Digitized by the Internet Archive in 2010 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/elementsofplanesOOsnow PLANE AND SPIIEEICAL T R I G N ]\I E T R Y. THE ELEMENTS OP PLANE AND SPHERICAL TRIGONOMETRY ; WITH THE CONSTRUCTION AND USE OF TABLES OF LOGARITHMS, BOTH OF NUMBERS, AND FOR ANGLES. BY J. C. SNOWBALL, MA."" LATE FELLOW OF ST JOHN'S COLLEGE, CAMBRIDGE. SbSTOU COLLEGE MBBABT ^■j7tfc«^'»^'^'^ niLL, MASS. MATH, DEPT, ILonbon : MACMILLAN AND CO. AND NEW YORK. 1891 [The Hight 0/ Translation is reserved.] Originally published elsnvhere. First Edition printed for Maonillan &^ Co. 1852. Crown Svo. Second Edition, 1857. Third Edition, 1863. Fourth Edition, 1876. Globe %vo. /Reprinted \^-j%, 1880, 1885, 1891. 150044 INDEX. PLANE TRTGOXOMETRT. CHAPTER I. AKTICLKS. On the Methods of Measuring Lines and Angles 1 — 12 CHAPTER II. Definitions of the Gonioaietrical Eatios, and Formula con- necting THEM WITH each OTHER 13 — 33 Definitions -^ l-i Changes of the Sine, &c. in magnitude and sign 20, 21 Formulae involving one angle only 22 — 29 Sines, &c. of 4:,^, 30^, GO^^ 32 Examples 33 CHAPTER III. FoRMULiE INVOLVING MORE ANGLES THAN ONE 3-i — 67 Sines and Cosines of A±B 34 — 36 Sin2^ and Cos 2A in terms of Sin J and Cos J, and conversely... 38—41 Formulae deduced from the Sines and Cosines of A±B 43 — 56 Sines and Cosines of 18°, 72o, 36", 54^ 58 Increments of the Sine, &c of an angle 59 — 63 Inverse Gouiometric Eatios 64, 65 Examples 67 CHAPTER IV. The Solution of Triangles and of Plane Eectilineal Figures 69 — 96 Subsidiary Angles 97, 98 CHAPTER Y. Analytical Trigonometry 99 — 125 The Circumference of a Circle varies as the Eadius 99 Circular Measure of an angle 100 — 103 ^>sin e, and < tan^ 104 Smd>d-id^ 105 vi INDEX. AUTICLES Number of seconds in 6=^-. — ry, 107 sm 1 Area of Circle of Eadius r = 7rr"^ 108 Demoivre's Theobem 109, 110 Sine and Cosine in terms of the Sines and Cosines of tlie multiples of the angle Ill, 112 Tann^ in terms of Tan ^ 113 Sin a and Cos a in terms of a 114, 115 Series for finding the value of ir '. 116 — 118 Miscellaneous Peopositions 119 — 135 CHAPTER YI. On the Solution of Equations by means of the Tbigonometric Tables, and the resolution op x^J=1, Bin 6, and Cos ^ into Factoes 126 — 133 APPENDIX I. On the Logarithms of Numbers, and the Construction and use of Tables of Logarithms of Numbers. APPENDIX II. On the Construction of the Tables of Natural Goniometrical Ratios. APPENDIX III. On the Construction of the Logarithmic Tables of the Goniometrical Eatios. APPENDIX lY. The General Proof of the FoRMUL^aB for Sin (^±.5) and Cos(i±B). A Collection of Examples for Practice. The Student is recommended to confine his attention at first to the Articles, 1_27 ; 30—32 ; 34 ; 38—43 ; 46 ; 50 ; 51 ; 58 ; 64 Appendix I. 1—13 ; II. 1—8 ; III. 1, 4, 5, 12. Ai-ts. 69—76; 78; 81—83; 86; 87; 90; 99-109 omitting Cor.; 110 ; 113—115 ; 118. INDEX. vii SPHEPJCAL TRIGONOMETRY. CHAPTER I. ♦ RTICLFA Explanation of the objects of Spherical Trigonometry 1—5 Certain preliminary general properties of Spherical Triangles esta- blished by means of Solid Geometry 6 — 26 Recapitulation of the results estabhshed in this Chapter 27 CHAPTER II. General Formula connecting the sides and angles of Tri- angles WITH ONE another 28 — 85 Values of the Angles in terms of the Sides of a Triangle, and vice versd, adapted to Logarithmic Computation 31, 32 Napier's Analogies 33 Gauss' Theorem 35 CHAPTER III. The Solution of right-angled, and QUADiuNT.Ui, Triangles ... 36 — 43 CHAPTER IV. On the Solution of oblique-angled Triangles 44 — 55 The radii of the small circles found which can be described within and about a given Triangle 56—59 CHAPTER V. On the Areas of spherical Triangles ; and the Solution of Triangles whose bides are small compared with the radius of the sphere 60 — 02 Cagnoli's and Lhuillier's Theorems for finding the Spherical Excess 63, 64 Solution of Triangles whose sides are small compared with the radius of the Sphere 66 — 76 Legendre's Theorem 73 The Chordal Trl\ngle 74, 75 Vlll INDEX. CHAPTER Vr. ARTICl.ES. On Geodetic Meastjkements ; the Instruments employed, and THE MANNEE OF USING THEil 77 86 CHAPTER YII. On THE SMALL CORRESPONDING VARIATIONS OF THE PARTS OF A Spherical Triangle ; and the connexion existing be- tween CERTAIN Formulae op Spherical Trigonometry and ANAIiOGOUB formula OF PlANE TRIGONOMETRY 87 — 96 CHAPTER YIII. The Eegular Polyhedrons : certain properties of them inves- tigated 92—103 Examples for Practice. The Student is recommended to confine his attention at first to these Ai'ticles; 1—41; 44—55; 60—62. PLANE TEIGONOMETEY. CHAPTER I. ox THE METHOD OF REPRESENTING LINES AND ANGLES, AND ON THE DIVISIONS OF ANGLES. 1. The magnitudes of lines may he represented by Alge- braical quantities. The length of a line is measured by finding how many times it contains a fixed and definite line, as an inch or a yard, which has been previously fixed on as the unit of length. Thus, if an inch be taken as the unit of length, a line is said to be 30, if it contain an inch thirty times ; and in like manner a line is called a, if it contain the lujit of lenirth a times. o C B 2. If lines draiun in a given direction from a fixed line he represented by 'positive quantities, then those u'hich are drawn from it in a contrary direction will be represented by negative quantities. If to a line AB measured at right angles from A towards the right of the fixed line Ay i% be required to add a given line, it . is evident that AB must be pro- duced to a point C such that BC shall be equal to that given line : and if from J^ a given line has to be taken, a pai*t BC must be cut ofi* from BA equal to the given line, and AC will be the required remainder. Let AB be represented by a, and BC by 5, and make BC = BC ; then AC + CB = AB ; .-. AC = AB-CB^a-b. S. T. 1 PLANE TEIGONOMETRY. C lies on the other side of V, Now if 6 be greater than A. AndAC" = a-b = -{b-a), which is a negative quantity equal in magnitude to the dif- ference of the lines h and a; ^ which difference, it is seen, lies in this case to the left of Ay. Hence, if a line be represented by a negative quantity, it is meant that it is measui^ed from Ay in. Si dii^ection contrary to that in which the lines represented by positive quantities were measured. 3. Let X'AX, F'^r be two fixed lines at right angles to each other, and produced, if necessary, indefimtely. Then the position of any point F^ in the plane of these lines is known, if the magnitudes of the perpen- diculars be known which are let fall from P^ upon the lines XAX and Y'AT, viz. F^N^ and F M (or AN^, which is equal to P^M). A!N"j, NjPi, are called " the co-ordinates of P^," and are said to be ^^ referred to the rectangular axes AX and AY." With respect to lines measured from A in the direction of the axis XAX, it is usual to call those ijositive which are measured to the right of the axis TA Y, and therefore those will be negative which are measured to the left of Y'A Y. With respect to lines measured in the direction of Y'AY, it is usual to call those positive which lie on the wpper ^side of X' AX and therefore those will be negative which Ke on the lower side of X'AX. Thus if Pj be a point in the space contained by AX and A Y, its ordinate AN ^ which is measured in the direction of XAX is positive, because it lies to the right of Y'AY; and its ordinate iVjPj measured in the direction of Y'A Y is j^ositive, because it lies on the upper side of X'AX. But if F^ be the point, its ordinate AX measured in the direction of X'AX is negative, because it lies to the left of Y'AY ; and the ordinate NJP^ measured in PLANE TRIGONOMETRY. the direction of F'AY is positive, because it lies on the upper side of X'AX. Similarly, the ordinates of P^ and P^ measured in the direction of Z'JXare negative and positive respectively; and the ordinates measured in the direction of Y'A 7 are negative in both cases. 4. Def. If a straight line revolve in one plane round its extremity A from a given position^. as AB, into any other position, as AC, the inclination of ylC to ^-5 is called an a'ogle ( ^ ) ; and the angle is signified by the letters BAG or CAB, the middle letter being that placed at the point in which the two lines meet. By continuing this revolving motion, the angle may be sup- posed to become of any magnitude whatever. 5. Def. If AD be equally incliued to the pai-ts AB, AB' of the straight line BAB', each of the angles BAD, BAD is called a right angle. 6. Def. An acute angle is less, and an obtuse angle is greater, than a right angle. 7. If the angles formed hy AC revolving in one direction, (as BCD) from the fixed line AB, he considered positive, then if AC revolve in the contrary direction from AB, it will trace out negative angles. If to the angle BAG, Fig. Art. 4, it be required to add a given angle, GA must move in the direction BCD through an angle GAE equal to the given angle, and the whole BAE will be the angle required. And if it be requii'ed to take a given angle from BAG, GA must evidently move in a contrary direction till it come into a position E'A, such that l GAE' is equal to the angle to be subtracted. Then L GAE' + z E'AB = z BAG ; .'. L E'AB = z BAG- L GAE'. Now if L GAE' be greater than l GAB, E'A lies on the other side oi AB, and L E'AB =lBAG-lGAE' = -(zGAE'-zBAG), a negative quantity, whose magnitude is the difference between the angles GAE' 1—2 4 PLANE TRIGONOMETRY. and BAC ; wMcli difference lies in this case on the lower side of AB. Hence, when an angle is called negative, it is meant that it is formed by the revolving line moving from the fixed line in a di- rection contrary to that in which it revolved to trace out positive angles. 8. A light angle is divided by the English into 90 equal angles which are called degrees ; a degree is subdivided into 80 minutes, and a minute into 60 seconds. And the magnitude of an angle is expressed by stating how many degrees and subdi- visions of a degree are contained in the angle. If great accuracy be requii-ed, the parts of an angle which are less than a second are expressed in decimal parts of a second. A degree and its subdivisions are thus indicated, 24", 50', 34" '7, which denotes an angle containing 24 degrees, 50 minutes, 34 seconds, and seven tenths of a second. 9. By the French and other Continental Mathematicians, a right angle is divided into 100 equal angles called grades, a grade into 100 minutes, and a minute into 100 seconds; and the divi- sions are thus marked, 26^, 24\ 32''-47. Now since T = ^^ = -Ol', and 1" = ^^ = Jl = -OOOP ; the above angle might have been written thus, 26^*243247. Whence it appears that if the French division be adopted, arithmetical operations can be performed on angles in the same manner as on any other decimal fractions; an advantage which does not attend the English division. 10. To find the relation betiueen E and F, the number of Degrees and of Grades contained in the same angle BAC. (Art. 14. Fig. 1.) T ^, T^ T , T . . ^" ^iven z BAG In the English division, 7-:-^ = - — r-^ : * ' 90° right z ' T XI T^ 1 T • • F^ given z BAG In the -b rench division, ^r^-^ = ^ — ^^ ; ' 100^ right z ' L 9 ^^9-"W- PLANK TRIGOXOMETRY. 5 Note. In employing the latter of these formulae it will be necessary to express the minutes and seconds of the angle in decimal pai'ts of a degree, since E represents the number of degrees in the angle. Ex. 1. To find how many degrees, minutes and seconds are contained in the angle 42", 34\ 56'\ i5' = 42-3456 --P- F 10" 4-23456 .-. E^ F "lu" : 38-11104 60 6-6624 60 30-744 Or BS**, 6', 39' -74, retaining the tenths and hundredths and neglecting the thousandth parts of a second. Ex. 2. Find how many grades, minutes and seconds are con- tained in the angle 24'', 51', 45". First reducing the minutes and seconds to the decimal parts of a degree, 60;45" .-. £ = 24-8625 60j 51-75 :|= 2-7625 •8625; .-. i;+:^ =27-6250 V and .-. i='=27', 62', 50". 11. Def. The Complement of an angle is its defect from a right angle. Thus, 900- 240, 32' = 650, 28', is the complement of 240, 32'. gO^-llO^, 15' = - (200, 15'), is the complement of 110^, 15'. 