ■ sin A 7. sinA = -, C = 2(90° — A), c = 2acosA. 8. tanA = — , C = 2(90°-A), a = — — c sin A 9. A = 67° 22' 50", C = 45° 14' 20", h = 13.2. 10. c = 0.21943, h = 0.27384, S = 0.03004. 11. a = 2.05.5, h = 1.6852, S = 1.9819. 12. a = 7.706, c = 3.6676, S = 13.725. 13. A = 25° 27' 47", 0 = 129° 4' 26", a = 81.41, A = 35. 14. A = 81° 12' 9", C = 17° 35' 42", a = 17, c = 5.2. 15. c = 14.049, A = 26.649, 5 = 187.2. 16. S = a2sin^OcosiC. 19.28.284 ft.; 21. 94° 20'. 24. 37.699 sq. in. 17. S = sin A cos A. 4525.44 sq. ft. 22. 2.7261. 25. 0.8775. 18. 5 = A^tan^^C. 20.0.76536. 23. 38° 56' 33". Exercise 33. Page 72 1. r= 1.618, A = 1.5388, 5 = 7.694. 4. r = 1.0824, c = 0-82842, 5 = 3.3137. 2. A = 0.9848, p = 6.2514, 5 = 3.0782. 5. r = 2.5942, A = 2.4891, c = 1 .461. 3. A = 19.754, c = 6.257, 5 = 1236. 6. r = 1.5994, A = 1.441, p = 9.716. 7. 0.51764 in. 9. 0.2238 sq. in. 13.6.283. g _ c 10. 0.310 in. 14. 0.635 sq. in ~ 90°' 11. 1.0235 in. 12. 0.062821 ; 6.2821. ANSWERS 11 Exercise 34. Page 73 2. 29.76 sq. in. 13. 52° 35' 42". 25. 362.09 ft. 36. 2675.8 mi. 3. 104.07 sq.ft. 14. 60° 36' 58". 26. 59° 2' 10". 37. 25.775 ft.; 4. 36.463 sq. in. 15. 6.3509 in. 27. 14.772 in. ; 19.45 ft. 5. 20.284 in. 16. 20 in. 15.595 in. 38. 10.941ft.; 7. 37.319 ft. 17. 7.7942 in. 28. 73.21 ft. 20.141 ft. 8. 342.67 ft. 18. 40° 7' 6". 29. 25° 36' 9". 39. 55.406 ft. 9. 36.602 ft.; 19. 77° 8' 31". 30. 26.613 in. 40. Between 131 86.602 ft. 20. 94.368 ft.; 31. 7.5 ft. and 132'. 10. 120.03 ft. 25° 42' 58". 32. 59° 58' 54"; 41. 43° 18' 48". 11. 2.9101 mi.; 21. 24.652 ft. 173.08 ft. 42. 2.6068 in. 3.531 mi. 22. 196.93 ft. 33. 7.2917 ft. 43. 14.542 in. ; 12. 11° 47"; 23. 220.8 ft. 34. 19.051. 26.87 in. 49.206 ft. 24. 1915.8 ft. 35. 1.732 in. 44. 6471.7 ft. Exercise 35. Page 80 29. 10. 33. 11. 37. 0. 41. 5.10. 45. 28^ in. 49. |V3. 30. 15. 34. 3|. 38. 7. 42. 5.10. 46. 9.43 in. 50. Yes. 31. 13. 35. 3. 39. 5. 43. 8.24. 47. 2. 51. Octagon 32. 21. 36. 5. 40. 15. 44. 4.24. 48. 3 Vs. 2.829.' Exercise 36. Page 84 16. I. 18. II. 20. III. 22. I. 24. III. 26. I. 28. III. 17. I. 19. II. 21. IV. 23. II. 25. IV. 27. II. 29. On OF'. 30. On OX. 64. sin = i V2 ; cos = — i V2 ; tan = — 1 ; 61. ^V3; j V6. CSC = V2 ; sec = - V2 ; cot = — 1. 62. 90°. 65. sin =: 0 ; cos = - 1 ; tan = 0 ; 63. 60°. CSC = 00 ; sec = 1; cot - 00 . Exercise 37. Page 88 62. 2 ; one in Quadrant I, one in Quadrant II. 83. 4 ; two in Quadrant I, two in Quadrant IV. 54. 2; 1; 1; 1; 1. 65. Between 90° and 270° ; between 0° and 90° or between 180° and 270® ; between 0° and 90° or between 270° and 360° ; between 180° and 360°. 57. 1 ; 0 ; 0 ; c» ; 1 ; CO ; 1 ; 0. 69. Ill; II. 60. 40; 20. 61. 0. 62. 0. 63. 0. 64. 4a5, 65. - 2(a2 + 62). 66 . 0 . 67. {'. 76. 30° ; 150° ; 390° ; 510°. 77. 30° ; 330° ; 390° ; 690°. 78. 60° ; 120° ; 420° ; 480°. 79. 60° ; 300° ; 420° ; 660°. 80. 30° ; 210° ; 390° ; 570°. 81. 60° ; 240° ; 420°; 600°. 82. 210°; 330°. 83. 120°; 240°. 84. 225°; 315°. 85. 135°; 225°. 86. 185°; 315°. 87. 135°; 315°. 12 PLANE TRIGONOMETRY Exercise 38. Page 91 1. sin 10°. 9. tan 78°. 17. - cot 65°. 25. - sin 7° 10' 3". 2. — cos 20°. 10. cot 82°. 18. - cot 1.3°. 26. cos 8.5° 54' 46". 3. — tan 32°. 11. - sin 85°. 19. — sin 0°. 27. - tan 37° 51' 4.5" 4. — cot 24°. 12. — sin 15°. 20. COS 0°. 28. cotl5°10'3". 5. sin 0°. 13. — tan 78°. 21. sin 31° 50'. 29. sin 32.2.5°. 6. — tan 0°. 14. — tan 35° 22. — COS 12° 20'. 30. — cos 52.25°. 7. — sin 20°. 15. cos 70°. 23. tan 85° 30'. 8. — cos 45°. 16. cos 10°. 24. - cot 72° 20'. Exercise 39. Page 93 1. cos 10°. 10. — cot 9°. 19. - sin 86°. 28. - cot 9.1°. 2. cos 30°. 11. - cot 29°. 20. cos 75°. 29, 0.0262. 3. cos 20°. 12. - cot 39°. 21. cos 87°. 30. - 0.5483. 4. cos 40°. 13. - tan 4° V. 22. — sin 5°. 31. - 0.7729. 5. - sin 5°. 14. — tan 7° 2'. 23. tan 80°. 32. 0.5040. 6. — sin 7°. 15. - tan 8° 3'. 24. tan 30°. 33. - 0.1304. 7. — sin 21°. 16. - tan 9° 9'. 25. — tan 20°. 34. 0.8686. 8. — sin 37°. 17. - sin 3°. 26. - cot 1.5°. 35. 0.1357. 9. — cot 1°. 18. — sin 9°. 27. - cot 7.8°. 36. - 0.1354. 37. 9.89947 - 10. 40. -(10.52286-10). 43. 10.147-53 - 10. 38. - (9.83861 - 10). 41. -(9.91969-10). 44. -(9.82489- 10). 39. -(9.79916- 10). 42. 9.92401 - 10. 46. 225°; 315°; 585°; 675° Exercise 40. Page 95 6. .«?in T. — -1- 1 19. 45°. 27. 60°. V COt^ X + 1 20. 30°. 28. 60° or 180°. 7. ms r. — -J- 1 21. 60°. 29. 4-5°. V tan'^ X + 1 22. 45°. 30. 30°. 8. .SPP. O'. — 4- 1 23. 45°. 31. 4-5°. Vi — sin^ X 24. 45°. 32. 4 V5 ; 4 V5. 9. CSC X 4- 1 25. 60°. 33. lVl5;Vl5 vr — COS^ X 26. 45°. 34. 4; 5. 35. 36. 37. 38. 39. 40. 53. 54. 65. 56. 57. 58. 59. 60 . = I Vs, cosx = ^ Vs, tanx = 2 ; cscx = ^ Vs, secx = Vs, cot x = 45. 270° or SO'’. 46. 30° or 150° 47. 45°, 135°, 225°, or 315°. 48. 60°. t'LVT 7; ^iyVl7. 41. 45° or 225°. ,4; 42. 45°, 135°, 225°, When X = 0°. or 315°. 0° or 180°. 43. 45° or 225°. 38° 10'. U. 0° or 60°. cosA=^-V5, tanA=|-V5, cscA= |, _ secA=:|-V5, cotJ.= i V5. sinA=lV7, tanA=^V7, cscA=iV7, secA=^, cotA= SV7. sinA= -y^jVlS, cosA = Vl3, cscA= ;jVl3, secA = |Vl3, cotA= sinA=^, cosA=3, tanA=^, _ cscA = |, secA = §. sinA=^V5, cosA=^, tanA = i^Vs, cscA = |Vs, cotA=|%5- oosA = tan A = esc A = -{-f , sec A = -G-, cot A = cosA tan A = f , esc A = |, sec A — cot A = |. sinA = tan A = cscA = sec A = cot A = answers 13 5 A = -2^, cot A = 61. sin A = If, tan A = csc^ = 62. sin^ = i, cos^=|, cscJ. = |, sec^ = cot^ 63. sin A = V2, cos A = I V2, tan Jl = 1, esc A = V2,_ sec A = V2 . 64. sin A = i V5, cos A = I VB, tan A = 65. sin^ = Vs, cos tan 4 = 66. sin ^ i Vi, cos A = ^ Vi, tan A = 67. cosJ. = Vl— m?, tan.4 ^ 2, CSC .<4 = ^ Vs, sec.4 = V6._ vi, csc^ = f Vi, cot 5 Vi. 1, sec^ = V2, cot^ = l. 2m Vl- y COt^ = 1 .1 1 CSC A = —, sec A = — --- m Vl— m2 70. cos 0° = 1, tan 0° = 0, esc 0“^ = oo, 71. cos 90° = 0, tan 90° = oo, esc 90° = 1, 72. sin 90° = 1, cos 90° = 0, esc 90° = 1, 73. sin 22° 30' = — ^ , cos 22° 30' V4 + 2 V 2 CSC 22° 30' = V 4 + 2 Vi, sec 22° 30' 1 — cos2 A 74. [■ cos A Vl — 5 68 . 69. 1 — m2 m2 + n2 m 2 mn sec 0° = 1, cot 0° = 00 . sec 90° = cx), cot 90° = 0. sec 90° = 00 , cot 90° = 0. , tan 22° 30' = V2 — 1, 1 V 4 - 2 V 2 y/i- 2 Vi. COs2.4 1 — cos2 A Exercise 41. Page 98 1. 0.25875. 5 . 1. 9. O.i 366. 13. 0.5. 2. 0.96575. 6. 0. 10. - 0.5. 14. - 0.866. 3. 0.96575. 7. 0.96575. 11. 0. 707. 15. 0.25875.^ 4. 0.25875. 8. - 0 .25875. 12. - 0.707 16. - 0.96575 Exercise 42. Page 99 1. 0.268. 5. CO. 9. - 1.732 13. - 0.577. 2. 3.732. 6. 0. 10. - 0.577 14. - 1.732. 3. 3.732. 7. - 3 .732. 11. - 1. 15. - 0.268. 4. 0.268. 8. - 0 .268. 12. - 1. 16. - 3.732. Exercise 43. Page 102 1. tl- 14. — COS y. 27. 1 — tan y 2. H- 15. — sin y 1 + tan y 3. 3 3 ^5* 16. sin y. 28. Vs cot y - - 1 4. fl- 17. sin X, cot y + Vs 5. 111- 18. — cos X. 29. ^ Vi cot 1 / + 1 6. M- 19. — sin X. cot 2 / „ 1 Vi 7. COS y. 20. — cotx. 30. tanw. 8. sin y. 21. tan X. 31. 0.8571 ; 0 . 2222 . 9. coty. 22. — tanx. 32. 3.732 ; 0.268. 10. COS y. 23. cotx. 33. 1 . 4 9 ^ y 7 I ; 45°. 11. sin y. 24. — sin y 34. x+y = 90°, 270° in 12. — sin y. 25. ^■V2(cos2/ — sin y) . the three cases. 13. — cosy. 26. Vi (cos y + sinw). 37. 135°, 405° 14 PLANE TRIGONOMETRY 5. 1. - 6. > Vs. 8. Exercise 44. Page 103 9. 0.8492. 11. - 1.1776. 13. 10. 0.5827. 12. 1.7161. 14. -Vf 15. 3 sin a: — 4 sin®a;. 16. 4 cos^i — 3 cos X. Exercise 45. Page 104 1. 0.2588. 3. 0.2679. 5. 7.5928. 7. 0.9239. 9. 2.4142. 2. 0.9659. 4. 3.7321. 6. 0.3827. 8. 0.4142. 10. 5.0280. 11. 0.10051; 0.99493. 12. 0.38730; 0.92196 ; 0.42009; 2.3805. 8 . 9. 15. 0 . iV3. 2 sin 2x 16. 2 cot2x. cos (x — y) 17. sin X cos y Exercise 46. Page 105 18. cos (x + y) sin X cos y 19. tan^x. cos (x — y) 20 . 21 . cos X cos y cos (x 4- y) cos X cos y gg cos(x-y) sin X sin y cos(x + y) sin X sin y 24. tan x tan y. 27. Exercise 47. Page 109 1. a = 6 sin A ; 6 = a sin R ; a = b-, sin A = sin R. 4. Sin. 5. 1000 ft. Exercise 48 1. C = 123° 12', b = 2061.6, c = 2362.6. 2. G = 66° 20', b = 667.^9, c = 663.99. 3. C = 36° 4', b = 677.31, c = 468.93. 4. G=26°12', 6 = 2276.6, c = 1673.9. 5. C = 47° 14', a = 1340.6, b - 1113.8. 6. A = 108° 60', a = 63.276, c = 47.324. 7. R = 66° 66', b = 5685.9, c = 5357.5. 8. R = 77°, a = 630.77, c = 929.48. 9. a = 5 ; c = 9.659. 10. a = 7; 6 = 8.573. 19. 8 and 5.4723. 20. 4.6064 mi. ; 4.4494 mi.; 3.7733 mi. 21. 5.4709 mi. ; 5.8013 mi. ; 4.3111 mi. 6. 8.5450 in.; 4.2728 in. 7. 27.6498 in. 8. 9.1121 in. Page 110 11. Sides, 600 ft. and 1039.2 ft. ; altitude, 519.6 ft. 12. 855: 1607. 13. 5.438; 6.857. 14. 15.588 in. 15. AR= 59.564 mi.; AG = 54.285 mi. 16. 4.1365 and 8.6416. 17. 6.1433 mi. and 8.7918 mi. 18. 6.4343 mi. and 5.7673 mi. Exercise 50 1. Two. 3. No solution. 2. One. 4. One. 9. R = 12° 13' 34' 10. R = 57° 23' 40' 11. R = 41° 12' 56' 12. A = 54° 31', 13. R = 24° 57' 26 R'= 155° 2' 34' Page 115 5. One. 6. Two. G = 146° 15' 26", c = 1272.1. G = 2° 1' 20", c = 0.38525. G = 87° 38' 4", c = 116.83. G = 47° 45', c = 50.496. 7. No solution. 8. One. G = 133° 48' 34", c = 615.7 ; G'= 3° 43' 26", c'= 55.414. ANSWERS 15 14. A = 51° 18' 27", A'= 128° 41' 33" 15. A = 147° 27' 47" A'= 0° 54' 13", 16. 5 =44°!' 28", B'= 135° 58' 32" 17. .B=90°, 18. 420. 19. C = 98° 21' 33" C'= 20° 58' 27". B = 16° 43' 13" B'= 163° 16' 47' C = 97° 44' 20", C'= 5° 47' 16", C = 32° 22' 43", 124.62. , c = 43.098; , c'= 15.593. , a = 35.519; ', a'= 1.0415. , c = 13.954 ; c'= 1.4202. c = 2.7901. 20. 3.2096 in. 21. AS =3.8771 in.; BC = 2.3716 in. ; CD = 3.7465 in. ; AD = 6.1817 in. 22. C = 125°6', J)= 93°24'; AB = 4.3075 in. ; BC = 3.1288 in.; CB = 5.431 in. ; BE = 4.4186 in. ; AE = 5.0522 in. Exercise 51. Page 117 2. b = a cos C + c cosA ; „ , 6- + — a- r ^ , T, °. cosA = ; , a = 0 COSU + c cosB; 2 6c c = 6cosA. 14. AC = 8.499 in. ; 4. Impossible. BB = 3.1254 in. 5. 5. 15. BC= 5.9924 in.; 6. 7.655. BB= 8.3556 in. 7. 7. 90° 16. AB = 1.9249 in. ; CB = 4.4431 in. ; A = 109° 26'; B = 112° 13' 40" C = 88° 11' 40"; B = 50°8'40". 17. 13.3157 in. Exercise 52. a — b 1 . a + 6 2. tan J (A 3. a = b. tan (A - 45°). B) = 0. 4. a + 6 = (ct — 6) (2 + Vs). Page 119 0 _ 0 13. ^ = «,V3. 0 14. tan ^ (A — B) = 0 ; A = B. 11. 2 sin A tan A 17. 5. — . or 00 = CO. 0 0 18. SidesAB, BC, AE ; diagonal AB ; angles B. , CAB , BAE. Exercise 53. Page 121 1. A = 51° 15', B = 56° 30', c = 95.24. 2. B = 60° 45' 2", C = 39° 14' 58", , a = 984. 83. 3. A = 77° 12' 53", B = 43° 30' 7", c = 14.987. 4. B = 93° 28' 36", C = 50° 38' 24", a = 1.3131. 5. A = 132° 18' 27", B = 14° 34' 24", , C = 0.67 75. 6. A = 118° 55' 49", C = 4.5° 41' 3.5", b = 4.1.554. 7. B = 65° 13' 51", C= 28° 42' 5", a = 3297.2. 19. 6. 8. A = 68° 29' 15", B = 45° 24' 18", c = 4449. 20. 10.392. 9. A 117 ° 24' 32", , B= 32° 11' 28", c = 31.431. 21. A = B = 90° - J C, 10. A = 2° 46' 8", B = 1° 54' 42", c = 81.066. a sin C 11. A 116° 33' 54", B = 26° 33' 54", c = 140.87. sinA 12. A := 6°1 ' 55", B = 108° 58' 5", c =862.5. 22. 8.9212. 13. A = 45° 14' 20", B= 17° 3' 40", c = 510.02. 23. 25. 14. A = 41° 42' 33", B = 32° 31' 1.5", c = 9.0398. 24. 3800 yd. 15. A izr 62° 58' 26", B = 21° 9' 58", c = 4151.7. 25. 729.67 yd. ^ 16. A = 84° 49' 58", B = 28° 48' 26", c = 42.374. 26. 430.85 yd. 17. B — 24° ir 20 ", C = 144° 55' 52". , a = 205. 27. 10.266 mi. 18. B = 20° 36' 34", C= 102° 10' 14", , a -37.5. 28. 2.3385 and 5.0032. 16 PLANE TRIGONOMETRY Exercise 54. Page 125 1. ^[log(s — 6)+ log(s— c)+ cologs+ colog(s— a)]. 4. log r + colog (s — a) . 2. i[log(s- b) + log(s — c) + colog b + 1 colog c]. 5. log(s — a) + log taniA, 3. ^[log(s- a) + log(s — b) + log (s — c) + cologs]. 6. The second. 7. Vj, or 0.37796 ; 41° 24' 34". 9. A = 60°. Exercise 55. Page : 127 1. 38° 52' 48" '; 126° 52' 12 "; 14° 15'. 17. 45°; 120°; 15°. 2. 32° 10' 65" ■; 136° 23' 50 "; 11° 25' 15". 18. 45° ; 60° ; 7 ■5°. 3. 27° 20' 32' '; 143° 7' 48" ; 9° .31' 40". 19. 84° 14' 34". 4. 42° 6' 13" ; ; 56° 6' 36"; 81° 47' 11". 20. 54° 48' 54". 5. 16° 25' 36" '; 30° 24'; 133° 10' 24". 21. 10.5°; 15°; 60°. 6. 46° 49' 35" '; 57° 59' 44" ; 75° 10' 41". 22. 54.516. 7. 26° 29"; 43° 25' 20"; 110° 34' 11". 23. O * O 8. 49° 34' 58" '; 58° 46' 58" ; 71° 38' 4". 24. 12.434 in. 9. 51° 53' 12" ' ; 59° 31' 48" ; 68° 35'. 25. 4° 23' 2" W . of N. or W. of S. 10. 36° 52' 12" ■; 53° 7' 48"; 90°. 26. A = 90° 37' 3"; 11. 36° 52' 12" • ; 53° 7' 48" ; : 90°. B = 104° 28 ;' 41"; 12. 33° 33' 27" '; 33° 33' 27" '; 112° 53' 6". C = 96° 55' 44"; 13. 60°; 60°; 60°. D = 67° 58' 32". 14. 28° 57' 18" '; 46° 34' 6"; 104° 28' 36". 27. 82° 49' 10". 15. 36° 52' 12" '; 53° 7' 48"; 90°. 28. 36° 52' 11"; 16. 8° 19' 9" ; 33° 33' 36" ; 1—* CO 00 0 53° 7' 49". 1. 277.68. 2. 452.87. 3. 8.0824. Exercise 56. Page 128 4. 27.891. 7. 10,280.9. 5. 139.53. 6. 1380.7. 8. 82,362. 9. 409.63. Exercise 57. Page 129 10. 1,067,750. 12. 10.0067 sq. in. 13. 18.064 sq. in./ 14. 13.41 sq. in. 1. 85.926. 3. 436,540. 5. 7,408,200. 7. 1 .76,384. 9. 92.963. 2. 23,531. 4. 157.63. 6. 398,710. 8. 25,848. 10. 3176.7. 11. 5.729 sq. in. Exercise 58. Page 131 1. 6. 14. 8160. 29. 13.93 ch., 23.21 ch., 32.50 ch. 2. 150. 15. 26,208. 30. 14 A. 5.54 sq. cll. 3. 43.301. 16. 17.3206. 31. 30° ; 30° ; 120°. 4. 1.1367. 17. 10.392. 32. 2,421,000 sq. : ft. 5. 10.279. 18. 365.68. 33. 199 A. 8 sq. ch. 6. 16.307. 19. 29,450; 6982.8. 34. 210 A. 9.1 sq. ch. 7. 1224.8 sq. rd. ; 20. 15,540. 35. 12 A. 9.78 sq. ch. 7.655 A. 21. 4,333,600. 37. 876.34 sq. ft. 8. 3.84. 22. 13,260. 38. 1229.5 sq. ft. 9. 4.8599. 24. 3 A. 0.392 sq. ch. 39. 9 A. 0.055 sq. ch. 10. 10.14. 25. 12 A. 3.45 sq. ch. 41. 1075.3. 11. 62.354. 26. 4 A. 6.634 sq. ch. 42. 2660.4. 12. 0.19975. 27. 61 A. 4.97 sq. ch. 43. 16,281. 13. 240. 28. 4 A. 6.633 sq. ch. 45. Area == ah sin A. ANSWERS 17 Exercise 59. Page 133 1. 20 ft. 13. 260.21 ft. ; 25. 50° 29' 35"; o, 2. 37° 34' 5". 3690.3 ft. 39° 30' 25". a + b 3. 30°. 14. 2922.4 mi. 26. 74° 44' 14". 36. 30°. 4. 199.70 ft. 15. 60°. 27. 3.50.61 in. 37. 97.86 in.; 5. 106.69 ft. ; 16. 3.2068. 28. 115.83 in. 153.3 in. ; 142.85 ft. 17. 6.6031. 29. 388.62 in. 159.31 in. 6. 43.12 ft. 18. 2.38,410 mi. 30. 83° 37' 40". 38. 1302.5 ft.; 7. 78.-36 ft. 19. 1..3438 mi. 31. 97°liq 33° 6' 51". 8. 75 ft. 20. 861,860 mi. 32. 89° 50' 18". 39. 0.9428. 9. 1.4442 mi. 21. 235.81 yd. 33. 0.2402; 41.45 ft. 10. 56.649 ft. 22. 26° 34'. 1.9216 in.; 43. 0.9524. 11. 2159.5 ft. 23. 69.282 ft. 33.306 in. 2hVl^ + w^ 12. 7912.8 mi. 24. 49° 18' 42" : ; 34. 1.7 in. ; — !r — vfi 40° 41' 18". 0.588 in. Exercise 60. Page 137 4. 460.46 ft. 8. 422.11yd. 12. 255.78 ft. 16. 210.44 ft. 5. 88.936 ft. 9. 41.411ft. 13. 529.49 ft. 18. 19.8; 35.7; 6. 56.564 ft. 10. 234.51 ft. 14. 294.69 ft. 44.5. 7. 51.595 ft. . 11. 12,492.6 ft. 15. 101.892 ft. 19. 13.657 mi. perjiour. - . OB sin 0 24. X — ; 28. 658.361b. ; 22° 23' 47' 20. N. 76°56'E. ; sin a ■with first force. 13.938 mi. per hour. sin a : 29. 88.3261b.; 45° 37' 16' 21. 3121.1 ft. ; 90°; B = 90° ; ■with known force. 3633.5 ft. Z a = : 90° - 0. 30. 757.50 ft. 22. 25.433 mi. 25. 288.67 ft. 31. 520.01yd. 23. 6.3397 mi. 26. 11.314 mi. per hour. 32. 1366.4ft. 35., 536.28 ft. ; 500.16 ft. 36. 345.46 yd. 37. 61.23 ft. 1. i9,647 sq. ft. 2. 27.527 sq. in. Exercise 61. Page 141 3. 41.569 sq. in. 4. 6. 6. I; iVi. 9. 6. 11. 40,320 sq. ft. Exercise 62. Page 142 1. 11.124A. 2. 21.617A. 3. 15.129A. 4. 14A. 5. 13.77A. 6. 10.026A. 7. lOA. 8. 4.5.348 A. ; 10.4652A. 9. 36.38A. 10. 20.07 A. 11. 3.766A. 12. 2.485A. Exercise 1. 0.5223 sq. in. 2. 66.2343 sq. in. 3. 3.583sq. in. ; 27.6565 sq. in. 63. Page 144 4. 8.6965 sq. in. 6. 112.26 sq. in.; 201.9 sq. in. 6. 0.14279. 7. 116.012 sq. in. 8 . |. 18 PLANE TRIGONOMETRY Exercise 64. Page 147 1. 18' 23" ; 5. 13' 53" ; 10. 101.44 mi. 18.385 mi. 20.787 mi. 11. 11.483 mi. — 2. 37' 29" ; 6. 19' 52" ; 12. 44.5 mi..-^ 37.4775 mi. 12° 57' 8" S. 13. S. 75° 31' 20" E.; 3. 51' 33" ; 7. 35.207 mi. 23.2374 mi. 34.445 mi. 8. 16.6296 mi. ; 14. N. 17° 6' 14" W. ; 4. 37' 16" ; 11' 6.7". 32° .50' 30" N. 7.4135 mi. 9. 59.155 mi. 15. 23.8.54 mi.; 16. 27.803 mi.; N. 52° 18' 21" W. S. 56° 58' 34" E. Exercise 65. Page 148 1. 42°16'N.; 68° 54' 39" W. 2. 103.57 mi. 3. 60°15'N.; 62° 15' 55" W. 1. 31° 26' 16" N.; 41° 44' 23" W. 2. S.63°26'W.; 42.486 mi. ; 16° 14' 52" W. Exercise 66. Page 149 3. 41° 50' 5" N.; 68° 15' 1" W. 4. 16.727 mi. ; 30° 16' 19" W. 5. N. 77° 9' 38" W. ; 3:3° 11' W. Exercise 67. Page 150 6. 40° 4' 16" N.; 72° 44' 56" W. 7. 42° 47' 4.3" N. ; 70° 48' 25" W. 1. 35° 49' 10" S.; 22° 2' 44" W.; N. 61°42'W.; 183.16 mi. 2. 42° 15' 29" N. ; 69° 6' 11" W. ; 44.939 mi. 3. 32° 53' 34" S. ; 13° 1' 53" E ; 287.16 mi. 4. 41° 1' 40" N. ; 69° 54' 1" W. 5. 57' 19"; 21.4 mi. 6. 1°37'8"; 45.652 mi. Exercise 68. Page 152 1. fir. 2 . 3. ^\lT. 4. 25. 216°, fir. 26. 300°, I IT. 27. 120°, §7T. 5. if-J-TT. 9. 270°. 13. 6. 8 IT. 10. 240°. 14. 7. J^ir. 11. 210°. 15. 8. -5/77. 12. 225°. 16. 28. 33° 45', IT. 29. 0.017453; 0.0002909. 7° 30'. 17. II. 21. n. 540°. 18. II. 22. II. 1080°. 19. III. 23. I. 1800°. 20. IV. 24. III. 30. 3437.75'; 206,265". 31. IT radians. 32. ^ IT radians. Exercise 69. Page 154 1. 16°, 164°, 376°, 524°. 2. 30°, 150°, 390°, 610°, 750°, 870°. 3. 30°, 150°, 390°, 510°, 750°, 870°, 1110°, 1230°. 4. 67° 30', 112° 30', 427° 30', 472° 30'. 9. 0.00058177632. 10. 0.000582. 5. 18°, 162°, 378°, 522°. 6. 0.99999995769. 7. 0.00029088820. 8. 0.00029088821. 11. 0.0175. ANSWERS 19 Exercise 70. Page 155 1. 60^, 300=’. 5. 45°, 225°. 9. 26° 34', 206° 34', 2. - 60=, - 300°. 6. _ 135°, - 315°. 386° 34', 566° 34'. 3. 25°, 335°, 7. 60°, 240°, 10, - 116° 34', - 296° 34', 385°, 695°. 420°, 600°. - 476° 34', - 656° 34'. 4. 60°, 300°, 8. 30°, 210°, 420°, 660°. 390°, 570°. Exercise 71, Page 156 5. 60°, 120°. 7. 30°, 210°. 9. 60°, 300°. 11. -iVs. 13. iV2. 6. 45°, 135°. 8. 90°, 270°. 10. 135°, 225°. 12. |. 14. |V2. 19. 60°, 240°, 22. 19°, 161°, 25. 19° 28' 17", 420°, 600°. 379°, 521°. 160° 31' 43". 20. 58°, 238°, 23. 15° 24' 30", 195° 24' 30", 26. ± lV2. 418°, 598°. 375° 24' 30", 555° 24' 30" 27. ± |VS or 0. 21. 74°, 106°, 24. 19°, 341°, 434°, 466°. 379°, 701°. Exercise 74. Page 161 2. 360° or 2 tt. 6. 180° or •7T. 9. 