12. Def. The Supplement of an angle is its defect from two right angles. Thus, 1800- 560, 20'=1230, 40', is the supplement of 56o, 20'. 1800-1860, 12'=- (60, 12'), is the supplement of I860, 12'. A Collection of Examples and Problems is placed after the fourth Appendix. CHAPTER II. THE GONIOMETKICAL RATIOS, AND SOME FORMULA CONNECTING THEM WITH EACH OTHER. 13. Def. Plane Trigonometry , in its original meaning, implies the measuring of plane triangles ; in its extended sig- nification it treats of the formulae connecting the relations of ano-les with each other, and of the determination of the parts of plane rectilineal figures from sufficient data. 14. Let a straight line revolve from the fixed line AB round the point A, in the direction of the lettei-s -5, Z), B\ D' , and come into the position AC. B' »' W B' N ^ BIN. R B' N B ly From any point (7 in ^(7 draw CN at right angles to AB,-~ produced either way if necessary j and through A draw DAD' at right angles to AB. Now, for the reasons given in Arts. 2 and 3, in these figures the signs of NC are +, +, -, - respectively, and those of AN are +, -, -, + respectively. 1. 15. Definitions. -r-P, is the 8ine of the l BAG ^ AU See the figures of the last Article. or, Sinz^4C=-T^. AG PLANE TRIGONOMETRY. 7 AN AN 2. — =, is tlie Cosine of the z BAC ; or, Cos l BAG= -j^ . NC NC 3. y^ is the Tangent of the i BA G ; or, Tan z ^^ C = j-^ . 4. —t:^ is the SecaiU of the z BAC \ or, Sec z J5^C = -t-^^ • 5. 1 - cos z ^JLC is the Versed-sine of the z ^^C ; or, Versin z ^^6' = 1 - cos z BAC. G. The Tangent of the Complement of the z BAC is called the Cotanyent of the z BAC ; or, Cot lABC^ tan (90" - z ^J6'). Cor. If 90" - z BAC be the oi-iginal angle, its Complement is z iy^C, Art 11, .-. Cotan (90"- z BAC) = inn z BAC, or, Tan z i)M6' = cotan (90° - z BAC). 7. The Secant of the Complement of the z BAC is called the Cosecant of the z i^.lC ; or, Cosec l BAC = sec (90" - z BAC). Cor. If 90"-/ i?vlC be the original angle, its Complement isz^^C, Art. 11, .-. Cosec (90"- z BAC) = ^Qc z BAC', or, Sec z BAC = cosec (90" - z ^.4 C). 16. The cosine of iBAC might have been defined to be the Sine of the Complement of lBAC. A N ForCosZ5^(7=^ = sinZiCiV=sm(9(y>-z5^C). Art. 14: Fig. 1. NC Also, Sin Z^-4C= 7^ = cos z ^(7iV= cos (900 -z^^C), A. m the Sine of an angle is equal to the Cosine of its Complement. For the sake of convenience an angle will generally hereafter be indicated by a single letter, as Sin^, Cos ^, Tan -5, where A, B respectively represent the number of degrees contained in *he angle. 8 PLANE TEIGONOMETRY. 17. So long as the magnitude of tlie angle is unaltered, its Sine, Cosine, Tangent, &c. remain the same, whatever be the magnitude of J. (7. For let D be any other point in ^C, and DM perpendicular to AB. Then, by definition, Sin 4 = — -^ ; or Sin ^ = -j^ . ^ . -, .•-... , NO MD a- A ■ But by similar triangles -j-p — -777 ' ®^ ^"^ -^ ^^ the same wherever in the line A O the point C be situated. Similarly it may be shewn that Cos 4, Tan^, Sec^ ... are invariable quantities so long as J^ }^^ M~ the magnitude of A remains unaltered. Hence, if any of the quantities Sin A, Cos A, Tan A, Sec A... be given, the angle A may be determined. 18. Def. The Ratios which are called the Sine, Cosine, Tangent, &c., of any angle are termed " The Goniometrical Ratios," because when any one of them is given the angle may be determined to which it belongs. These Ratios are also called " Trigonometrical Functions " of angles. 19. To express Yersin A, Cot A, Cosec A, in terms of the sides of the triangle ANC. (Fig. 1. Art. 20.) AN (1 ) Yersin j4 = 1-cosJ. = 1- — ^ . A (2) Cot A = tan (90'' -A) = tan ACJV NA = y^pj^ , by def. of the tangent. (3) Cosec A = sec (90" -A)= sec ACJ^ CA = 7:7^^, by def. of the secant. 20. To trace the variation in the algebraic signs o/SinA, Cos A, Tan A, Sec A, as A increases from 0** to 860^ (1) Sin A = -jyy , and therefore has the same sign in any case as ]^C has ; for A G which lies in the direction of neither of the PLAXE TRIGONOMETRY. 9 lines AB and AD, cannot change its sign, and is always to be reckoned as positive. Hence (14) Sin^ is positive if ^ be an angle between 0° and 180" (figs. 1, 2); and is negative if ^ be between 180" and 360°. (Figs. 3, 4.) A V (2) Cos A = — ^, , and therefore has the same sign as AN. ALf Hence (14) Cos ^ is positive if ^ be between 0° and 90°, or between 270° and 360" (tigs. 1, 4); and is negative if ^ be be- tween 90" and 270". (Figs. 2, 3.) B' D' A K B' I* B B'NL V. K N D liy NC (3) Tan -4 = ^ tv, and is therefore positive or negative accord- ing as NC and AN have the same or different signs. Hence (14) Tan ^ is positive if A be between 0" and 90", or between 180" and 270" (figs. 1, 3); and it is negative if A be between 90" and ISO'', or between 270" and 360". (Figs. 2, 4.) AC (4) Sec A = j-T^, and therefore has the same sign as AN. Hence Sec J is positive if A be between 0" and 90", or between 270" and 360"; and it is negative if ^ be between 90" and 270". 21. To trace the variations in the magnitudes of the Sine, Cosine, Tangent, and Secant, as the angle increases from 0" to 360^ (Figs. Art. 20.) Since (17) the values of the Sine, Cosine, Tangent, and Secant are not affected by the magnitude of AC, suppose this line to remain of the same magnitude while the angle A increases from 0" to 360". Kow (fig. 1) as AC revolves from the position AB into the position AD, NC increases in magnitude from to ^C, and is positive; and AN decreases from AC to 0, and is positive. 10 PLANE TEIGONOMETRY. As (fig. 2) AG revolves from tlie position AD into the position A^ , NO decreases in magnitude from AG to 0, and is positive; and AISF increases from to AG, and is negative. As (fig. 2) AG revolves from AB' to AD', NG increases in magnitude from to AG, and is negative; and AN decreases from -4C to 0, and is negative. As (fig. 4:) AG revolves from AD' to AB, NG decreases in magnitude from ^C to 0, and is negative; and ^iV' increases from to AG, and is positive. Hence it appears, that as A changes from Sm A {j^ Cos A (^ Tan^(JJ «-^© 0" to 90° +AG AG^^'^V +AG AG"'AG +AG +AG'" AG AG +AG'" 90° to 180° 180° to 270° 270° to 360° +AG ^ to AG AG AG"'~AC +A^ _0_ '"-AG AG JlG^ '"-AG -AG AG^''^X! -AG _0_ ~A'G"'AG -AG'" AV AG -AG"' -AG^ to AG AG +AG AG' -AG AG '" + AG AG +AG AG These changes in sign and magnitude of the Sine, Cosine, Tangent, and Secant may be thus exhibited; the signs which belong to them in each right angle being written in a bracket. The symbol go indicates an infinitely large quantity. A being) between) Sin^ Cos A Tan^ Sec J. 0° and 90° and 1, (+) 1 ... 0, (+) ... oo, (+) 1 ... CC, (+) 90° and 180° 1 and 0, (+) ...-i,(-) a. ... 0, (-) 00 ...-1,(-) 180° and 270° Oand-1, (-) -1 0,(-) a>,(-) -1 <^,{-) 270° and 360° - 1 and 0, (-) ... 1,(+) a> ... 0,(-) 00 PLANE TRIGOXOMETEY. 11 Since Cos A is never greater than unity, Yersin A (or 1 - cos A) Is always positive; and its gi^eatest value is when A becomes 180", when Cos^ becomes- 1, and Yersin^ becomes 2. 22. To shew that Sin A=sin (180°- A), or =-sin (180'+ A), 07' =- sin (360" -A), or =-sin(-A); where A is an angle less than a I'ight angle*. Let BAC^=A = B'AC^ = B'AC^ = BAC ; 2indiAC^ = AG^ = AC^ = AC\. Join C,C,, and Cfi^. It may easily be shewn that the angles at M and N are right angles; B and that NC^, MC\, MC^, NC^, are equal in magnitude, — as are also AN and AM. Now,Sm^=---=-^^» D c 1 C, , ]M \v ^y^ \ C 8 C4 D' sin^^C„ = sin {BAD + BAB' - B'AC^ = sin(180"-^) (1). Again, Sin ^ = -^ = -j^q- , since NC^ = - MC^, = - sin {BAD + DAB'+ B'A C^) = -sin(180° + J) AC/ .(2). Again, S"i^ = 20 "IcT 1 4 but ^7^" is either the sine of the positive angle {BAD + i)^^' + B'AD' + {D'AB - BACy^, or the sine of the negative angle BAC^, (7) ; .-. Sin^=-sin(360''-^) (3), or = — sin (— ^) (4). • In strictness these angles ought to be written thus, 4", (180-^)", (180 + ^)0, &c. 12 PLANE TKIGONOMETRY. 23. In like manner it might be shewn that (1) Cos^=-cos(180''-^),=-cos(180V^),=cos(360*'-^),= cos(-^). (2) Tan^=-tan(180''-^),=tan(180V^),=-tan(360*'-^),=-tan(-^). (3) Sec^=-sec(180°-^),=-sec(180''+^),=sec(360''-^), = sec(-yl). 24. If any angle, as ^^C^, be increased by 360", the line which bounds it will come into the same position again, and the sine of the angle will therefore remain unaltered. Wherefore sin ^ is in all cases the same with sin (360" + J.); and in like manner, sin (360" + A) = sin (2 x 360" + A), and so on. If, there- fore, n be any positive integer, Sin^=sin(?^.360"+^) = sin(2^.180+^) (1). Similarly, Sin ^ = sin (180"-^), Art. 22, (1), = sin{2n. 180" + (180"-^)} = sin{(27^ + l). 180"-^} (2). In like manner it appears from (2) and (4) of Art. 22, that Sin j; = - sin {(2^ + 1). 180" + ^} (3), Sin^=-sin(2^^.180"-^) (4). 25. By the same process of reasoning it may be proved from the formulae of (23) that Cos^ = cos(2^.180"+^), or = -cos{(2w+l).180"-^}, or = -cos{(27^+l).180" + J^}, or= cos (2^. 180"-^). And, Tan J[ = tan (2^. 180" + ^), or = -tan{(2?^ + 1). 180"-^}, or = tan {{2n + 1) . 180" + A}, or = - tan {2n . 180" - A). PLAXE TRIGONOMETRY. 13 Similarly it might be proved that Sec^- sec{2n. 180" + ^), or = - sec {(2r6 + 1). 180°-^}, or = - sec {{2n + 1) . 180" + ^}, or = sec {2n . 180" - A)*. * The relations, similar to those given in Arts. 22 — 25, between the trigo- nometrical functions of m.l80±^ and those of A may be established directly, whatever be the magnitude of A, as follows. [See figui-e to Art. 22,] All angles are supposed (Art. 4) to be described by a line revolving from the initial position ^^ to some other position AC. Let AB be called the initial line, and AC the terminal line of the angle BAC. Then it will be seen that 1. The Sines of all angles whose terminal lines lie on the same side of B'A B will have the same algebraical sign ; 2. The Cosines will have the same sign for all angles whose terminal lines lie on the same side of D'A D ; 3. The Tangents will have the same sign for all angles whose terminal lines lie in the same quadrant, or in the alternate (or opposite) quadrant. Now ^ and 2n . 180'' + J , (where n is any integer, positive or negative), have the same terminal line, and therefore all the trigonometrical functions of 2n . ISO** + A are the same as those of A . Again, the terminal line of (2n + l). 180''+^ will be the terminal line of A produced, and will therefore be in the alternate quadrant and on the sides of both B'AB and DAD that are opi)osite to that in which the terminal Una of A lies. Hence, the magnitudes of the trigonometrical ratios remaining the same, Sin {(2n+l).180« + .-I}=- sin^, Cos {(2/i + 1).1800 +^; =- cos.4, Tan{(2n + l).180*' + J}=tanJ. Again, A and -A will have their terminal lines in the adjacent quadrants that are on the same side of D'AD but on opposite sides of B'AB. Hence, Sin (- ^) = - sin^, Cos{-A)—cosA, Tan (- J) = - tan.4. Also (- A) and 2n. 180" - A will have the same terminal line, and therefore Sin (2n. 180« -A) = -smA, Cos (2« . 180° - ^) = cos ^ , T..n (2n . 180" -A) = -tanA. Again, A and (2n + l).180'' -^4 have their terminal lines in the adjacent quadrants which lie on the same side of B'AB and on opposite sides of D'AD. Hence Sin{(2n + l).180''-^}=sinJ, Cos {(2n + l).1800-^ } =-cos^. Tan { (2;i + 1). 180" - ^ ; = - tan A. Collecting the above results. Sin ^ = sin (2n . 180" + A) = sin {{2n + 1) . 180" - A } = - sin (2n . 1 80" - ^) = - sin {{2n + 1) . 180^ + A] ; Cos A ^ cos (2n . 180" + ^) = cos (2n . 180" - A ) = -cos{(2n + l).180" + ^} = -cos {(2;?. + l).180"-^}; Tan 4 = tan (2?i . 180" - ^) = - tan { (27i + 1) . 180" - A } = - tan (2n . 180" - ^) = tan } (2n + 1) . 180" + A}. 14 PLANE TRIGONOMETKr. 26. From Arts. 16, 22, 23, it will appear that Sill ^ = 008(90"-^) Cos^= sin (90' -^) Sin ^ = sin (180"- ^). Cos ^ = -cos (180"- ^). Tan^= cot (90°-^) Sec^= 00860(90"-^) Tan^ = - tan (180" - A). Sec ^ = - sec (180° -^). That is, The Sine of an angle = cosine of its complement, or, = sine of its supplement. Cosine of an angle = sine of its complement, or, = — cosine of its supplement. Tangent of an angle = cotangent of its complement, or, = — tangent of its supplement. Secant of an angle = cosecant of its complement, or, = — secant of its supplement. Note. These relations between the sine, cosine, tangent, and secant of an angle and the sine, cosine, tangent, and secant of its complement and supplement, are perpetually occurring in practice, and will often be made use of in the following pages. These two formulae also are often useful ; Sin ^ = cos (90"-^)= -cos {180" -(90"-^)} = -cos (90" + J); Cos A = sin (90" -A)= sin {180" - (90" - A)} = sin (90" + A). 27. It will be found necessary to carry in memory the following expressions, c ,^. ^ . HO AG sinJ (1) Tan ^=-7-^=--— = r. ^ ^ AJ}i AN^ cos A AC A N B PLANE TRIGONOMETRY. 15 /9\ Q J _ _ ■*• _ •'■ . . n i _ ^^^ ~ AJ^~ A^~ cos A' ' ' "sec J.' Jo AN (.3) ^otA-^^^,-^-^^^. AG (i) Cot A = ^-^ = -77, = -. : ^ ' NO NO tau^' AN .-. Tan^ = -^ , . cot A (5) CoseC J = .^p, = -rrrr = -r— . \ .'. Sill J^ = . . ^ ' NO JS a sin ^ ' cosec A fNCK- fAN"- AC) ' AO (6) AC = NC + AN' ; • '• 1 = (^^T -^ ( or 1 = (sin ^)* + (cos Ay ; * .'. Sin ^ =^(1 — cos^ Af and Cos A = ^^(1 — sin' A). (7) AC' = AN' + A^C'; .• Y^^j = 1 + ^ — ^^V or Sec' ^ = 1 + tan'' yl ; .'. Sec A = ^(1 + tan'' A) ; and Tan A = J {sec- A - 1). (8) AC"' = AN' + NC' ; * '• ( "Wn ) ~ \ 'Wr ) + 1 ^ ^1' Cosec" A = cotan" J. + 1 ; . \ Cosec ^ = ;^(1 + cot'' A) ; and Cot A = ^(cosec^ ^ - 1). * The powers of the goniometrical ratios, as (sin^)-, (cos^)^, {tan^i}'*, are generally written thus, sin^^, cos*^, tan"' J.. IG PLANE TRIGONOMETRY. 28. Bj means of the expressions proved in the last Article the value of any one of the quantities defined in (15) may be found in terms of any other of them. For example : (1) Tan^ = sin J. V(l-sin^^) • For Tan^ = ^^, Art. 27,(1) j = cos J. ^ \ / ^ sin J. V(l-sin^^) ; (27, 6). (2) Tan^^g^ = ^(^-^f ^). cos A cos A (3) Sin A = , . cos ^ = tan A . tan^ cos^ sec A J {I + tan^ A) ' 29. The formula proved in the last Article will often be found useful to the analyst. The same method of proof is applicable to all other questions of the same kind. Thus, required to express the cosine of an angle in terms of the cosecant, and the cosecant in terms of the versed sine : (l)-Cos^ = v'(l-rin^^) = y;i-,^| = V(cosec^^-l) cosec A (2) Cosec ^=- sin A \/(l - cos^^) 1 Vl — (1 - versin J.)^ ^1 "" \/(2 versin A - versin^^) * 30. It will be found useful to remember the following values of Sin yl, Cos A, Tan A, Sec A. Sin A Cos A Tan 4 - Sec^ J{l-co^'A) tauudL ^(1-sin^^) 1 sin A 1 V(l-sin^^) V(l-cosM) cos^ ^(secM-1) V(l-sin^^) 1 cos^ V(l+tanM) ^(1+tanM) ^(secM-1) sec- A _V(l+tanM) 1 sec A PLANE TRIGONOMETRY. 17 31. If K he less tlmn half a right angle, or 45°, Cos A ii greater than Sin A. Let z NAC be less than 45". Then, since / NAG + z NO A = 90°, z NO A c is greater than 45". And in every triangle the greater side is opposite to the greater a- angle (End. i. 19) ; "" AN NO .*. AN> NO ; .'. -— ■ > -, , or Cos .-1 > Sin A. AO AC Similarly, for angles between 45° and 90°, it may be shewn that the Cosine is less than tlie Sine. 32. To fiiul tJie Sines, Cosines, and Tangents of 45°, 30°, and 60°. (1) (Fig. Art. 27) Let z ^.1^ = 45°; .-. z NCA^^O' - z .y.4C=45°; ^AN NG ^. ,.0 ^ ,., ,'. -779 = -jYi ') 01' Sin 4o = Cos 4a . AO AO Also, yl6'' = AN' + CN' = 2AN' ; J V 1 1 yV^ ... Sin45° = 477 = -7-; Cos45°=4-; Tan45°=^ = l. AC J-2 J-2 AN (2) Let ABC be an equilateral and equi- angular triangle; each of its angles, therefore, being ^ of two right angles, contains 60°. Let AD he perpendicular to £C ; .:BD = DC = IBC = IAB; and z BAD = i DAC = 30' ; •Sin 30°--^^-*^^-^ .Sm30 -Jj^--j^-2' Cos30°=V{l-sm^30}=V(l-i) = x/3 Tan 30° = S.T. sin 30° __ 1 cos30°~;yl' 18 PLANE TRIGONOMETRY. (3) Sill 60" = cos (90" - 60"), by (16), = cos 30° - ^^, A Cos 60° = sin (90° - 60») = sin 30" = i Tan60«=!^«°:=y3. COS 60 V 33. Equations like the following may often be solved by means of the relations established in (27) between the different Goniometrical Ratios. Ex. 1. From the equation, Siii^A + 5 cos^A = 3, required the value of Sin A. Since Cos^^ = 1 - sin^^, the equation becomes Sin^^ + 5 {1 - sin^^} = 3 ; whence 4sin2J = 2: and Sin^= — 3. V2 Ex. 2. From the equations Sin A = m . sin B, and Tan A = n_. tan B, required the values of Sin A and Cos B. For Sin A put x, and for Cos B put y\ .'. shiB = Jl- y\ TanB = i^il^i^=i^i^, (30). cos^ y Making these substitutions, the equations become r^ — -9 ;j ^ J'^-y'^ x = vijl-y^ 1 and , ^^ =^ = n.— =^ ; ^1-x' y ■whence, £c=sm^= . / — ^ ; and v = cos B — — , / -. r, . \/ 1 - %2 ^ m -y^ l-n^ -17, o ry- [m = cosec A — sill Al . 7 ^ /• 7 Ex. 6. Given < _ . a _ a r 5 'i^^quired to jina an equa- tion between m and n in which the angle A shall not ajjpear. rrv, A- n • COS^^ Sin^^ The equations severally give, m = —. — r , and n = . ; ^ J e. J sm 4 ' COS 1 ' n sin^J. , „ . A. A n^ .'. — — — 5— = tan3.4 ; .*. tan^=— . m COS-* J. ^j „ cos*^ eos^^ 1 Also, m^ = ~.—^-r = r— TT = T 5-; TT • sin^A tan^^ tan^^ sec' -4 .-. mnanM.(l + tan2J) = l; whence, by substituting its value for tan^, m3;i3(w3 + 7i3) = l. CHAPTEK III. GONIOMETRICAL FORMULA INVOLVING MORE THAN ONE ANGLE. 34. Given the Sines and Cosines of two an^fles, to find the Sines and Cosines of the awjles equal to their Sum and to their Difference. Let BAC, CAD be two angles repre- sented by A and B respectively. From D, any point in AD, draw DB and DC perpendiculars on AB and AC \ and from C draw CE and CF perpen- diculars on AB and DB. Then FE is a rectangle; FB=CE, and FC = BE. t CDF = 90° = z DCF- i FCA = A, since FC is parallel to AE. Now, Sin {A + B) = BD BF+FD EC FD AD AD AD "^ AD Also, Cos (^1 +B) = ec .ac fd dc ''^aCad^dc'Td = sin A co%B ■¥ cos A sin B AB AE-EB AE FC ,(1). AD AD AD AD ^AE AC_^FG CD ~ AC AD CD'aD = cos ^1 cos B - sin .1 sin B (2). 2—2 20 PLANE TEIGONOMETEY. Again; let /.BAC=^A, and lCAD = B. From D, any point in AD. draw DB and DC perpendiculars on AB and AC, CE a perpendicular from G on AB, DF perpendicular to GE. FB is a rectangle; FE = DB, and FD = EB', I EGF= 90'- I AGE = A. m a- /A 7?s -^^ EG-CF _^ ^ _CF CD ~AG'AD~GB AD = sin ji cos ^ COS. sin^ (3). ., ^, .. ^. AB AE + EB Also, Cos(^-^)=-^= -^^— AE FD AD'^ AD AD _AE AV FD GD ~AG'AD'^ GD' AD = cos A cos B + sin A sin 5. .(4). Note, The four formulae of this Article are proved gene- rally for all values of A and B in Appendix iv. Ex. Given the Sines and Cosines of 45** and 30°, required the Sine and Cosine of 75°, and of 1 5°. Sin 450 = cos 450 =i=; sm30»=A cos300=*^, (32). Sin 750 = sin (45o + 300) = sin 45o. cos 300 + cos 45». sin "30» ^2 '^ ^2 2 2^2 ^ Similarly, Cos 75o = cos {450 + 300)= -^(^/3_1). Sin 150 = sin (450 - 300) = -^( ^y3"- 1). Cos 150 = cos (450 - 300) ^ _L^ (^+ 1) , 2 V 2 PLANE TRIGONOMETRY. 21 35. In the figures attached to the last Article, each of the simple angles A and B was represented as less than a right angle, — as was also their sum. But of whatever magnitude these simple angles are, if the same construction be made, and proper attention be paid to the signs of the sines and cosines of A and B, the same result will invariably be arrived at. For example, let it be required to prove the formula Sin {A-B)= sin A cos B — cos A sin B from the annexed figure, where BAC'=A, and C'AD = B; each angle being greater than a right angle. From Z>, any point m AD, draw DC perpendicular to C'A produced ; let CBF be pei-peudicular to AB, DF parallel to AB, DB perpendicular to AB. Then FB is a rectangle, and EF ^' = DB. BD EF Now, Sin {A-B) = AD AD CF - CE AD W ' but _ C^ CD _CE AC^ ~ CD' AD AC' AD = cos FCD .BmDAC-smCAE . cos DA (7, CoaFCD = co& EAC = - cos {180'^ -CAB), by Art. 2G, = - cos C'A B = - cos A . Bin D AC = Bin {180'^ -D AC), Art. 26, ^BmCAD = smB, sin CA E= sin C"^ 5*= sin A , COB DAC=- COB {180'^ - D AC) = -cos B; Sin {A- B)=- cos AsinB + sin A cos B = sin A cos B - cos A sin B. 36. If any one of the formulce of Art. 34, as Sin (A + B) = sin A . cos B + cos A . sin B, he given, tJie others may he deduced from it. For let B become (-5), then Sin(^ -5) = sin{^ + (-^)}=sin^.cos(-5) + cos^ .sin(- B), But Cos (- B) = cos B, (23), and Sin {-B) = - sin B, (22) ; .♦. Sin {A-B) = BinA cos B - cos A sin B. 22 PLANE TRIGONOMETRY. Again, Cos(4 +5) = sm{900- (J +5)}, (16) =sin {(900-^) + (- 5)} = sin (900 _ ^) . cos (- B) + cos (900 - A) . sin (- B) = cos^ cos5-sin^ sin5. Similarly, Cos {A - B) may be proved =cos A cos 5 + sin 4 sin B. 37. From tlie formulsB of (34) the Sine or Cosine of the sum of three or more angles may easily be found in terms of the Sines and Cosines of the simple angles. Given the Sines and Cosines of the angles A, B, C, required the Sine o/ (A + B + C). Sin(^ + ^+C) = sin{(^ + 5) + {7} = sin(^+5)cos(7+cos(^ + 5) sin (7 = (sin ^ cos B + cos A sin B) cos 0+ (cos A cos 5 - sin ^ sin 5) sin C = sin A cos B cos C+ sin B cos A cos 0+ sin G cos A cos ^ - sin A sin 5 sin C. In like manner Sin(Ji5±(7), and Cos(4±5±(7), may be found in terms of the Sines and Cosines of A, B, C; and the same method may be applied to the sum of any number of simple angles. CoK. li A + B + G— (2w + 1) . 180", where n is an integer, smce sm (2/1 + 1) 180'^ = 0, the above equation becomes Sin -4 sin 5 sin C= sin A cos B cos G + sin B cos A cos (7+ sin G cos A cos B. If n = 0, A +5+ C=1800, and therefore this equation expresses a relation which exists between the sines and cosines of the three angles of any plane triangle. 88. To shew that Sin 2A = 2 sin A . cos A. Sin (A +B)= sin J^ cos ^ + cos A sin B, which becomes, by writing A for B, Sia 2A = sin A cos ^ + cos ^ sin A = 2 sin A cos A. 89. To shew that (1) Cos 2A = cosM - sinM ; (2) Cos 24 = 2 cosM-1; (3) Cos24 = l -2 sin'^. PLANE TRIGONOMETRY. 23 (1) Cos {A + £)= cos A cos ^ - sin A sin B, and writing A for £, Cos 2 A = cos ^ cos ^ - sin ^ sin A = cos' A - sin- J. A^'ain : Cos 2 A = cos^ A - sin" A , and 1 = cos- ^4 + sin-yl ; .-. 1 + cos 2A = 2 cosM, and 1 - cos 2 A = 2 sin' .4. (2) Therefore, Cos 2 A = 2 cos' A -I. (3) And, Cos2xl-l-2sinM ^ , , rCosA + sin A= + v^(l + sin2A), 40. To shew tfiat \ ^ . • \ . /n ci.. .')a\ (Cos A — sm A = i \/[l — sin ^A). Since, Sin 2/1=2 sin .4 . cos A, and 1 = cos" A + sin' A ; .: by addition and subtraction, 1 + sin 2A = cos'^ +2 sin A cos ^ + sin-.l, 1 - sin 2/1 = C0S-/1 - 2 sin A cos /I + sin'.-l ; .'. Cos A + suiA=^ J{\ + sin 2/1), and Cos /I - sin /I = ± ^/(l - sin 2/1). 41. To shew that if A he less than 45°, rCos^ = 1 {J{1 + sin 2/1) + J{1 - sin 2/1)}, ^^'''' tsin A=l {J[l + sin 2.4) -J{1- sin 2/1)}. By (31) if ^<450, Cos^ is >Sin.-l, and they are both positive; therefore Cos ^4-siu-4 and Cos^ -sin^ are both positive quantities when A is 3x450 but <4x45'', (i.e. if it be any angle comprehended under the form 180^- 5, where B is <450,) Cos A is negative, and greater in magnitude than Sin^, which is a positive quantity. In this case, therefore, Cos^ +sin^ = -V(l + sin2^), Cos ^ - sin 4 = - \/[l - sin 2 J ) ; .-. Cos 4 = - 1 W{1 + sin 2A ) + \/{l - sin 2^)}, Sin^= ^{V(l-sin24)- V(l + sin2^)}. If at first sight this value of Sin A appear to be negative, it is to be remem- bered that sin 2A is a negative quantity, (24 being between 270" and 360") and therefore 1 - sin 2A is greater than 1 + sin 2 J. ; wherefore the value of sin A is here a positive quantity, as it ought to be. So (2), If 4 be a negative angle which is between -45" and -90*', Cos A is a positive quantity, and is less in magnitude than Sin A, which is a negative quantity ; wherefore the equations to be taken of Art. 40 are Cos 4 + sin 4 = - V(l + sin 24 ) , Cos 4 - sin 4 = + V(l - sin 24).* * Problem. To determine the limits hetioeen lohich the values of A must lie, which satisfy the equations, Sin 4 + cos 4 = - \/(l + sin 24), Cos4-sin4= -\/(l-sin24). For positive values of 4 , The former equation is fulfilled if 4 be between 90" + 45" and 180" : for the value of Cos 4, which is negative, is greater for such angles than the value of Sin 4, which is positive. So A may lie between 180" and 270"; and between 270" and 270" + 45". Wherefore, if 4 be between 135" and 315", the former equation is fulfilled. And it may be shewn, in Uke manner, that the latter equation is fulfilled if 4 lie between 45" and 225". Wherefore both equations will be fulfilled if A be between 135" and 225". And as the Sine and Cosine of any angle remain the same if the angle itself be increased by 360", it follows that all the positive values of 4 which satisfy both the equations, lie between n . 