180° and 360°. 4. 180° or 7T. 8. 360° or 2 tt. 10. Complements. Exercise 75. Page 162 1. 270.63. 9. 40' 9". 13. ^ radian ; 2. 416.65. 10. - 175°, 185°, 19° 5' 55". 3. 2695.8. 535°, 545°. 22. 30°, 210°, 4. 4.163. 11. - 200°, 160°, 390°, 570°. 5. Impossible. 560°, 520°. 23. 60°, 240°, 6. Impossible. 12. 2 radians ; 420°, 600°. 7. 345.48 ft. 114° 35' 30". Exercise 77. Page 166 1. l-n-or %TT. 16. 26° 34' or 206° 34'. 2. 90° or 270°. 17. 30° or 150°. 3. 21° 28' or 158° 32'. 18. 4.5° or 135°. 4. 0° or 90°. 19. 60°, 90°, 270°, or 300°. 5. 30°, 150°, 199° 28', or 340° 32'. 20. 60°, 90°, 120°, 240°, , 270°, or 6. 51° 19', 180°, or 308° 41'. 300°. 7. 30°, 150°, or 270°. 21. 32° 46', 147° 14', 212° 46', ,01-327° 14'. 8. 35° 16', 144° 44', 215° 16', or 324° 44 9. 75° 58' or 255° 58'. Ldll • 2 a 10. 60°, 180°, or 300°. 23. . / — a + Va'^ + 8 a + 8\ 11. 90° or 143° 8'. cos--^ V 4 ' )■ 13. 30°, 150°, 210°, or 330°. 24. 1. 13. 0°, 120°, 180°, or 240°. 25. 1. 14. 45°, 161° 34', 225°, or 341° 34'. 26. 0°, 45°, 90°, 180°, 225°, or 270°. 15. 60°, 120°, 240°, or 300°. 27. 30°, 150°, 210°, or 330°. 20 PLANE TRIGONOMETRY 28. 30°, 60°, 120°, 150°, 210°, 240°, CO o o o 60. 60°, 90°, 120°, 240°, 270°, or 300°. or 330°. 61. 0°, 90°, 180°, or 270°. 29. 0°, 65° 42^ 180°, or 204° 18'. 63. 0°, 90°, 120°, 180°, 240°, or 270°. 30. 14° 29', 30°, 150°, or 165° 31'. 63. 0°, 74° 5', 127° 25', 180 °, 232° 35', 31. 0°, 20°, 100°, 140°, 180°, 220°, 260°, or 285° 55'. or 340°. 64. 0°, 180°, 220° 39', or 319° 21'. 32. 45°, 90°, 135°, 225°, 270°, or 315°. 65. 8° or 168°. 33. 30°, 150°, or 270°. 66. 40° 12', 1.39° 48', 220° 12', or 319° 48'. 34. 26° 34', 90°, 206° 34', or 270°. 67. 0°, 60°, 120°, 180°, 240°, or 300°. 35. 45°, 135°, 225°, or 315°. 68’. 30° or 330°. 36. 45°, 135°, 22.5°, or 315°. 69. 60°, 120°, 240°, or 300°. 37. 15°, 7.5°, 135°, 195°, 255°, or 31-5°. 70. 18°, 90°, 162°, 2.34°, 270°, , or 306°. 38. 45°, 135°, 225°, or 315°. 71. 30°, 00°, 120°, 150°, 210°, , 240°, 300°, 39. 0°, 45°, 180°, or 22-5°. or 330°. 40. 0°, 90°, 120°, 240°, or 270°. 72. 53° 8', 126° 52', 233° 8', or 306° 52'. 41. 0°, 36°, 72°, 108°, 144°, 180°, 216°, 73. 30°. 252°, 288°, or 324°. 74. 22° 37' or 143° 8'. 42. 120°. 75. 0°, 20°, 30°, 40°, 60°, 80°, 90°, 100°, 43. 54° 44', 125° 16', 234° 44', .30.5° 16'. 120°, 140°, 1.50°, 160°, 180°, 200°, 44. 30°, 60°, 90°, 120°, 150°, 210°, 240°, 210°, 220°, 240°, 260°, 270°, 280°, 270°, 300°, or 330°. 300°, 320°, 330°, or 340°. 45. ■ 1 I* - 1 76. 221°, 45°, 671°, 90°, 1121°, 1350^ 2 ■ 157i°, 2021°, 22.5°, 247^°, 270°, 46. 90°, 216° 52', or 323° 8'. 292 i°, 315°, or 337^' 47. 30°, 90°, 150°, 210°, 270°, or 330°. 77. 45° or 22.5°. 48. 0°, 45°, 180°, or 225°. 78. ± 1 or ± 1 ViT. 49. 45°, 60°, 120°, 135°, 225°, 240°, , 300°, 79. ^ V3 or — J Vs. or 315°. 80. 0 or ± 1. 50. 0°, 45°, 135°, 225°, or 315°. 81. 0°, 30°, 90°, 150°, 180°, 210°, 270°, 51. 90° or 270°. or 330°. 52. lV3. 82. 120° or 240°. 53. i. 83. 60°, 120°, 240°, or 300°. 54. 6°, 45°, 90°, 180°, 225°, or 270°. 84. 10° 12', 34° 48', 190° 12', or 214° 48'. 55. 30°, 150°, 210°, or 330°. 85. 29°19', 105°41', 209°19', or285°41'. 56. 60°. 86. 0°, 45°, 90°, 180°, 225°, or 270°. 57. 105° or 345°. 87. 0°, 45°, 135°, 225°, or 315°. 58. 135°, 315°, or -3, sin-i(l - a). 88. 0°, 60°, 120°, 180°, 240°, , or 300°. 59. 30°, 60°, 120°, 150°, 210°, 240°, , 300°, 89. 27° 58', 135°, 242° 2', or 315°. or 330°. Exercise 78. Page 170 1. X = a, y = 0; or x = 0, y = a. 4. X = 100, y = 200. in —71 + 1 1 la — b 5. X=:Sin-l±A ■9 2. x-sin ^ , \ 2 m + 71 — 1 ■ 1 , /« + ^ 2 ■ y -sin 1 ^ ■ 6. X = 90°, 3. a; ^ 76° 10', y = 15° 30'. y-0° or 180°. ANSWERS 21 1.x — cos-1 j- V6 — a- + 2) ; y = cos-i l(a ±V6 — a- + 2). X = tan- tan a + 1 cos-i [2 — {2 m- — 2 n-) cos- a — 1] ; y = tan-i ^ tan a — 1 cos-i [2 m'^ — (2 — 2n^) cos^a — 1]. 9, a: = tan-i- + cos-i i- Va^ + 6^ ; ?/ = tan-i - — cos-i 1 Va^ + 6^. b - b " 10. 2 = 24° 13', r = 225.12; 2 = 204° 13', r=- 225.12. 11. 2 = 42° 28', r = 151 ; 2 = 222° 28', r=- 151. Exercise 79. Page 171 1. ^ = 30° or 1-50°; 2 = 0.134 or 1.866. 2. B = sin-1 (a — i) ; x = 2 — a. 3. X = 45°, 135°, 225°, or 315°; 4. 0 sin-1 31218 31 239 31260 31281 31 302 31323 31345 31366 206 387 40^ 429 450 471 492 513 534 555 576 207 597 618 639 660 681 702 723 744 765 785 208 806 827 848 869 890 911 931 952 973 994 209 32 015 32 035 32 056 32 077 32 098 32 118 32 139 32 160 32 181 32 201 210 32 222 32 243 32 263 32 284 32 305 32 325 32 346 32 366 32 387 32 408 211 428 449 469 490 510 531 552 572 593 613 212 634 654 675 695 715 736 756 777 797 818 213 838 858 879 899 919 940 960 980 33 001 33 021 214 33 041 33 062 33 082 33 102 33 122 33 143 33 163 33 183 203 224 216 33 244 33 264 33 284 33 304 33 325 33 345 33 365 33 385 33 405 33 425 216 445 465 486 506 526 546 566 586 606 626 217 646 666 686 706 726 746 766 786 806 826 218 846 866 885 905 925 945 965 985 34 005 34 025 219 34 044 34 064 34 084 34 104 34 124 34 143 34 163 34 183 203 223 220 34 242 34 262 34 282 34 301 34 321 34 341 34 361 34 380 34400 34 420 221 439 459 479 498 518 537 557 577 596 616 222 635 655 674 694 713 733 753 772 792 811 223 830 850 869 889 908 928 947 967 986 35 005 224 35 025 35 044 35 064 35 083 35 102 35 122 35 141 35 160 35 180 199 226 35 218 35 238 35 257 35 276 35 295 35 315 35 334 35 353 35 372 35 392 226 411 430 449 468 488 507 526 545 564 583 227 603 622 641 660 679 698 717 736 755 774 228 793 813 832 851 870 889 908 927 946 965 229 984 36 003 36 021 36 040 36 059 36 078 36097 36116 36 135 36 154 230 36173 36 192 36 211 36 229 36 248 36 267 36 286 36 305 36 324 36 342 231 361 380 399 418 436 455 474 493 511 530 232 549 568 586 60S 624 642 661 680 698 717 233 736 754 773 791 810 829 847 866 884 903 234 922 940 959 < 977 996 37 014 37 033 37 051 37 070 37 088 236 37107 37125 37144 37162 37181 37 199 37 218 37 236 37 254 37 273 236 291 310 328 346 365 383 401 420 438 457 237 475 493 511 530 548 566 585 603 621 639 238 658 676 694 712 731 749 767 785 803 822 239 840 858 876 894 912 931 949 967 985 38 003 240 38 021 38 039 38 057 38 075 38 093 38112 38 130 38 148 38 166 38 184 241 202 220 238 256 274 292 310 328 346 364 242 382 399 417 435 453 471 489 507 52i 543 243 561 578 596 614 632 650 668 686 703 721 244 739 757 775 792 810 828 846 863 . 881 899 246 38 917 38 934 38 952 38 970 38987 39 005 39023 39 041 39 058 39 076 246 39 094 39111 39129 39 146 39 164 182 199 217 235 252 247 270 287 305 322 340 358 375 393 410 428 248 445 463 480 498 515 533 550 568 585 602 249 620 637 655 672 690 707 724 742 759 777 260 39 794 39 811 39 829 39 846 39 863 39 881 39 898 39 915 39 933 391950 N O 1 2 3 4 6 6 7 8 9 200-250 250-300 31 N O 1 2 3 4 5 6 7 8 9 250 39 79+ 39 811 39 829 39 S +6 39 863 39 881 39 898 39 915 39 933 39 950 251 967 985 +0 002 40 019 +0 037 40 05+ 40 071 40 088 40106 40 123 252 +0 1+0 +0157 175 192 209 226 243 261 278 295 ^ 253 312 329 3+6 36+ 381 398 415 432 449 466 25+ +83 500 . 518 535 552 569 586 603 620 637 255 +0 65+ 40 671 +0688 40 705 +0 722 40 739 40 756 40 773 40 790 40807 256 82+ 8+1 858 875 892 909 926 943 960 976 257 993 41 010 41027 410+4 41 061 41 078 41 095 41 111 41 128 41 145 25 S +1 162 179 196 212 229 246 263 280 296 313 259 330 3+7 363 380 397 414 430 4+7 464 481 260 +1497 +151+ +1 531 +1 5+7 +1 56+ 41 581 41 597 41 614 41631 41 647 261 66+ 681 697 71+ 731 747 764 780 797 814 262 830 8+7 863 880 896 913 929 946 963 979 263 996 42 012 42 029 +2 0+5 +2 062 42 078 42 095 42 111 42 127 42144 26+ 42 160 177 193 210 226 2+3 259 275 292 308 265 42 325 42 3+1 42 357 42 37+ 42 390 42 406 42 423 42 439 42 455 42 472 266 +88 50+ 521 537 553 570 586 602 619 635 267 651 667 68+ 700 716 732 749 765 781 797 26 S 813 830 8+6 862 878 894 911 927 943 959 269 975 991 43 008 43 02+ 43 040 43 056 43 072 43 088 43 104 43 120 270 +3 136 +3 152 43 169 43 185 43 201 43 217 43 233 43 249 43 265 43 281 271 297 313 329 3+5 361 377 393 409 425 441 272 457 +73 +89 505 521 537 553 569 584 600 273 616 632 648 66+ 680 696 712 727 743 759 27+ 775 791 807 823 838 854 870 886 902 917 275 43 933 43 9+9 43 965 43 981 43 996 44 012 44 028 44 044 44 059 44 075 276 ++ 091 4+107 4+122 4+ 138 44 154 170 185 201 217 232 277 2+8 26+ 279 295 311 326 342 358 373 389 278 +0+ +20 436 451 467 483 498 514 529 545 279 560 576 592 607 623 638 654 669 685 700 280 4+716 4+ 731 4+7+7 44 762 44 778 44 793 44 809 44 824 44 840 44 855 281 871 886 902 917 932 948 963 979 994 45 010 282 45 025 +5 040 +5 056 +5 071 45 086 45 102 45 117 45 133 45 148 163 283 179 19+ 209 225 240 255 271 286 301 317 28+ 332 3+7 362 378 393 408 423 439 454 469 285 +5 +8+ 45 500 45 515 +5 530 45 545 45 561 45 576 45 591 45 606 45 621 286 637 652 667 682 697 712 728 743 758 773 287 788 803 818 83+ 849 864 879 894 909 924 288 939 ' 95+ 969 98+ 46 000 46015 46 030 46 0+5 46 060 46 075 289 +6090 +6 105 46120 46 135 150 165 180 195 210 225 290 46 2+0 46 255 46 270 +6 285 46 300 46 315 46 330 46 345 46 359 46 374 291 389 +0+ 419 43+ 449 46+ 479 494 509 523 292 538 553 568 583 598 613 627 642 657 672 293 687 702 716 731 746 761 776 790 805 820 29+ 835 850 86+ 879 894 909 923 938 953 967 295 +6982 +6 997 47 012 47 026 47 041 47 056 47 070 47 085 47 100 47 114 296 +7 129 47 1+4 159 173 188 202 217 232 246 261 297 276 290 305 319 334 349 363 378 392 407 298 422 +36 +51 +65 480 49+ 509 524 538 553 299 567 582 596 611 625 640 654 669 683 698 300 +7 712 47 727 +7 7+1 47 756 47 770 47 784 47 799 47 813 47 828 47 842 N O 1 2 3 4 5 6 7 8 9 250-300 32 300-350 N 0 1 2 3 4 6 6 7 8 9 300 47 712 47 727 47 741 47 756 47 770 47 784 47 799 47 813 47 828 47 842 301 857 871 885 900 914 929 943 958 972 986 302 48 001 48 015 48 029 48 044 48058 48 073 48087 48101 48116 48 130 303 144 159 173 187 202 216 230 244 259 273 304 287 302 316 330 344 359 373 387 401 416 305 48 430 48 444 48 458 48 473 48 487 48 501 48 515 48 530 48 544 48 558 306 572 586 601 615 629 643 657 671 686 700 307 714 728 742 756 770 785 799 813 827 841 308 855 869 883 897 911 926 940 954 968 982 309 996 49 010 49 024 49 038 49 052 49 066 49 080 49 094 49 108 49122 310 49 136 49150 49164 49 178 49192 49 206 49 220 49 234 49 248 49 262 311 276 290 304 318 332 346 360 374 / 388 402 312 415 429 443 457 471 485 499 513 527 541 313 554 568 582 596 610 624 638 651 665 679 314 693 707 -1 721 734 748 762 776 790 803 817 315 49 831 49 845 49 859 49 872 49 886 49 900 49 914 49 927 49 941 49955 316 969 982 996 50 010 50 024 50 037 SO 051 50 065 50 079 50092 317 SO 106 50120 50133 147 161 174 188 202 215 229 318 243 256 270 284 297 311 325 338 352 365 319 379 393 406 420 433 447 461 474 488 SOI 320 50 515 50 529 SO 542 SO 556 50 569 50 583 50 596 50 610 50 623 50 637 321 651 664 678 691 70S 718 732 745 759 772 322 786 799 813 826 840 853 866 880 893 907 323 920 934 947 961 974 987 51001 51014 51028 51041 324 51055 51068 51081 51095 51 108 51 121 135 148 162 175 325 Si 188 51 202 51 215 51228 51 242 51 255 51 268 51 282 51295 51308 326 322 335 348 362 375 388 402 415 428 441 327 455 468 481 495 508 521 534 548 561 574 328 587 601 614 627 640 654 667 680 693 706 329 720 733 746 759 772 786 799 812 825 838 330 51851 51865 51878 51 891 51904 5191’7 51930 51943 51957 51970 331 983 996 52 009 52 022 52 035 52 048 52 061 52 075 52 088 52 101 332 52 114 52 127 140 153 166 179 192 205 218 231 333 244 257 270 284 297 310 323 336 349 362 334 375 388 401 414 427 440 453 466 479 492 335 52 504 52 517 52 530 52 543 52 556 52 569 52 SS2 52 595 52 608 52 621 336 634 647 660 673 686 699 711 724 737 750 337 763 776 789 802 815 827 840 853 866 879 338 892 90S 917 930 943 956 969 982 994 S3 007 339 53 020 53 033 S3 046 53 058 53 071 53 084 53 097 S3 no 53 122 135 340 S3 148 53 161 53 173 53 186 53 199 53 212 S3 224 53 237 S3 250 53 263 341 275 288 301 314 326 339 352 364 377 390 342 403 415 428 441 453 466 479 491 504 517 343 529 542 555 567 580 593 605 618 631 643 344 656 668 681 694 706 719 732 744 757 769 345 53 782 S3 794 53 807 S3 820 53 832 S3 845 53 857 53 870 53 882 53 895 346 90S 920 933 945 958 970 983 995 54 008 54 020 347 54 033 54-045 54 058 54 070 54 083 54 095 54 108 54120 133 145 348 158 170 183 195 208 220 233 245 258 270 349 283 295 307 320 332 34i 357 370 382 394 350 54 407 54 419 54 432 54 444 54 456 54 469 54 481 54 494 54 506 54 5iS O 1 2 3 4 6 6 7 8 9 300-350 350-400 33 N O 1 2 3 4 5 6 7 8 9 350 54 407 54 419 54 432 54 444 54 456 54 469 54 481 54 494 54 506 54 518 351 531 543 555 568 580 593 605 617 630 642 352 654 667 679 691 704 716 728 741 753 765 353 777 790 802 814 827 839 851 864 876 888 354 900 913 925 937 949 962 974 986 998 55 on 355 55 023 55 035 55 047 55 060 55 072 55 084 55 096 55 108 55 121 55 133 356 145 157 169 182 194 206 218 230 242 255 357 267 279 291 303 315 328 340 352 364 376 358 388 400 413 425 437 449 461 473 485 497 359 509 522 534 546 558 570 582 594 606 618 360 55 630 55 642 55 654 55 666 55 678 55 691 55 703 55 715 55 727 55 739 361 751 763 775 787 799 811 823 835 847 859 362 871 883 895 907 919 931 943 955 967 979 363 991 56003 56015 56027 56 038 56 050 56 062 56 074 56 086 56 098 364 56110 122 134 146 158 170 182 194 205 217 365 56 229 56 241 56 253 56 265 56 277 56 289 56 301 56 312 56 324 56 336 366 348 360 372 384 396 407 419 431 443 455 367 467 478 490 502 514 526 538 549 561 573 368 585 597 608 620 632 644 656 667 679 691 369 703 714 726 738 750 761 773 785 797 808 370 56 820 56 832 56 844 56 855 56 867 56 879 56 891 56 902 56 914 56 926 371 937 949 961 972 984 996 57 008 57 019 57 031 57 043 372 57 054 57 066 57 078 57 089 57101 57113 124 136 148 159 373 171 183 194 206 217 229 241 252 264 276 374 287 299 310 322 334 345 357 368 380 392 375 57 403 57 415 57 426 57 438 57 449 57 461 57 473 57 484 57 496 57 507 376 519 530 542 553 565 576 588 600 611 623 377 634 646 657 669 680 692 703 715 726 738 378 749 761 772 784 795 807 818 830 841 852 379 864 875 887 898 910 921 933 944 955 967 380 57 978 57 990 58 001 58 013 58 024 58 035 58 047 58 058 58 070 58 081 381 58092 58 104 115 127 138 149 161 172 184 195 382 206 218 229 240 252 263 274 286 297 309 383 320 331 343 354 365 377 388 399 410 422 384 433 444 456 467 478 490 501 512 524 535 385 58 546 58 557 58 569 58 580 58 591 58 602 58 614 58 625 58 636 58 647 386 659 670 681 692 704 715 726 737 749 760 387 771 782 794 805 816 827 838 850 861 872 388 883 894 906 917 928 939 950 961 973 984 389 995 59006 59 017 59 028 59 040 59 051 59062 59 073 59084 59 095 390 59 106 59118 59129 59 140 59151 59 162 59173 59 184 59 195 59 207 391 218 229 240 251 262 273 284 295 306 318 392 329 340 351 362 373 384 395 406 417 428 393 439 450 461 472 483 494 506 . 517 528 539 394 550 561 572 583 594 605 616 627 638 649 395 59 660 59 671 59 682 59 693 59 704 59 715 59 726 59 737 59 748 59 759 396 770 780 791 802 813 824 835 846 857 868 397 879 890 901 912 923 934 945 956 966 977 398 988 999 60 010 60 021 60 032 60 043 60 054 60 065 60 076 60086 399 60 097 60108 119 130 141 152 163 173 184 195 400 60 206 60 217 60 228 60 239 60 249 60 260 60 271 60 282 60 293 60 304 N 0 1 2 3 4 6 6 7 8 9 350-400 34 400-450 N O 1 2 3 4 o G 7 8 9 400 60 206 60 217 60 228 60 239 60 249 60 260 60 271 60 282 60 293 60 304 401 314 325 336 347 558 369 379 390 401 412 402 423 433 444 455 466 477 487 498 509 520 403 531 541 552 563 574 584 595 606 • 617 627 404 638 649 660 670 681 692 703 713 724 735 405 60 746 60 756 60 767 60 778 60 788 60 799 60 810 60 821 60 831 60 842 406 853 863 874 885 895 906 917 927 938 949 407 959 970 981 991 61 002 61 013 61 023 61 034 61045 61 055 408 61 066 61 077 61 087 61098 109 119 130 140 151 162 409 172 183 194 204 215 225 236 247 257 268 410 61 278 61 289 61 300 61310 61321 61 331 61 342 61 352 61 363 61374 411 384 395 405 416 426 437 448 458 469 479 412 490 500 511 521 532 542 553 563 574 oS4 413 595 606 616 627 637 648 65S ' 669 679 690 414 700 711 721 731 742 752 763 773 784 794 415 61 805 61815 61 826 61 836 61 847 61 857 61 868 61 878 61 888 61899 416 909 920 930 941 951 962 972 982 993 62 003 417 62 014 62 024 62 034 62 045 62 055 62 066 62 076 62 086 62 097 107 418 118 128 138 149 159 170 ISO 190 201 211 419 221 232 242 252 263 273 284 294 304 315 420 62 325 62 335 62 346 62 356 62 366 62 377 62 387 62 397 62 408 62 418 421 428 439 449 459 469 480 490 500 511 521 422 531 542 552 562 572 583 593 603 613 624 423 634 644 655 665 675 685 696 706 716 726 424 737 747 757 767 778 788 798 SOS 818 829 425 62 839 62 849 62 859 62 870 62 880 62 890 62 900 62 910 62 921 62 931 426 941 951 961 972 982 992 63 002 63 012 63 022 63 033 427 63 043 63 053 63 063 63 073 63 083 63 094 104 114 124 134 428 144 155 165 175 185 195 205 215 225 236 429 246 256 266 276 286 296 306 317 327 337 430 63 347 63 357 63 367 63 377 63 387 63 397 63 407 63 417 63 428 63 438 431 448 458 468 478 488 498 508 518 528 538 432 548 558 568 579 589 599 609 619 629 639 433 649 659 669 679 689 699 709 719 729 739 434 749 759 769 779 789 799 809 819 829 839 435 63 849 63 859 63 869 63 879 63 889 63 899 63 909 63 919 63 929 63 939 436 949 959 969 979 988 998 64 008 64 018 64 028 64 038 437 64 048 64 058 64 068 64 078 64 088 64 098 108 118 128 137 438 147 157 167 177 187 197 207 217 227 237 439 246 256 266 276 286 296 306 316 326 335 440 64 345 64 355 64 365 64 375 64 385 64 395 64 404 64 414 64 424 64 434 441 444 454 464- 473 483 493 503 513 523 532 442 542 552 562 572 582 591 601 611 621 631 443 640 650 660 670 680 689 699 709 719 729 444 738 748 758 768 777 787 797 807 816 826 445 64 836 64 846 64 856 64 865 64 875 64 885 64 895 64 904 64 914 64924 446 933 943 953 963 972 982 992 65 002 65 011 65 021 447 65 031 65 040 65 050 65 060 65 070 65 079 65 089 099 108 118 448 128 137 147 157 167 176 186 196 205 215 449 225 234 244 254 263 273 283 292 302 312 450 65 321 65 331 65 341 65 350 65 360 65 369 65 379 65 389 65 398 65 408 N O 1 2 3 4 5 6 7 8 9 400-450 450-500 35 If O 1 2 3 4 5 6 7 S 9 450 65 321 65 331 65 341 65 350 65 360 65 369 65 379 65 389 65 398 ■ 65 408 451 418 427 437 447 456 466 475 485 495 504 452 514 523 533 543 552 562 571 581 591 600 453 610 619 629 639 648 658 667 677 686 696 454 ' 706 715 725 734 744 753 763 772 782 792 455 65 801 65 811 65 820 65 830 65 839 65 849 65 858 65 868 65 877 65 887 456 896 906 916 25 935 944 954 963 973 982 457 992 66 001 66 011 66020 66 030 66 039 66 049 66 058 66 068 66 077 458 66 087 096 106 115 124 134 143 153 162 172 459 181 191 200 210 219 229 238 247 J 66 332 66 342 257 266 460 66 276 66 285 66 295 66 304 66 314 66 323 66 351 66 361 461 370 380 389 398 408 417 427 436 445 455 462 464 474 483 492 502 511 521 530 539 549 463 558 567 577 586 596 605 614 624 633 642 464 652 661 671 680 689 699 708 717 727 736 465 66 745 66 755 66 764 66 773 66 783 66 792 66 801 66 811 66 820 66 829 466 839 848 857 867 876 885 894 904 913 922 467 932 941 950 960 969 978 987 997 67 006 67 015 468 67 025 67 034 67 043 67 052 67 062 67 071 67 080 67 089 099 108 469 117 127 136 145 154 164 173 182 191 201 470 67 210 67 219 67 228 67 237 67 247 67 256 67 265 67 274 67 284 67 293 471 302 311 321 330 339 348 357 367 376 385 472 394 403 413 422 431 440 449 459 468 477 473 486 495 504 514 523 532 541 550 560 569 474 578 587 596 . 