360" + 135" and w. 360" + 225", {i.e. between (8?i + 3) . 45" and (8n + 5) . 45"}, where w is or any positive integer. And in the same manner the negative values of A which satisfy both the equations may be shewn to he between - m . 360" - 135" and - m . 360" - 225", m being or any positive integer ; i. e. between - (8m + 3) . 45" and - (8m + 5) . 45", which are of the same form as the limits obtained for the positive values of 4. PLANE TRIGONOMETRY. 25 43. Given the Tangents of tvjo angles, to find the Tangents of tlieir Sum and their Difference. ^ , , „, sin (A + B) sin A cos B + cos .-1 sin B Tan (^ + ^) = \-^ -^ = ; TT : — —-. — = , ^ cos (^ + B) cos ^ cos B - sin J. sm B and dividing the numerator aud tlie denominator by cos A cos B, sin A sin ^ m / i r.\ COS A cos ^ tan A + tan i? Tan (il + i?) = r— : t; = .; =: . ^ ' sin A sm /> 1 - tan A tan ^ cos A ' cos /? tan A + tan 7? Similarly, Tan (.-1 - B) 1 + tan A tan ^ Cor. 1. \iB = A, Tan 2.1 = ?- --^. ' 1-tanM Cor. 2. If B = i5'^, since Tan45«' = l (32) ; .,TauU^4o")=*JSliii (1). 1 - tail ^ sin^ , cos J sin A + cos /I >iu .-1 cos J^ — tiiu A cos J. (2). Cor. 3. Similarly, Tan (^ - 45") = J— ^-^ (3)' 18,11 ^ T" J. sin 4 -cos J ,,, or =^ — (4). sm A + cos A Cob, 4 Tan ^ + 45') + tan (^ - 45") ='^2Hill + ^_i^ 4tan^ , ^ 2 tan 2^, by Cor. 1 (o). 1 - tau- J. If ^ be < 45" ; Since Tan(^ + 450) = tan-(450-^) = - tan (45°- J), (23), the last expression becomes Tan (450 + ^) -tan (450-^) = 2 tan 2^.. (6). 26 PLANE TEIGONOMETRY. Tan A+tsiJiB sin (A + S) 44. To shew that Tan^-tan^ sm(^-^)* sin A sin B Tan A + tan 5 _ cos A cos ^ Tan A — tan ^ sui A sm ^ cos JL cos B _ sin J. cos 5 + cos ^ sin 5 _ sin {A + B) ~ sin J. cos B - cos ^ sin B ~ sin (ii - B) 45. 6^^ve?^ Tcan A, Tan B, Tan C, to find Tan (A + B + 0). '...^.B.O^UnU.BHO)^^^^^^^, (43), tan A + tan 5 . ,^ 1 -; TT 77 + tan 6' 1 - tan A tan B tan 4 + tan B , ^ 1 .tanC/ 1-tan^ tan£ __ tan ul + tan ^ + tan (7-tan ^ tan .g tan C "~ i -tan^ tan^ - tan A tan C- tan 5 tan C ' In the same manner the tangent of the sum of four or more angles might be found in terms of the tangents of the simple angles. Cor. If A + B + C^-{2'n + l).180'^, n being or an integer, Tan{A + B+C)=-0; .'. tan ^ + tan if + tan C-tan A tan B tan (7=0, or, Tan A H- tan B + tan C = tan A tan B tan C. And since if 7i = 0, A + B + C=180^, this relation between the tangents of A, B, C, is one which exists between the tangents of the angles of any plane triangle. 46. To find the values of Sin 2A, and Cos 2A, in terms of Tan A. " J Sin 2^ = 2 sin ^ . cos A, (38), = " ^^^^ . cosM. ^ ' cos 4 = 2W, (27.2). =T^^,, (27.7). socM ^ ^ 1 + tanM ' \ ^ PLANE TRIGONOMETRY. 27 Again, Cos 2A = 2 cos' A - 1, (39. 2), = 2 sec'^l -1. 1 +taiiM 1 = 1 — tan-^ 1 4- tan" J. * 47. The following values of Sin2.-1, Cos 2^, Tan 2.1 are of frequent occurrence, and necessary to be remembered ; those which have not been proved already are easily found after the method of the last Article. 1. Sin2i4 = 2sin^cos i4. 2 tan A " 1 + tan" J. * 3 _ 27(secM-l) sec^yl 1. Cos 2.4 = cos" -4 — sin" -4. 4. Tan 2^ = 2 tan A 1 - tan-i 2. 3. 4. 5. = 2 cos"^ — 1. = 1-2 sin-^. _l-tan-yl 1 +tan-J * 2 - secM sec^^ 48. In the same way the following values of Sin 2A and Cos '2 A can be found in terms of Cot J, Cosec A, and Versing. Sm 2A = , -7,— , or = — ^^-^ —. , l + cot^^ ' cosec-M or =2 (1- versing) .■v'lS vers i4 -vers^J). ^ ... cot2.4-l cosec- J -2 i o /o a Cos 2^= ^„ - — -, or = 5-7— > or =1-2 2 ver?i-vers2J). cot^^ + l' cosec^^ ' ' 49. The easiest method of deducing such formulae as these i& first to ex2)ress Sin 2.1 and Cos 2 J. in terms of Sin .4 and Cos .4. eosee2.4 -2 Thus let it be required to prove that Cos 2A = cosec''^ A Since Cos 2il=cos-^ - sia-^ =1-2 siu-^ ; and Sin 4 = cosec A ' (27); .-. Cos2^=l- cosec- A cosec- A -2 cosec- A 50. Since Sin (^ 4- ^) = sin A cos B + cos A sin B, and Sin (A — B)- sin .4 cos B - cos .4 sin B ; 28 PLANE TRIGONOMETRY. .'. by adding and subtracting, Sin {A + £) + sin (A-JB) = 2 sin A cos B (1). Sin(^ + ^)-sin(^-^) = 2cos^sin^ (2). Similarly, Cos(7l + ^)+cos(^-^) = 2cos^cos^ (3). Cos(^-^)-cos.(^ + ^) = 2sin^sin^ (4). 51. To find the values of Sin A + sinB, and Cos A + cos B, in terms of the Sines arid Cosines q/" -^A + B) and J (A — B). Since A = l{A + B) + l{A-B\ 2indB = i{A+£)-i{A-B); .\BmA = sm^{A + B) cos ^(A-B) + cos i(A + B) sin ^{A-B)', Sin ^ = sin 1 (^ + B) cos J (A -B) -cos ^{A + B) sin I {A- B), ,'.BmA + shiB = 2 sin i{A + B} cos J {A ~ B) (1). Sin^-sin^ = 2cosi(^+^)sin^(^-5) (2). Similarly, Cos^ + cos^ = 2 cos 1{A+B) cosj (^ -^) (3). Cos^-cos^ = 2sinJ(J +^)sinl(^-j5) (4). These four formulae {v^hich are of the very greatest utility) might have been deduced from the formulae of the last Article, by making A + B = jS and A-B = D 'j in which case A = ^(JS+ B) 2indB = i{S-I)). 52. Dividing (2) of Art. 51 by (1), Sin ^ - sin 5 _ 2 cos ^{A+ B) smlJA- B) _ tan k{A- B ) SinT+sSTB" 2 sin 4 [A+B) cos ^{A-B)~ tan kk^-^B)' Similarly, dividing (4) by (3), Cos B — cos A , -i I A . -nx 1. 1 I A T>\ 7^ : = =tan ^(A + B) tan UA- B). Cos^+cos-B *^ / 2\ „ , Sin^isin^ . , , ^ . m ;i Sin4=fcsin5 „^tov, i /" j = r\ S^^^^^' Cos^ + cosJ9 =*"^^(^^-^^- ^^^ C3^3^^=°^t^^M^=P^)- (The upper of the double signs going together, and the lower together). PLANE TRIGONOMETRY. 29 53. Tan ii tan 5= ± „ cos A cos B __ sin A cos 5 ± cos A sin 5 _ sin (A ± i?) cos -4 cos B cos ^4cos^ ' Similarly, Cot 5 ± cot .i = ^^\ ^ I . '' sin A sm B (The upper signs going together, and the lower together). 64. Sin(4 + j5)sin(4-5) = sin2J cos^^-cos^J sin^^ =^8in2 4 . (l-sin2/?)-(l-sin'M).siu2fi = siQ2^-sm2 B. Similarly, Sin (A + B) sin (A-B) = cos^ B - cos- A. And in like maimer it may be shewn that Cos {A + B) cos {A- B) = cos'^A - sin-5 ; or =cos2^ - sin". 4. 55. To prove that rSiii nA + sin {n- 2) A = 2 sin (n- I) A . cosA^ (Cos nA + cos (n-2)A = 2 cos {n-l)A . coaA. Sinn^ = Bin{(w-l) J +-4} = sin (n- 1) j; cos .4 +cos(/t- 1).4 siu^. So, Sin (n - 2) ^ = sin {71 -1) A cos A -cos(?i-l)^ sin .4; .'. Sin nA + sin (n - 2) ^ = 2 sin (n -1)A cos A (1). Cosn^ =cos(n- 1)^ cos ^- sin (»- 1)^ sia^, and Cos {n - 2) ^4 = cos (n - 1) J cos u4 + sin (n - 1) 4 sin ^ ; .-. Cos 71^ + cos (?i-2).4 = 2 cos {n~l)A cos A (2). CoR. If n = 2, then, from (1) , Sin 2^ = 2 sLa ^ cos ^, from (2), Cos2^ + l = 2cos24, or, Cos2^ =2cos2^-l. If 71 = 3; from (1), Sin 3^ =2 sin 24 cos ^ -sin J. =4 sin ^ cos^^ -sin J, = 4 sin^ (l-siii2^)-sin J. = 3 sin A -4: siu^^. from (2), Cos 3J, = 2 cos 2A cos A - cos J. = 2 cos^(2cos2j4- 1) -cos^ = 4cos34-3cos J ; and, by successive substitutions, Sin 4^4, Sin5^... Cos AA, Cos 5A... might be found in terms of the Sines and Cosines of A respectively. 56. Tn like manner, SinnJ. -sin (n- 2)^ = 2cos (w- 1) -4 . sin^ (1), Cos {n-2) A - cos nA = 2 sin {n— I) A . sin A (2). so PLANE TRIGONOMETRY. 57. From tlie latter of these formulae, by means of successive substitu- tions, the Cosine of nA may be found in terms of the Sines of A and its multiples. Suppose n to be an even positive integer and =2m. .'. Cos 2{m-l)A -cos2mJ. = 2 sin {2m- 1) A sin 4, so cos 2{m-2)A- cos 2 (m - 1) -4 = 2 sin {2m - 3) ^ sin A, cos2 (wi-3) J.-COS 2 {m-2)A = 2 sin(2?n-5)J. sinA, &c. = &G. COS 2 {m-m) A -cos 2 {m — {m - 1)} A -2 sin A sin A. Whence, by addition, Since cos 2 (m-m) 4, or cos 0, =1,.. 1-cos 2mA = 2 sin^, {sin (2m -1) jl+sin(2m-3)4 + ... +sin 3J. + sini4}. .-. Cos2m^=l-2sin^.{(sin 2m-l)4 + sin(2»i-3)4+...+ sin3^ + sin ^}...(1). In like manner, if n were odd and =2m+ 1, it would appear that Cos(27/i+l)^=cosJ.-2sin^.{sin2»iJ,+sin2(m-l)J.+...+sin4J.+sin2^}...(2). Cor. If m = l, these formula give, Cos 2^ = 1 - 2 sin.4 sin^l = 1 - 2 sinM. Cos 3^ = cos A -2 sin A sin 2A = cos J. - 2 sin ^ . 2 sin A cos A = cos A -4: cos A (l-cos2^)=4cos34-3cos^. 58. To find the Sines and Cosines of 18°, 72°, SQ'^, and 54°. Sin 36° = cos (90° - 36°) = cos 54°, or, if 18°=^, sin 2^ = cos 3^; .'. 2 sin A cos A = 2 cos 2 A cos A — cos A, (55) ; .-. 2 sin^ = 2 cos 2^ - 1 = 2 (1 - 2 sin'^)- 1 ; .-. 4 Bin^A + 2 sin ^ = 1. And, solving this equation, Sin^ - "^ sj ^ - ^ ^f -^liidi the posi- 4 tive sign is to be taken, because sin 18° is a positive quantity. ••• i (x/5 - 1) = sin 18° = cos (90° - 18°) = cos 72° (1). PLANE TRIGONOMETRY. 31 A in 21Q0 n • 21Q0 1 6-27-5 10 + 2^5 And Cos^lS = 1 -sin^lS =1 t ,. = ., ^ j lb lo .•.CoslS' = iJ{lO + 2jo)=sm72' (2). Again; Sin 54" = cos 36° = cos 2 x 18" = cosn 8"- sin" 18" _ 10 + 2^ 3 _ 6-275 16 16 = i(l + v/5) (3). Cos^54" = 1 - sin^ 51" = 1 - ^^ = 16 ^^^ " ^^^) > .•.Cos51" = iV(10-2V5) = sin36" (1). 59. If an angle receive any increment^ to find tite corresponding increment of the tSine of the angle. Let the augle A receive an iucrement a, and let the corresponding incre- ment of the Sine of A be represented by A sin A. Then, a sin .l = sin (.1 +a) -sin^ = sin A cos a + cos .1 sin a - sin A = cos A sin a - sin ^ (1 - cos a) = cos ^ sin a ( 1 - tan A . —~. — ^ ) (89) \ sina / . . / ^ . 2sin2Aa \ = cos ^ sin o 1 - tan A . tt—. — ^ — r— I \ 2smiacos4a/ = cos A sin a (1 - tan A tan ^a). Cor. If a be very small, Tan-^a is very small. In this case, if Tan J. be not exceedingly large, (that is, if A be not nearl// equal to {2n + 1} DO", oi being or an integer), tan J. tan |a is a very small quantity, and may be neglected in comparison with unity. When, therefore, a is very smally and A is also not nearly equal to {'2n + 1) 90", A sin A = cos A sin a, very nearly. Hence it appears that when A is an angle of a triangle, this result cannot be aj^plied to determine the corresponding incre- ment of Sin -4 which results from A receiving a small given incre- ment, if A be nearly equal to a right angle. 82 PLANE TRIGONOMETRY. 60. If an angle receive an increment, to find the correspondi7ig decrement of the Cosine of the angle. A COS A — COS (^ + a) - cos A = cos A cos a - sin A sin a - cos A ■ A • . /-, ^ ■ A • (^ ^ A 2sin2ia\ = - sin A sina — cos J. (1-cosa) =-sin J. sin a 1 + cot J. . . -■— I ' \ sma J = - sin A sin a (1 + cot A tan ^a). Cor. 1. And as before, if a be very small, and also Cot A be not exceedingly large (i.e. if' A be not = 2n . 90^ nearly), cot A tan ^a may be neglected with respect to unity, and A cos A = — sin A sin a, very nearly. Note. Hence it follows, that unless (1st) a be a very small angle, and also (2nd) A be an angle which is not nearly equal to 0" or 180'^, this result cannot be applied to any particular case where A is an angle of a triangle. Cor. 2. If A be less than 90", Cos J. is positive, and Sin^ being also positive, in this case a cos A is necessarily negative. Wherefore, in angles Less than a right angle, as the angle increases its cosine decreases. If A be greater than one right angle but less than two, Cos A is negative, and. Sin A being positive, a cos A is negative. Wherefore, when the augie is greater than one right angle but less than two, as the angle increases the cosine also increases in magnitude, but is negative. 61. If an angle receive any increment, to find the corresponding increment of the Secant of the angle. , , , , 1 1 cos A - cos [A + a) A sec ^ = sec iA-\-a)- sec A = ■ r- = -^^ '- cos (A + a) cos A cos A cos {A + a) _ sin ^ sin a {1 + cot 4 tan 4 a} by (60) cos A (cos A cos a - sin A sin a) sin A 1 sin a 1 + cot J. tan i a ^ , ^ , 1 + cot ^ tan A a = T. — , . . ^i — -7 — 2— = tan J. sec ^ tana. ^; — ; — =^ . cos A cos A cos a 1 - tan A tan a 1 - tan A tan a Cor. If a be very small, and neither Tan^ nor Cot^ be very large (that is, if A he not nearly equal to n . 90°, when 7i is or any positive or negative integer), cot A tan |a and tan A tan a will both be so small that they may be neglected when compared with unity. In this case therefore A sec A = tan A sec A tan a, very nearly. PLANE TRIGONOMETRY. 33 Note. It is to be remarked that before tliis result can be applied in any case where ^ is an angle of a triangle, a must be a very small angle, and A must also not be nearly equal to 0, or 90^ or 180^ 62. If an angle receive any increment, to find the correspond' ing increment of the Tangent of the angle. A X / i .X . sin (^ + o) sin A L tan A = tan (A+a)- tan A = — 7 cos(J[ + a) cos 4 sin {A + a) cos A - cos (A + a) sin A ~ cos 4 (cos ^ cos a — sin ^ sin a) But Sin {A + a) cos A - cos (4 + a) sin J = sin {(.-1 + a) - J} = sin a ; A sin a „ ^ . ■■ .-. Atan^ = — s— ; 71 — : rr = sec-vltaua 008^-4 cos a . (1 - tau A tan a) ' 1 - tan A tan a Cor. If a be very small, and also Tan A be noc very large, (that is, if K be not (2n+l)00'^ iiearbj, ...n bt^hig 0, or any integer, ...}, then A tan A = sec* A tan a, very nearly. Note. If A be an angle of a triangle, this result will not hold when A is nearly equal to a right angle. 63. For a given small increment of A, the increment of the Sine of the angle is >, =, < the decrement of tlte Cosine, according as Cos A is >, =, < Sin A ; A iwt being very small, or nearly a multijile q/'90^ For A sin ^ = cos A sin a, if ^4 be not nearly {2n + 1) 90<' ; Art. 59. Cor. A cos ^ = - sin ^ sin a, if A be not nearly 2n . 90*' ; Art. 60. Cor. 1. Wherefore (w being 0, or any integer), if A be not very small, or nearly a multiple of go**, Asin^ is >, =, <(- A cos .4), as cos A is >, =, < sin A. Cob. In angles less than 90^, A sin ^ is > or <{- A cos ^) as A is < or > 45°. ..(31). S. T. S .3 4 PLANE TRIGONOMETRY. 64. Def. By Ta,n~'^ tlie angle is indicated of whicli the tangent is ^ ; i. e. if ^ = tan A, then A = tan~^^. ^_ So Sin~^s, and Cos~^c, &c. respectively indicate the angle of which the sine is s, and that of which the cosine is c, &c. 65. To sheio that Tan~''tj + tan~'t^ = tan~^ .. ^ ^ , and 1 2 Tan-'t, - tan"'t, = tan"^ ^'^rf- • 1 + t,t^ Let TsinA = t{, and Tan^ = ;2. Then, by definition (64), A—ta,n~^t^, and ^^tan"^^^* XT rr / 4 Dx tan ^ + tan B Now Tan(^ + 5)= , ,. „; ^ 1 - tan A tan B , tan A + tan B .-. by def. 4 + 5 = tan-i- — ^ . •^ 1 - tan A tan ^ Or, Tan-ifi + tan-ie2=:tan-J Il±k. (i). J. ti tiy '1'2 Similarly, Tan-i t^ - tan-i t^ = tan"'- -^ ^J (2). 66. 7/" tj, tg, ,...tn 6e ^/ie tangents of any angles, then Tan-^^,-tan-^^ =tan-^^^-^ + tan"^ ,^^^ + ...+ tan-\^"-^~'^" . For, Tan-i f, - tan"! U = tan-i -^-2, tan-i f, - tan-^ U = tan"! ,^~ ^ , fv>— 1 ~ fll tan-i?„_i-tan-^?„ = tan-i:j^=^ '; ; by addition, Tan~i f^ - tan~i ^„ J PLANE TRIGONOMETRY. 35 67. Examples of questions solved by the application of formulae proved in this Chapter and the preceding. (1 ) To prove that -p^ — r . — . = sec 2 A + tan 2 A. ^ ' CosA-smA [It is here required to bring the proposed fraction into one of which the denominator shall be cos 2^, or cos-^-sin^^. Multiply, therefore, the nu- merator and denominator by the numerator, and] Cos ^ + sin -4 (cos ^ + sin ^ )2 _ cos^^d + sin^^ + 2 cos A sin A Cos^-siujl cos^^-sin^^ ~ cos-'^-sin-J 1 -i- sin 2A cos 2^ ; by (27, 6), and (38), (3'J). 1 sin 2^ rt . X rt i H — — sec 2 A -t- tan 2A . cos 2A cos 2 A 1 (2) To prove that Cos 2 A = 1 + tan 2A . Uiu A [The equation becomes, when inverted, — - = 1 + tan 2A tan A ; where cos 2A if the right-hand side were expressed in terms of the sines and cosines of A and 2.1, and then put into a fractional form, the denominator would be cos 2A . cos A, and the numerator would involve the sines and cosines of the same angles. The first step therefore is to express r- in a fraction of COS 2A such a form.] „ 1 _ cos .4 _cob(2A-A) Cos 2vl cos 2^ cos ^ cob2AcobA cos 2 A cos A + sin 2 A sin A , sin 2A sin A ^ . ^ = — :ri 1 = 1 + — ^rr • 1 = 1 + tan 2A tan A : cos 2^ cos A cos 2 A cos A ' .'. Cos 2^= ^ 1 + tan 2 A tan A ' Note. In the following Examples it is required to determine an angle from some given relation between its Goniometrical Ratios and those of either a multiple of the angle sought, or of some given angle ; and conversely. (3) Determine a value of A that will satisfy the equation Sin2A = sinA. Sin^ = sm2^=2 sua^ cos^ '. (38); .*. 2cos^=l, and cos^=i. Wherefore vl = 60* (32). 3—2 86 PLANE TRIGONOMETRY. (4) Determine values qf^ that will satisfy the equation, Sin A + sin (2B + A) - sin (2B - A) = sin (B + A) - sin (B - A) ; .-. sm^ + 2cos2.Bsin^=2cos^sin^ ; Art. 50, (2). .-. 1 + 2 cos 25 = 2 cos 5; .'. l + 2(2cos2^-l)=2cos^ .(39). Whence Cos^-^ (li^^) ; aud, Art. 58, (3) , ^ (1 + V5) = cos 36o. Art. 68,(1), ^(l-V5)=-i(V5-l)=-cos720=cos(1800-720)=cosl080. .-. B is 36^ or lOSO. (5) To iwove that 2 cos 1 1", 15' = ^2 + ^{2 + V^}. Cos 45° =-^ ; .-. 2 cos 45° = ^2, .-. 2.{2cos^i.45°-l} = V2, .•.2cosi.45° = V2T72; 45 2^ .-. 2. {2 cos^l|^- 11 = ^2 + ^2; 45° / 7 .-. 2 cos-— , or 2 cos 11°, 15^ = n/2 + ^'1 + J% A Cor. By repeatiug the same process n times, it would appear in like manner that 2 cos -^ =;y'2 + ^(2 +(fec.), where 2, with the sign of a square root over it, appears ?z+ 1 times in the second member of the equation, the square root every time reaching to the end of the expression. (6) If X . tan A = (^ 1 + x — 1) . (^^ 1 — x + 1), required to prove that X = sin 4 A satisfies the equation, i. A I lr~, — i\ / /i , i\ \/l + a; + l Jl-x + \ X. tan^=(/>/l + a;-l) . {^1-x + l). , - — = x. , = ; ^l-t-ic+l Vl+a' + l .-. Jl-x+l = {Jl + x-\-l)X&uA; .'. (l-tan^)2 = {^iT^tani4-^l-a}2; PLANE THIGONOMETEY. 37 Whence, 2 tan 1 = a; {1 - tan2^ } + 2^/1- ^' tan .1 ; ... i=.^,lzp^^jr^\ ... Vr3^=l.-xcot2^...(47); 2 tan A .'. l-a;2 = l-2a;cot24+x2cot2 2.4; .♦. x2(i + cot2 24)-2a;cot2^=0; 2 cot 24 - cot 24 o • o / o i • w .-. 5c = 0: or x = q ^., ^ , = 2 . — — --2 sin 24 cos 24 = sm 44. 1 + cot^ 24 cosec' 24 (7) To prove that Tan"' ^ + tan"' - + tan"' - + tan"' ^ = 45". 1 1 Tan"' - + tan"' - = tan"' ^ — 7 , (65). = tan*' - , 3 o 1 _ * 3*5 1 1 1 1 7 8 3 So Tan"' - + tan"' - = tan"' , .. = tan"' ~- ; t o , i 1 11 ~ 7 ' 8 .-. Tan"' :y + tan"' - + tan"' ^ + tan"' -r = tan"' ^ -\r tan"' — - = tan"' , -^ tan"' -^= tan"' 1 = 45°. 7 11 1 _ ^ A ^^ ^"7*11 68. The Appendices I, II, ill, on the Logarithmic Tables of Numbers and of Goniometrical Ratios, ought to be read before entering on the next Chapter. CHAPTER TV. ON THE SOLUTION OF TRIANGLES. 69. A TRIANGLE consists of six parts, namely tliree sides and three angles. When three of these parts are given, (except they be the three angles), it will be shewn that the other three can, generally, be determined. The number of degrees in the angles of a triangle will be designated by the letters A, £, C, which are placed at the angular points of the triangle ; and the length of the sides respectively opposite to the angles A, £, C, by the letters a, b, c. 70. The Sines of the Angles of a Triangle are proportional to the Sides respectively op^posite to them. Let ABC be the triangle. From C draw CD perpendicular to AB, ov AB produced either way. c ^c c A D Then Sin CAB = smCAD = CA' (With reference to the third figure, see Art. 26.) DP Also, Sin CBA = sin CBB = ~ ; CB si n CAB _ CB sinCBA ~ CA' Sin^ a 'EmB^b' or. Sinyl Sin^ a PLANE TRIGONOMETRY. o9 In like maniier, if a perpendicular were let fall from B upon the side opposite to it, or that side produced, it might be proved that Sill A Sin C a c Wherefore, — = — ; — = : — the magnitudes of the lines a, 6, c being represented by the number of units of length they respectively contain : for otherwise Sin .1 and a would not be quantities of the same kind, and consequently no ratio could exist between them. 71. Since (Euclid, i. 32,) the interior angles of a triangle are together equal to two right angles \ .\ A + B -\-C = 180". . , Sin A a , Sin B b Also, ^T— 7 = J ; and ;tt— r, = - . Sin B b Sui 6 c And three of the parts of the triangle being given, the remaining three parts may be determined by these three equations. One, at least, of the three given }>arts must be a side, or the ratios only are given which a, b, c bear to each other, and their itiagnitudes cannot be determined, because there are but two equa- , Sin A a , Sin B b „ . tions given, namely, ^^ — ~ = 7 , and _^^ — -^ = - , lor determining om B oin C c the three unknown quantities a, b, c. And this is also apparent from the consideration that an in- definite number of triangles having the same angles, of all ])os- sible degrees of magnitude, may be formed by drawing lines parallel to the sides of a given triangle. 72. There is however one case, commonly called " the am- biguous case^^ in which the equations of the last Article are not sufficient to determine the triangle when three of the parts aie Sfiven. 40 PLANE TPJGOXOMETRT. If two Sides he given, and an Angle opposite to one of them fa, b, A), the triangle can he determined only when the side opposite to the given angle is the greater of the two given sides; i.e. ivhen a is greater than b. The equations of the last Article are ii) B^C=\m'-A, (ii) Siii^ = -, sin^, (iii) c = 6.^^. If B can be determined from (ii), € and c are known from (i) and (iii), and the triangle is determined. But since the sine of an angle is equal to the sine of its supplement, there are tv:o values of the angle B which satisfy (ii), the one greater and the other less than 90". (1) Let a be greater than h ', .'. A> B. Eucl. i. 18. Now B cannot be greater than 90° ; for if it could, then A + B would be greater than 180'^, which is impossible, (Eucl. i. 17) : .'. ^<90'^, and the lesser angle which satisfies (ii) is to be taken for the value of B. (2) Let a be less than h ] .'. A<.B. Li this case, since the only limitations, B-^A < 180", and B> A, may be satisfied whether B be greater or less than 90", it is impossible to determine from (ii) whether £ be > or < 90°. Thus, — taking the annexed C figure, — if CB = CB', it is evi- dent that both the triangles ABC, AB'C have a, h, A of the same A^alues ; also, in this case, z. ^ is less than the exterior ^"^--.,._ .,---''' angle CBB', or the exterior an- gle CB'D, i.e. A is less than CB'A, aud therefore is less than CBA ; and also a <.h. Which agrees with what has been asserted above. 73. To find the Cosine of an angle of a Triangle in terms of the Sides. (Fig. Art. 70.) Let ABC be a triangle, and from C draw Cl> perpendicular to AB, or AB produced either way. PLANE TRIGONOMETRY. 41 Then, figs. 1, 2 of Art. 70, CB' = AC + AB' - 2AB . AD, Eucl. II. 13. fig. 3, CB' = AC' + AB' + 2AB . AD, Eucl. ii. 12. AD And -jri ~ ^^^ CAD, = cos CAB in figs. 1, 2. = -cosC^i?mfig. 3. ..(26). Therefore, in each of these cases, CB' .= AC + AB'-2AB. AC. cos CAB ; ,, ^.^ AC + AB'-CB^ h' + c'-a' .: Cos CAB = — .--r — 7-p^ : Or, Cos A --. . 'lAB . AC 26c 74. To shew that CosJA=./ — — ^ -, where S = ^(a + b + c). For 1 + cos .( = 1 + 26c 26c _ (6 + c)^ - a" _ (6 + c + a){b + c-a) ~ 26^^ " 26c * Now, 1 4- cos J = 2 cos" ^A, (39, 2). And >S'-a = ^ (rt + 6 + c)-a = ^ (6 + c-a); • • -^ '^' ^^^ ^26c 26^ ' ... Cos.U= /?A^, ^ \ be 75. To shew that Sin J A = / (S-b).(S-c) ^ By Art. 39, (3), 2 sin^ i ^1 = 1 - cos ^ ^ 1 r^ = ^^ -^- 26c 26c _(a -^c-b)(a-\-b- c) " 26^ • 42 PLANE TEIGONOMETRY. Now S-b = ^{a + b + c)-h = ^{a + c-b), and so S — c = ^(a + h — c) ; The positive sign of the root must be taken both here and in the last Article ; because A, being an angle of a triangle, is less than 180*', and therefore Cos ^A and Sin ^A are necessarily positive quantities. 76. Sin^4 = 1 - cos'^ = (1 + cos ^) . (1 - cos A) _C h' 4- c' - a' ] f b' + c'-a' ] ~[ 26c J ' I 26c j — ■ 2 {b + c — a){b + c + a) (a + b — c) [a + G - b) W-c 1 46V 2S.2{S-a),2{S-b).2{S-c) = ^^S(S-a){S-b){S-c):, .'. BmA^-^J{S{S-a){S-b){S-c)}. So, Tanji.^= /^^^W^K ' 2 COS J^ V '^ {'^ - ^) 11 . To explain the meaning of the double sign, of the second member of the equation cosp=.yfc'). Since Cos (2n. 1800i^) = cos^, (25); .-. cos [In . ISO" J.A)= '^^^~ ^ . Whence it may be proved in the same manner as in (74), that Cos(».180«±i^)=i^^l^'. Now the first member of this equation ought to furnish two sets of angles whose cosines are of the same magnitude but are of different signs. PLANE TrJGOXOMETRY. 43 First. Let n be even, and =2w. Then Cos (n. 180<'±^4) = cos(2m. 180''±^^) = cos fm. 360"^^^); which, by making m equal to 0, 1, 2, 3... successively, gives the series of angle? ±^^, 360'^±^4, 2x3600±4^, 3x360o±4J, all the cosines of which are of the same magnituf^e as that of ^ ^A, or of + h A. Second. Let n be odd, and = 2m + 1. Then Cos {n. 1800±^4 = cos (2/;i. 180"+ ISO^i^ J)} ; which, by making m equal to 0, 1, 2, 3 successively, gives the series of angles 180'^ri-i.