605 614 624 633 642 651 660 475 67 669 67 679 67 688 67 697 67 706 67 715 67 724 67 733 67 742 67 752 476 761 770 779 788 797 806 815 825 834 843 477 852 861 870 879 888 897 906 916 925 934 478 943 952 961 970 979 988 997 68 006 68 015 68 024 479 68 034 68043 68 052 68 061 68070 68 079 68 088 097 106 115 480 68124 68 133 68 142 68 151 68 160 68 169 68178 68187 68 196 68 205 481 215 ' 224 233 242 251 260 269 ' 278 287 296 482 305 314 323 332 341 350 359 368 377 386 483 395 404 413 422 431 440 449 458 467 476 484 485 494 502 511 520 529 538 547 556 565 485 68 574 68 583 68 592 68 601 68 610 68 619 68 628 68 637 68 646 68 655 486 664 673 681 690 699 708 717 726 735 744 487 753 762 771 780 789 797 806 815 824 833 488 842 851 860 869 878 886 895 904 913 922 489 931 940 949 958 966 975 984 993 69 002 69 011 490 69 020 69 028 69 037 69 046 69055 69 064 69 073 69 082 69 090 69 099 491 108 117 . 126 135 144 152 161 170 179 188 492 197 205 214 223 232 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601 887 895 902 909 916 924 931 938 945 952 602 960 967 974 981 988 996 78 003 78 010 78 017 78 025 603 78 032 78 039 78 046 78 053 78 061 78 068 075 082 089 097 604 104 111 118 125 132 140 147 154 161 168 605 78 176 78 183 78 190 78 197 78 204 78 211 78 219 78 226 78 233 78 240 606 247 254 262 269 276 283 290 297 305 312 607 319 326 333 340 347 355 362 369 376 383 608 390 398 405 412 419 426 433 440 447 455 609 462 469 476 483 490 497 504 512 519 526 610 78 533 78 540 78 547 78 554 78 561 78 569 78 576 78 583 78 590 78 597 611 604 611 618 625 633 640 647 654 661 668 612 675 682 689 696 704 711 718 725 732 739 613 746 753 760 767 774 781 789 796 803 810 614 817 824 831 838 845 852 859 866 873 880 615 78 888 78 895 78 902 78 909 78 916 78 923 78 930 78 937 78 944 78 951 616 958 965 972 979 986 993 79 000 79 007 79 014 79 021 617 79 029 79 036 79 043 79 050 79 057 79 064 071 078 085 092 618 099 106 113 120 127 134 141 148 155 162 619 169 176 183 190 197 204 211 218 225 232 620 79 239 79 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982 82 988 676 995 S3 001 83 008 83 014 83 020 S3 027 83 033 83 040 83 046 S3 052 677 83 059 065 072 078 085 091 097 104 110 117 678 123 129 136 142 149 155 161 168 174 181 679 187 193 200 206 213 219 225 232 238 245 680 83 251 83 257 83 264 83 270 83 276 83 283 83 289 83 296 83 302 83 308 681 315 321 327 334 340 347 353 359 366 372 682 378 385 391 398 404 410 417 423 429 436 683 442 448 455 461 467 474 480 487 493 499 684 506 512 518 525 531 537 544 . 550 556 563 685 83 569 83 575 83 582 S3 588 83 594 83 601 83 607 S3 613 83 620 83 626 686 632 639 645 651 658 664 670 677 683 689 687 696 702 70S 715 721 727 734 740 746 753 688 759 765 771 778 784 790 797 803 809 816 689 822 828 835 841 847 853 860 866 872 879 690 83 885 83 891 83 897 83 904 .3 910 83 916 S3 923 83 929 83 935 83 942 691 948 954 960 967 973 979 985 992 998 84 004 692 84 011 84 017 84 023 84 029 84 036 84 042 84 048 84 055 84 061 067 693 073 080 086 092 098 105 111 117 123 130 694 136 142 148 155 161 167 173 180 186 192 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87 529 87 535 87 541 87 547 87 552 87 558 N O 1 2 3 4 .5 6 7 8 9 700-750 750-800 41 N O 1 2 3 4 5 6 7 8 9 760 87 506 87 512 87 518 87 523 87 529 87 535 87 541 87 547 87 552 87 558 751 564 570 576 581 587 593 599 604 610 616 752 622 628 633 639 645 651 656 662 668 674 753 679 685 691 697 703 70S 714 720 726 731 754 737 743 749 754 760 766 772 777 783 789 765 87 795 87 800 87 806 87 812 87 818 87 823 87 829 87 835 87 841 87 846 756 852 858 864 869 875 881 887 892 898 904 757 910 915 921 927 933 938 944 950 955 961 758 967 973 978 984 990 996 88 001 88 007 88 013 88 018 759 88 024 88 030 88 036 88 041 88 047 88 053 058 064 070 076 760 88 081 88 087 88 093 88 098 88 104 88110 88116 88121 88 127 88 133 761 138 144 150 156 161 167 173 178 184 190 762 195 201 207 213 218 224 230 235 241 247 763 252 258 264 270 275 281 287 292 298 304 764 309 315 321 326 332 338 343 349 355 360 766 88 366 88 372 88 377 88 383 88 389 88 395 88 400 88 406 88 412 88 417 766 423 429 434 440 446 451 457 463 468 474 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535 823 540 545 551 556 561 566 572 577 582 587 824 593 598 603 609 614 619 624 630 635 640 825 91645 91 651 91 656 91 661 91 666 91 672 91 677 91 682 91 687 91693 826 698 703 709 714 719 724 730 735 740 745 827 751 756 761 766 772 777 782 787 793 798 828 803 808 814 819 824 829 834 840 845 850 829 855 861 866 871 876 882 887 892 897 903 830 91 908 91 913 91 918 91 924 91 929 91934 91 939 91 944 91 950 91 955 831 960 965 971 976 981 986 991 997 92 002 92 007 832 92 012 92 018 92 023 92 028 92 033 92 038 92 044 92 049 054 059 833 065 070 075 OSO 085 091 096 101 106 111 834 117 122 127 132 137 143 148 153 158 163 835 92 169 92 174 92179 92 184 92 189 92 195 92 200 92 205 92 210 92 215 836 221 226 231 236 241 247 252 257 262 267 837 273 278 283 288 293 298 304 309 314 319 838 324 330 335 340 345 350 355 361 366 371 839 376 381 387 392 397 402 407 412 418 423 840 92 428 92 433 92 438 92 443 92 449 92 454 92 459 92 464 92 469 92 474 841 480 485 490 495 500 505 511 516 521 526 842 531 536 542 547 552 557 562 567 572 578 843 583 588 593 598 603 609 614 619 624 629 844 634 639 645 650 655 660 665 670 675 681 845 92 686 92 691 92 696 92 701 92 706 92 711 92 716 92 722 92 727 92 732 846 737 742 747 752 758 763 768 773 778 783 847 788 793 799 804 809 814 819 824 829 834 848 840 845 850 855 860 865 870 875 SSI 886 849 891 896 901 906 911 916 921 927 932 937 850 92 942 92 947 92 952 92 957 92 962 92 967 92 973 92 978 92 983 92 9SS 0 1 2 3 4 5 6 7 8 9 800-850 850-900 43 N O 1 2 3 4 5 6 7 8 9 850 92 942 92 947 92 952 92 957 92 962 92 967 92 973 92 978 92 983 92 988 851 993 998 93 003 93 008 93 013 93 018 93 024 93 029 93 034 93 039 852 93 044 93 049 054 059 064 069 075 080 085 090 853 095 100 105 no 115 120 125 131 136 141 854 146 151 156 161 166 171 176 181 186 192 855 93197 93 202 93 207 93 212 93 217 93 222 93 227 93 232 93 237 93 242 856 247 252 258 263 268 273 278 283 288 293 857 298 303 308 313 318 323 328 334 339 344 858 349 354 359 364 369 374 379 384 389 394 859 399 404 409 414 420 425 430 435 440 445 860 93 450 93 455 93 460 93 465 93 470 93 475 93 480 93 485 93 490 93 495 861 500 505 510 515 520 526 531 536 541 546 862 551 556 561 566 571 576 581 586 591 596 863 601 606 . 611 616 621 626 631 636 641 646 864 651 656 661 666 671 676 682 687 692 697 865 93 702 93 707 93 712 93 717 93 722 93 727 93 732 93 737 93 742 93 747 866 752 757 762 767 772 777 782 787 792 797 867 802 807 812 817 822 827 832 837 842 847 868 852 857 862 '867 872 877 882 887 892 897 869 902 907 912 917 922 927 932 937 942 947 870 93 952 93 957 93 962 93 967 93 972 93 977 93 982 93 987 93 992 93 997 871 94 002 94 007 94 012 94 017 94 022 94 027 94 032 94 037 94 042 94 047 872 052 057 062 067 072 077 082 086 091 096 873 101 106 111 116 121 126 131 136 141 146 874 151 156 161 166 171 176 181 186 191 196 875 94 201 94 206 94 211 94 216 94 221 94 226 94 231 94 236 94 240 94 245 876 250 255 260 265 270 275 280 285 290 295 877 300 305 310 315 320 325 330 335 340 345 878 349 354 359 364 369 374 379 384 389 394 879 399 404 409 414 419 424 429 433 438 443 880 94 448 94 453 94 458 94 463 94 468 94 473 94 478 94 483 94 488 94 493 881 498 503 507 512 517 522 527 532 537 542 882 547 552 557 562 567 571 576 581 586 591 883 596 601 606 611 616 621 626 630 635 640 884 645 650 655 660 665 670 675 680 685 689 885 94 694 94 699 94 704 94 709 94 714 94 719 94 724 94 729 94 734 94 738 886 743 748 753 758 763 768 773 778 783 787 887 792 797 802 807 812 817 822 827 832 836 888 841 846 851 856 861 866 871 876 880 885 889 890 895 900 905 ^ 910 915 919 924 929 934 890 94 939 94 944 94 949 94 954 94 959 94 963 94 968 94 973 94 978 94 983 891 988 993 998 95 002 95 007 95 012 95 017 95 022 95 027 95 032 892 95 036 95 041 95 046 051 056 061 066 071 075 080 893 085 090 095 100 105 109 114 119 124 129 894 134 139 143 148 153 158 163 168 173 177 895 95 182 95 187 95 192 95 197 95 202 95 207 95 211 95 216 95 221 95 226 896 231 236 240 245 250 255 260 265 270 274 897 279 284 289 294 299 303 308 313 318 323 898 328 332 337 342 347 352 357 361 366 371 899 376 381 386 390 395 400 405 410 415 419 900 95 424 95 429 95 434 95 439 95 444 95 448 95 453 95 458 95 463 95 468 N 0 1 2 3 4 5 6 7 8 9 850-900 44 900-950 N O 1 2 3 4 6 6 7 8 9 900 95 424 95 429 95 434 95 439 95 444 95 448 95 453 95 458 95 463 95 468 901 472 477 482 487 492 497 501 506 511 516 902 521 525 530 535 540 545 550 554 559 564 903 569 574 578 583 588 593 598 602 607 612 904 617 622 626 631 636 641 646 650 655 660 905 95 665 95 670 95 674 95 679 95 684 95 689 95 694 95 698 95 703 95 708 906 713 718 722 727 732 737 742 746 751 756 907 761 766 770 775 780 785 789 794 799 804 908 809 813 818 823 828 832 837 842 847 852 909 856 861 866 871 875 880 885 890 895 899 910 95 904 95 909 95 914 95 918 95 923 95 928 95 933 95 938 95 942 95 947 911 952 957 961 966 971 976 980 985 990 995 912 999 96 004 96 009 96 014 96 019 96 023 96 028 96 033 96 038 96 042 913 96 047 052 057 061 066 071 076 080 085 090 914 095 099 104 109 114 118 123 128 133 137 915 96 142 96 147 96152 96156 96 161 96166 96171 96175 96 180 96185 916 190 194 199 204 209 213 218 223 227 232 917 237 242 246 251 256 261 265 270 275 280 918 284 289 294 298 303 308 313 317 322 327 919 332 336 341 346 350 355 360 365 369 374 920 96 379 96 384 96 388 96 393 96 398 96 402 96 407 96 412 96 417 96 421 921 426 431 435 440 445 450 454 459 464 468 922 473 478 483 487 492 497 501 506 511 515 923 520 525 530 534 539 544 548 553 558 562 924 567 572 577 581 586 591 595 600 605 609 925 96 614 96 619 96 624 96 628 96 633 96 638 96 642 96 647 96 652 96 656 926 661 666 670 675 680 685 689 694 699 703 927 70S 713 717 722 727 731 736 741 745 750 928 755 759 764 769 774 778 783 788 792 797 929 802 806 811 816 820 825 830 834 839 844 930 96 848 96 853 96 858 96 862 96 867 96 872 96 876 96 881 96 886 96 890 931 895 900 904 909 914 918 923 928 932 937 932 942 946 951 956 960 965 970 974 979 984 933 988 993 997 97 002 97 007 97 011 97 016 97 021 97 025 97 030 934 97 035 97 039 97 044 049 053 058 063 067 072 077 935 97 081 97 086 97 090 97 095 97 100 97 104 97 109 97114 97118 97 123 936 128 132 137 142 146 151 155 160 165 169 937 174 179 183 188 192 197 202 206 211 216 938 220 225 230 234 239 243 248 253 257 262 939 267 271 276 280 285 290 294 299 304 308 940 97 313 97 317 97 322 97 327 97 331 97 336 97 340 97 345 97 350 97 354 941 359 364 368 373 377 382 387 391 396 400 942 405 410 414 419 424 428 433 437 442 447 943 451 456 460 465 470 474 479 483 488 493 944 497 502 506 511 516 520 525 529 534 539 945 97 543 97 548 97 552 97 557 97 562 97 566 97 571 97 575 97 580 97 585 946 589 594 598 603 607 612 617 621 626 630 947 635 640 644 649 653 658 663 667 672 676 948 681 685 690 695 699 704 70S 713 717 722 949 727 731 736 740 745 749 754 759 763 768 950 97 772 97 777 97 782 97 786 97 791 97 795 97 800 97 80+ 97 809 97 813 N O 1 2 3 4 5 6 7 8 9 900-950 950-1000 45 N o 1 2 3 4 5 6 7 8 9 950 97 772 97 777 97 782 97 786 97 791 97 795 97 800 97 804 97 809 97 813 951 818 823 827 832 836 841 845 850 855 859 952 864 868 873 877 882 886 891 896 900 905 953 909 914 918 923 928 932 937 941 946 950 954 ,955 959 964 968 973 978 982 987 991 996 965 / 98 000 98 005 98 009 98 014 98 019 98 023 98 028 98 032 98 037 98 041 956 046 050 055 059 064 068 073 078 082 087 957 091 096 100 105 109 114 118 123 127 132 958 137 141 146 150 155 159 164 168 173 177 959 182 186 191 195 200 204 209 214 218 223 960 98 227 98 232 98 236 98 241 98 245 98 250 98 254 98 259 98 263 98 268 961 272 277 281 286 290 295 299 304 308 313 962 318 322 327 331 336 340 345 349 354 358 963 363 367 372 376 381 385 390 394 399 403 964 408 412 417 421 426 430 435 439 444 448 965 98 453 98 457 98 462 98 466 98 471 98 475 98 480 98 484 98 489 98 493 966 498 502 507 511 516 520 525 529 534 538 967 543 547 552 556 561 565 570 574 579 583 968 588 592 597 601 605 610 614 619 623 628 969 632 637 641 646 650 655 659 664 668 673 970 98 677 98 682 98 686 98 691 98 695 98 700 98 704 98 709 98 713 98 717 971 722 726 731 735 740 744 749 753 758 762 972 767 771 776 780 784 789 793 798 802 807 973 811 816 820 825 829 834 838 843 847 851 974 856 860 865 869 874 878 883 887 892 896 976 98 900 98 905 98 909 98 914 98 918 98 923 98 927 98 932 98 936 98 941 976 945 949 954 958 963 967 972 976 981 985 977 989 994 998 99 003 99 007 99 012 99 016 99 021 99 025 99 029 978 99 034 99 038 99 043 047 052 056 061 065 069 074 979 078 083 087 092 096 100 105 109 114 118 980 99123 99 127 99 131 99136 99 140 99 145 99149 99154 99158 99 162 981 167 171 176 180 185 189 193 198 202 207 982 211 216 220 224 229 233 238 242 247 251 983 255 260 264 269 273 277 282. 286 291 295 984 300 304 308 313 317 322 326 330 335 339 985 99 344 99 348 99 352 99 357 99 361 99 366 99 370 99 374 99 379 99 383 986 388 392 396 401 405 410 414 419 423 427 987 432 436 441 445 449 454 458 463 467 471 988 476 480 484 489 493 498 502 506 511 515 989 520^ 524 528 533 537 542 546 550 555 559 990 99 564 99 568 99 572 99 577 99 581 99 585 99 590 99 594 99 599 99 603 991 607 612 616 621 625 629 634 638 642 647 992 651 656 660 664 669 673 677 682 686 691 993 695 699 704 708 712 717 721 726 730 734 994 739 743 747 752 756 760 765 769 774 778 996 99 782 99 787 99 791 99 795 99 800 99 804 99 808 99 813 99 817 99 822 996 826 830 835 839 843 848 852 856 861 865 997 870- 874 878 883 887 891 896 900 904 909 998 913 917 922 926 930 935 939 944 948 952 999 V 957 961 965 970 974 978 983 987 991 996 lOOO 00 000 00 004 00 009 00 013 00 017 00 022 00 026 00 030 00 035 00 039 N 0 1 2 3 4 5 6 7 8 9 950-1000 46 PKOPOETIONAL PARTS OF DIFFERENCES 47 This table contains the proportional parts of differences from 1 to 100. For example, if the difference between two numbers is 73, 0.7 of this difference is 51.1. D 1 2 3 4 5 a 7 8 9 51 5.1 10.2 15.3 20.4 25.5 30.6 35.7 40.8 45.9 52 5.2 10.4 15.6 20.8 26.0 31.2 36.4 41.6 46.8 53 5.3 10.6 15.9 21.2 26.5 31.8 37.1 42.4 47.7 54 5.4 10.8 16.2 21.6 27.0 32.4 37.8 43.2 48.6 55 5.5 11.0 16.5 22.0 27.5 33.0 38.5 44.0 49.5 56 5.6 11.2 16.8 22.4 28.0 33.6 39.2 44.8 50.4 57 5.7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51.3 58 5.8 11.6 17.4 23.2 29.0 34.8 40.6 46.4 52.2 59 5.9 11.8 17.7 23.6 29.5 35.4 41.3 47.2 53.1 60 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 61 6.1 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 62 6.2 12.4 18.6 24.8 31.0 37.2 43.4 49.6 55.8 63 6.3 12.6 18.9 25.2 31.5 37.8 44.1 50.4 56.7 64 6.4 12.8 19.2 25.6 32.0 38.4 44.8 51.2 57.6 65 6.5 13.0 19.5 26.0 32.5 39.0 45.5 52.0 58.5 66 6.6 13.2 19.8 26.4 33.0 39.6 46.2 52.8 59.4 67 6.7 13.4 20.1 26.8 33.5 40.2 46.9 53.6 60.3 68 6.8 13.6 20.4 27.2 34.0 40.8 47.6 54.4 61.2 69 6.9 13.8 20.7 27.6 34.5 41.4 48.3 55.2 62.1 70 7.0 14.0 21.0 28.0 35.0 42.0 49.0 56.0 63.0 71 7.1 14.2 21.3 2S.4 35.5 42.6 49.7 56.8 63.9 72 7.2 14.4 21.6 28.8 36.0 43.2 50.4 57.6 64.8 73 7.3 14.6 21.9 29.2 36.5 43.8 51.1 58.4 65.7 74 7.4 14.8 22.2 29.6 37.0 44.4 51.8 59.2 66.6 75 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 76 7.6 15.2 22.8 30.4 38.0 45.6 53.2 60.8 68.4 77 7.7 15.4 23.1 30.8 38.5 46.2 53.9 61.6 69.3 78 7.8 15.6 23.4 31.2 39.0 46.8 54.6 62.4 70.2 79 7.9 15.8 23.7 31.6 39.5 47.4 55.3 63.2 71.1 80 8.0 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 81 8.1 16.2 24.3 32.4 40.5 48.6 56.7 64.8 72.9 82 8.2 16.4 24.6 32.8 41.0 49.2 57.4 65.6 73.8 83 8.3 16.6 24.9 33.2 41.5 49.8 58.1 66.4 74.7 84 8.4 16.8 25.2 33.6 42.0 50.4 58.8 67.2 75.6 85 8.5 17.0 25.5 34.0 42.5 51.0 59.5 68.0 76.5 86 8.6 17.2 25.8 34.4 43.0 51.6 60.2 68.8 77.4 87 8.7 17.4 26.1 34.8 43.5 52.2 60.9 69.6 78.3 88 8.8 17.6 26.4 35.2 44.0 52.8 61.6 70.4 79.2 89 8.9 17.8 26.7 35.6 44.5 53.4 62.3 71.2 80.1 90 9.0 18.0 27.0 36.0 45.0 54.0 63.0 72.0 81.0 91 9.1 18.2 27.3 36.4 45.5 54.6 63.7 72.8 81.9 92 9.2 18.4 27.6 36.8 46.0 55.2 64.4 73.6 82.8 93 9.3 18.6 27.9 37.2 46.5 55.8 65.1 74.4 83.7 94 9.4 18.8 28.2 37.6 47.0 56.4 65.8 75.2 84.6 95 9.5 19.0 28.5 38.0 47.5 57.0 66.5 76.0 85.5 96 9.6 19.2 28.8 38.4 48.0 57.6 67.2 76.8 86.4 97 9.7 19.4 29.1 38.8 48.5 58.2 67.9 77.6 87.3 98 9.8 19.6 29.4 39.2 49.0 58.8 68.6 78.4 88.2 99 9.9 19.8 29.7 39.6 49.5 59.4 69.3 79.2 89.1 100 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 1 2 3 4 5 6 7 8 9 48 TABLE V. LOGARITHMS OF CONSTANTS Number Loo Number Log Circle = 360° 2.55630 772 9.86960 0.99430 = 21,600' = 1,296,000" 4.33445 6.11261 ^ = 0.10132 TT ^ 9.00570 - 10 IT = 3.14159 0.49715 = 1.77245 0.24857 2 77 = 6.28319 4 77 = 12.56637 0.79818 1.09921 = 0.56419 Vtt 9.75143 - 10 4 77 — = 4.18879 3 0.62209 = 1.12838 0.05246 - = 0.78540 4 9.89509 - 10 = 1.46459 0.16572 - = 0.52360 6 9.71900 - 10 = 0.68278 Vt7 9.83428 - 10 - = 0.31831 7T 9.50285 - 10 = 0.62035 \477 9.79264 - 10 — = 0.15915 2 7T 9.20182 - 10 = 0.80600 9.90633 - 10 V2 = 1.41421 0.15052 V2 = 1.25992 0.10034 Vs = 1.73205 0.23856 Vs = 1.44225 0.15904 V5 = 2.23606 0.34949 Vb = 1.70997 0.23299 Ve = 2.44948 0.38908 Ve = 1.81712 0.25938 1 radian = 7T 1° = VIL radians 180 = 57.2958° 1.75812 1° = 0.01745 radians 8.24188 - 10 = 3437.75' 3.53627 1' = 0.00029 radians 6.46373 - 10 = 206,264.81" 5.31443 1" = 0.000005 radians 4.68557 — 10 Base of natural logs., e log .