^, 360" + (180"±4J), 2x,360'' + (180<>±4^), all the cosines of which are of the same magnitude as that of 180'' ± A , which = cos{1800±4^) = - cos \A. Wlierefore the first member of the equation does furnish two sets of angles whose cosines are of the same magnitudes but are of different signs. And in like manner sets of angles may be determined corresponding to each of the signs affecting the expressions which have been «»btained as the values of Sin 4 <^1, Tan I A, and Sin A. ON THE SOLUTION OF RIGHT-ANGLED TRIANGLES. 78. The ri(jht angle, a side, and another part heinj given, to determine the remaining parts. Let ABC be a right-angled ti-iangle, C being the right angle. (1) Let c and k. he the other given parts. Then - = cos A : and - = sin A ; c c (\^J) — \^^c -^- L cos ^ — 10 ; which determines 6, Uio^^ = IiqC + L sin ^4 — 10 ; which determines «. Also, L B= 90° — ^ ; which determines B. In like manner, if B were the given angle, A might be determined. 4i PLANE TRIGONOMETRY. (2) Let A and b he given. C Cd Then ^ = sec -4 ; and -7 = tan A ; 6 C\^^c = \Jb + L sec J. — 10 ; which determines c, Q^^a = 1^^6 + L tan ^ — 10 ; which determines a. Also, jB = 90" — J. ; which determines B. (3j ie^ A and a &g given. Then - = tan ^; .-.6 = ; 7; and l^^S = l^^a - Z tan J + 1 0. tan ^ Aorain, - = sm J. : .*. c=-. — 7, or=a.cosec^: ° ' c ' sin ^ ' ^ Andl^,c = lj„a-Zsin^ + 10. Also, ^ = 90"- 4. (4) Let a and b 6e given. Then, tan^=-; .-. Ztanjl= l^^a - 1,^6 + 10. Again, ^ = 90" -A Again, y = sec ^ j . •. l^^c = \^J) + Z sec -4 - 10. The equation c=v'(a^ + &^) would give c; but the operation of determining c is tedious, particularly if a and 6 be large numbers. (5) Let c and a he given. Then, sin ^ = - j and . •. Z sin ^ = \^a - \^^g +10, Again, - = cos A ; ,'. \^J) = l^^c + Z cos ^4 — 10. c Or W = c^-a^ = {G + a) (c-a) ; which determines b without previously finding A. PLANE TRIGONOMETRY. 45 79. Different methods must be used in different cases to deter- mine the unknown quantities ; for what is said in Appendix in. 11, must always be carefully borne in mind, and such a formula must be selected in each case as is calculated to give the result with the greatest practical degree of accuracy. Thus in the last case, if h be very small compared with a and c. the anj^^le A is nearly a right angle, and the increment of sin A corresponding to a ^mall given increment a of the angle, [i. e. cos A . sin a . {1 - tan A . tan ha], ...(59)], is inconsiderable, and does not, besides, vary nearly as the increment of the angle. In this case therefore the valne of sin J cannot be determined from the Tables with any great degree of accui-acy. (App. iii. 11.) A better way of determining A in such a case is first to find the value of b, and then to determine ^4 from its cosine. Thus Cos^= ; .'. LcoaA-l^Jj-l^tiC + lO, = lioV(c''-a=)-W + 10, = 4{lio(c + «) -•■ lio(c -a)} + 10- Iio^. 80. Example. Given c = 3G5-1, and a = 348'3, to find A. (The Logarithms employed in this Example and others, will generally be found in the three pages of Logarithms subjoined to Appendices i. and ii. j Here, c + a = 713*4, and c-a = 16 "8. Performmg the operations indicated in the last line of the last Article, lio713-4 = 2-8533331 lio 16 -8 = 1 •2253093 2; 4 -0786424 2-0393212 10 12-0393212 lio365-l = 2 -5624118 9-4769094 And L cos 720, 33' = 9-4769380 See App. iii. 9, Ex. 4. Difference = 286 Now Diff. for 1" is in this case 67-016 ; .-. J = 72°, 33', 4" -27, neai'ly. 286 and ^^^^ = 4-267 = 4-27 nearly, 4G PLANE TRIGONOMETRY. II. ON THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES. 81. Let two angles and the side between them be given.. (A, C, b.) wliicli determines £. ^^ / sill A =* ' sm^ A. Ys ' \ a — \ 6 + Zsiii^— X sin B : wliicli determines a. * . , sin C Again, c = o . -. — .^ ; ^ ' sm ^ . '. 1 c = 1 6 + Z sin (7 — X sin 5 ; which determines c. If ^ + C be < 90", the value of £ is not required in order to determine a and c ; For since Sin ^ = sin {180° - (^ + C)] = sm(A + C); ,'. l^^a = l^J) + LsinA-L sin {A + (7), and lj„c = \Jb + Lsm.C -L sin {A + C). 82. Let two angles a^id a side opposite to one of them be given, (A, C, a.) Then, B=lSO'-{A+C). Again, b = a.~ — •, .:l.Jb=\.^a + L s'mB- LsinA, o 7 Sin J. or = \^a + L sin (A + C) — L sin A . /^ Again, c = a. -. — r ; • '• Lo<^ = lio^ + X sin (7 - Z sin A. ^ ' smA ^° ^" 83. Let two sides and the angle included between them be given, (c, A, b.) First, to determine B and G, ^ + (7 = 180"-^; .-. J(5 + C) = i(180°-J). PLANE TRIGONOMETRY. 47 h sin 5 h sin B c _ sin C c sin C^ 6 - c sin ^ - sin C _ tan \{B -C) ,^, •'• 6Tc " sin B + sin (7 " ta^J (B + C)' ^ '^'' And Tan i- (5 + C) - tan J ( 1 80" - J ) = cot ^ .1 ; -^ ^ ' b + c " r. Lt^ii\ {B-C) = X,{h-c)-X^{h + c)^L cot lA. (2). J (// - C) being thus determined, and ^{B+ C) being known, and6'=J(i?-fQ-i(^-6';i"''^^"""^ And a - c . -^ — —, determines a. sm C 84. JTie 5i(Ze a can he determined without previously deter^ mining B and C. (c, A, b.) For a' = b' + G^ - 2bc . cos A - 6^' + c- - 26c (1 - 2 sin' Jvl) 39, (3). = (6 - c)" + 46c sin^ ^A Now *^^^^^ — . sin ^2 A may be of any magnitude and sign, and b — c therefore there is some angle of which the tangent is equal to this quantity. Let 6 be the angle. Then T-^nO=^^ sin U (1). ^8 PLANE TRIGONOMETRY. And a^ - (6 - cf (1 + tan'^) = (6 - cf sec'O ; . '. a = (6 - c) sec (9 (2). From (1), Z tan ^ = 1^„2 + J IJ + J l^^c - l^{h - c) + L sin ^A, = i {IJ + lio^) + l:o2 + Z sin 1.4 - l^b - c) ■ whidi determines 0. From (2), l^^a = \^(b -c) +L sec 0-10 ; which determines a. '2 k/ he' 85. If 6 = c nearly, 6-c is very small, and -^ — .sini^, the value of tan e, is very large, (unless ^A, and therefore sin hA, be very small.) And since the tangents of angles which are nearly right angles are very large, 6 in this case is nearly a right angle. Now if it be required to find from its tangent, an angle which does not consist of a certain number of degrees and minutes exactly, the additional seconds have to be determined on the principle that the increment of the tabular logarithm varies as the increment of the angle. But when the angle is equal to {'2n+ 1) . 90° nearly, this principle does not obtain for the tangent, 62, Cor., App. Tii. 11; and therefore 6 cannot in this case be determined near enough to find a with any great degree of exactness from the equation a = {b-c) sec d. "^Tien therefore c is nearly equal to b, and A is not a very small angle, the following method will give a with more exactness than the last Article does. a^=:P + c^ -2bc cos A = b'^ + c^-2bc {2 cos^^ A- 1} 39, (2) = (62 + cH 26c) - 46c cos2 1 4 = (6 + c)^ 1 1 - (j^ • cos 4 aYI . Now {sjb-\jc)^, or b-2jbc + c, being a square, is necessarily a positive quantity ; therefore the positive part of it is greater than the negative part. ^^^ /7l^* • 5l -la -f VQ f>+l /->v> o 1 on/I Jit -Frtf-fi/wi N' tionaL or h + c>2jbc; .'. -^ — is fractional, and a fortiori -J^ ^ . cos M is frac- Let therefore be the angle whose sine =~ — . cos I A (1\ h + c ^ ^ '■ Then, a2 = (6+c)2{l-sin2 0), and.*. a = (6 + c)cos0 (2). From (1), ism0=lio2+ilio* + 41ioC-lio(S + c)+Zcos 4^, = \ dio^ + lio c) + lio2 + Z cos 4 ^ - lio (5 + c). (2), lio«=lio(& + c)+^ cos 0-10: which give and a respectively. PLANE TRIGONOMETRY. 49 86. Let two sides he given and an angle opposite to one of them, (a, b, A.) It has been shewn in (72) that with these data the solution is ambiguous unless a be greater than h. But if a be greater than h, then Sin jB = - . sin A, where ^ is an angle less than 90*'. a Also, G= 180'- (A + B); and, c = a . -?^ . 87. Let all the sides he given, (a, b, c.) _2 he '(S-h)(S-c) Now, SmA = ^^{S{S-a){S-h){S-c)], Sin iA = y < he -a) be ' * "When the three sides of a triangle are given, the angle A is more easily found in practice by dividing the triangle into two right-angled tri- angles in the following manner, than from the formulas in the text. If a perpendicular CD be let fall from the vertex on the base or the base produced, it may be shewn from the results arrived at in Euclid ii. 12, 13, that Base : Sum of the other two sides :: Difference of the sides : Difference, or Sum, of the segments of the base ; the fourth term of the proportion being the Difference of the segments of the base, or their Sum, according as the perpendicular cuts the base or the base produced. And the fourth term of this proportion being found, AD { = ABh BD) is known, and thence Cos A ( -r-p ) niay be determined. S. T. 4 50 PLANE TRIGONOMETRY. 88. Obs. If A he nearly 90<>, the first formula of (87) wiU not give the value of A very exactly, because the increment of sin A does not in that case vary as the increment of A, and it is also very small ; App. m. 11. In this case any one of the last three forms may be used, and the second or the third form must be taken according as cos^A or sin^^ is the greater, i. e. as ^A is less or greater than 45'', (63). The fourth form is applicable in all cases except where ^A is nearly 90°, (62). 89. Examples. 1. If £G be a perpendicular object stand- ing on a horizontal plane, its height may be deter- ^q mined by measuring in that plane a line AC, which is called a base, and observing the angle BAG with a proper instrument. For £0= AC. t&n BAG; .-. \^f^BC=\^QAC+LtanBAC-10. 2. If it be not practicable to come to the foot of the object, let a base DA be measured, such that the points D, A, C may be in the same straight line; and let the angles BDA, BAG he observed. Here two angles and a side of the isBAB are known. "^ By first determining the side BA the height BC can be found from the right-angled triangle BAC. Thus, sin BDA BA _ sin BDA AD ~ sin DBA ~ sm {BA C - BDA) And BC = BA. sin BAC= AD. sin BDA sin. BAG _ sm [BAG -BDA) ' .-. \..BC=ho^^+L^^-BDA + LsinBAG-Lsin{BAC- BDA) -10. 3. If D be not in the line AG, the height BG. can still be determined. At A let the angles BAG and BAD be observed, and at D the angle BDA. Then in the ^BDA, the angles BDA, BAD and the side AD are given. If then BA be determined from these data, BG can be found from the right-angled triangle BAG. ^- BA sin BDA Thus, —j: = sin BDA AD sin DBA sin { 180° - {BDA + BAD)]' PLANE TKIGONOMETRY. 51 Bin B DA BA=AD. sin {BI)A+ BAD) ' sin BDA sin BA C And BC=BA.sinBAC=AD. . .^.j.. „ , ^- , Bm.{BDA +BAD) l^^BC =IiqAD + L sin BDA +L sin BAC -L sin (BDA + BAB) -10. [It is evident that this determination of BC is not affected by the circum- stance of D lying out of the horizontal plane which passes through A and (7. Hence it follows that if a straight base AD he measured in any direction from A, and the angles BAC, BA D, BDA be observed, these data will be sufficient for finding the height of B above the horizontal plane passing through A and C] 4. Kequired to find the breadth of a river AD, from ob- servations made from the top of a tower BC of which the height is known. (Figure to Ex. 2.) At B let there be observed the angles of depression of the points Z) and A below a horizontal line passing through B and parallel to CD. The angles BBC and BAC are equal to these angles of depression ; and ^, „, sin DBA „, sin {BAC -BDA) JJA = BA . —. y—, — r = BA . . -^-^r-. sin BDA sm BI)A BC BJajBAC- BDA) ~sin~BAC' sinBBA 5. Kequired the error in height arising from a small given error in an observation of the angle in Example 1. Let BC = h, AC = a, iBAC = A. Let ^• be the error in height produced by an error a of the observed angle. Then h = a . tan A, h + k — a. tan (^ + a) ; T .i. /i N i. ^> ^sin(^+a) sin J) .*. k = a . {tan (4 +a)-tan.d} =a. \ r , \ ' ^ ' (cos(^ + a) cos J.) _ sin{^ +a) cos^-cos{4 + a) sin.4 _ sin{(4+a)-^} ~ * cos (J, + a) cos ^ ' cos (^ + a) cos J. — a . — 2~7 » since cos ^ =cos [A + a) nearly, when a is very small, (60) COS ^ Cor. Hence it can be determined when the error in height, arising from a small given error in the observed angle, will be the least. 4—2 52 PLANE TRIGONOMETRY. For h = a. sma A sma Asiu a 2Asma cos 2A sin a ^ ~ tan ^ * cos^-il ~ sin J. cos J. ~ 2 sin A cos J^ ~ sin 2Jr ' Now 7i being constant, and a being given, this expression, (wMch is the error in altitude,) will be the least when sin 2 A is the greatest ; that is, when 24=900 or 4 18 450. The observer, therefore, ought to move along the base until ^ BAC = 4:6\ and then by measuring AG he will determine GB (which is in this case equal to AG) with the least chance of error. 