^0 e = log^o 2.71828 0.43429 e = 2.71828 0.43429 l:logjo 6 = 2.302585 0.36222 1 m. = 39.3708 in. 1.59517 1 knot = 6080.27 ft. 3.78392 = 1.0936 yd. 0.03886 = 1.1516 mi. 0.06130 = 3.2809 ft. 0.51599 1 lb. At. = 7000 gr. 3.84510 1 km. = 0.6214 mi. 9.79336 - 10 1 bu. = 2150.42 cu. in. 3.33252 1 mi. = 1.6093 km. 0.20664 1 U.S. gal. = 231 cu. in. 2.36361 1 oz. Av. = 28.3495 g. 1.45254 1 Brit. gal. = 277.463 cu. in. 2.44320 11b. Av. = 453.5927 g. 2.65666 Earth’s radii 1 kg. = 2.2046 lb. 0.34333 = 3963 mi. 3.59802 11. = 1.0567 liq. qt. 0.02396 and 3950 mi. 3.59660 1 liq. qt. = 0.9463 1. 9.97603 - 10 1 ft. /lb. = 0.1383 kg./m. 9.14082 - 10 49 TABLE YI THE LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS From 0° to 0° 3', and from 89° 57' to 90°, for every second From 0° to 2°, and from 88° to 90°, for every ten seconds From 1° to 89°, for every minute To each logarithm — 10 is to be appended log sin 0° log tan = log sin log cos = 10.00 000 ff O' 1' 2' rr ff O' 1' 2' ff 0 6. 46 373 6. 76 476 60 30 6. 16 270 6. 63 982 6. 86 167 30 1 4.68 557 6. 47 090 6. 76 836 59 31 6. 17 694 6. 64 462 6. 86 455 29 2 4. 98 660 6. 47 797 6. 77 193 58 32 6. 19 072 6. 64 936 6. 86 742 28 3 5.15 270 6. 48 492 6. 77 548 57 33 6. 20 409 6. 65 406 6. 87 027 27 4 5. 28 763 6. 49 175 6. 77 900 56 34 6. 21 70S 6. 65 870 6. 87 310 26 5 5. 38 454 6. 49 849 6. 78 248 55 35 6. 22 964 6. 66 330 6. 87 591 26 6 5. 46 373 6. 50 512 6. 78 595 54 36 6. 24 188 6. 66 785 6. 87 870 24 7 5. 53 067 6. 51 165 6. 78 938 S3 37 6. 25 378 6. 67 235 6. 88 147 23 8 5. 58 866 6. 51 808 6. 79 278 52 38 6. 26 536 6. 67 680 6. 88 423 22 9 5. 63 982 6. 52 442 6. 79 616 51 39 6. 27 664 6. 68 121 6. 88 697 21 10 5.68 557 6. 53 067 6. 79 952 50 40 6. 28 763 6. 68 557 6. 88 969 20 11 5. 72 697 6. 53 683 6. 80 285 49 41 6. 29 836 6. 68 990 6. 89 240 19 12 5. 76 476 6. 54 291 6. 80 615 48 42 6. 30 882 6. 69 418 6. 89 509 18 13 5. 79 952 6. 54 890 6. 80 943 47 43 6.31904 6. 69 841 6. 89 776 17 14 5. 83 170 6. 55 481 6. 81 268 46 44 6. 32 903 6. 70 261 6. 90 042 16 15 5. 86 167 6. 56 064 6. 81 591 45 45 6. 33 879 6. 70 676 6. 90 306 16 16 5.88 969 6. 56 639 6.81911 44 46 6. 34 833 6. 71 088 6. 90 568 14 17 5. 91 602 6. 57 207 6. 82 230 43 47 6. 35 767 6. 71 496 6. 90 829 13 18 5. 94 OSS 6. 57 767 6. 82 545 42 48 6. 36 682 6. 71 900 6. 91 088 12 19 5. 96 433 6. 58 320 6. 82 859 41 49 6. 37 577 6. 72 300 6. 91 346 11 20 5. 98 660 6. 58 866 6. 83 170 40 50 6. 38 454 6. 72 697 6. 91 602 10 21 6. 00 779 6. 59 406 6. 83 479 39 51 6. 39 315 6. 73 090 6. 91 857 9 22 6. 02 800 6. 59 939 6. 83 786 38 52 6. 40 158 6. 73 479 6. 92 110 8 23 6. 04 730 6. 60 465 6. 84 091 37 53 6. 40 985 6. 73 865 6. 92 362 7 24 6. 06 579 6. 60 985 6. 84 394 36 54 6. 41 797 6. 74 248 6. 92 612 6 25 6. 08 351 6. 61 499 6. 84 694 35 55 6. 42 594 6. 74 627 6. 92 861 6 26 6. 10 OSS 6. 62 007 6. 84 993 34 56 6. 43 376 6. 75 003 6. 93 109 4 27 6. 11 694 6. 62 509 6. 85 289 33 57 6. 44 145 6. 75 376 6. 93 355 3 28 6. 13 273 6. 63 006 6. 85 584 32 58 6. 44 900 6. 75 746 6. 93 599 2 29 6. 14 797 6. 63 496 6. 85 876 31 59 6. 45 643 6.'76 112 6. 93 843 1 30 6. 16 270 6. 63 982 6. 86 167 30 60 6. 46 373 6. 76 476 6. 94 085 0 tt 59' 58' 57' ff ff 59' 68' 67' ff log cot = log cos log sin=^ 10.00 000 89 ° log cos 50 O' r ff log sin log cos log tan t ft f ff log sin log cos log tan f ff 0 0 10.00000 60 0 10 0 7. 46 373 10.00000 7.46373 50 0 10 5.68 557 10.00000 5. 68 557 SO 10 7. 47 090 10.00000 7. 47 091 SO 20 5.98 660 10.00000 5.98 660 40 20 7. 47 797 10.00000 7. 47 797 40 30 6. 16 270 10.00000 6. 16 270 30 30 7. 48 491 10.00000 7. 48 492 30 40 6. 28 763 10.00000 6. 28 763 20 40 7. 49 175 10.00000 7. 49 176 20 50 6. 38 454 10.00000 6. 38 454 10 SO 7. 49 849 10.00000 7. 49 849 10 1 0 6. 46 373 10.00000 6. 46 373 59 0 11 0 7. 50 512 10.00000 7. 50 512 49 0 10 6. S3 067 10.00000 6. 53 067 50 10 7.51 165 10.00000 7. 51 165 50 20 6. 58 866 10.00000 6. 58 866 40 20 7. 51 808 10.00000 7.51809 40 30 6. 63 982 10.00000 6. 63 982 30 30 7. 52 442 10.00000 7. 52 443 30 40 6. 68 557 10.00000 6. 68 557 20 40 7. S3 067 10.00000 7. 53 067 20 SO 6. 72 697 10.00000 6. 72 697 10 50 7. 53 683 10.00000 7. 53 683 10 2 0 6. 76 476 10.00000 6. 76 476 58 0 12 0 7. 54 291 10.00000 7. 54 291 48 0 10 6. 79 952 10.00000 6. 79 952 50 10 7. 54 890 10.00000 7. 54 890 50 20 6. 83 170 10.00000 6. 83 170 40 20 7. 55 481 10.00000 7. 55 481 40 30 6. 86 167 10.00000 6. 86 167 30 30 7. 56 064 10.00000 7. 56 064 30 40 6. 88 969 10.00000 6. 88 969 20 40 7. 56 639 10.00000 7. 56 639 20 50 6. 91 602 10.00000 6. 91 602 10 50 7. 57 206 10.00000 7. 57 207 10 3 0 6. 94 OSS 10.00000 6. 94 OSS 57 0 13 0 7. 57 767 10.00000 7. 57 767 47 0 10 6. 96 433 10.00000 6. 96 433 50 10 7. 58 320 10.00000 7. 58 320 50 20 6, 98 660 10.00000 6. 98 661 40 20 7. 58 866 10.00000 7. 58 867 40 30 7. 00 779 10.00000 7.00 779 30 30 7. 59 406 10.00000 7. 59 406 30 40 7. 02 800 10.00000 7. 02 800 20 40 7. 59 939 10.00000 7. 59 939 20 SO 7. 04 730 10.00000 7. 04 730 10 50 7. 60 465 10.00000 7. 60 466 10 4 0 7. 06 579 10.00000 7. 06 579 56 0 14 0 7. 60985 10.00000 7. 60 986 46 0 10 7. 08 351 10.00000 7.08 352 50 10 7. 61 499 10.00000 7. 61 500 50 20 7. 10 055 10.00000 7. 10 055 40 20 7. 62 007 10.00000 7. 62 OOS 40 30 7. 11 694 10.00000 7. 11 694 30 30 7. 62 509 10.00000 7. 62 510 30 40 7. 13 273 10.00000 7. 13 273 20 40 7. 63 006 10.00000 7. 63 006 20 50 7. 14 797 10.00000 7. 14 797 10 SO 7. 63 496 10.00000 7. 63 497 10 5 0 7. 16 270 10.00000 7. 16 270 55 0 15 0 7. 63 982 10.00000 7. 63 982 45 0 10 7. 17 694 10.00000 7. 17 694 50 10 7. 64 461 10.00000 7.64 462 50 20 7. 19 072 10.00000 7. 19 073 40 20 7. 64 936 10.00000 7. 64 937 40 30 7. 20 409 10.00000 7. 20 409 30 30 7. 65 406 10.00000 7. 65 406 30 40 7. 21 705 10.00000 7. 21 70S 20 40 7. 65 870 10.00000 7. 65 87l 20 50 7. 22 964 10.00000 7. 22 964 10 50 7. 66 330 10.00000 7. 66 330 10 6 0 7. 24 188 10.00000 7. 24 188 54 0 16 0 7. 66 784 10.00000 7. 66 785 440 10 7. 25 378 10.00000 7. 25 378 50 10 7. 67 235 10.00000 7. 67 235 50 20 7. 26 536 10.00000 7. 26 536 40 20 7. 67 680 10.00000 7. 6/ 680 40 30 7. 27 664 10.00000 7. 27 664 30 30 7. 68 121 10.00000 7.68121 30 40 7. 28 763 10.00000 7. 28 764 20 40 7. 68 557 9.99999 7.68 558 20 SO 7. 29 836 10.00000 7. 29 836 10 50 7. 68 989 9.99999 7.68 990 10 7 0 7. 30 882 10.00000 7. 30 882 53 0 170 7.69 417 9.99 999 7. 69 418 43 0 10 7. 31 904 10.00000 7. 31 904 50 10 7. 69 841 9. 99 999 7. 69 842 50 20 7. 32 903 10.00000 7. 32 903 40 20 7. 70 261 9. 99 999 7. 70 261 40 30 7. 33 879 10.00000 7. 33 879 30 30 7. 70 676 9. 99 999 7 . 70 t )/ / 30 40 7. 34 833 10.00000 7. 34 833 20 40 7. 71 OSS 9. 99 999 7. 71 OSS 20 50 7. 35 767 10.00000 7. 35 767 10 50 7. 71 496 9. 99 999 7. 71 496 10 8 0 7. 36 682 10.00000 7. 36 682 52 0 18 0 7. 71 900 9. 99 999 7. 71 900 42 0 10 7.37 577 10.00000 7.37 577 SO 10 7. 72 300 9. 99 999 7. 72 301 50 20 7. 38 454 10.00000 7. 38 455 40 20 7. 72 697 9. 99 999 7. 72 697 40 30 7. 39 314 10.00000 7.39 315 30 30 7. 73 090 9. 99 999 7. 73 090 30 40 7. 40 158 10.00000 7. 40 158 20 40 7. 73 479 9. 99 999 7. 73 480 20 50 7. 40 985 10.00000 7. 40 985 10 50 7. 73 865 9. 99 999 7. 73 866 10 9 0 7. 41 797 10.00000 7. 41 797 51 0 190 7. 74 248 9. 99 999 7. 74 248 41 0 10 7. 42 594 10.00000 7. 42 594 SO 10 7. 74 627 9. 99 999 7. 74 628 50 20 7. 43 376 10.00000 7.43 376 40 20 7. 75 003 9.99 999 7. 75 004 40 30 7. 44 145 10.00000 7. 44 145 30 30 7. 75 376 9. 99 999 V . / ^ 0 / / 30 40 7. 44 900 10.00000 7. 44 900 20 40 7- 75 745 9.99 999 7. 7^ 746 20 SO 7. 45 643 10.00000 7. 45 643 10 50 7. 76 112 9. 99 999 7. 76 113 10 lOO 7. 46 373 10.00000 7. 46 373 50 0 20 0 7. 76475 9. 99 999 7. 76 476 40 0 log cos log sin log cot / ff f ff log cos log sin log cot ! ft 89 ** O' 51 f ff log sin log cos log tan f ff t ff log sin log cos log tan / ff 200 7. 76 475 9. 99 999 7. 76 476 40 0 30 0 7. 94 084 9. 99 998 7. 94 086 30 0 10 7. 76 836 9. 99 999 7. 76 837 50 10 7. 94 325 9. 99 998 7. 94 326 50 20 7. 77 193 9. 99 999 7. 77 194 40 20 7. 94 564 9. 99 998 7. 94 566 40 30 7. 77 548 9. 99 999 7. 77 549 30 30 7. 94 802 9. 99 998 7. 94 804 30 40 7. 77 899 9. 99 999 7. 77 900 20 40 7. 95 039 9. 99 998 7. 95 040 20 50 7. 78 248 9. 99 999 7. 78 249 10 50 7. 95 274 9. 99 998 7. 95 276 10 210 7. 78 594 9. 99 999 7. 78 595 39 0 31 0 7. 95 508 9. 99 998 7.95 510 29 0 10 7. 78 938 9. 99 999 7. 78 938 50 10 7. 95 741 9. 99 998 7. 95 743 50 20 7. 79 278 9. 99 999 7. 79 279 40 20 7. 95 973 9. 99 998 7. 95 974 40 30 7. 79 616 9. 99 999 7.79 617 30 30 7. 96 203 9. 99 998 7. 96 205 30 40 7. 79 952 9. 99 999 7. 79 952 20 40 7. 96 432 9. 99 998 7. 96 434 20 50 7. 80 284 9. 99 999 7. 80 285 10 50 7. 96 660 9. 99 998 7. 96 662 10 220 7. 80 615 9. 99 999 7. 80 615 38 0 32 0 7. 96 887 9. 99 998 7. 96 889 28 0 10 7. 80 942 9. 99 999 7. 80 943 50 10 7. 97 113 9. 99 998 7. 97 114 50 20 7. 81 268 9. 99 999 7. 81 269 40 20 7. 97 337 9. 99 998 7. 97 339 40 30 7. 81 591 9. 99 999 7. 81 591 30 30 7. 97 560 9. 99 998 7. 97 562 30 40 7. 81 911 9. 99 999 7. 81 912 20 40 7. 97 782 9. 99 998 7. 97 784 20 50 7. 82 229 9. 99 999 7. 82 230 10 50 7. 98 003 9. 99 998 7. 98 005 10 230 7. 82 545 9. 99 999 7. 82 546 37 0 33 0 7. 98 223 9. 99 998 7. 98 225 27 0 10 7. 82 859 9. 99 999 7. 82 860 50 10 7. 98 442 9. 99 998 7. 98 444 50 20 7. 83 170 9. 99 999 7. S3 171 40 20 7. 98 660 9. 99 998 7. 98 662 40 30 7. 83 479 9. 99 999 7. S3 480 30 30 7. 98 876 9. 99 998 7. 98 878 30 40 7. 83 786 9. 99 999 7. S3 787 20 40 7. 99 092 9. 99 998 7. 99 094 20 50 7. 84 091 9. 99 999 7. 84 092 10 50 7. 99 306 9. 99 998 7. 99 308 10 240 7. 84 393 9. 99 999 7. 84 394 36 0 34 0 7. 99 520 9. 99 998 7. 99 522 26 0 10 7. 84 694 9. 99 999 7. 84 695 50 10 7. 99 732 9. 99 998 7. 99 734 50 20 7. 84 992 9. 99 999 7. 84 994 40 20 7. 99 943 9. 99 998 7. 99 946 40 30 7. 85 289 9. 99 999 7. 85 290 30 30 8. 00 154 9. 99 998 8. 00 156 30 40 7. 85 583 9. 99 999 7. 85 584 20 40 8. 00 363 9. 99 998 8.00365 20 50 7. 85 876 9. 99 999 7. 85 877 10 50 8.00 571 9. 99 998 8. 00 574 10 250 7. 86 166 9. 99 999 7. 86 167 35 0 35 0 8. 00 779 9. 99 998 8. 00 781 25 0 10 7. 86 455 9. 99 999 7. 86 456 50 10 8. 00 985 9. 99 998 8. 00 987 50 20 7. 86 741 9. 99 999 7. 86 743 40 20 8. 01 190 9. 99 998 8. 01 193 40 30 7. 87 026 9. 99 999 7.87 027 30 30 8. 01 395 9. 99 998 8. 01 397 30 40 7. 87 309 9. 99 999 7. 87 310 20 40 8. 01 598 9. 99 998 8. 01 600 20 50 7. 87 590 9. 99 999 7. 87 591 10 50 8. 01 SOI 9. 99 998 8. 01 803 10 260 7. 87 870 9. 99 999 7. 87 871 340 36 0 8. 02 002 9. 99 998 8. 02 004 24 0 10 7. 88 147 9. 99 999 7. 88 148 50 10 8. 02 203 9. 99 998 8. 02 205 50 20 7. 88 423 9. 99 999 7. 88 424 40 20 8. 02 402 9. 99 998 8. 02 405 40 30 7. 88 697 9. 99 999 7. 88 698 30 30 8. 02 601 9. 99 998 8. 02 604 30 40 7. 88 969 9. 99 999 7. 88 970 20 40 8, 02 799 9. 99 998 8. 02 SOI 20 50 7. 89 240 9. 99 999 7.89 241 10 50 8. 02 996 9. 99 998 8. 02 998 10 270 7. 89 509 9. 99 999 7.89 510 33 0 37 0 8. 03 192 9. 99 997 8. 03 194 23 0 10 7. 89 776 9. 99 999 7.89 777 50 10 8. 03 387 9. 99 997 8. 03 390 50 20 7.90 041 9. 99 999 7. 90 043 40 20 8. 03 581 9. 99 997 8. 03 584 40 30 7. 90 305 9. 99 999 7.90 307 30 30 8. 03 775 9. 99 997 8. 03 777 30 40 7. 90 568 9. 99 999 7. 90 569 20 40 8. 03 967 9. 99 997 8. 03 970 20 50 7. 90 829 9. 99 999 7. 90 830 10 50 8. 04 159 9. 99 997 8. 04 162 10 280 7. 91 088 9. 99 999 7. 91 089 32 0 38 0 8. 04 350 9. 99 997 8. 04 353 22 0 10 7.91346 9. 99 999 7. 91 347 50 10 8. 04 540 9. 99 997 8. 04 543 50 20 7. 91 602 9. 99 999 7. 91 603 40 20 8. 04 729 9. 99 997 8. 04 732 40 30 7.91 857 9. 99 999 7. 91 858 30 30 8.04 918 9. 99 997 8. 04 921 30 40 7.92 no 9. 99 998 7. 92 111 20 40 8. 05 105 9. 99 997 8. 05 108 20 50 7. 92 362 9. 99 998 7. 92 363 10 50 8. 05 292 9. 99 997 8. 05 295 10 290 7. 92 612 9. 99 998 7. 92 613 31 0 39 0 8. 05 478 9. 99 997 8. 05 481 21 0 10 7.92 861 9. 99 998 7. 92 862 50 10 8. 05 663 9. 99 997 8. 05 666 50 20 7. 93 108 9. 99 998 7. 93 110 40 20 8. 05 848 9. 99 997 8. 05 851 40 30 7. 93 354 9. 99 998 7. 93 356 30 30 8.06 031 9. 99 997 8. 06 034 30 40 7. 93 599 9. 99 998 7. 93 601 20 40 8. 06 214 9. 99 997 8. 06 217 20 50 7. 93 842 9. 99 998 7. 93 844 10 50 8. 06 396 9. 99 997 8. 06 399 10 300 7. 94 084 9. 99 998 7. 94 086 30 0 40 0 8.06 578 9. 99 997 8. 06 581 20 0 f 9f log cos log sin log cot / ff f ff log cos log sin log cot f ff 89 ' 62 0 r tt log sin log C03 log tan t ff f tt log sin log cos log tan t tt 400 8. 06 578 9. 99 997 8. 06 581 20 0 50 0 8. 16 268 9. 99 995 8. 16 273 10 0 10 8. 06 758 9. 99 997 8. 06 761 50 10 8. 16 413 9. 99 995 8. 16417 50 20 8. 06 938 9. 99 997 8. 06 941 40 20 8. 16 557 9. 99 995 8. 16 561 40 30 8. 07 117 9. 99 997 8. 07 120 30 30 8. 16 700 9. 99 995 8. 16 705 30 40 8. 07 295 9. 99 997 8. 07 299 20 40 8. 16 843 9. 99 995 8. 16 848 20 50 8. 07 473 9. 99 997 8. 07 476 10 50 8. 16 986 9. 99 995 8. 16 991 10 410 8. 07 650 9. 99 997 8. 07 653 19 0 51 0 8. 17 128 9. 99995 8. 17 133 9 0 10 8. 07 826 9. 99 997 8. 07 829 50 10 8. 17 270 9. 99 995 8. 17 275 50 20 8. 08 002 9. 99 997 8. 08 005 40 20 8.17 411 9. 99995 8. 17 416 40 30 8. 08 176 9. 99 997 8. 08 180 30 30 8.17 552 9. 99 995 8.17 557 30 40 8. 08 350 9. 99 997 8. 08 354 20 40 8. 17 692 9. 99 995 8. 17 697 20 50 8. 08 524 9. 99 997 8. 03 527 10 50 8. 17 832 9. 99 995 8.17 837 10 420 8. 08 696 9. 99 997 8. 08 700 18 0 52 0 8.17 971 9. 99 995 8.17 976 8 0 10 8. 08 868 9. 99 997 8. 08 872 50 10 8. 18110 9. 99 995 8. 18115 50 20 8. 09 040 9. 99 997 8. 09 043 40 20 8. 18 249 9.99 995 8. 18 254 40 30 8. 09 210 9. 99 997 8. 09 214 30 30 8. 18 387 9. 99 995 8. 18 392 30 40 8. 09 380 9. 99 997 8. 09 384 20 40 8. 18 524 9. 99 995 8. 18 530 20 50 8. 09 550 9. 99 997 8. 09 553 10 50 8. 18 662 9. 99 995 8. IS 667 10 430 8. 09 718 9. 99 997 8. 09 722 17 0 53 0 8. 18 798 9. 99 995 8. 18 804 70 10 8. 09 886 9. 99 997 8. 09 890 50 10 8. 18 935 9. 99 995 8. 18 940 50 20 8. 10 054 9. 99 997 8. 10 057 40 20 8. 19 071 9. 99 995 8. 19 076 40 30 8. 10 220 9. 99 997 8. 10 224 30 30 8. 19 206 9. 99 995 8. 19 212 30 40 8. 10 386 9. 99 997 8. 10 390 20 40 8. 19 341 9. 99 995 8. 19 347 20 50 8. 10 552 9. 99 996 8. 10 555 10 50 8. 19 476 9. 99 995 8. 19 481 10 440 8. 10 717 9. 99 996 8. 10 720 16 0 54 0 8. 19 610 9. 99 995 8. 19 616 60 10 8. 10 881 9. 99 996 8. 10 884 50 10 8. 19 744 9. 99 995 8. 19 749 50 20 8. 11 044 9. 99 996 8. 11048 40 20 8. 19 877 9. 99995 8. 19 883 40 30 8. 11 207 9. 99 996 8. 11211 30 30 8. 20 010 9. 99 995 8. 20 016 30 40 8.11370 9. 99 996 8.11373 20 40 8. 20 143 9. 99 995 8. 20 149 20 50 8.11531 9. 99 996 8. 11 535 10 50 8. 20 275 9. 99 994 8. 20 281 10 450 8. 11 693 9. 99 996 8. 11 696 15 0 65 0 8. 20 407 9.99 994 8. 20 413 o 0 10 8. 11853 9. 99 996 8. 11857 50 10 8. 20 538 9. 99 994 8. 20 544 50 20 8. 12 013 9. 99 996 8. 12 017 40 20 8. 20 669 9. 99 994 8. 20 675 40 30 8. 12 172 9. 99 996 8. 12 176 30 30 8. 20 800 9. 99 994 8. 20 806 30 40 8.12 331 9. 99 996 8. 12 335 20 40 8. 20 930 9. 99 994 8. 20 936 20 50 8. 12 489 9. 99 996 8. 12 493 10 50 8. 21 060 9. 99 994 8. 21 066 10 460 8. 12 647 9. 99 996 8. 12 651 14 0 66 0 8. 21 189 9. 99 994 8. 21 195 40 10 8. 12 804 9. 99 996 8. 12 808 50 10 8.21319 9. 99 994 8.21 324 50 20 8. 12 961 9. 99 996 8. 12 965 40 20 8. 21 447 9. 99 994 8. 21 453 40 30 8. 13 117 9. 99 996 8. 13 121 30 30 8.21 576 9. 99 994 8. 21 581 30 40 8. 13 272 9. 99 996 8. 13 276 20 40 8. 21 703 9. 99 994 8. 21 709 20 50 8. 13 427 9. 99 996 8. 13 431 10 50 8. 21 831 9. 99 994 8. 21 837 10 470 8. 13 581 9. 99 996 8. 13 585 13 0 67 0 8. 21 958 9. 99 994 8. 21 964 30 10 8. 13 735 9. 99 996 8. 13 739 50 10 8. 22 085 9. 99 994 8. 22 091 50 20 8. 13 888 9. 99 996 8. 13 892 40 20 8. 22 211 9. 99 994 8. 22 217 40 30 8. 14 041 9. 99 996 8. 14 045 30 30 8. 22 337 9. 99 994 8. 22 343 30 40 8. 14 193 9. 99 996 8. 14 197 20 40 8. 22 463 9. 99 994 8. 22 469 20 50 8. 14 344 9. 99 996 8. 14 348 10 50 8. 22 588 9. 99 994 S. 22 595 10 480 8. 14 495 9. 99 996 8. 14 500 12 0 58 0 8. 22 713 9. 99 994 8. 22 720 2 0 10 8. 14 646 9. 99 996 8. 14 650 50 10 8. 22 838 9. 99 994 8. 22 844 50 20 8. 14 796 9. 99 996 8. 14 800 40 20 8. 22 962 9.99 994 8. 22 968 40 30 8. 14 945 9. 99 996 8. 14 950 30 30 8. 23 086 9. 99 994 8. 23 092 30 40 8. 15 094 9. 99 996 8. 15 099 20 40 8. 23 210 9. 99 994 8. 23 216 20 50 8. 15 243 9. 99 996 8. 15 247 10 50 8. 23 333 9. 99 994 8. 23 339 10 490 8. 15 391 9. 99 996 8. 15 395 11 0 59 0 8. 23 456 9. 99 994 8. 23 462 1 0 10 8. 15 538 9. 99 996 8. 15 543 50 10 8. 23 578 9. 99 994 8. 23 585 50 20 8. 15 685 9. 99 996 8. 15 690 40 20 8. 23 700 9. 99 994 8. 23 707 40 30 8. 15 832 9. 99 996 8. 15 836 30 30 8. 23 822 9. 99 993 8. 23 829 30 40 8. 15 978 9. 99 995 8. 15 982 20 40 8. 23 944 9. 99 993 S. 23 950 20 50 8. 16 123 9. 99 995 8. 16 128 10 50 8. 24 065 9. 99 993 8. 24 071 10 500 8. 16 268 9. 99 995 8. 16 273 lOO 60 0 8. 24 186 9.99 993 8. 24 192 OO f tf log eo3 log sin log cot r n f tf log cos log sin log cot / // 89 " 1 53 r >> log sin log cos log tan f // t ff log sin log cos log tan f ft 0 0 8. 24 186 9. 99 993 8. 24 192 60 0 10 0 8. 30 879 9. 99 991 8. 30 SSS 50 0 10 8. 24 306 9. 99 993 8.24 313 50 10 8. 30 983 9. 99 991 8. 30 992 SO 20 8. 24 426 9. 99 993 8. 24 433 40 20 8. 31 0S6 9. 99 991 8. 31 095 40 30 8. 24 546 9. 99 993 8. 24 553 30 30 8. 31 188 9. 99 991 8. 31 198 30 40 8. 24 665 9. 99 993 8. 24 672 20 40 8. 31 291 9. 99 991 8. 31 300 20 SO 8. 24 785 9. 99 993 8. 24 791 10 SO 8.31393 9. 99 991 8. 31 403 10 1 0 8. 24 903 9. 99 993 8. 24 910 59 0 11 0 8.31495 9. 99 991 8. 31 505 49 0 10 8. 25 022 9. 99 993 8. 25 029 50 10 8. 31 597 9. 99 991 8. 31 606 50 20 8. 25 140 9. 99 993 8. 25 147 40 20 8. 31 699 9. 99 991 8. 31 708 40 30 8. 25 258 9. 99 993 8. 25 265 30 30 8. 31 800 9. 99 991 8. 31 809 30 40 8. 25 375 9. 99 993 8. 25 382 20 40 8. 31 901 9. 99 991 8.31911 20 50 8. 25 493 9. 99 993 8. 25 500 10 50 8. 32 002 9. 99 991 8. 32 012 10 2 0 8. 25 609 9. 99 993 8. 25 616 58 0 12 0 8. 32 103 9. 99 990 8. 32 112 48 0 10 8. 25 726 9. 99 993 8. 25 733 50 10 8. 32 203 9. 99 990 8. 32 213 50 20 8. 25 842 9. 99 993 8. 25 849 40 20 S- 32 303 9. 99 990 8. 32 313 40 30 8. 25 958 9. 99 993 8. 25 965 30 30 8. 32 403 9. 99 990 8.32 413 30 40 8. 26 074 9. 99 993 8. 26 081 20 40 8. 32 503 9. 99 990 8.32 513 20 50 8. 26 189 9. 99 993 8. 26 196 10 50 8. 32 602 9. 99 990 8. 32 612 10 3 0 8. 26 304 9. 99 993 8. 26 312 57 0 13 0 8. 32 702 9. 99 990 8. 32 711 47 0 10 8. 26 419 9. 99 993 8. 26 426 50 10 8. 32 SOI 9. 99 990 8. 32 811 50 20 8. 26 533 9. 99 993 8. 26 541 40 20 8. 32 899 9. 99 990 8. 32 909 40 30 8. 26 648 9. 99 993 8. 26 655 30 30 8. 32 998 9. 99 990 8. 33 008 30 40 8. 26 761 9. 99 993 8. 26 769 20 40 8. 33 096 9. 99 990 8. 33 106 20 50 8. 26 875 9. 99 993 8. 26 882 10 50 8. 33 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180 971 029 37 24 259 803 456 544 36 24 166 169 92 996 07 004 36 25 80 274 88 793 91482 08 518 35 25 81 180 88 158 93 022 06 978 35 26 290 782 507 493 34 26 195 148 048 952 34 27 305 772 533 467 33 27 210 137 073 927 33 28 320 761 559 441 32 28 225 126 099 901 32 29 336 751 585 415 31 29 240 115 124 876 31 30 80351 88 7-W 91 610 08 390 30 30 81 254 88 105 93150 06 850 30 31 366 730 636 364 29 31 269 094 175 825 29 32 382 720 662 338 28 32 284 083 201 799 28 33 397 709 688 312 27 33 299 072 227 773 27 34 412 699 713 287 26 34 314 061 252 748 26 35 80428 88 688 91 739 08 261 25 35 81 328 88 051 93 278 06 722 26 36 443 678 765 235 24 36 343 040 303 697 24 37 458 668 791 209 23 37 358 029 329 671 23 38 473 657 816 184 22 38 372 018 354 646 22 39 489 647 842 158 21 39 387 88 007 380 620 21 40 80 504 88 636 91 868 08 132 20 40 81 402 87 996 93 406 06 594 20 41 519 626 893 107 19 41 417 985 431 569 19 42 534 615 919 081 18 42 431 975 457 543 18 43 550 605 945 055 17 43 446 964 482 518 17 44 565 594 971 029 16 44 461 953 508 492 16 45 80 580 88 584 91 996 08 004 16 45 81 475 87942 93 533 06 467 16 46 595 573 92 022 07 978 14 46 490 931 559 441 14 47 610 563 048 952 13 47 505 920 584 416 13 48 625 552 073 927 12 48 519 909 610 390 12 49 641 542 099 901 11 49 534 898 636 364 11 50 80 656 88 531 92 125 07 875 10 60 81 549 87 887 93 661 06 339 lO 51 671 521 150 850 9 51 563 877 687 313 9 52 686 510 176 824 8 52 578 866 712 288 8 53 701 499 202 798 7 53 592 855 738 262 7 54 716 489 227 773 6 54 607 844 763 237 6 66 80 731 88 478 92 253 07 747 6 65 81 622 87 833 93 789 06 211 6 56 746 468 279 721 4 56 636 822 814 186 4 57 762 457 304 696 3 57 651 811 840 160 3 58 777 447 330 670 2 58 665 800 865 C'135 2 59 792 436 356 644 1 59 680 789 891 109 1 60 80 807 9 88 425 9 92 381 9 07 619 10 O 60 81694 9 87 778 9 93 916 9 06 084- lO O f log cos log sin log cot log tan f f log cos log sin log oot log tan f 50 ° 49 ° 76 4r 42 “ f log sin 9 log 003 9 log tan 9 log cot lO 9 f log sin 9 log coa 9 log tan 9 log cot 10 f 0 81694 87 778 93 916 06 084 60 0 82 551 87 107 95 444 04 556 60 1 709 767 942 058 59 1 565 096 469 531 59 2 723 756 967 033 58 2 579 085 495 505 58 3 738 745 93 993 06 007 57 3 593 073 520 480 57 4 752 734 94 018 05 982 56 4 607 062 545 451 56 5 81767 87 723 94 044 05 956 65 6 82 621 87 050 95 571 04 429 55 6 781 712 069 931 54 6 635 039 596 404 54 7 796 701 095 905 S3 7 649 028 622 378 53 8 810 690 120 880 52 8 663 016 647 353 52 9 825 679 146 854 51 9 677 87 005 672 328 51 lO 81 839 87 668 94 171 05 829'' 60 lO 82 691 86 993 95 698 04 302 50 11 854 657 197 803 49 11 70S 982 723 277 49 12 868 646 222 778 48 12 719 970 748 252 48 13 882 635 248 752 47 13 733 959 774 226 47 14 897 624 273 727 46 14 747 947 799 201 46 15 81911 87 613 94 299 05 701 46 15 82 761 86 936 95 825 04175 45 16 926 601 324 676 44 16 775 924 850 150 44 17 940 590 350 650 43 17 788 913 875 125 43 18 955 579 375 625 42 18 802 902 901 099 42 19 969 568 401 599 41 19 816 890 926 074 41 20 81 983 87 557 94 426 OS 574 40 20 82 830 86 879 95 952 04 048 40 21 81998 546 452 548 39 21 844 867 95 977 (M023 39 22 82 012 535 477 523 38 22 858 855 96 002 03 998 38 23 026 524 503 497 37 23 872 844 028 972 37 24 041 513 528 472 36 24 885 832 053 947 36 25 82 055 87 501 94 554 05 446 86 26 82 899 86 821 96078 03 922 35 26 069 490 579 421 34 26 913 809 104 896 34 27 084 479 604 396 33 27 927 798 129 871 33 28 098 468 630 370 32 28 941 786 155 845 32 29 112 457 655 345 31 29 955 771 180 820 31 30 82 126 87 446 94 681 05 319 30 30 82 968 86 763 96 205 03 795 30 31 141 434 706 294 29 31 982 752 231 769 29 32 155 423 732 268 28 32 82 996 740 256 744 28 33 169 412 757 243 27 33 83 010 728 281 719 27 34 184 401 783 217 26 34 023 717 307 693 26 35 82 198 87 390 94 808 05 192 26 35 83 037 86 70S 96 332 03 668 25 36 212 378' 834 166 24 36 051 694 357 643 24 37 226 367 859 141 23 37 061 682 383 617 23 38 240 356 884 116 22 38 078 670 408 592 22 39 255 345 910 090 21 39 092 659 433 567 21 40 82 269 87 334 94 935 OS 065 20 40 83 106 86 647 96 459 03 541 20 41 283 322 961 039 19 41 120 635 484 516 19 42 297 311 94 986 05 014 18 42 133 624 510 490 IS 43 311 300 95 012 04 988 17 43 147 612 531 465 17 44 326 288 037 963 16 44 161 600 560 440 16 45 82 340 87 277 95 062 04 938 16 45 83 174 86 589 96 586 03 414 15 46 354 266 088 912 14 46 188 577 611 389 14 47 368 255 113 887 13 47 202 565 636 364 13 48 382 243 139 861 12 48 215 554 662 338 12 49 396 232 164 836 11 49 229 542 687 313 11 50 82 410 87 221 95 190 04 810 lO 50 83 242 86 530 96 712 03 288 10 51 424 209 215 785 9 51 256 518 738 262 9 52 439 198 240 760 8 52 270 507 763 237 8 53 453 187 266 734 7 53 283 495 788 212 7 54 467 175 291 709 6 54 297 483 814 186 6 55 82 481 87 164 95 31^ 04 683 6 55 83 310 86472 96 839 03161 5 56 495 153 342: 658 4- 56 324 460 864 136 4 57 509 141 368 632 3 57 338 448 890 no 3 58 523 130 393 607 2 58 351 436 915 085 2 59 537 119 418 582 1 59 365 421 940 060 1 60 82 551 9 87 107 9 95 444 9 04 556 lO O 60 83 378 9 86413 9 96 966 9 03 034 lO 0 9 log oos log 3m log cot log tan f t log 003 log sin log oot log tan t 48 ' 47 43 " 77 44 " f log sin 9 log 003 9 log tan 9 log cot 10 f r log sin 9 log cos 9 log tan 9 log oot 10 t 0 83 378 86 413 96 966 03 034 60 0 84 177 85 693 98 484 01516 60 1 392 401 96 991 03 009 59 1 190 681 509 491 59 2 405 389 97 016 02 984 58 2 203 669 534 466 58 3 419 377 042 958 57 3 216 657 560 440 57 4 432 366 067 933 56 4 229 645 585 415 56 6 83 446 86 354 97 092 02 908 55 5 84 242 85 632 98 610 01 390 55 6 459 342 118 882 54 6 255 620 635 365 54 7 473 330 143 857 53 7 269 608 661 339 53 8 486 318 168 832 52 8 282 596 686 314 52 9 500 306 193 807 51 9 295 583 711 289 51 10 83 513 86 295 97 219 02 781 50 10 84 308 85 571 98 737 01 263 50 11 527, 540 283 244 756 49 11 321 559 762 238 49 12 271 269 731 48 12 334 547 787 213 48 13 554 259 295 705 47 13 347 534 812 188 47 14 567 247 320 680 46 14 360 522 838 162 46 15 83 581 86 235 97 345 02 655 45 15 84 373 85 510 98 863 01 137 45 16 594 223 371 629 44 16 385 497 888 112 44 17 608 211 396 604 43 17 398 485 913 087 43 18 621 200 421 579 42 18 411 473 939 061 42 19 634 188 447 553 41 19 424 460 964 036 41 20 83 648 86176 97 472 02 528 40 20 84 437 85 448 98 989 01 on 40 21 661, / 164 497 503 39 21 450 436 99 015 00 985 39 22 674 152 523 477 38 22 463 423 040 960 38 23 688 140 548 452 37 23 476 411 065 935 37 24 701 128 573 427 36 24 489 399 090 910 36 25 83 715 86116 97 598 02 402 35 25 84 502 85 386 99116 00 884 35 26 728 104 624 376 34 26 515 374 141 859 34 27 741 092 649 351 33 27 528 361 166 834 33 28 755 080 674 326 32 28 540 349 191 809 32 29 768 068 700 300 31 29 553 337 217 783 31 30 83 781 86 056 97 725 02 275 30 30 84 566 85 324 99 242 00 758 30 31 795 044 750 250 29 31 579 312 267 733 29 32 808 032 776 224 28 32 592 299 293 707 28 33 821 020 801 199 27 33 605 287 318 682 27 34 834 86 008 826 174 26 34 618 274 343 657 26 35 83 848 85 996 97 851 02149 25 35 84 630 85 262 99 368 00 632 25 36 861 984 877 123 24 36 643 250 394 606 24 37 874 972 902 098 23 37 656 237 419 581 23 38 887 960 927 073 22 38 669 225 444 556 22 39 901 948 953 047 21 39 682 212 469 531 21 40 83 914 85 936 97978 02 022 20 40 84 694 85 200 99 495 00 505 20 41 927 924 98 003 01997 19 41 707 187 520 480 19 42 940 912 029 971 18 42 720 175 545 455 18 43 954 900 054 946 17 43 733 162 570 430 17 44 967 888 079 921 16 44 745 150 596 404 16 45 83 980 85 876 98 104 01 896 15 45 84 758 85 137 99 621 00 379 15 46 83 993 864 130 870 14 46 771 125 646 354 14 47 84 006 851 155 845 13 47 784 112 672 328 13 48 020 839 180 820 12 48 796 100 697 303 12 49 033 827 206 794 11 49 809 087 722 278 11 50 84 046 85 815 98 231 01 769 10 50 84 822 85 074 99 747 00 253 10 51 059 803 256 744 9 51 835 062 773 227 9 52 072 791 281 719 8 52 847 049 798 202 8 53 085 779 307 693 7 53 860 037 823 177 7 ,54 098 766 332 668 6 54 873 024 848 152 6 55 84112 85 754 98 357 01643 5 55 84 885 85 012 99 874 00126 5 56 125 742 383 617 4 56 898 84 999 899 101 4 57 138 730 408 592 3 57 911 986 924 076 3 58 151 718 433 567 2 58 923 974 949 051 2 59 164 706 458 542 1 59 936 961 975 025 1 60 84177 9 85 693 9 98 484 9 01516 10 0 60 84 949 9 84 949 9 00 000 10 00 000 10 0 f log cos ' log sin log cot log tan f log cos log sin log cot log tan f 46 45 78 TABLE YII FOR DETERMINING THE FOLLOWING WITH GREATER ACCURACY THAN CAN BE DONE BY MEANS OF TABLE VI 1. log sin, log tan, and log cot, when the angle is between 0° and 2° ; 2. log cos, log tan, and log cot, when the angle is between 88° and 90° ; 3. The value of the angle when the logarithm of the function does not lie between the limits 8.54 684 and 11. 45 316. FOKMULAS FOE THE USE OF THE NUMBEES S AND T I. When the angle a is between 0° and 2° : log sin a = log a" + S. log tan a = log a" + T. log cot a = colog tan a. log ct" = log sin a — S = log tan a — T = colog cot a — T. II. When the angle a is between 88° and 90° : log cos a = log (90° — a)" + S. log cot or = log (90° — a)" + T. log tan a — colog cot a. log (90° — a)" = log cos a — S = log cot a — T = colog tan a — T; O' 90° - (90° - a). Values of S and T 0 2 409 3 417 3 823 4190 4 840 5 414 5 932 6 408 6 633 6 851 7 267 4. 68 557 4. 68 556 4. 68 555 4. 68 551 4. 68 554 4. 68 553 4. 68 552 4. 68 551 4. 68 550 4. 68 550 4. 68 549 S log sin a 8. 06 740 8. 21 920 8. 26 795 8. 30 776 8. 37 038 8. 41 904 8. 45 872 8. 49 223 8. 50 721 8. 52 125 8. 54 684 log sin a 0 200 1 726 2 432 2 976 3 434 3 838 4 204 4 540 4 699 4 853 5 146 4. 68 557 4. 68 558 4. 68 559 4. 68 560 4. 68 561 4. 68 562 4. 68 563 4. 68 564 4. 68 561 4. 68 565 4. 68 566 log tan a 6. 98 660 7. 92 263 8. 07 156 8. 15 924 8. 22 142 8. 26 973 8. 30 930 8. 34 270 8. 35 766 8. 37 167 8. 39 713 log tan a 5 146 5 424 5 689 5 941 6184 6 417 6 642 6 859 7 070 7 173 7 274 4. 68 567 4. 68 568 4. 68 569 4.68 570 4.68 571 4. 68 572 4. 68 573 4. 68 574 4. 68 571 4. 68 575 log tana 8. 39 713 8. 41999 8. 44 072 8. 45 955 8. 47 697 8. 49 305 S. 50 802 8. 52 200 8. 53 516 8. 54 145 8. 54 753 log tan a 0 79 TABLE YIII NATURAL FUNCTIONS Owing to the rapid change in the functions, interpolation is not accurate for the cotangents from 0° to 3°, nor for the tangents from 87° to 90°. For the same functions interpolation is not accurate, in general, in the last figure from 3° to 6° and from 84° to 87°, respectively. 0 ° 0 ° f sin cos tan cot f t sin cos tan cot r 0 0.0000 1.0000 0.0000 Infinite 60 30 0.0087 1.0000 0.0087 114.589 30 1 03 00 03 3437.75 59 31 90 00 90 110.892 29 2 06 00 06 1718.87 58 32 93 00 93 107.426 28 3 09 00 09 1145.92 57 33 96 00 96 104.171 27 4 12 00 12 859.436 56 34 99 1.0000 99 101.107 26 5 0.0015 1.0000 0.0015 687.549 55 35 0.0102 0.9999 0.0102 98.2179 25 6 17 00 17 572.957 54 36 05 99 05 95.4895 24 7 20 00 20 491.106 S 3 37 08 99 08 92.9085 23 8 23 00 23 429.718 52 38 11 99 11 90.4633 22 9 26 00 26 381.971 51 39 13 99 13 88.1436 21 10 0.0029 1.0000 0.0029 343.774 50 40 0.0116 0.9999 0.0116 85.9398 20 11 32 00 32 312.521 49 41 19 99 19 83.8435 19 12 35 00 35 286.478 48 42 22 99 22 81.8470 18 13 38 00 38 264.441 47 43 25 99 25 79.9434 17 14 41 00 41 245.552 46 44 28 99 28 78.1263 16 15 0.0044 1.0000 0.0044 229.182 45 45 0.0131 0.9999 0.0131 76.3900 15 16 47 00 47 214.858 44 46 34 99 34 74.7292 14 17 49 00 49 202.219 43 47 37 99 37 73.1390 13 18 52 00 52 190.984 42 48 40 99 40 71.6151 12 19 55 00 55 180.932 41 49 43 99 43 70.1533 11 20 0.0058 1.0000 0.0058 171.885 40 50 0.0145 0.9999 0.0145 68.7501 10 21 61 00 61 163.700 39 51 48 99 48 67.4019 9 22 64 00 64 156.259 38 52 51 99 51 66.1055 8 23 67 00 67 149.465 37 S 3 54 99 54 64.8580 7 24 70 00 70 143.237 36 54 57 99 57 63.6567 6 25 0.0073 1.0000 0.0073 137.507 35 o5 0.0160 0.9999 0.0160 62.4992 5 26 76 00 76 132.219 34 56 63 99 63 61.3829 4 27 79 00 79 127.321 33 57 66 99 66 60.3058 3 28 81 00 81 122.774 32 58 69 99 69 59.2659 2 29 84 00 84 118.540 31 59 72 99 72 58.2612 1 30 0.0087 1.0000 0.0087 114.589 30 60 0.0175 0.9998 0.0175 57.2900 O / COS sin cot tan r t COS sin cot tan f 89 89 ' 80 1 ° / sin cos tan cot / o 0.0175 0.9998 0.0175 57.2900 60 1 77 98 77 56.3506 59 2 80 98 80 55.4415 58 3 83 98 83 54.5613 57 4 86 98 86 53.7086 56 5 0.0189 0.9998 0.0189 52.8821 55 6 92 98 92 52.0807 54 7 95 98 95 51.3032 53 8 0198 98 0198 50.5485 52 9 0201 98 0201 49.8157 51 10 0.0204 0.9998 0.0204 49.1039 50 11 07 98 07 48.4121 49 12 09 98 09 47.7395 48 13 12 98 12 47.0853 47 14 15 98 15 46.4489 46 15 0.0218 0.9998 0.0218 45.8294 45 16 21 98 21 45.2261 44 17 24 97 24 44.6386 43 18 27 97 27 44.0661 42 19 30 97 30 43.5081 41 20 0.0233 0.9997 0.0233 42.9641 40 21 36 97 36 42.4335 39 22 39 97 39 41.9158 38 23 41 97 41 41.4106 37 24 44 97 44 40.9174 36 25 0.0247 0.9997 0.0247 40.4358 35 26 50 97 50 39.9655 34 27 53 97 53 39.5059 33 28 56 97 56 39.0568 32 29 59 97 59 38.6177 31 30 0.0262 0.9997 0.0262 38.1885 30 31 65 96 65 37.7686 29 32 68 96 68 37.3579 28 33 70 96 71 36.9560 27 34 73 96 74 36.5627 26 35 0.0276 0.9996 0.0276 36.1776 25 36 79 96 79 35.8006 24 37 82 96 82 35.4313 23 38 85 96 85 35.0695 22 39 88 96 88 34.7151 21 40 0.0291 0.9996 0.0291 34.3678 20 41 94 96 94 34.0273 19 42 0297 96 0297 33.6935 18 43 0300 96 0300 33.3662 17 44 02 95 03 33.0452 16 45 0.0305 0.9995 0.0306 32.7303 15 46 08 95 08 32.4213 14 47 11 95 11 32.1181 13 48 14 95 14 31.8205 12 49 17 95 17 31.5284 11 50 0.0320 0.9995 0.0320 31.2416 10 51 23 95 23 30.9599 9 52 26 95 26 30.6833 8 53 29 95 29 30.4116 7 54 32 95 32 30.1446 6 55 0.0334 0.9994 0.0335 29.8823 5 56 37 94 38 29.6245 4 57 40 94 40 29.3711 3 58 43 94 43 29.1220 2 59 46 94 46 28.8771 1 60 0.0349 0.9994 0.0349 28.6363 0 f COS sin cot tan / 2 “ t sin cos tan cot / o 0.0349 0.9994 0.0349 28.6363 60 1 52 94 52 28.3994 59 2 55 94 55 28.1664 58 3 58 94 58 27.9372 57 4 61 93 61 27.7117 56 5 0.0364 0.9993 0.0364 27.4899 55 6 66 93 67 27.2715 54 7 69 93 70 27.0566 53 8 72 93 73 26.8450 52 9 75 93 75 26.6367 51 lO 0.0378 0.9993 0.0378 26.4316 oO 11 81 93 81 26.2296 49 12 84 93 84 26.0307 48 13 87 93 87 25.8348 47 14 90 92 90 25.6418 46 15 0.0393 0.9992 0.0393 25.4517 45 16 96 92 96 25.2644 44 17 0398 92 0399 25.0798 43 18 0401 92 0402 24.8978 42 19 04 92 05 24.7185 41 20 0.0407 0.9992 0.0407 24.5418 40 21 10 92 10 24.3675 39 22 13 91 13 24.1957 38 23 16 91 16 24.0263 37 24 19 91 19 23.8593 36 25 0.0422 0.9991 0.0422 23.6945 35 26 25 91 25 23.5321 34 27 27 91 28 23.3718 33 28 30 91 31 23.2137 32 29 33 91 34 23.0577 31 30 0.0436 0.9990 0.0437 22.9038 30 31 39 90. 40 22.7519 29 32 42 90 42 22.6020 28 33 45 90 45 22.4541 27 34 48 90 48 22.3081 26 35 0.0451 0.9990 0.0451 22.1640 25 36 54 90 54 22.0217 24 37 57 90 57 21.8813 23 38 59 89 60 21.7426 22 39 62 89 63 21.6056 21 40 0.0465 0.9989 0.0466 21.4704 20 41 68 89 69 21.3369 19 42 71 89 72 21 .2049 IS 43 74 89 75 21.0747 17 44 77 89 77 20.9460 16 45 0.0480 0.9988 0.0480 20.8188 15 46 83 88 83 20.6932 14 47 86 88 86 20.5691 13 48 88 88 89 20.4465 12 49 91 88 92 20.3253 11 50 0.0494 0.9988 0.0495 20.2056 10 51 0497 88 0498 20.0872 9 52 0500 87 0501 19.9702 8 53 03 87 04 19.8546 7 54 06 87 07 19.7403 6 55 0.0509 0.9987 0.0509 19.6273 5 56 12 87 12 19.5156 4 57 15 87 15 19.4051 3 58 18 87 18 19.2959 2 59 20 86 21 19.1879 1 60 0.0523 0.9986 0.0524 19.0811 0 f COS sin cot tan 88 " 87 ' 3 4 81 f sin cos tan cot f 0 0.0523 0.9986 0.0524 19.0811 60 1 26 86 27 18.9755 59 2 29 86 30 18.8711 58 3 32 86 33 18.7678 57 4 35 86 36 18.6656 56 5 0.0538 0.9986 0.0539 18.5645 55 6 41 85 42 18.4645 54 7 44 85 44 18.3655 53 8 47 85 47 18.2677 52 9 50 85 50 18.1708 51 10 0.0552 0.9985 0.0553 18.0750 50 11 55 85 56 17.9802 49 12 58 84 59 17.8863 48 13 61 84 62 17.7934 47 14 64 84 65 17.7015 46 15 0.0567 0.9984 0.0568 17.6106 45 16 70 84 71 17.5205 44 17 73 84 74 17.4314 43 18 76 83 77 17.3432 42 19 79 83 80 17.2558 41 20 0.0581 0.9983 0.0582 17.1693 40 21 84 83 85 17.0837 39 22 87 83 88 16.9990 38 23 90 83 91 16.9150 37 24 93 82 94 16.8319 36 25 0.0596 0.9982 0.0597 16.7496 35 26 0599 82 0600 16.6681 34 27 0602 82 03 16.5874 33 28 05 82 06 16.5075 32 29 08 82 09 16.4283 31 30 0.0610 0.9981 0.0612.. 