90. To find the Area of a triangle, the sides heing given. Let ABG be the triangle, and from G draw GD perpendicular to AB, or to AB produced either way. Then Area of the ti^iangle ABG, being half of the rectangle on the same base and between the same parallels, = lAB.GD = \AB.AG.BhLGAB = ich.^^J{S(S-a){S-h){S-c)}, by (76). = J{S{S-a){S-h){S-c)}. sin B sin C 91. The Area of the triangle also = Ja'^ For Area = IAB.GD = ^AB . BG . sin B sin(B + C)* = iBG . i"-^ . BG. sin^ = 1.1 P^^, ; ^ sm j1 ^ (sin B + G) since Sin A = sin {180° - (^ + C)} = (sin B + G). PLANE TRIGOXOMETRY. 53 92. To find the Radii of the circles described within and about a regular polygon of any number of sides. Let AB be a side of a regular polygon of n sides. Since the polygon is regular, it may have a circle described in and about it ; and each of the sides will subtend the same angle at the common center C of these circles. Draw CD perpendicular to AB. Then AD = DB, and CD is the radius of the inscribed circle. Let r=CD, andi2 = (7J. Now the sum of all the angles which the sides subtend at (7= n x lA Ci? = 360^; LACB = -—; .: lACD = \lACB = r, =CD, =AD. cot ACD=hAB.cot^~ n 1800 , AD ^ ,^^ "nT ' ^ = tan^(7i); 1800 (1). Again, R, =AC, = — AD sin A CD },AB ISQO 180° = lAB . cosec (2). sm 93. To find the Area of a regular polygon of any number of sides which is described within or about a circle of given radius. Let AEB be an arc of the circle whose center is (7; AB a side of the inscribed regular polygon of n sides ; CE at right angles to A B, and therefore bisecting it ; FG a tangent through E, meeting CA and CB in F and ; then FG is a side of the circum- scribed regular polygon of n sides. Let CA^r. AB = 2AD = 2r^mACD = 2rBm.~=2rshx'^^ 2n n FG = 2FE = 2CE. tan ECF = 2r tan 1800 A, = area of inscribed polygon, =n . A CAB = n. CA . CB. sin ACB sm 3600 A', =area of circumscribed polygon, = w . A CFG 1800 n = n.CE.FE=nrHan 54> PLANE TRIGONOMETRY. CoE. These areas may be thus compared ; Area of inscribed polygon Area of circumscribed polygon A CFE CE^ = TTTT? > smce the As are similar ; - (^ ~\CA = cos^ 1800 94. To find the Area of a regular polygon of n sides in terms of a side of the polygon. (Fig. Art. 92.) AB being a side of the polygon, Area = n . A CAB = n. I AB . DC = In . AB .AD .cot ACD 1 QAO 180*' = hi . AB . lAB . cot-^ = ln(ABf cot . 95. To find the Radii of the circles described in and about a triangle of which the sides are given. ^^ Let the lines bisecting the angles A and B meet in 0, and from draw OB, OE, OF perpendiculars to the sides. Then, Euclid iv. 4, is the center of the inscribed circle ; and its radius r=OD = 0E= OF. Now, Area of lABC=^AOB + LBOC + ti COA\ .-. ^J{S{S-a)[S-h){S-c)} '-a){S-h){S-c) -v— s Again ; bisect the sides of the triangle in D, E, F, and draw perpendiculars from these points which will meet in a point which is the center of the circum- scribed circle ; R, its radius, =0A = 0B = 00, Euclid IV. 5. And .-. A = \B0C, EucUd iii. 20; BE .'. SmA = sm^BOC=sinBOE = ^ i2' ty{76), g^{S{S-a){S-b){S-c)}=^ abc .'. R= 4.^/{S[S-a){S-b){S-c)} .(2). PLANE TRIGONOMETRY. 55 96. To find the Area of a quadrilateral figure of ichich the opposite angles are supplements to each other. Let A BCD be the quadrilateral. Let ^5 = a, BC^h, CD = c, DA=-d. Join^C. Then Area ABCB^^ ABC + A ADC = ^ab sin B + ^cd sm D = 4 («6 + cd) sin B, since Sini) = sin (1800-i)) = sin ^. Now from A 4^C, a'^ + b'^ - AC^ = 2abcoaB ; and from ^ACD, c'^ + d^-AC^ = 2cd cos 2) = - 2cd cos B ; .'., by subtracting, a^ + b"^ - c^-d^ =2 {ab + cd) cos B; .*. Sm2i,' = l-co62^ = l- • — --—, ; — } i 2{ab + cd) ) _ 4(a& + crf )2 - (a2 + 52 _c2_^2 ~ MabTcdf • And {&TeaABCJ))'^ = i{ab + cd)'&myB; .'. = ^ {4 {ab + cd)2 - (a^ + 6- - c^ - d'f-} = ^{2{ab + cd) + {a^ + b-'-c''-d^)}.{2{ab + cd)-{a^ + V--c''~d')} = ^ {(a + &)2 - (c - rf)'^} {(c + d)2 - (a - 6)2} = j-(a + 6 + c-c?)(a + 6 + c?-c)(c + d + a-6)(c + £Z + 6-a), and if S=ii{a + b + c + d), Area ^^Ci) = ViC-S^ - a) [S -b){S- c) {S - d)}. 97. The detenmnation of the unknown quantity in an equa- tion may often be facilitated by breaking the equation up into two others by means of the Goniometrical Ratios. This artifice has been employed in Arts. 84, So, and the following are addi- tional examples of its use. Examples. (1) Having given a, a, and B, to determine \ in a f(yrm adapted to logarithmic calculation from the equation, Siaa = cosZcos 5 cos a + sin Z sin §. {Hymers" Astronomy , Art. 154.) 56 PLANE TRIGONOMETRY. The equation may be thus written, r.. . * Z' . 7 , cos 5 cos a\ Sm a = sm 5 sin I + cos I . : — r — | . \ sm 6 / Now since there are tangents of every magnitude and sign, there is some angle 0, such that _, , sin cos 5 cos a , - TanA, or -. = ; — r — =cotScosa (1); ^* COS0' sm 5 ^ " o- • «. / • 7 , sm 0^ Sm a=sm 5 ( sm Z + cos I . cos (pj sm3 , . , , , , , sm5 . ,, = (sm I cos d> f cos lsin.d))= . sm (l + A), cos ^ ^' cos _., „ ,, sin a COS0 ,^. .-. Bm {l + (p) = r-r— ^ (2). ^ ^' sm3 ^ ' From (1), L tan c^ = Z cot S + L cos a — 10, which gives <;^. (2), L sin (I + (fi) = L sin a + L COB (f) - L sin. 8, which gives l + cf>, and thence ?. (2) To express a . cos ^ + b . cos (0 + a) under the form A . cos (B + 0). Let /3 = acos ^ + 6 cos (^ + a) = a cos ^^ ' & result of the required form. PLANE TEIGONOilETRY. 57 (3) Adapt I r + A / 1: ^^ logarithmic computation. la-h /a + h _a-h+a + 'b _ 2a _ 2 52 Now — is necessarUy a positive quantity, and it may be of any magnitude ; let therefore 6 be an angle such that Tan2^ = - (1). 2 2 2 cos ^ ; an expression of the required form (2). _ V cos- d - sin- d aV \/ V \cosdJ J 2cos^ cos^ 2d 98. Def. The anfjies introduced to assist the solution of an equation, by breaking it up into other two equations or more, are called Subsidiary/ Angles. CHAPTER y. ANALYTICAL TRIGONOMETRY. 99. TliG Circumference of a Circle varies as its Radius. Let P, _p be the perimeters of two regular polygons of n sides each, which are inscribed in two cii'cles whose radii are R, r and circumferences C, c respectively. Let X=C-P, x=c-p. Now when n increases, P and p increase ; therefore X and x are variable quantities dependent on the value of n. Let AB, ah be sides of the polygons ; 0, o the centers of the circles. 360" Then IA0B= = Laoh ; and A OB, aoh are similar triangles : n B AB _ nxAB _P _ C-X ' ' r ab n>^ab p ~ c -x ^ .'. Ilxc-E'xx=r>iC-rxX. Now since J? x c, rxC are constant, and Rxx, rxX are variable quanti- ties, (by Francoeufs Pure Mathematics, Art. 167,) Bxc=rx C, and Exx=rx X; c _ r •*• C~B' or ccc r. PLANE TEIGONOMETEY. 59 _, ^ „. c f the circumference of a circleN . Cor. 1. Since cocr; .*. - , or ^. , is acon- r \ radius j BtfJTit quantity. This constant quantity is always represented by 2Tr ; the approximate value of TT, (3"14159...), will be determined hereafter. CoR. 2. Since -=:27r; .'. c=27rr; or 27rr represents the circumference of r a circle whose radius is r. 100. The Circular Measure of an Angle. If an arc he traced out by a point in the line CB, by the revolu- tion of tuhich from the i^osition CA an angle ACB is de- scribed, the angle ACB may he properly measured by the ratio arc AB radius AC * Since in equal circles (and therefore in the same circle) angles at the centre have the same ^^ ratio to each other as the arcs on which they stand, Euclid vi. 33, z ACB arc AB 4 right angles whole circumference ABDA 2^' (99, Cor. 2); .-. ^ACB = 4 right angles A B &' 27r ' CA ,T . 4 right angles . ^ . AB . jNow, since ^—^ — is a constant quantity, ■-^-- increases 2i7r CA or decreases in the same ratio as the angle ACB increases or de- creases, and therefore — ,. — is a proper measure of the magni- radius ^ ^ ° tude of an angle. ,Tn £■ 1 1 • 1 i 1 7 J t^G subtending arc Whereiore an angle may be said to be equal to rr . radius Cor. 1. If ^ = — :r-' — ) tlien Arc = r$, radius 60 PLANE TRIGONOMETRY. Cor. 2. Since the circular measure of an angle (arc -r- radius) becomes unity when the arc is equal to the radius, the angle which is subtended by an arc that is equal to the radius is the U7iit of the Circular measuremeiit of angles*. * The theory of Angular Units may perhaps be rendered more inteUigible in the following manner. From Art. 100, it appears that Angleoe — ^. — : wherefore Angle = c . — ^. — , ' ^ radias radius where c is a constant quantity, whose value may be determined if a particular angle be taken for the unit of measurement ; and, conversely, if a particular value be assigned to the constant quantity c, the magnitude of that angle may be determined which, in consequence of such assumption, will become the unit of measm'ement. For the purposes of analytical calculations it is convenient that the above equation should be of the simplest form possible, and this will be the case if c be taken = 1 ; the equation then becomes Angle = arc -f- radius. Now to determine the angular unit impHed in the assumption c = l, make in this last equation, the Angle = 1, the arc then which subtends this angular unit is equal to the radius. In the case therefore, that any angle is repre- sented by the ratio arc -r- radius, the unit of measure is that angle which is subtended by an arc which is equal to the radius with which it is described. Again, if c have any particular value (as a) assigned to it, then the angle that is the unit of measurement, is such that its arc=- . radius ; for in that a case the general equation becomes, Angle = ax - = 1. Example. The unit of measurement being the fourth of a right angle, find the relation between any angle, its arc, and the radius. Generally Angle =c. -^. — . In this case, when the angle taken is the unit of measurement, the subtending arc _ subtending arc of j of a right angle ' radius radius arc subtending two right angles _ ^ ^ radius ~^ 8 "Wherefore l = c.47r, and .*. c- TT .*. the general equation would become, on the supposition that 3 of a right arc angle is the Angular Unit, Angle =- . — -^. — PLANE TRIGONOMETRY. 61 101. In the preceding Chapters the magnitude of an angle has been measured by the number of times it contains a fixed and definite angle (which is the ninetieth part of a right angle, and is called a degree), and its sub- divisions ; for several analytical investigations, however, the circular measure — ^. — is much more convenient. The circular measures of angles will, radius generally speaking, be denoted by the letters of the Greek Alphabet. —Y- — of an radius/ *' angle, to determine how many degrees the angle contains; and conversely. Let be the circular measure of the angle which contains A degrees. circumference Since ^^ = 27r, and the cii'cumference subtends four radius right angles, therefore 27r is the cii'cular measure of four right angles. A° ffiven angle 6 Now ■ = ~ ~ — = — • 360» 4 right angles 27r' = -017453... x^. And ^ = 360.^ = — .6 (2) = 57-29577... x^. Ex. 1. If ^ = 60, 60 ^ = _9-.7r=i X 3-14159... = 1-04719. 180 "* 62 PLANE TEIGONOMETRY. Ex. 2. Required the numher of degrees, d'c. in the angle which is the unit of measurement when the circular measure is used ; i. e. the degrees, (he. subtended hy the arc which is equal to the radius. (100. Cor. 2.) •TT « arc , Here 6= — r^— = 1; radius . A^]^ e=-^^ 570-29577 60 17-7462 60 44-772 And the degrees, minutes, and seconds required are 57°, 17', 4A"-77. CoE. 1. The number of seconds subtended by an arc which is equal to the radius with which it is described, is 57-29577 x 60 x 60 = 206264-772 = 206265 nearly. The number of minutes subtended by the same arc is 3438 nearly. CoE. 2. Since the angle subtended by an arc which is equal to the radius contains a degree 57*29577 times, the unit of measurement when the magni- tude of an angle is estimated by the circular measure is 57-29577 times as great as the unit of measurement when the angle is expressed in degrees. 103. Four right angles being represented by 27r, and therefore two right angles by tt, if the angle A^ he represented, according to the circular mea- sure, by 6, it follows from 24, 25, that (Sin 6= sin(2n7r+^), or-sin(2mr-^). (Sin ^= -sin{(2?n-l)7r + ^}, or sin{(2r4 + l)7r-^}. (Cos 6=: cos{2mr+9), or cos(2n7r-^). (Cos ^= -cos{(2ft + l)7r + ^}, or-cos{('27i+l)7r-6'}. jTan^= tan(2n7r+6'), or-tan(2n7r- e"). lTan(?