16.3499 30 31 13 81 15 16.2722 29 32 16 81 17 16.1952 28 33 19 81 20 16.1190 27 34 22 81 23 16.0435 26 35 0.0625 0.9980 0.0626 15.9687 25 36 28 80 29 15.8945 24 37 31 80 32 15.8211 23 38 34 80 35 15.7483 22 39 37 80 38 15.6762 21 40 0.0640 0.9980 0.0641 15.6048 20 41 42 79 44 15.5340 19 42 45 79 47 15.4638 18 43 48 79 50 15.3943 17 44 51 79 53 15.3254 16 45 0.0654 0.9979 0.0655 15.2571 15 46 57 78 58 15.1893 14 47 60 78 61 15.1222 13 48 63 78 64 15.0557 12 49 66 78 67 14.9898 11 50 0.0669 0.9978 0.0670 14.9244 10 51 71 77 73 14.8596 9 52 74 77 76 14.7954 8 53 77 77 79 14.7317 7 54 80 77 82 14.6685 6 55 0.0683 0.9977 0.0685 14.6059 5 56 86 76 88 14.5438 4 57 89 76 90 14.4823 3 58 92 76 93 14.4212 2 59 95 76 ' 96 14.3607 1 60 0.0698 0.9976 0.0699 14.3007 O / COS sin cot tan t t sin cos tan cot f 0 0.0698 0.9976 0.0699 14.3007 60 1 0700 75 0702 2411 59 2 03 75 05 1821 58 3 06 75 08 1235 57 4 09 75 11 0655 56 5 0.0712 0.9975 0.0714 14.0079 55 6 15 74 17 13.9507 54 7 18 74 20 8940 53 8 21 74 23 8378 52 9 24 74 26 7821 51 10 0.0727 0.9974 0.0729 13.7267 50 11 29 73 31 6719 49 12 32 73 34 6174 48 13 35 73 37 5634 47 14 38 73 40 5098 46 15 0.0741 0.9973 0 0743 13.4566 45 16 44 72 46 4039 44 17 47 72 49 3515 43 18 50 72 52 2996 42 19 53 72 55 2480 41 20 0.0756 0.9971 0.0758 13.1969 40 21 58 71 61 1461 39 22 61 71 64 0958 38 23 , 64 71 67 13.0458 37 24 67 71 69 12.9962 36 25 0.0770 0.9970 0.0772 12.9469 35 26 73 70 75 8981 34 27 76 70 78 8496 33 28 79 70 81 8014 32 29 82 69 84 7536 31 30 0.0785 0.9969 0.0787 12.7062 30 31 87 69 90 6591 29 32 90 69 93 6124 28 33 93 68 96 5660 27 34 96 68 0799 5199 26 35 0.0799 0.9968 0.0802 12.4742 25 36 0802 68 05 4288 24 37 05 68 08 3838 23 38 08 67 10 3390 22 39 11 67 13 2946 21 40 0.0814 0.9967 0.0816 12.2505 20 41 16 67 19 2067 19 42 19 66 22 1632 18 43 22 66 25 1201 17 44 25 66 28 0772 16 45 0.0828 0.9966 0.0831 12.0346 15 46 31 65 34 11.9923 14 47 34 65 37 9504 13 48 37 65 40 9087 12 49 40 65 43 . 8673 11 50 0.0843 0.9964 0.0846 11.8262 10 51 45 64 49 7853 9 52 48 64 51 7448 8 53 51 64 54 7045 7 54 54 63 57 6645 6 55 0.0857 0.9963 0.0860 11.6248 5 56 60 63 63 5853 4 57 63 63 66 5461 3 58 66 62 69 5072 2 59 69 62 72 4685 1 60 0.0872 0.9962 0.0875 11.4301 0 / COS sin cot tan f 86 85 82 5 ° / sin cos tan cot t o 0.0872 0.9962 0.0875 11.4301 60 1 74 62 78 3919 59 2 77 61 81 3540 58 3 80 61 84 3163 57 4 83 61 87 2789 56 5 0.0886 0.9961 0.0890 11.2417 55 6 89 60 92 2048 54 7 92 60 95 1681 53 8 95 60 0898 1316 52 9 0898 60 0901 0954 51 lO 0.0901 0.9959 0.0904 11.0594 oO 11 03 59 07 11.0237 49 12 06 59 10 10.9882 48 13 09 59 13 9529 47 14 12 58 16 9178 46 15 0.0915 0.9958 0.0919 10.8829 45 16 18 58 22 8483 44 17 21 58 25 8139 43 18 24 57 28 7797 42 19 27 57 31 7457 41 20 0.0929 0.9957 0.0934 10.7119 40 21 32 56 36 6783 39 22 35 56 39 6450 38 23 38 56 42 6118 •37 24 41 56 45 5789 36 25 0.0944 0.9955 0.0948 10.5462 35 26 47 55 51 5136 34 27 50 55 54 4813 33 28 53 55 57 4491 32 29 56 54 60 4172 31 30 0.0958 0.9954 0.0963 10.3854 30 31 61 54 66 3538 29 32 64 53 69 3224 28 33 67 53 72 2913 27 34 70 53 75 2602 26 35 0.0973 0.9953 0.0978 10.2294 25 36 76 52 81 1988 24 37 79 52 83 1683 23 38 82 52 86 1381 22 39 85 51 89 1080 21 40 0.0987 0.9951 0.0992 10.0780 20 41 90 51 95 0483 19 42 93 51 0998 10.0187 18 43 96 50 1001 9.9893 17 44 0999 50 04 9601 16 45 0.1002 0.9950 0.1007 9.9310 15 46 05 49 10 9021 14 47 08 49 13 8734 13 48 11 49 16 8448 12 49 13 49 19 8164 11 50 0.1016 0.9948 0.1022 9.7882 lO 51 19 48 25 7601 9 52 22 48 28 7322 8 53 25 47 30 7044 7 54 28 47 33 6768 6 55 0.1031 0.9947 0.1036 9.6493 5 56 34 46 39 6220 4 57 37 46 42 5949 3 58 39 46 45 5679 2 59 42 46 48 5411 1 60 0.1045 0.9945 0.1051 9.5144 0 9 COS sin cot tan / 6 ° / sin cos tan cot f o 0.1045 0.9945 0.1051 9.5144 60 1 48 45 54 4878 59 2 51 45 57 4614 58 3 54 44 60 4352 57 4 57 44 63 4090 56 5 0.1060 0.9944 0.1066 9.3831 OO 6 63 43 69 3572 54 7 66 43 72 3315 53 8 68 43 75 3060 52 9 71 42 78 2806 51 lO 0.1074 0.9942 0.1080 9.2553 50 11 77 42 83 2302 49 12 80 42 86 2052 48 13 83 41 89 1803 47 14 86 41 92 1555 46 15 0.1089 0.9941 0.1095 9.1309 4:0 16 92 40 1098 1065 44 17 94 40 1101 0821 43 18 1097 40 04 0579 42 19 1100 39 07 0338 41 20 0.1103 0.9939 0.1110 9.0098 40 21 06 39 13 8.9860 39 22 09 38 16 9623 38 23 12 38 19 9387 37 24 15 38 22 9152 36 25 0.1118 0.9937 0.1125 8.8919 35 26 20 37 28 8686 34 27 23 37 31 8455 33 28 26 36 33 8225 32 29 29 36 36 7996 31 30 0.1132 0.9936 0.1139 8.7769 30 31 35 35 42 7542 29 32 38 35 45 7317 28 33 41 35 48 7093 27 34 44 34 51 6870 26 35 0.1146 0.9934 0.1154 8.6618 25 36 49 34 57 6427 24 37 52 33 60 6208 23 38 55 33 63 5989 22 39 58 33 66 5772 21 40 0.1161 0.9932 0.1169 8.5555 20 41 64 32 72 5340 19 42 67 32 75 5126 18 43 70 31 78 4913 17 44 72 31 81 4701 16 45 0.1175 0.9931 0.1184 8.4490 lo 46 78 30 87 4280 14 47 81 30 89 4071 13 48 84 30 92 3863 12 49 87 29 95 3656 11 50 0.1190 0.9929 0.1198 8.3450 10 51 93 29 1201 3245 9 52 96 28 04 3041 8 53 1198 28 07 2838 7 54 1201 28 10 2636 6 55 0.1204 0.9927 0.1213 8.2434 5 56 07 27 16 2234 4 57 10 27 19 2035 3 58 13 26 22 1837 2 59 16 26 25 1640 1 60 0.1219 0.9925 0.1228 8.1443 0 f COS sin cot tan f 84 ® 83 ® 7 ' 8 83 sin cos tan cot f o 0.1219 0.9925 0.1228 8.1443 60 1 22 25 31 1248 59 2 24 25 34 1054 58 3 27 24 37 0860 57 4 30 24 40 0667 56 5 0.1233 0.9924 0.1243 8.0476 55 6 36 23 46 0285 54 7 39 23 49 8.0095 53 8 42 23 51 7.9906 52 9 45 22 54 9718 51 \o 0.1248 0.9922 0.1257 7.9530 50 11 50 22 60 9344 49 12 53 21 63 9158 48 13 56 21 66 8973 47 14 59 20 69 8789 46 15 0.1262 0.9920 0.1272 7.8606 45 16 65 20 75 8424 44 17 68 19 78 8243 43 18 71 19 81 8062 42 19 74 19 84 7883 41 20 0.1276 0.9918 0.1287 7.7704 40 21 79 18 90 7525 39 22 82 17 93 7348 38 23 85 17 96 7171 37 24 88 17 99 6996 36 25 0.1291 0.9916 0.1302 7.6821 35 26 94 16 05 6647 34 27 97 16 08 6473 33 28 1299 15 11 6301 32 29 1302 15 14 6129 31 30 0.1305 0.9914 0.1317 7.5958 30 31 08 14 19 5787 29 32 11 14 22 5618 28 33 14 13 25 5449 27 34 17 13 28 5281 26 35 0.1320 0.9913 0.1331 7.5113 25 36 23 12 34 4947 24 37 25 12 37 4781 23 38 28 11 40 4615 22 39 31 11 43 4451 21 40 0.1334 0.9911 0.1346 7.4287 20 41 37 10 49 4124 19 42 40 10 52 3962 18 43 43 09 55 3800 17 44 46 09 58 3639 16 45 0.1349 0.9909 0.1361 7.3479 15 46 51 08 64 3319 14 47 54 08 67 3160 13 48 57 07 70 3002 12 49 60 07 73 2844 11 50 0.1363 0.9907 0.1376 7.2687 10 51 66 06 79 2531 9 52 69 06 82 2375 8 53 72 05 85 2220 7 54 74 05 88 2066 6 55 0.1377 0.9905 0.1391 7.1912 5 56 80 04 94 1759 4 57 83 04 97 1607 3 58 86 03 1399 1455 2 59 89 03 1402 1304 1 60 0.1392 0.9903 0.1405 7.1154 O f COS sin cot tan f sin COS tan cot f o 0.1392 0.9903 0.1405 7.1154 60 1 95 02 08 1004 59 2 1397 02 11 0855 58 3 1400 01 14 0706 57 4 03 01 17 0558 56 5 0.1406 0.9901 0.1420 7.0410 55 6 09 00 23 0264 54 7 12 9900 26 7.0117 53 8 15 9899 29 6.9972 52 9 18 99 32 9827 51 10 0.1421 0.9899 0.1435 6.9682 50 11 23 98 38 9538 49 12 26 98 41 9395 48 13 29 97 44 9252 47 14 32 97 47 9110 46 15 0.1435 0.9897 0.1450 6.8969 45 16 38 96 53 8828 44 17 41 96 56 8687 43 18 44 95 59 8548 42 19 46 95 62 8408 41 20 0.1449 0.9894 0.1465 6.8269 40 21 52 94 68 8131 39 22 55 94 71 7994 38 23 58 93 74 7856 37 24 61 93 77 7720 36 25 0.1464 0.9892 0.1480 6.7584 35 26 67 92 83 7448 34 27 69 91 86 7313 33 28 72 91 89 7179 32 29 75 91 92 7045 31 30 0.1478 0.9890 0.1495 6.6912 30 31 81 90 1497 6779 29 32. 84 89 1500 6646 28 33 87 89 03 6514 27 34 90 88 06 6383 26 35 0.1492 0.9888 0.1509 6.6252 25 36 95 88 12 6122 24 37 1498 87 15 5992 23 38 1501 87 18 5863 22 39 04 86 21 5734 21 40 0.1507 0.9886 0.1524 6.5606 20 41 10 85 27 5478 19 42 13 85 30 5350 18 43 15 84 33 5223 17 44 18 84 36 5097 16 45 0.1521 0.9884 0.1539 6.4971 15 46 24 83 42 4846 14 47 27 83 45 4721 13 48 30 82 48 4596 12 49 33 82 51 4472 11 50 0.1536 0.9881 0.1554 6.4348 lO 51 38 81 57 4225 9 52 41 80 60 4103 8 53 44 80 63 3980 7 54 47 80 66 3859 6 55 0.15.50 0.9879 0.1569 6.3737 5 56 53 79 72 3617 4 57 56 78 75 3496 3 58 59 78 78 3376 2 59 61 77 81 3257 1 60 0.1564 0.9877 0.1584 6.3138 0 t COS sin cot tan f 82 ' 81 84 9 “ f sin cos tan cot f 0 0.1564 0.9877 0.1584 6.3138 60 1 67 76 87 6.3019 59 2 70 76 90 6.2901 58 3 73 76 93 783 57 4 76 75 96 666 56 5 0.1579 0.9875 0.1599 6.2549 65 6 82 74 1602 432 54 7 84 74 05 316 53 8 87 73 08 200 52 9 90 73 11 6.2085 51 lO 0.1593 0.9S72 0.1614 6.1970 50 11 96 72 17 856 49 12 1599 71 20 742 48 13 1602 71 23 628 47 14 05 70 26 515 46 15 0.1607 0.9870 0.1629 6.1402 45 16 10 69 32 290 44 17 13 69 35 178 43 18 16 69 38 6.1066 42 19 19 68 41 6.0955 41 20 0.1622 0.9868 0.1644 6.0844 40 21 25 67 47 734 39 22 28 67 50 624 38 23 30 66 53 514 37 24 33 66 55 405 36 25 0.1636 0.9865 0.1658 6.0296 35 26 39 65 61 188 34 27 42 64 64 6.0080 33 28 45 64 67 5.9972 32 29 48 63 70 865 31 30 0.1650 0.9863 0.1673 5.9758 30 31 53 62 76 651 29 32 56 62 79 545 28 33 59 61 82 439 27 34 62 61 85 333 26 35 0.1665 0.9860 0.1688 5.9228 25 36 68 60 91 124 24 37 71 59 94 5.9019 23 38 73 59 1697 5.8915 22 39 76 59 1700 811 21 40 0.1679 0.9858 0.1703 5.8708 20 41 82 58 06 605 19 42 85 57 09 502 18 43 88 57 12 400 17 44 91 56 15 298 16 45 0.1693 0.9856 0.1718 5.8197 15 46 96 55 21 5.8095 14 47 1699 55 24 5.7994 13 48 1702 54 27 894 12 49 05 54 30 794 11 50 0.1708 0.9853 0.1733 5.7694 lO 51 11 53 36 594 9 52 14 52 39 495 8 53 16 52 42 396 7 54 19 51 45 297 6 55 0.1722 0.9851 0.1748 5.7199 5 56 25 50 51 101 4 57 28 50 54 5.7004 3 58 31 49 57 5.6906 2 59 34 49 60 809 1 60 0.1736 0.9848 0.1763 5.6713 0 / COS sin cot tan / 10 “ r sin cos tan cot / 0 0.1736 0.9848 0.1763 5.6713 CO 1 39 48 66 617 59 2 42 47 69 521 58 3 45 47 72 425 57 4 48 46 75 330 56 5 0.1751 0.9846 0.1778 5.6234 o5 6 54 45 81 140 54 7 57 45 84 5.6045 53 8 59 44 87 5.5951 52 9 62 43 90 857 51 10 0.1765 0.9843 0.1793 5.5764 50 11 68 42 96 671 49 12 71 42 1799 578 48 13 74 41 1802 485 47 14 77 41 05 393 46 15 0.1779 0.9840 0.1808 5.5301 45 16 82 40 11 209 44 17 85 39 14 118 43 18 88 39 17 5.5026 42 19 91 38 20 5.4936 41 20 0.1794 0.9838 0.1823 5.4845 40 21 97 37 26 755 39 22 1799 37 29 665 38 23 1802 36 32 575 37 24 05 36 35 486 36 25 0.1808 0.9835 0.1838 5.4397 35 26 11 35 41 308 34 27 14 34 44 219 33 28 17 34 47 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COS sin cot tan / 80 ‘ 79 ir f sin cos tan cot / 0 0.1908 0.9816 0.1944 5.1446 60 1 11 16 47 366 59 2 14 15 50 286 58 3 17 IS S3 207 57 4 20 14 56 128 56 5 0.1922 0.9S13 0.1959 5.1049 55 6 25 13 62 5.0970 54 7 28 12 65 892 53 8 31 12 68 814 52 9 34 11 71 736 51 10 0.1937 0.9811 0.1974 5.0658 50 11 39 10 77 581 49 12 42 10 80 504 48 13 45 09 83 427 47 14 48 08 86 350 46 15 0.1951 0.9808 0.1989 5.0273 45 16 54 07 92 197 44 17 57 07 95 121 43 18 59 06 1998 5.0045 42 19 62 06 2001 4.9969 41 20 0.1965 0.9805 0.2004 4.9894 40 21 68 04 07 819 39 22 71 04 10 744 38 23 74 03 13 669 37 24 77 03 16 594 36 25 0.1979 0.9802 0.2019 4.9520 35 26 82 02 22 446 34 27 85 01 25 372 33 28 88 00 28 298 32 29 91 9800 31 225 31 30 0.1994 0.9799 0.2035 4.9152 30 31 97 99 38 078 29 32 1999 98 41 4.9006 28 33 2002 98 44 4.8933 27 34 05 97 47 860 26 35 0.2008 0.9796 0.2050 4.8788 25 36 11 96 53 716 24 37 14 95 56 644 23 38 16 95 59 573 22 39 19 94 62 SOI 21 40 0.2022 0.9793 0.2065 4.8430 20 41 25 93 68 359 19 42 28 92 71 288 18 43 31 92 74 218 17 44 34 91 77 147 16 45 0.2036 0.9790 0.2080 4.8077 15 46 39 90 83 4.8007 14 47 42 89 86 4.7937 13 48 45 89 89 867 12 49 48 88 92 798 11 50 0.2051 0.9787 0.2095 4.7729 10 51 54 87 2098 659 9 52 56 86 2101 591 8 53 59 86 04 522 7 54 62 85 07 453 6 55 0.2065 0.9784 0.2110 4.7385 5 56 68 84 13 317 4 57 71 83 16 249 3 58 73 83 19 181 2 59 76 82 23 114 1 60 0.2079 0.9781 0.2126 4.7046 O t COS sin cot tan / 12 ° 85 f sin COS tan cot / o 0.2079 0.9781 0.2126 4.7046 60 1 82 81 29 4.6979 59 2 85 80 32 912 58 3 88 80 35 845 57 4 90 79 38 779 56 5 0.2093 0.9778 0.2141 4.6712 55 6 96 78 44 646 54 7 2099 77 47 580 S3 8 2102 77 SO 514 52 9 05 76 53 448 51 10 0.2103 0.9775 0.2156 4.6382 50 11 10 75 59 317 49 12 13 74 62 252 48 13 16 74 65 187 47 14 19 73 68 122 46 15 0.2122 0.9772 0.2171 4.6057 45 16 25 72 74 4.5993 44 17 27 71 77 928 43 18 30 70 80 864 42 19 33 70 83 800 41 20 0.2136 0.9769 0.2186 4.5736 40 21 39 69 89 673 39 22 42 68 93 609 38 23 45 67 96 546 37 24 47 67 2199 483 36 25 0.2150 0.9766 0.2202 4.5420 35 26 53 65 05 357 34 27 56 65 08 294 33 28 59 64 11 232 32 29 62 64 14 169 31 30 0.2164 0.9763 0.2217 4.5107 30 31 67 62 20 4.5045 29 32 70 62 23 4.4983 28 33 73 61 26 922 27 34 76 60 29 860 26 35 0.2179 0.9760 0.2232 4.4799 25 36 81 59 35 737 24 37 84 59 33 676 23 38 87 58 41 615 22 39 90 57 44 555 21 40 0.2193 0.9757 0.2247 4.4494 20 41 96 56 51 434 19 42 2193 55 54 374 18 43 2201 55 57 313 17 44 04 54 60 253 16 45 0.2207 0.9753 0.2263 4.4194 15 46 10 53 66 134 14 47 13 52 69 075 13 48 15 51 72 4.4015 12 49 18 51 75 4.3956 11 50 0.2221 0.9750 0.2278 4.3897 10 51 24 50 81 838 9 52 27 49 84 779 8 53 30 48 87 721 7 54 33 48 90 662 6 55 0.2235 0.9747 0.2293 4.3604 5 56 38 46 96 546 4 57 41 46 2299 488 3 58 44 45 2303 430 2 59 47 44 06 372 1 60 0.2250 0.9744 0.2309 4.3315 O t COS sin cot tan f 78 77 ° 86 13 ° / sin cos tan cot f 0 0.2250 0.9744 0.2309 4.3315 60 1 52 43 12 257 59 2 55 42 15 200 58 3 58 42 18 143 57 4 61 41 21 086 56 6 0.2264 0.9740 0.2324 4.3029 65 6 67 40 27 4.2972 54 7 69 39 30 916 53 8 72 38 33 859 52 9 75 38 36 803 51 10 0.2278 0.9737 0.2339 4.2747 50 11 81 36 42 691 49 12 84 36 45 635 48 13 86 35 49 580 47 14 89 34 52 524 46 15 0.2292 0.9734 0.2355 4.2468 46 16 95 33 58 413 44 17 2298 32 61 358 43 18 2300 32 64 303 42 19 03 31 67 248 41 20 0.2306 0.9730 0.2370 4.2193 40 21 09 30 73 139 39 22 12 29 76 084 38 23 15 28 79 4.2030 37 24 17 28 82 4.1976 36 25 0.2320 0.9727 0.2385 4.1922 35 26 23 26 88 868 34 27 26 26 92 814 33 28 29 25 95 760 32 29 32 24 2398 706 31 30 0.2334 '0.9724 0.2401 4.1653 30 31 37 23 04 600 29 32 40 22 07 547 28 33 43 22 10 493 27 34 46 21 13 441 26 35 0.2349 0.9720 0.2416 4.1388 25 36 51 20 19 335 24 37 54 19 22 282 23 38 57 18 25 230 22 39 60 18 28 178 21 40 0.2363 0.9717 0.2432 4.1126 20 41 66 16 35 074 19 42 68 15 38 4.1022 18 43 71 15 41 4.0970 17 44 74 14 44 918 16 45 0.2377 0.9713 0.2447 4.0867 15 46 80 13 50 815 14 47 83 12 53 764 13 48 85 11 56 713 12 49 88 11 59 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3.8667 30 31 07 81 89 621 29 32 09 80 92 575 28 33 12 79 95 528 27 34 15 79 2599 482 26 35 0.2518 0.9678 0.2602 3.8436 25 36 21 77 05 391 24 37 24 76 08 345 23 38 26 76 11 299 22 39 29 75 14 254 21 40 0.2532 0.9674 0.2617 3.8208 20 41 35 73 20 163 19 42 33 73 23 118 18 43 40 72 27 073 17 44 43 71 30 3.8028 16 45 0.2546 0.9670 0.2633 3.7983 15 46 49 70 36 938 14 47 52 69 39 893 13 48 54 68 42 848 12 49 57 67 45 804 11 50 0.2560 0.9667 0.2648 3.7760 10 51 63 66 51 715 9 52 66 65 55 671 8 53 69 65 58 627 7 54 71 64 61 583 6 55 0.2574 0.9663 0.2664 3.7539 5 56 77 62 67 495 4 57 80 62 70 451 3 58 S3 61 73 408 - 2 59 85 60 76 364 1 60 0.2588 0.9659 0.2679 3.7321 0 COS sin cot tan t 76 75 15 “ 16 87 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 0.2588 91 94 97 2599 0.2602 05 08 11 13 0.2616 19 22 25 28 0.2630 33 36 39 42 0.2644 47 50 53 56 0.2658 61 64 67 70 0.2672 75 78 81 84 0.2686 89 92 95 2698 0.2700 03 06 09 12 0.2714 17 20 23 26 0.2728 31 34 37 40 0.2742 45 48 51 54 0.2756 cos 0.9659 59 58 57 56 0.9655 55 54 53 52 0.9652 51 50 49 49 0.9648 47 46 46 45 0.9644 43 42 42 41 0.9640 39 39 38 37 0.9636 36 35 34 33 0.9632 32 31 30 29 0.9628 28 27 26 25 0.9625 24 23 22 21 0.9621 20 19 18 17 0.9617 16 15 14 13 0.9613 tan 0.2679 83 86 89 92 0.2695 2698 2701 04 08 0.2711 14 17 20 23 0.2726 29 33 36 39 0.2742 45 48 51 54 0.2758 61 64 67 70 0.2773 76 80 83 86 0.2789 92 95 2798 2801 0.2805 08 11 14 17 0.2820 23 27 30 33 0.2836 39 42 45 49 0.2852 55 58 61 64 0.2867 cot cot 3.7321 277 234 191 148 3.7105 062 3.7019 3.6976 933 3.6891 848 806 764 722 3.6680 638 596 554 512 3.6470 429 , 387 346 305 3.6264 222 181 140 100 3.6059 3.6018 3.5978 937 897 3.5856 816 776 736 696 3.5656 616 576 536 497 3.5457 418 379 339 300 3.5261 222 183 144 105 3.5067 3.5028 3.4989 951 912 3.4874 tan 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 74 “ 73 “ 88 17 “ 18 " 72 71 19 20 ° 89 70 ' 69 ° 90 21 22 f sin cos tan cot / t sin cos tan cot •• o 0.3584 0.9336 0.3839 2.6051 60 0 0.3746 0.9272 0.4040 2.4751 60 1 86 35 42 028 59 1 49 71 44 730 59 2 89 34 45 2.6006 58 2 51 70 47 709 58 3 92 33 49 2.5983 57 3 54 69 50 689 57 4 95 32 52 961 56 4 57 67 54 668 56 5 0.3597 0.9331 0.3855 2.5938 55 5 0.3760 0.9266 0.4057 2.4648 55 6 3600 30 59 916 54 6 62 65 61 627 54 7 03 28 62 893 53 7 65 64 64 606 53 8 05 27 65 871 52 8 68 63 67 586 52 9 08 26 69 848 51 9 70 62 71 566 51 lO 0.3611 0.9325 0.3872 2.5826 50 10 0.3773 0.9261 0.4074 2.4545 50 11 14 24 75 804 49 11 76 60 78 525 49 12 16 23 79 782 48 12 78 59 81 504 48 13 19 22 82 759 47 13 81 58 84 484 47 14 22 21 85, 737 46 14 84 57 88 464 46 15 0.3624 0.9320 0.3889 2.5715 45 15 0.3786 0.9255 0.4091 2.4443 45 16 27 19 92 693 44 16 89 54 95 423 44 17 30 18 95 671 43 17 92 53 4098 403 43 18 33 17 3899 649 42 18 95 52 4101 383 42 19 35 16 3902 627 41 19 3797 51 05 362 41 20 0.3638 0.9315 0.3906 2.5605 40 20 0.3800 0.9250 0.4108 2.4342 40 21 41 14 09 583 39 21 03 49 11 322 39 22 43 13 12 561 38 22 05 48 15 302 38 23 46 12 16 539 37 23 08 47 18 282 37 24 49 11 19 517 36 24 11 45 22 262 36 25 0.3651 0.9309 0.3922 2.5495 35 25 0.3813 0.9244 0.4125 2.4242 35 26 54 08 26 473 34 26 16 43 29 222 34 27 57 07 29 452 33 27 19 42 32 202 33 28 60 06 32 430 32 28 21 41 35 182 32 29 62 05 36 408 31 29 24 40 39 162 31 30 0.3665 0.9304 0.3939 2.5386 30 30 0.3827 0.9239 0.4142 2.4142 30 31 68 03 42 365 29 31 30 38 46 122 29 32 70 02 46 343 28 32 32 37 49 102 28 33 73 01 49 322 27 33 35 35 52 083 27 34 76 9300 53 300 26 34 38 34 56 063 26 35 0.3679 0.9299 0.3956 2.5279 25 35 0.3840 0.9233 0.4159 2.4043 25 36 81 98 59 257 24 36 43 32 63 023 24 37 84 97 63 236 23 37 46 31 66 2.4004 23 38 87 96 66 214 22 38 48 30 69 2.3984 22 39 89 95 69 193 21 39 51 29 73 964 21 40 0.3692 0.9293 0.3973 2.5172 20 40 0.3854 0.9228 0.4176 2.3945 20 41 95 92 76 150 19 41 56 27 80 925 19 42 3697 91 79 129 18 42 59 25 83 906 18 43 3700 90 83 108 17 43 62 24 87 886 17 44 03 89 86 086 16 44 64 23 90 867 16 45 0.3706 0.9288 0.3990 2.5065 15 45 0.3867 0.9222 0.4193 2.3S47 15 46 08 87 93 044 14 46 70 21 4197 828 14 47 11 86 3996 023 13 47 72 20 4200 808 13 48 14 85 4000 2.5002 12 48 75 19 04 789 12 49 16 84 03 2.4981 11 49 78 18 07 770 11 50 0.3719 0.9283 0.4006 2.4960 10 50 0.3881 0.9216 0.4210 2.3750 10 51 22 82 10 939 9 51 83 15 14 731 9 52 24 81 13 918 8 52 86 14 17 712 8 53 27 79 17 897 7 53 89 13 21 693 7 54 30 78 20 876 6 54 91 12 24 673 6 55 0.3733 0.9277 0.4023 2.4855 5 55 0.3894 0.9211 0.4228 2.3654 5 56 35 76 27 834 4 56 97 10 31 635 4 57 38 75 30 813 3 57 3899 OS 34 616 3 58 41 74 33 792 2 58 3902 07 38 597 2 59 43 73 37 772 1 59 05 06 41 578 1 60 0.3746 0.9272 0.4040 2.4751 0 60 0.3907 0.9205 0.4245 2.3559 0 / COS sin cot tan t / COS sin cot tan f 68 67 23 24 ‘ 91 / sin cos tan cot / o 0.3907 0.9205 0.4245 2.3559 60 1 10 04 48 539 59 2 13 03 52 520 58 3 15 02 55 501 57 4 18 9200 58 483 56 5 0.3921 0.9199 0.4262 2.3464 55 6 23 98 65 445 54 7 26 97 69 426 53 8 29 96 72 407 52 9 31 95 76 388 51 10 0.3934 0.9194 0.4279 2.3369 50 11 37 92 83 351 49 12 39 91 86 332 48 13 42 90 89 313 47 14 45 89 93 294 46 15 0.3947 0.9188 0.4296 2.3276 45 16 50 87 4300 257 44 17 53 86 03 238 43 18 55 84 07 220 42 19 58 83 10 201 41 20 0.3961 0.9182 0.4314 2.3183 40 21 63 81 17 164 39 22 66 80 20 146 38 23 69 79 24 127 37 24 71 78 27 109 36 25 0.3974 0.9176 0.4331 2.3090 35 26 77 75 34 072 34 27 79 74 38 053 33 28 82 73 41 035 32 29 85 72 45 2.3017 31 30 0.3987 0.9171 0.4348 2.2998 30 31 90 69 52 980 29 32 93 68 55 962 28 33 95 67 59 944 27 34 3998 66 62 925 26 35 0.4001 0.9165 0.4365 2.2907 25 36 03 64 69 889 24 37 06 62 72 871 23 38 09 61 76 853 22 39 11 60 79 835 21 40 0.4014 0.9159 0.4383 2.2817 20 41 17 58 86 799 19 42 19 57 90 781 18 43 22 55 93 763 17 44 25 54 4397 745 16 45 0.4027 0.9153 0.4400 2.2727 15 46 30 52 04 709 14 47 33 51 07 691 13 48 35 50 11 673 12 49 38 48 14 655 11 50 0.4041 0.9147 0.4417 2.2637 10 51 43 46 21 620 9 52 46 45 24 602 8 53 49 44 28 584 7 54 51 43 31 566 6 55 0.4054 0.9141 0.4435 2.2549 5 56 57 40 38 531 4 57 59 39 42 513 3 58 62 38 45 496 2 59 65 37 49 478 1 60 0.4067 0.9135 0.4452 2.2460 0 1 COS sin cot tan 1 66 ° / sin cos tan cot / 0 0.4067 0.9135 0.4452 2.2460 60 1 70 34 56 443 59 2 73 33 59 425 58 3 75 32 63 408 57 4 78 31 66 390 56 5 0.4081 0.9130 0.4470 2.2373 55 6 83 28 73 355 54 7 86 27 77 338 53 8 89 26 80 320 52 9 91 25 84 303 51 10 0.4094 0.9124 0.4487 2.2286 50 11 97 22 91 268 49 12 4099 21 94 251 48 13 4102 20 4498 234 47 14 05 19 4501 216 46 15 0.4107 0.9118 0.4505 2.2199 45 16 10 16 08 182 44 17 12 15 12 165 43 18 15 14 15 148 42 19 18 13 19 130 41 20 0.4120 0.9112 0.4522 2.2113 40 21 23 10 26 096 39 22 26 09 29 079 38 23 28 08 33 062 37 24 31 07 36 045 36 25 0.4134 0.9106 0.4540 2.2028 35 26 36 04 43 2.2011 34 27 39 03 47 2.1994 33 28 42 02 50 977 32 29 44 01 54 960 31 30 0.4147 0.9100 0.4557 2.1943 30 31 50 9098 61 926 29 32 52 97 64 909 28 33 55 96 68 892 27 34 58 95 71 876 26 35 0.4160 0.9094 0.4575 2.1859 25 36 63 92 78 842 24 37 65 91 82 825 23 38 68 90 85 808 22 39 71 89 89 792 21 40 0.4173 0.9088 0.4592 2.1775 20 41 76 86 96 758 19 42 79 85 4599 742 18 43 81 84 4603 725 17 44 84 83 07 708 16 45 0.4187 0.9081 0.4610 2.1692 15 46 89 80 14 675 14 47 92 79 17 659 13 48 95 78 21 642 12 49 4197 77 24 625 11 50 0.4200 0.9075 0.4628 2.1609 10 51 02 74 31 592 9 52 05 73 35 576 8 53 08 72 38 560 7 54 10 70 42 543 6 55 0.4213 0.9069 0.4645 2.1527 5 56 16 68 49 510 4 57 18 67 52 494 3 58 21 66 56 478 2 59 24 64 60 461 1 60 0.4226 0.9063 0.4663 2.1445 0 / COS sin cot tan f 66 ° 92 26 26 f sin cos tan cot / 9 sin cos tan cot / o 0.4226 0.9063 0.4663 2.1445 60 o 0.4384 0.89S8 0.4877 2.0503 60 1 29 62 67 429 59 1 86 87 81 488 59 2 31 61 70 413 58 2 89 85 85 473 58 3 34 59 74 396 57 3 92 84 88 458 57 4 37 58 77 380 56 4 94 83 92 443 56 5 0.4239 0.9057 0.4681 2.1364 65 5 0.4397 0.8982 0.4895 2.0428 55 6 42 56 84 348 54 6 4399 80 4899 413 54 7 45 54 88 332 53 7 4402 79 4903 398 53 8 47 53 91 315 52 8 05 78 06 383 52 9 50 52 95 299 51 9 07 76 10 368 51 lO 0.4253 0.9051 0.4699 2.1283 50 10 0.4410 0.8975 0.4913 2.0353 50 11 55 50 4702 267 49 11 12 74 17 338 49 12 58 48 06 251 48 12 15 73 21 323 48 13 60 47 09 235 47 13 18 71 24 308 47 14 63 46 13 219 46 14 20 70 28 293 46 15 0.4266 0.9045 0.4716 2.1203 46 15 0.4423 0.8969 0.4931 2.0278 45 16 68 43 20 187 44 16 25 67 35 263 44 17 71 42 23 171 43 17 28 66 39 248 43 18 74 41 27 155 42 18 31 65 42 233 42 19 76 40 . 31 139 41 19 33 64 46 219 41 20 0.4279 0.9038 0.4734 2.1123 40 20 0.4436 0.8962 0.4950 2.0204 40 21 81 37 38 107 39 21 39 61 53 189 39 22 84 36 41 092 38 22 41 60 57 174 38 23 87 35 45 076 37 23 44 58 60 160 37 24 89 33 48 060 36 24 46 57 64 145 36 25 0.4292 0.9032 0.475 2 2.1044 35 25 0.4449 0.8956 0.4968 2.0130 35 26 95 31 55 028 34 26 52 55 71 115 34 27 4297 30 59 2.1013 33 27 54 53 75 101 33 28 4300 28 63 2.0997 32 28 57 52 79 086 32 29 02 27 66 981 31 29 59 51 82 072 31 30 0.4305 0.9026 0.4770 2.0965 30 30 0.4462 0.8949 0.4986 2.0057 30 31 08 25 73 950 29 31 65 48 89 042 29 32 10 23 77 934 28 32 67 47 93 028 28 33 13 ‘ 22 - 80 918 27 33 70 45 4997 2.0013 27 34 16 21 84 903 26 34 72 44 5000 1.9999 26 35 0.4318 0.9020 0.4788 2.0887 25 35 0.4475 0.8943 0.5004 1.9984 25 36 21 18 91 872 24 36 78 42 08 970 24 37 23 17 95 856 23 37 80 40 11 955 23 38 26 16 4798 840 22 38 83 39 15 941 22 39 29 IS 4802 825 21 39 85 38 19 926 21 40 0.4331 0.9013 0.4806 2.0809 20 40 0.448S 0.8936 0.5022 1.9912 20 41 34 12 09 794 19 41 91 35 26 897 19 42 37 11 13 778 18 42 93 34 29 883 18 43 39 10 16 763 17 43 96 32 33 868 17 44 42 08 20 748 16 44 4498 31 37 854 16 45 0.4344 0.9007 0.4823 2.0732 16 45 0.4501 0.8930 0.5040 1.9840 15 46 47 06 27 717 14 46 04 28 44 825 14 47 50 04 31 701 13 47 06 27 48 811 13 48 52 03 34 686 12 48 09 26 51 797 12 49 55 02 38 671 11 49 11 25 782 11 50 0.4358 0.9001 0.4841 2.0655 10 50 0.4514 0.8923 0.5059 1.9768 10 51 60 8999 45 640 9 51 17 22 62 754 9 52 63 98 49 625 8 52 19 21 66 740 8 S3 65 97 52 609 7 S3 22 19 70 725 7 54 68 96 56 594 6 54 24 18 73 711 6 55 0.4371 0.8994 0.4859 2.0579 6 65 0.4527 0.8917 0.5077 1.9697 5 56 73 93 63 564 4 56 30 15 81 683 4 57 76 92 67 549 3 57 32 14 84 669 3 58 78 90 70 533 2 58 35 13 88 654 2 59 81 89 74 518 1 59 37 11 92 640 1 60 0.4384 0.8988 0.4877 2.0503 O 60 0.4540 0.8910 0.5095 1.9626 0 9 COS sin cot tan / r COS sin cot tan 9 64 ‘ 63 27 “ 28 ' 93 / sin cos tan cot / o 0.4540 0.8910 0.5095 1.9626 60 1 42 09 5099 612 59 2 45 07 5103 598 58 3 48 06 06 584 57 4 50 05 10 570 56 6 0.4553 0.8903 0.5114 1.9556 55 6 55 02 17 542 54 7 58 8901 21 528 S3 8 61 8899 25 514 52 9 63 98 28 500 51 lO 0.4566 0.8897 0.5132 1.9486 50 11 68 95 36 472 49 12 71 94 39 458 48 13 74 93 43 444 47 14 76 92 47 430 46 15 0.4579 0.8890 0.5150 1.9416 45 16 81 89 54 402 44 17 84 88 58 388 43 18 86 86 61 375 42 19 89 85 65 361 41 20 0.4592 0.8884 0.5169 1.9347 40 21 94 82 72 333 39 22 97 81 76 319 38 23 4599 79 80 306 37 24 4602 78 84 292 36 25 0.4605 0.8877 0.5187 1.9278 35 26 07 75 91 265 34 27 10 74 95 251 33 28 12 73 5198 237 32 29 15 71 5202 223 31 30 0.4617' 0.8870 0.5206 1.9210 30 31 20 69 09 196 29 32 23 67 13 183 28 33 25 66 17 169 27 34 28 65 20 155 26 35 0.4630 0.8863 0.5224 1.9142 25 36 33 62 28 128 24 37 36 61 32 115 23 38 38 59 35 101 22 39 41 58 39 088 21 40 0.4643 0.8857 0.5243 1.9074 20 41 46 55 46 061 19 42 48 54 50 047 18 43 51 S3 54 034 17 44 54 51 58 020 16 45 0.4656 0.8850 0.5261 1.9007 15 46 59 49 65 1.8993 14 47 61 47 69 980 13 48 64 46 72 967 12 49 66 44 76 953 11 50 0.4669 0.8843 0.5280 1.8940 10 51 72 42 84 927 9 52 74 40 87 913 8 53 77 39 91 900 7 54 79 38 95 887 6 55 0.4682 0.8836 0.5298 1.8873 5 56 84 35 5302 860 4 57 87 34 06 847 3 58 90 32 10 834 2 59 92 31 13 820 1 60 0.4695 0.8829 0.5317 1.8807 0 ! COS sin cot tan / o / sin cos tan cot / o 0.4695 0.8829 0.5317 1.8807 60 1 4697 28 21 794 59 2 4700 27 25 781 58 3 02 25 28 768 57 4 05 24 32 755 56 5 0.4708 0.8823 0.5336 1.8741 55 6 10 21 40 728 54 7 13 20 43 715 S3 8 15 19 47 702 52 9 18 17 51 689 51 lO 0.4720 0.8816 0.5354 1.8676 50 11 23 14 58 663 49 12 26 13 62 650 48 13 28 12 66 637 47 14 31 10 69 624 46 15 0.4733 0.8809 0.5373 1.8611 45 16 36 08 77 598 44 17 38 06 81 585 43 18 41 05 84 572 42 19 43 03 88 559 41 20 0.4746 0.8802 0.5392 1.8546 40 21 49 8801 96 533 39 22 51 8799 5399 520 38 23 54 98 5403 507 37 24 56 96 07 495 36 25 0.4759 0.8795 0.5411 1.8482 35 26 61 94 15 469 34 27 64 92 18 456 33 28 66 91 22 443 32 29 69 90 26 430 31 30 0.4772 0.8788 0.5430 1.8418 30 31 74 87 33 405 29 32 77 85 37 392 28 33 79 84 41 379 27 34 82 83 45 367 26 35 0.4784 0.8781 0.5448 1.8354 25 36 87 80 52 341 24 37 89 78 56 329 23 38 92 77 60 316 22 39 95 76 64 303 21 40 0.4797 0.8774 0.5467 1.8291 20 41 4800 73 71 278 19 42 02 71 75 265 18 43 05 70 79 253 17 44 07 69 82 240 16 45 0.4810 0.8767 0.5486 1.8228 15 46 12 66 90 215 14 47 15 64 94 202 13 48- 18 63 5498 190 12 49 20 62 5501 177 11 50 0.4823 0.8760 0.5505 1.8165 10 51 25 59 09 152 9 52 28 57 13 140 8 53 30 56 17 127 7 54 33 55 20 115 6 55 0.4835 0.8753 0.5524 1.8103 5 56 38 52 28 090 4 57 40 50 32 078 3 58 43 49 35 065 2 59 46 48 39 053 1 60 0.4848 0.8746 0.5543 1.8040 0 COS sin cot tan 61 ° 62 94 29 ° f sin cos tan cot t 0 0.4848 0.8746 0.5543 1.8040 60 1 51 45 47 028 59 2 53 43 51 016 58 3 56 42 55 1.8003 57 4 58 41 58 1.7991 56 5 0.4861 0.8739 0.5562 1.7979 55 6 63 38 66 966 54 7 66 36 70 954 53 8 68 35 74 942 52 9 71 33 77 930 51 10 0.4874 0.8732 0.5581 1.7917 50 11 76 31 85 905 49 12 79 29 89 893 48 13 81 28 93 881 47 14 84 26 5596 868 46 15 0.4886 0.8725 0.5600 1.7856 45 16 89 24 04 844 44 17 91 22 08 832 43 18 94 21 12 820 42 19 96 19 16 808 41 20 0.4899 0.8718 0.5619 1.7796 40 21 4901 16 23 783 39 22 04 15 27 771 38 23 07 14 31 759 37 24 09 12 35 747 36 25 0.4912 0.8711 0.5639 1.7735 35 26 14 09 42 723 34 27 17 OS 46 711 33 28 19 06 50 699 32 29 22 05 54 687 31 30 0.4924 0.8704 0.5658 1.7675 30 31 27 02 62 663 29 32 29 8701 65 651 28 33 32 8699 69 639 27 34 34 98 73 627 26 35 0.4937 0.8696 0.5677 1.7615 25 36 39 95 81 603 24 37 42 94 85 591 23 38 44 92 88 579 22 39 47 91 92 567 21 40 0.4950 0.8689 0.5696 1.7556 20 41 52 88 5700 544 19 42 55 86 04 532 18 43 57 85 08 520 17 44 60 83 12 508 16 45 0.4962 0.8682 0.5715 1.7496 15 46 65 81 19 485 14 47 67 79 23 473 13 48 70 78 27 461 . 12 49 72 76 31 449 11 50 0.4975 0.8675 0.5735 1.7437 10 51 77 73 39 426 9 52 80 72 43 414 8 53 82 70 46 402 7 54 85 69 50 391 6 55 0.4987 0.8668 0.5754 1.7379 5 56 90 66 58 367 4 57 92 65 62 355 3 58 95 63 66 344 2 59 4997 62 70 332 1 60 0.5000 0.8660 0.5774 1.7321 O / COS sin cot tan ! 30 ° f sin cos tan cot / 0 0.5000 0.8660 0.5774 1.7321 60 1 03 59 77 309 59 2 05 57 81 297 58 3 08 56 85 286 57 4 10 54 89 274 56 5 0.5013 0.8653 0.5793 1.7262 5o 6 15 52 5797 251 54 7 18 50 5801 239 53 8 20 49 05 228 52 9 23 47 08 216 51 10 0.5025 0.8646 0.5812 1.7205 50 11 28 44 16 193 49 12 30 43 20 182 48 13 33 41 24 170 47 14 35 40 28 159 46 15 0.5038 0.8638 0.5832 1.7147 45 16 40 37 36 136 44 17 43 35 40 124 43 18 45 34 44 113 42 19 48 32 47 102 41 20 0.5050 0.8631 0.5851 1.7090 40 21 53 30 55 079 39 22 55 28 59 067 38 23 58 27 63 056 37 24 60 25 67 045 36 25 0.5063 0.8624 0.5871 1.7033 35 26 65 22 75 022 34 27 68 21 79 1.7011 33 28 70 19 83 1.6999 32 29 73 18 87 988 31 30 0.5075 0.8616 0.5890 1.6977 30 31 78 15 94 965 29 32 80 13 5898 954 28 33 S3 12 5902 943 27 34 85 10 06 932 26 35 0.5088 0.8609 0.5910 1.6920 25 36 90 07 14 909 24 37 93 06 18 898 23 38 95 04 22 887 22 39 5098 03 26 875 21 40 0.5100 0.8601 0.5930 1.6864 20 41 03 8600 34 853 19 42 05 8599 38 842 18 43 08 97 42 831 17 44 10 96 45 820 16 45 0.5113 0.8594 0.5949 1.6808 lo 46 15 93 53 797 14 47 18 91 57 786 13 48 20 90 61 775 12 49 23 88 65 764 11 50 0.5125 0.8587 0.5969 1.6753 lO 51 28 85 73 742 9 52 30 84 77 731 8 53 33 82 81 720 7 54 35 81 85 709 6 55 0.5138 0.8579 0.5989 1.6698 5 56 40 78 93 687 4 57 43 76 5997 676 3 58 45 75 6001 665 2 59 48 73 05 654 1 60 0.5150 0.8572 0.6009 1.6643 O f COS sin cot tan t 60 59 3r 95 f sin 008 tan cot t 0 0.5150 0.8572 0.6009 1.6643 60 1 53 70 13 632 59 2 55 69 17 621 58 3 58 67 20 610 57 4 60 66 24 599 56 5 0.5163 0.8564 0.6028 1.6588 55 6 65 63 32 577 54 7 68 61 36 566 53 8 70 60 40 555 52 9 73 58 44 545 51 10 0.5175 0.8557 0.6048 1.6534 50 11 78 55 52 523 49 12 80 54 56 512 48 13 83 52 60 501 47 14 85 51 64 490 46 15 0.5188 0.8549 0.6068 1.6479 45 16 90 48 72 469 44 17 93 46 76 458 43 18 95 45 SO 447 42 19 5198 43 84 436 41 20 0.5200 0.8542 0.6088 1.6426 40 21 03 40 92 415 39 22 05 39 6096 404 38 23 08 37 6100 393 37 24 10 36 04 383 36 25 0.5213 0.8534 0.6108 1.6372 35 26 15 32 12 361 34 27 18 31 16 351 33 28 20 29 20 340 32 29 23 28 24 329 31 30 0.5225 0.8526 0.6128 1.6319 30 31 27 25 32 308 29 32 30 23 36 297 28 33 32 22 40 287 27 34 35 20 44 276 26 35 0.5237 0.8519 0.6148 1.6265 25 36 40 17 52 255 24 37 42 16 56 244 23 38 45 14 60 234 22 39 47 13 64 223 21 40 0.5250 0.8511 0.6168 1.6212 20 41 52 10 72 202 19 42 55 08 76 191 18 43 57 07 80 181 17 44 60 05 84 170 16 45 0.5262 0.8504 0.6188 1.6160 15 46 65 02 92 149 14 47 67 8500 6196 139 13 48 70 8499 6200 128 12 49 72 97 04 118 11 50 0.5275 0.8496 0.6208 1.6107 10 51 77 94 12 097 9 52 79 93 16 087 8 53 82 91 20 076 7 54 84 90 24 066 6 55 0.5287 0.8488 0.6228 1.6055 5 56 89 87 33 045 4 57 92 85 37 034 3 58 94 84 41 024 2 59 97 82 45 014 1 60 0.5299 0.8480 0.6249 1.6003 O COS sin cot tan / 32 ° / sin cos tan cot f o 0.5299 0.8480 0.6249 1.6003 60 1 5302 79 53 1.5993 59 2 04 77 57 983 58 3 07 76 61 972 57 4 09 74 65 962 56 5 0.5312 0.8473 0.6269 1.5952 55 6 14 71 73 941 54 7 16 70 77 931 53 8 19 68 81 921 52 9 21 67 85 911 51 10 0.5324 0.8465 0.6289 1.5900 50 11 26 63 93 890 49 12 29 62 6297 880 48 13 31 60 6301 869 47 14 34 59 05 859 46 15 0.5336 0.8457 0.6310 1.5849 45 16 39 56 14 839 44 17 41 54 18 829 43 18 44 53 22 818 42 19 46 51 26 808 41 20 0.5348 0.8450 0.6330 1.5798 40 21 51 48 34 788 39 22 53 46 38 778 38 23 56 45 42 768 37 24 58 43 46 757 36 25 0.5361 0.8442 0.6350 1.5747 35 26 63 40 54 737 34 27 66 39 58 727 33 28 68 37 63 717 32 29 71 35 67 707 31 30 0.5373 0.8434 0.6371 1.5697 30 31 75 32 75 687 29 32 78 31 79 677 28 33 80 29 S3 667 27 34 83 28 87 657 26 35 0.5385 0.8426 0.6391 1.5647 25 36 88 25 95 637 24 37 90 23 6399 627 23 38 93 21 6403 617 22 39 95 20 08 607 21 40 0.5398 0.8418 0.6412 1.5597 20 41 5400 17 16 587 19 42 02 15 20 577 18 43 05 14 24 567 17 44 07 12 28 557 16 45 0.5410 0.8410 0.6432 1.5547 15 46 12 09 36 537 14 47 15 07 40 527 13 48 17 06 45 517 12 49 20 04 49 507 11 50 0.5422 0.8403 0.6453 1.5497 10 51 24 8401 57 487 9 52 27 8399 61 477 8 53 29 98 65 468 7 54 32 96 69 458 6 55 0.5434 0.8395 0.6473 1.5448 5 56 37 93 78 438 4 57 39 91 82 428 3 58 42 90 86 418 2 59 44 88 90 408 1 60 0.5446 0.8387 0.6494 1.5399 O COS sin cot tan 58 ' 57 96 33 ° 34 ° sin cos tan cot f t sin cos tan cot t 0 0.5446 0.8387 0.6494 1.5399 60 o 0.5592 0.8290 0.6745 1.4826 60 1 49 85 6498 389 59 1 94 89 49 816 59 2 51 84 6502 379 58 2 97 87 54 807 58 3 54 82 06 369 57 3 5599 85 58 798 57 4 56 80 11 359 56 4 5602 84 62 788 56 5 0.5459 0.8379 0.6515 1.5350 55 5 0.5604 0.8282 0.6766 1.4779 oo 6 61 77 19 340 54 6 06 81 71 770 54 7 63 76 23 330 53 7 09 79 75 761 53 8 66 74 27 320 52 8 11 77 79 751 52 9 68 72 31 311 51 9 14 76 83 742 51 10 0.5471 0.8371 0.6536 1.5301 50 lO 0.5616 0.8274 0.6787 1.4733 50 11 73 69 40 291 49 11 18 72 92 724 49 12 76 68 44 282 48 12 21 71 6796 715 48 13 78 66 48 272 47 13 23 69 6800 705 47 14 80 64 52 262 46 14 26 68 05 696 46 15 0.5483 0.8363 0.6556 1.5253 45 15 0.5628 0.8266 0.6809 1.4687 45 16 85 61 60 243 44 16 30 64 13 678 44 17 88 60 65 233 43 17 33 63 17 669 43 18 90 58 69 224 42 18 35 61 22 659 42 19 93 56 73 214 41 19 38 59 26 650 41 20 0.5495 0.8355 0.6577 1.5204 40 20 0.5640 0.8258 0.6830 1.4641 40 21 5498 53 81 195 39 21 42 56 34 632 39 22 5500 52 85 185 38 22 45 54 39 623 38 23 02 50 90 175 37 23 47 53 43 614 37 24 05 48 94 166 36 24 50 51 47 605 36 25 0.5507 0.8347 0.6598 1.5156 35 25 0.5652 0.8249 0.6851 1.4596 35 26 10 45 6602 147 34 26 54 48 56 586 34 27 12 44 06 137 33 27 57 46 60 577 33 28 15 42 10 127 32 28 59 45 64 568 32 29 17 40 15 118 31 29 62 43 69 559 31 30 0.5519 0.8339 0.6619 1.5108 30 30 0.5664 0.8241 0.6873 1.4550 30 31 22 37 23 099 29 31 66 40 77 541 29 32 24 36 27 089 28 32 69 38 81 532 28 33 27 34 31 080 27 33 71 36 86 523 27 34 29 32 36 070 26 34 74 35 90 514 26 35 0.5531 0.8331 0.6640 1.5061 25 35 0.5676 0.8233 0.6894 1.4505 25 36 34 29 44 051 24 36 78 31 6899 496 24 37 36 28 48 042 23 37 81 30 6903 487 23 38 39 26 52 032 22 38 83 28 07 478 22 39 41 24 57 023 21 39 86 26 11 469 21 40 0.5544 0.8323 0.6661 1.5013 20 40 0.5688 0.8225 0.6916 1.4460 20 41 46 21 65 1.5004 19 41 90 23 20 451 19 42 48 20 69 1.4994 18 42 93 21 24 442 18 43 51 18 73 985 17 43 95 20 29 433 17 44 53 16 78 975 16 44 5698 18 33 424 16 45 0.5556 0.8315 0.6682 1.4966 15 45 0.5700 0.8216 0.6937 1.4415 15 46 58 13 86 957 14 46 02 15 42 406 14 47 61 11 90 947 13 47 05 13 46 397 13 48 63 10 94 938 12 48 07 11 50 388 12 49 65 08 6699 928 11 49 10 10 54 379 11 50 0.5568 0.8307 0.6703 1.4919 10 50 0.5712 0.8208 0.6959 1.4370 10 51 70 05 07 910 9 51 14 07 63 361 9 52 73 03 11 900 8 52 17 05 67 352 8 53 75 02 16 891 7 53 19 03 72 344 7 54 77 8300 20 882 6 54 21 02 76 335 6 55 0.5580 0.8299 0.6724 1.4872 o 55 0.5724 0.8200 0.6980 1.4326 5 56 82 97 28 863 4 56 26 8198 85 317 4 57 85 95 32 854 3 57 29 97 89 308 3 58 87 94 37 844 2 58 31 95 93 299 2 59 90 92 41 835 1 59 33 93 6998 290 1 «o 0.5592 0.8290 0.6745 1.4826 O 60 0.5736 0.8192 0.7002 1.4281 O f COS sin cot tan f COS sin cot tan 9 56 55 36 “ / sin cos tan cot / 0 0.5736 0.8192 0.7002 1.4281 60 1 38 90 06 273 59 2 41 88 11 264 58 3 43 87 15 255 57 4 45 85 19 246 56 6 0.5748 0.8183 0.7024 1.4237 56 6 50 81 28 229 54 7 52 80 32 220 53 8 55 78 37 211 52 9 57 76 41 202 51 10 0.5760 0.8175 0.7046 1.4193 50 11 62 73 50 185 49 12 64 71 54 176 48 13 67 70 59 167 47 14 69 68 63 158 46 15 0.5771 0.8166 0.7067 1.4150 45 16 74 65 72 141 44 17 76 63 76 132 43 18 79 61 80 124 42 19 81 60 85 115 41 20 0.5783 0.8158 0.7089 1.4106 40 21 86 56 94 097 39 22 88 55 7098 089 38 23 90 53 7102 080 37 24 93 51 07 071 36 25 0.5795 0.8150 0.7111 1.4063 35 26 5798 48 15 054 34 27 5800 46 20 045 33 28 02 45 24 037 32 29 05 43 29 028 31 30 0.5807 0.8141 0.7133 1.4019 30 31 09 39 37 on 29 32 12 38 42 1.4002 28 33 14 36 46 1.3994 27 34 16 34 51 985 26 35 0.5819 0.8133 0.7155 1.3976 25 36 21 31 59 968 24 37 24 29 64 959 23 38 26 28 68 951 22 39 28 26 73 942 21 40 0.5831 0.8124 0.7177 1.3934 20 41 33 23 81 925 19 42 35 21 86 916 18 43 38 19 90 908 17 44 40 17 95 899 16 45 0.5842 0.8116 0.7199 1.3891 16 46 45 14 7203 882 14 47 47 12 08 874 13 48 50 11 12 865 12 49 52 09 17 857 11 50 0.5854 0.8107 0.7221 1.3848 10 51 57 06 26 840 9 52 59 04 30 831 8 53 61 02 34 823 7 54 64 8100 39 814 6 55 0.5866 0.8099 0.7243 1.3806 6 56 68 97 48 798 4 57 71 95 52 789 3 58 73 94 57 781 2 59 75 92 61 772 1 60 0.5878 0.8090 0.7265 1.3764 0 t COS sin cot tan / 36° 9T t sin cos tan cot / o 0.5878 0.8090 0.7265 1.3764 60 1 80 88 70 755 59 2 83 87 74 747 58 3 85 85 79 739 57 4 87 83 83 730 56 5 0.5890 0.8082 0.7288 1.3722 55 6 92 80 92 713 54 7 94 78 7297 705 53 8 97 76 7301 697 52 9 5899 75 06 688 51 10 0.5901 0.8073 0.7310 1.3680 50 11 04 71 14 672 49 12 06 70 19 663 48 13 08 68 23 655 47 14 11 66 28 647 46 15 0.5913 0.8064 0.7332 1.3638 45 16 15 63 37 630 44 17 18 61 41 622 43 18 20 59 46 613 42 19 22 58 50 605 41 20 0.5925 0.8056 0.7355 1.3597 40 21 27 54 59 588 39 22 30 52 64 580 38 23 32 51 68 572 37 24 34 49 73 564 36 25 0.5937 0.8047 0.7377 1.3555 35 26 39 45 82 547 34 27 41 44 86 539 33 28 44 42 91 531 32 29 46 40 7395 522 31 30 0.5948 0.8039 0.7400 1.3514 30 31 51 37 04 506 29 32 53 35 09 498 28 33 55 33 13 490 27 34 58 32 18 481 26 35 0.5960 0.8030 0.7422 1.3473 25 36 62 28 27 465 24 iP 65 26 31 457 23 67 25 36 449 22 39 69 23 40 440 21 40 0.5972 0.8021 0.7445 1.3432 20 41 74 19 49 424 19 42 76 18 54 416 18 43 79 16 58 408 17 44 81 14 63 400 16 46 0.5983 0.8013 0.7467 1.3392 15 46 86 11 72 384 14 47 88 09 76 375 13 48 90 07 81 367 12 49 93 06 85 359 11 50 0.5995 0.8004 0.7490 1.3351 10 51 5997 02 95 343 9 52 6000 8000 7499 335 8 53 02 7999 7504 327 7 54 04 97 08 319 6 55 0.6007 0.7995 0.7513 1.3311 5 56 09 93 17 303 4 57 11 92 22 295 3 58 14 90 26 287 2 59 16 88 31 278 1 60 0.6018 0.7986 0.7536 1.3270 O / COS sin cot tan r 64' 63 98 37 ® / sin cos tan cot o 0.6018 0.7986 0.7536 1.3270 60 1 20 85 40 262 59 2 23 83 45 254 58 3 25 81 49 246 57 4 27 79 54 238 56 6 0.6030 0.7978 0.7558 1.3230 55 6 32 76 63 222 54 7 34 74 68 214 53 8 37 72 72 206 52 9 39 71 77 198 51 10 0.6041 0.7969 0.7581 1.3190 50 11 44 67 86 182 49 12 46 65 90 175 48 13 48 64 7595 167 47 14 51 62 7600 159 46 15 0.6053 0.7960 0.7604 1.3151 45 16 55 58 09 143 44 17 58 56 13 135 43 18 60 55 18 127 42 19 62 53 23 119 41 20 0.6065 0.7951 0.7627 1.3111 40 21 67 50 32 103 39 22 69 48 36 095 38 23 71 46 41 087 37 24 74 44 46 079 36 25 0.6076 0.7942 0.7650 1.3072 35 26 78 41 55 064 34 27 81 39 59 056 33 28 83 37 64 048 32 29 85 35 69 040 31 30 0.60SS 0.7934 0.7673 1.3032 30 31 90 32 78 024 29 32 92 30 83 017 28 33 95 28 87 009 27 34 97 26 92 1.3001 26 35 0.6099 0.7925 0.7696 1.2993 25 36 6101 23 7701 985 24 37 04 21 06 977 23 38 06 19 10 970 22 39 08 18 15 962 21 40 0.6111 0.7916 0.7720 1.2954 20 41 13 14 24 946 19 42 15 12 29 938 18 43 18 10 34 931 17 44 20 09 38 923 16 45 0.6122 0.7907 0.7743 1.2915 15 46 24 05 47 907 14 47 27 03 52 900 13 48 29 02 57 892 12 49 31 7900 61 884 11 50 0.6134 0.7898 0.7766 1.2876 10 51 36 96 71 869 9 52 38 94 75 861 8 53 41 93 80 853 7 54 43 91 85 846 6 55 0.6145 0.78S9 0.7789 1.2838 5 56 47 87 94 830 4 57 50 85 7799 822 3 58 52 84 7803 815 2 59 54 82 08 807 1 60 0.6157 0.7880 0.7813 1.2799 0 f COS sin cot tan f 38 ® t sin cos tan cot / 0 0.6157 0.7880 0.7813 1.2799 CO 1 59 78 18 792 59 2 61 77 22 784 58 3 63 75 27 776 57 4 66 73 32 769 56 5 0.6168 0.7871 0.7836 1.2761 55 6 70 69 41 753 54 7 73 68 46 746 53 8 75 66 50 738 52 9 77 64 55 731 51 10 0.6180 0.7862 0.7860 1.2723 50 11 82 60 65 715 49 12 84 59 69 708 48 13 86 57 74 700 47 14 89 55 79 693 46 15 0.6191 0.7853 0.7883 1.2685 45 16 93 51 88 677 44 17 96 50 93 670 43 18 6198 48 7898 662 42 19 6200 46 7902 655 41 20 0.6202 0.7844 0.7907 1.2647 40 21 05 42 12 640 39 22 07 41 16 632 38 23 09 39 21 624 37 24 11 37 26 617 36 25 0.6214 0.7835 0.7931 1.2609 35 26 16 33 35 602 34 27 18 32 40 594 33 28 21 30 45 587 32 29 23 28 50 579 31 30 0.6225 0.7826 0.7954 1.2572 30 31 27 24 59 564 29 32 30 22 64 557 28 33 32 21 69 549 27 34 34 19 73 542 26 35 0.6237 0.7817 0.7978 1.2534 25 36 39 IS 83 527 24 37 41 13 88 519 23 38 43 12 92 512 22 39 46 10 7997 504 21 40 0.6248 0.7808 0.8002 1.2497 20 41 50 06 07 489 19 42 52 04 12 482 18 43 55 02 16 475 17 44 57 7801 . 21 467 16 45 0.6259 0.7799 0.8026 1.2460 15 46 62 97 31 452 14 47 64 95 35 445 13 48 66 93 40 437 12 49 68 92 45 430 11 50 0.6271 0.7790 0.8050 1.2423 10 51 73 88 55 415 9 52 75 86 59 408 8 53 77 84 64 401 7 54 80 82 69 393 6 55 0.6282 0.7781 0.8074 1.2386 5 56 84 79 79 378 4 57 86 77 83 371 3 58 89 75 88 364 2 59 91 73 93 356 1 60 0.6293 0.7771 0.8098 1.2349 o f COS sin cot tan t 52 51 39 ® 40 ° 99 / sin cos tan cot t o 0.6428 0.7660 0.8391 1.1918 60 1 30 59 8396 910 59 2 32 57 8401 903 58 3 35 55 06 896 57 4 37 53 11 889 56 5 0.6439 0.7651 0.8416 1.1882 55 6 41 49 21 875 54 7 43 47 26 868 53 8 46 45 31 861 52 9 48 44 36 854 51 lO 0.6450 0.7642 0.8441 1.1847 50 11 52 40 46 840 49 12 55 38 51 833 48 13 57 36 56 826 47 14 59 34 61 819 46 15 0.6461 0.7632 0.8466 1.1812 45 16 63' 30 71 806 44 17 66 29 76 799 43 18 68 27 81 792 42 19 70 25 86 785 41 20 0.6472 0.7623 0.8491 1.1778 40 21 75 21 8496 771 39 22 77 19 8501 764 38 23 79 17 06 757 37 24 81 15 11 750 36 25 0.6483 0.7613 0.8516 1.1743 35 26 86 12 21 736 34 27 88 10 26 729 33 28 90 08 31 722 32 29 92 06 36 715 31 30 0.6494 0.7604 0.8541 1.1708 30 31 97 02 46 702 29 32 6499 7600 51 695 28 33 6501 7598 56 688 27 34 03 96 61 681 26 35 0.6506 0.7595 0.8566 1.1674 25 36 08 93 71 667 24 37 10 91 76 660 23 38 12 89 81 653 22 39 14 87 86 647 21 40 0.6517 0.7585 0.8591 1.1640 20 41 19 83 8596 633 19 42 21 81 8601 626 18 43 23 79 06 619 17 44 25 78 11 612 16 45 0.6528 0.7576 0.8617 1.1606 15 46 30 74 22 599 14 47 32 72 27 592 13 48 34 70 32 585 12 49 36 68 37 578 11 50 0.6539 0.7566 0.8642 1.1571 10 51 41 64 47 565 9 52 43 62 52 558 8 53 45 60 57 551 7 54 47 59 62 544 6 55 0.6550 0.7557 0.8667 1.1538 5 56 52 55 72 531 4 57 54 53 78 524 3 58 56 51 83 517 2 59 58 49 88 510 1 60 0.6561 0.7547 0.8693 1.1504 0 f COS sin cot tan 50 ' 49 100 41 ® 42 ® f sin cos tan cot f 0 0.6691 0.7431 0.9004 1.1106 60 1 93 30 09 100 59 2 96 28 15 093 58 3 6698 26 20 087 57 4 6700 24 25 080 56 5 0.6702 0.7422 0.9030 1.1074 55 6 04 20 36 067 54 7 06 18 41 061 53 8 09 16 46 054 52 9 11 14 52 048 51 10 0.6713 0.7412 0.9057 1.1041 50 11 15 10 62 035 49 12 17 08 67 028 48 13 19 06 73 022 47 14 22 04 78 016 46 15 0.6724 0.7402 0.9083 1.1009 45 16 26 7400 89 1.1003 44 17 28 7398 94 1.0996 43 18 30 96 9099 990 42 19 32 94 9105 983 41 20 0.6734 0.7392 0.9110 1.0977 40 21 37 90 15 971 39 22 39 88 21 964 38 23 41 87 26 958 37 24 43 85 31 951 36 25 0.6745 0.7383 0.9137 1.0945 35 26 47 81 42 939 34 27 49 79 47 932 33 28 52 77 53 926 32 29 54 75 58 919 31 30 0.6756 0.7373 0.9163 1.0913 30 31 58 71 69 907 29 32 60 69 74 900 28 33 62 67 79 894 27 34 64 65 85 888 26 35 0.6767 0.7363, 0.9190 1.0881 25 36 69 61 9195 875 24 37 71 59 9201 869 23 38 73 57 06 862 22 39 75 55 12 856 21 40 0.6777 0.7353 0.9217 1.0850 20 41 79 51 22 843 19 42 82 49 28 837 IS 43 84 47 33 831 17 44 86 45 • 39 824 16 45 0.6788 0.7343 0.9244 1.0818 15 46 90 41 49 812 14 47 92 39 55 805 13 48 94 37 60 799 12 49 97 35 66 793 11 50 0.6799 0.7333 0.9271 1.0786 10 51 6801 31 76 780 9 52 03 29 82 774 8 53 05 27 87 768 7 54 07 25 93 761 6 5o 0.6809 0.7323 0.9298 1.0755 5 56 11 21 9303 749 4 57 14 19 09 742 3 58 16 18 14 736 2 59 18 16 20 730 1 60 0.6820 0.7314 0.9325 1.0724 0 / COS sin cot tan f 48 47 43 ° t sin cos tan cot / o 0.6820 0.7314 0.9325 1.0724 60 1 22 12 31 717 59 2 24 10 36 711 58 3 26 08 41 705 57 4 28 06 47 699 56 5 0.6831 0.7304 0.9352 1.0692 55 6 33 02 58 686 54 7 35 7300 63 680 53 8 37 7298 69 674 52 9 39 96 74 668 51 lO 0.6841 0.7294 0.9380 1.0661 50 11 43 92 85 655 49 12 45 90 91 649 48 13 48 88 9396 643 47 14 50 86 9402 637 46 15 0.6852 0.7284 0.9407 1.0630 45 16 54 82 13 624 44 17 56 80 18 618 43 18 58 78 24 612 42 19 60 76 29 606 41 20 0.6862 0.7274 0.9435 1.0599 40 21 65 72 40 593 39 22 67 70 46 587 38 23 69 68 51 581 37 24 71 66 57 575 36 25 0.6873 0.7264 0.9462 1.0569 35 26 75 62 68 562 34 27 77 60 73 556 33 28 79 58 79 550 32 29 81 56 84 544 31 30 0.6884 0.7254 0.9490 1.0538 30 31 86 52 9495 532 29 32 88 50 9501 526 28 33 90 48 06 519 27 34 92 46 12 513 26 35 0.6894 0.7241 0.9517 1.0507 25 36 96 42 23 501 24 37 6898 40 28 495 23 38 6900 38 34 489 22 39 03 36 40 483 21 40 0.6905 0.7234 0.9545 1.0477 20 41 07 32 51 470 19 42 09 30 56 464 18 43 11 28 62 458 17 44 13 26 67 452 16 45 0.6915 0.7224 0.9573 1.0446 15 46 17 22 78 440 14 47 19 20 84 434 13 48 21 18 90 428 12 49 24 16 9595 422 11 50 0.6926 0.7214 0.9601 1.0416 10 51 28 12 06 410 9 52 30 10 12 404 8 53 32 08 18 398 7 54 34 06 23 392 6 55 0.6936 0.7203 0.9629 1.0385 5 56 38 7201 34 379 4 57 40 7199 40 373 3 58 42 97 46 367 2 59 44 95 51 361 1 60 0.6947 0.7193 0.9657 1.0355 0 COS sin cot tan / 44 ° 101 sin cos tan cot / o 0.6947 0.7193 0.9657 1.0355 60 1 49 91 63 349 59 2 51 89 68 343 58 3 53 87 74 337 57 4 55 85 79 331 56 5 0.6957 0.7183 0.9685 1.0325 55 6 59 81 91 319 54 7 61 79 9696 313 53 8 63 77 9702 307 52 9 65 75 08 301 51 10 0.6967 0.7173 0.9713 1.0295 50 11 70 71 19 289 49 12 72 69 25 283 48 13 74 67 30 277 47 14 76 65 36- 271 46 15 0.6978 0.7163 0.9742 1.0265 45 16 80 61 47 259 44 17 82 59 53 253 43 18 84 57 59 247 42 19 86 55 64 241 41 20 0.6988 0.7153 0.9770 1.0235 40 21 90 51 ,76 230 39 22 92 49 81 224 38 23 95 47 87 218 37 24 97 45 93 212 36 25 0.6999 0.7143 0.9798 1.0206 35 26 7001 41 9804 200 34 27 03 39 10 194 33 28 05 37 16 188 32 29 07 35 21 182 31 30 0.7009 0.7133 0.9827 1.0176 30 31 11 30 33 170 29 32 13 28 38 164 28 33 15 26 44 158 27 '34 17 24 50 152 26 35 0.7019 0.7122 0.9856 1.0147 25 36 22 20 61 141 24 37 24 18 67 135 23 38 26 16 73 129 22 39 28 14 79 123 21 40 0.7030 0.7112 0.9884 1.0117 20 41 32 10 90 111 19 42 34 08 9896 105 18 43 36 06 9902 099 17 44 38 04 07 094 16 45 0.7040 0.7102 0.9913 1.0088 15 46 42 7100 19 082 14 47 44 7098 25 076 13 48 46 96 30 070 12 49 48 94 36 064 11 50 0.7050 0.7092 0.9942 1.0058 10 51 53 90 48 052 9 52 55 88 54 047 8 53 57 85 59 041 7 54 59 83 65 035 6 55 0.7061 0.7081 0.9971 1.0029 5 56 63 79 77 023 4 57 65 77 83 017 3 58 67 75 88 012 2 59 69 73 94 006 1 60 0.7071 0.7071 1.0000 1.0000 0 t COS sin cot tan / 46 45 ° 102 TABLE IX CONVERSION TABLE— DEGREES TO RADIANS 1° = radians 1 radian = — degrees 180 7 t ^ 0 °- 45 ° o O' lO' 20' 30' 40' 50' o 0.0000 0.0029 0.0058 0.0087 0.0116 0.0145 1 0175 0204 0233 0262 0291 0320 2 0349 0378 0407 0436 0165 0495 3 0524 0553 0582 0611 0640 0669 4 0698 0727 0756 0785 0814 0844 5 0.0873 0.0902 0.0931 0.0960 0.0989 0.1018 6 1047 1076 1105 1134 1164 1193 7 1222 1251 1280 1309 1338 1367 8 1396 1425 1454 1484 1513 1542 9 1571 1600 1629 1658 1687 1716 lO 0.1745 0.1774 0.1804 0.1833 0.1862 0.1891 11 1920 1949 1978 2007 2036 2065 12 2094 2123 2153 2182 2211 2240 13 2269 2298 2327 2356 2385 2414 14 2443 2473 2502 2531 2560 2589 15 0.2618 0.2647 0.2676 0.2705 0.2734 0.2763 16 2793 2822 2851 2880 2909 2938 17 2967 2996 3025 3054 3083 3113 18 3142 3171 3200 3229 3258 3287 19 3316 3345 3374 3403 3432 3462 20 0.3491 0.3520 0.3549 0.3578 0.3607 0.3636 21 3665 3694 3723 3752 3782 3811 22 3840 3869 3898 3927 3956 3985 23 4014 4043 4072 4102 4131 4160 24 4189 4218 4247 4276 4305 4334 25 0.4363 0.4392 0.4422 0.4451 0.4480 0.4508 26 4538 4567 4596 4625 4654 4683 27 4712 4741 4771 4800 4829 4858 28 4887 4916 4945 4974 5003 5032 29 5061 5091 5120 5149 5178 5207 50 0.5236 0.5265 0.5294 0.5323 0.5352 0.5381 31 5411 5440 5469 5498 5527 5556 32 5585 5614 5643 5672 5701 5730 33 5760 5789 5818 5847 5876 5905 34 5934 5963 5992 6021 6050 6080 35 0.6109 0.6138 0.6167 0.6196 0.6225 0.6254 36 6283 6312 6341 6370 6400 6429 37 6458 6487 6516 6545 6574 6603 38 6632 6661 6690 6720 6749 6778 39 6807 6836 6865 6894 6923 6952 40 0.6981 0.7010 0.7039 0.7069 0.7098 0.7127 41 7156 7185 7214 7243 7272 7301 42 7330 ■ 7359 7389 7418 7447 7476 43 7505 7534 7563 7592 7621 7650 44 7679 7709 7738 7767 7796 7825 45 0.7854 0.7883 0.7912 0.7941 0.7970 0.7999 O O' 10' 20' 30' 40' 50' 103 In using this table, interpolations may be made as with other tables. Thus to find the number of radians corresponding to 49° 15', we have : 49° 10' = 0.8581 radians Tabular diff. = 0.0029 A of 0.0029 = 0.0015 Adding, 49° 15' = 0.8596 radians 45 °- 90 ° o O' 10' 20' 30' 40' 50' 45 0.7854 0.7883 0.7912 0.7941 0.7970 0.7999 46 8029 8058 8087 . 8116 8145 8i74 47 8203 8232 8261 8290 8319 8348 48 8378 8407 8436 8465 8494 8523 49 8552 8581 8610 8639 8668 8698 50 0.8727 0.8756 0.8785 0.8814 0.8843 0.8872 51 8901 8930 8959 8988 9018 9047 52 9076 9105 9134 9163 9192 9221 53 9250 9279 9308 9338 9367 9396 54 9425 9454 9483 9512 9541 9570 55 0.9599 0.9628 0.9657 0.9687 0.9716 0.9745 56 9774 9803 9832 9861 9890 9919 57 9948 9977 1.0007 1.0036 1.0065 1.0094 58 1.0123 1.0152 0181 0210 0239 0268 59 0297 0326 0356 0385 0414 0443 60 1.0472 1.0501 1.0530 1.0559 1.0588 1.0617 61 0647 0676 0705 0734 0763 0792 62 0821 0850 0879 0908 0937 0966 63 0996 1025 1054 1083 1112 1141 64 1170 1199 1228 1257 1286 1316 65 1.1345 1.1374 1.1403 1.1432 1.1461 1.1490 66 1519 1548 1577 1606 1636 1665 67 1694 1723 1752 1781 1810 1839 68 1868 1897 1926 1956 1985 2014 69 2043 2072 2101 2130 2159 2188 70 1.2217 1.2246 1.2275 1.2305 1.2334 1.2363 71 2392 2421 2450 2479 2508 2537 72 2566 2595 2625 2654 2683 2712 73 2741 2770 2799 2828 2857 2886 74 2915 2945 2974 3003 3032 3061 75 1.3090 1.3119 1.3148 1.3177 1.3206 1.3235 76 3265 3294 3323 3352 3381 3410 77 3439 3468 3497 3526 3555 3584 78 3614 3643 3672 3701 3730 3759 79 3788 3817 3846 3875 3904 3934 80 1.3963 1.3992 1.4021 1.4050 1.4079 1.4108 81 4137 4166 4195 4224 4254 4283 82 4312 4341 4370 4399 4428 4457 83 4486 4515 4544 4573 4603 4632 84 4661 4690 4719 4748 4777 4806 85 1.4835 1.4864 1.4893 1.4923 1.4952 1.4981 86 5010 5039 5068 5097 5126 5155 87 5184 5213 5243 5272 5301 5330 88 5359 5388 5417 5446 5475 5504 89 5533 5563 5592 5621 5650 5679 90 1.5708 1.5737 1.5766 1.5795 1.5824 1.5853 0 O' lO' 20' 30' 40' 60' 104 TABLE X. CONVERSION OF MINUTES AND SECONDS TO DECIMALS OF A DEGREE, AND OF DECIMALS OF A DEGREE TO MINUTES AND SECONDS t 0 n O 0 > and " o 1 and " O 0.0000 o 0.00000 0.000 0' 0" 0.50 30' 0 " 1 0167 1 028 001 0' 4" 51 30' 36" 2 0333 2 056 002 0' 7" 52 31' 12" 3 0500 3 083 003 0' 11" 53 31' 48" 4 0667 4 111 004 0' 14" 54 32' 24" 5 0.0833 5 0.00139 0.005 0' 18" 0.55 33' 0" 6 1000 6 167 006 O' 22" 56 33' 36" 7 1167 7 194 007 0' 25" 57 34' 12" 8 1333 8 222 008 0' 29" 58 34' 48" 9 1500 9 250 009 0' 32" 59 35' 24" lO 0.1667 lO 0.00278 0.00 0' 0" 0.60 36' 0" 11 1833 11 306 01 0' 36" 61 36' 36" 12 2000 12 333 02 1' 12" 62 37' 12" 13 2167 13 361 03 1' 48" 63 37' 48" 14 2333 14 389 04 2' 24" 64 38' 24" 15 0.2500 15 0.00417 0.05 3' 0" 0.65 39' 0" 16 2667 16 444 06 3' 36" 66 39' 36" 17 2833 17 472 07 4' 12" 67 40' 12" 18 3000 18 500 08 4' 48" 68 40' 48" 19 3167 19 528 09 5' 24" 69 41' 24" 20 0.3333 20 0.00556 0.10 6' 0" 0.70 42' 0" 21 3500 21 583 11 6' 36" 71 42' 36" 22 3667 22 611 12 7' 12" 72 43' 12" 23 3833 23 639 13 7' 48" 73 43' 48" 24 4000 24 667 14 8' 24" 74 44' 24" 25 0.4167 25 0.00694 0.15 9' 0" 0.75 45' 0" 26 4333 26 722 16 9' 36" 76 45' 36" 27 4500 27 750 17 10' 12" 77 46' 12" 28 4667 28 778 18 10' 48" 78 46' 48" 29 4833 29 806 19 11' 24" 79 47' 24" 30 0.5000 30 0.00833 0.20 12' 0" 0.80 48' 0" 31 5167 31 861 21 12' 36" 81 48' 36" 32 5333 32 889 22 13' 12" 82 49' 12" 33 5500 33 917 23 13' 48" 83 49' 48" 34 5667 34 944 24 14' 24" 84 50' 24" 35 0.5833 35 0.00972 0.25 15' 0" 0.85 51' 0" 36 6000 36 01000 26 15' 36" 86 51' 36" 37 6167 37 028 27 16' 12" 87 52' 12" 38 6333 38 056 28 16' 48" 88 52' 48" 39 6500 39 083 29 17' 24" 89 53' 24" 40 0.6667 40 0.01111 0.30 IS' 0" 0.90 54' 0" 41 6833 41 139 31 18' 36" 91 54' 36" 42 7000 42 167 32 19' 12" 92 55' 12" 43 7167 43 194 33 19' 48" 93 55' 48" 44 7333 44 222 34 20' 24" 94 56' 24" 45 0.7500 45 0.01250 0.35 21' 0" 0.95 57' 0" 46 7667 46 278 36 21' 36" 96 57' 36" 47 7833 47 306 37 22' 12" 97 58' 12" 48 8000 48 333 38 22' 48" 98 58' 48" 49 8167 49 361 39 23' 24" 99 59' 24" 50 0.8333 50 0.01389 0.40 24' 0" 1.00 60' 0" 51 8500 51 417 41 24' 36" 10 66' 0" 52 8667 52 444 42 25' 12" 20 72' 0" 53 8833 53 472 43 25' 48" 30 78' 0" 54 9000 54 500 44 26' 24" 40 84' 0" 55 0.9167 55 0.01528 0.45 27' 0" 1.50 90' 0" 56 9333 56 556 46 27' 36" 60 96' 0" . 57 9500 57 583 47 28' 12" 70 102' 0" 58 9667 58 611 48 28' 48" SO 108' 0" 59 9833 59 639 49 29' 24" 90 114' 0" 60 1.0000 60 0.01667 0.50 30' 0" 2.00 120' 0" f 0 tf 0 0 r and " 0 1 and " r ^c’fC^ryce /s /cc/^e> c-f ^ehcLk^/or* "\V\<;: ? VA/i-fc. O^ CJ'<''^ ■ . . • '^ ,— . ..a-^v — jri X- n 0 ^ \yij "to ^.^'y-'-cA tl!^-~a. , , (T^ IUaXoi->w^ V-/0 “ter- , •J ‘ <>Lj-«-P ^ o— ^ . 4) c^rv,^ c.o^^ ^ ^ Q^ t;, ^ "CU^ j 5 I ,-, ( VQ « /aXxo-Co- . 0“^ — - -*v V- ;/ K": ^'"- ■ # ' f> .’■. ■ 'T >’ r.., ' i^!"‘