= tan{(2?i + l)7r+^}, or-tan{(2?i + l)7r -^}. \ Sec 6 = sec {2nir + 6}, or sec {2mr - 6). I Sec ^ = - sec {(2n + 1) tt + 6}, or - sec {{2w + l)7r - d}. 104. The Circular Measure of an angle less than a right angle is greater than the Sine and less than the Tangent of the Angle. Also as the angle decreases, each of the quantities — — ^ and 7 — ^ a'pproaches to unity, and on 6 vanishing, each does become equal to unity. PLANE TRIGONOMETRY. 63 Let 6 be an angle less than a right angle, and = l BAG, = l CAB'. From any point 6' in ^C draw CB, CB' perpen- dicular \jQ) AB and AB' \ then the triangles CAB, CAB' are similar and equal in every respect. With center A and radius AB describe a circle ; this will pass through 1^, and BC, B'C will touch it at B and B' . Join BB\ cutting AG in N. Then (Legemire^s Geometry, iv. 9), arc BB' is > BXB' and — ,andis<-^; that is, ^ > sin 0, and is < tan 0. Again ; since lies between sin 6 and tan 0, — — 7, lies between 1 and —. — - sm & sin ' {'-'-^^ But when 6 is diminished, cos $ is increased ; and when is diminished indefinitely and vanishes, cos 6 becomes equal to unity. Wherefore —. — - becomes ultimately equal to unity, and conse- quently •- , which = -^ — ^ -f- r , likewise ultimately becomes tan d sin cos u equal to unity. [Le/ebure de Foiircij.'\ 105. If 6 he an angle less than a right angle, 2 sin 1(9 Sin 6 = 2 sin W . cos IB = \^^ . cos^ W ; cos ^d ^ COS *> \j 64 PLANE TEIGONOMETRT. .-. Sin > 2 (161) cos'J^ > ^ {1 - sin' J^} a fortiori, > ^ {1 - (1^)^} > ^ - i^^ 106. The Sine of a very small angle is equal to the Circular Measure of the angle, very nearly. Since, (105), 6 - {0' < sin 9, ,:6-sui0< iO\ ISTow if the angle be very small, ^ is a very small fraction and ^0^ a still less quantity, so that the circular measure may be written in the place of the sine in any numerical calculation into which such an angle enters without introducing errors of consequence by the substitution. CoE. If a and (B be circular measures of two very small angles, sin a a the number of seconds there is in the z a sin/3 ~ ^~ number of seconds in z /? sinr_2 sinr_3 ^^''^^ sinr~i' sinr"r And, generally, if w be any small number, hut not otherwise^ sin n" n ^ .: Sin 2" = 2 sin 1", Sin 3" = 3 sin 1", sin n" = nsm 1". 107. Required the number of seconds contained in the angle of which 6 is the Circular Measure. Let a be the required number of seconds. _ _ . . . T Arc subtending a" a !N ow, in the same circle, —. ^i — v- tt, = t ? ' Arc subtending 1 1 arc subtending a" radius .'. a = arc subtending 1" * radius PLANE TRIGONOMETEY. 65 ^ , , , , . arc subtending a" ^ , . , ,, . But, by hypothesis, . — = d ; and smco 1 is a very small angle, arc subtending 1" radius sin 1", nearly; (105). .2 sinl"^ a result which is often of great practical use. 108. The Area of a Circle of radius r is tti The Area of a Regular Polygon of n sides inscribed in the Cii-cle is n timea '2tr the area of a triangle two of whose sides are r, r and the included angle is — ; ,*. Area of the Polvgon — n . - sm — = irr- 2 n Ci) Now as the number of the sides increases the Area of the Polygon ap- proaches nearer and nearer to that of the Circle, and when n is infinite becomes identical with it : in which case — becomes an indefinitely small n angle, and therefore sin 1 becomes 1. (104.) n n .'. Area of the Circle = tt/--. 109. Demoivre's Theorem. To shew thcd for any value (cos ^ ± ^ - 1 sin BY = cos mO ± V^l sin 7n0 ; the upper signs being taken together, or the lower together. (Cos 6 ± y ^ sin 0) (cos 6 ± J'^ sin 6) = cos-6' - sin'^ =t 7^ (2 sin 6 cos 0). Or, (cos e ± V^ sin Oy = cos 20 ± J~^l sin 29. Again, (cos ^ =«» y^ - 1 sin ^)^ (cos ^ ± ^^ - 1 sin ^) = (cos 26 ± J^ sin 20) (cos 6 ± J^^ sin 0) = (cos 20 cos ^ - sin 2^ sin ^) ± ^ - 1 (sin 20 cos + cos 20 sin 0) ; . •, (cos =t J^ sin oy = cos 30 ± J~^ sin 3^. S. T. 5 QQ PLANE TRIGONOMETRY. Suppose tMs law to hold for m factors, so that (cos $^J-lsin6)'" = cos mO ± ^ - 1 sin m^ ;- then {coBO^J^BmO)"'^' = (cos m^ =±= ^ _ 1 sin mO) (cos 0^ J -I sin 6) = cos mO cos ^ - sin m^ sin ^ ± *y - 1 (sin mO cos 6 + cos m6 sin 0) = cos (m + 1) ^ ± ^ - 1 sin (m + 1 ) 6*. If therefore the law hold for m factors, it holds for m + 1 factors ; but it has been shewn to hold when m = 3, it therefore holds when m = 4, and so hj successive inductions it may be proved to be true when the index is any positive integer. Again, (cos ^ =t ^ ^ sin $)-"' = ( j=. V cos c/ + •sin"^ Y ^TshT^i .COS O^J = (cos ^ =F ^ - 1 sin O)"", hj actual division ; = cos md =F J - I sin m9 = cos (- mO) ± ^ - 1 sin (- mO) which proves the theorem for negative integral indices. Again, (cos O^J -I sin Gf = cos mO^J -I sin mO. But (cos-^±7-l sin-^)" = cosm^±y- 1 sinm^; .-. (cos ^± V - 1 sin e)"' = (coa~O^J- 1 sin- ^)'' ; .'. (cos^i J-1 sin^)" = cos-^±y- Isin-^; which proves the theorem for fractional indices. PLANE TRIGONOMETRY. 67 Cor. By the theorem just proved, (cos it ^y - 1 sin 0)"* = cos rrKp d= ^- 1 sin »i0, m being positive or negative, whole or fractional. Let (p = 2pir+9, where p is any integer ; .-. Cos = cos (2/>7r + ^) =cos ^, I . , Sin = sin (2i)7r + ^) = sin ^, i * " ' ^ ^' First, let the index of (cos 6 ± ,J~-1 sin 6) be integral, as m ; Then Cos m0 = cos(2mp7r + m0) = cosm5, Sin 7710 = sin (2 wpTT + m^) = sin ni^. TTt Second, let the index be fractional, as - ; Then p, being an integer, may be represented by qn + r, where q may be or any integer, and r may be or any integer less than n ; 771 771 771 .•. Cos -0 = cos- {2pir + e)=coa - {2{qn+r)Tr + d} 771 771 = cos {mq .2ir+ - {2nr + 6)] = cos - (2rir + 6). n n Similarly, Sin - = sin - {2rir + 6). From these two cases, therefore, it appears that the theorem might have been thus enunciated ; If the index he an integer, (cos ^ ± ^/ - 1 sin ^)'" = cos m^ ± ^ - 1 sin mB ; If the index be fractional, (cos e ± y ^ sin ^)^ = cos - (2r7r + ^) ± ^ 31 sin - (2r7r + 0) ; lohere r may he 0, or any integer less than n. [Note. It is to be observed that by giving r all values from to n. - 1 there will be obtained from the second member of the second equation the n different values of (cos^i^vTTi sin^)". Also these values will recur if r have values given it which are greater than n-1. For taking r =pn-\-q, where ^J is any integer, and 2 any integer less than n, 5—2 68 PLANE TRIGONOMETEY. Since - {2(pn + q)Tr+e] = 2rmr + - (2q7r + e), n ' n Tfl therefore the cosine and the sine ol — {2{'pn + q)Tr -{- 6] are respectively equal to the cosine and the sine of - {^qir+d), and the numbers q and pi + g when m substituted for r give the same value for (cos ^± \/^l sin 6>)".] 110. 7/ 2 COS 6 he repj^esented hy x-\--j then 2 V^^ sin 6 luill be represented hy x , 2 cos m^ hy x"" + -^ , and X X 2 V^ sin m^ &?/ x"" - — . For-siii'^ = cos'^- 1 = \ (a^ + -)'- 1 = i (a?'- 2 + -,") ; .*. 2 V — 1 sin ^ = a; — - : ^ a; And since 2 cos ^ = cc + - , .'.by addition and subtraction, 1 . . cc = cos ^ + ^- 1 sin ^ j and - = cos ^ — ^ — 1 sin ^ ; faj"* = (cos 6 + J- 1 sin 0)""= cos mO + J - 1 sin mO •'• \ 1 , , -^ = (cos ^ — ^y - 1 sin O)"" = cos m6 — ^^y — 1 sin m6, .'. cc"' + -^ = 2 cos mO ; and cc"* — - = 2 J — 1 sin m^. X"' X Cor. By taldng the equations of the corollary to (109), the following results are obtained ; (1) The index being an integer, 1 , 1 2cosm^=a5'^ + — ; 2 J -l&-mm9=x'^- —-i (2) The index being a fraction f — )> 2 cos- {2rTr + e)=x^ + ^; 2^/^l sin - (2r7r + ^) = a;"- — ; w ~ w - where r may be or any integer less than n. PLANE TRIGOXOMETRY. 69 111. To express any positive integral power of tlie Cosine of an angle in terms of the Cosines of the multiples of the a/ngle. Let 2cos(9 = a; + -; .*. 2cosnd = x^ + — (110). X x^ Now (2cos^)" If n be even ; the last term of (2), or the (in+1)"' term of (1), is n - 1 n— 2 n-hn + 1 2 6 In If n be odd ; the sum of the two middle terms of (1), viz. the {l{n - 1) + 1}"' and the {i(/i - 1) + 2i'^ (winch have the same coefficient, being terms equidistant from the ex- tremities of the series (1)), is n-1 n-Mn-l) + l n. 2 - i(n-l) '\'''x)' Hence (2) becomes \n + (2 cos d)'*' = 2 cos nd + n . 2 cos (/i - 2) ^ + ... n-1 n-hi + 1 , — -^— ... — -, ; when n is even, 2 hi n-1 n-Un-V) + l ^ « , • -.-i n . -^r— . . . f r- . 2 cos ^ ; when n is odd ; 2 i{n-l) n-1 ,: 2""icos"^ = cos7i^ + n cos {n-2)9 + n .- . cos (n-4)5+ ... the last term of the series being n-1 n-^n + l ,. + in . — ^— ... — = ; n bemg even, n-1 n-i(;i-l) + l ^ , . ,, + n. — 15— ... fr ty . cos ^ ; n being odd. Cob. Thus, when n = 2, 2 cos2^=cos 2^ + 1, n = 3, 4cos35 = cos3^ + 3cos^, n = 4, 8 003"*^ = cos 4^ + 4 cos 29 + 3. 70 PLANE TRIGONOMETRY. " 112. To express any positive integral power of the Sine of an angle in terms of the Sines and Cosines of the multiples of the angle. Let 2\/-lsin^=a;-- ; .'. 2J^smn9=x^- \ . And {2j—lsme)-=(x-iy = \x+(^-'^].'' 77-1 77 _ 1 f _ 1 \«-2 / 1 \n-l I 1 \n = x^-nx^-^ + n.'--^.x^-^-...+n.~}-^±^ 2 2 »"■ * x'^ ^ x^ 1. Let n be even ; then [J^r = {J^f ' ^= { (7^)' }^= (- 1)^ ; (n \*^^ - + 1 J , term of tlie expansion (1) is J^ n-1 n — ln+1 ~2~ \n ' Wherefore when n is even the series (2) becomes (- l)^2«sinn^= («.« + !) - n (^-H^,) +n.^ . [f-'-^^ - ... 2 ^71 n-1 = 2cosne-n2co&{n-2)e + n. —-— . 2 cos (n - 4) ^ - . . . 2 - _ 1 .*. (-l)'^2"-isin"^ = cosn^-»cos(n-2)^ + w. ^^-^. cos(w-4)^-... a + (_l)?i„.^.. .114^1. PLANE TRIGONOMETRY. 71 2. If n be odd, then (J^)^ = J^{^rZlf -''={- !)*("- V^ ; and the last binomial in (2) is Wlierefore, when n is an odd number, the series (2) becomes ( _ 1) i Cn-i) 21 si -^1 sin»* ^ =(--i»)-''(-=-.4)-."-^.(--^)-- (-)-'-^-^|[^5r-(-^)- = 2v/^sinM^-n2 v/ri sin (;i-2)^+... / iM^« ^^ '^-l H-i(n- 1)4-1 o / T • a TT- 1 ... (_ l)k»-i)2i-isin''^ = sin/i^-n.sin(n-2)^ + n.— r- .sin(?i-4)^+ 113. Having given Tan ^, to find Tan n^. Cos n^ + /^ ^ sin n^ = (cos ^ + ^ -Tl sin 0)" = (cos ^)" (1 + sT^ tan ^)'* = (cos^)"jl + n7^tan^-n.— ^.tan2^-«/-^.^.v/^tan3^ + ...j(l). And equating the possible and impossible parts of this equation, ^ ^ /, L «-l , „„ 71-1 «-2 n-3 , .„ ) Co87i^ = cos"^ l-».-jr-.tan2^ + n.— ^ . -s— . — j— . tan^^- ...... (2 . ( 2 2 3 4 ) • Sinn^ = eos"^ jwtan^-n.^— .-3— .tan«^+ ...j (3). * Cos nO can be expanded in terms of cos 6 alone, when w is a positive integer, by an algebraical calculation, but the problem is solved more easily by means of the Differential Calculus. [See Gregory's Examples.] 72 PLANE TRIGONOMETRY. ^ ntand-n.—^ — . — r — . ta.n^6+ ... ^ „ s,m7i9 2 3 .-. Tanw^= 2,= ., . 1-w. -^— .tan2^+,.. CoK. If n be a positive integer, tlie series (1) will terminate, and tiie last term is {J^^ tan e}\ 1. Let n be even ; then n n ( V^ tan ey = ij'^f ' 2 tan" ^ = ( - 1)^ tan« ; n therefore the last term is ( - 1)^ tan" 6, and n~l n-2 --1 nt&ne-n. -^r— . — ^ . tan=5^+ ... +(-1)2 ntari^-^e Tanw0 = ?i - 1 -' 1-n. -^tan2 6'+... + (-l)Han"^ 2 2. Let n be odd ; then (7^ tan dY = J'^ isT^T'^ tan" ^= ^^1 ( - l)'(«-i) tan" ^ ; therefore the last term of (3) is { - l)i(»»-i) tan" 6, and n tan ^ - w . ?^ . ^^ . tan^ ^+ ... + ( _ l)M«-i) tan" d Tan?ig= .* l-7i.^^tan2^+ + (-l)^(«-i)%tan"-ie 2 4 114. To shew that Cos a = 1 — ^j — ^ + ^ — - — ^— - — ... and Sma = a-^3+^-^-|l^-... By 113, (2) Cos?i0=cos"^|l-w.-^ .tan^^ + Ti. — — . — — . — — .tan^^- ...\ * The sign of n tan"~i 6 is the same as that of tan" 6, because every odd term in the expansion (1) has the same sign as the term immediately follow- ing it ; and when n is odd, n tan""^ 6 is an odd term, and has therefore the same sign as tan" 6. PLANE TRIGONOMETRY. 73 ct Let nd=a, and .*. n = -, u \ a d „ a. d 6 e ^ A ^ .'. Cosa = cos"^|l--.-~.taii2 5 + -.— -. -^ . -|-. tan^^-.. ( a- 9 /tan^\2 a-^ a- 2^ a -36 ftanOy ) Now this is true whatever be the value that is given to 6. Let then be written for 0. In this case —3— = !, and cos 9 = 1, Art. 104, a and with regard to the value of cos" 9 when ^ = and n is infinite, cos ) ^ l-sin2- )2 = 1 _- . sm--+n . -7^— . ^rr, . sm-*-- \ nj \ nj 2 71 2 2^ n • ct „ ^2 ■ 'J- A siu -\ 2 1 — , /sm -\ ^ a- I n\ ^ n a- I 7i 2?i \ ci y 2 2-yr \ a but the limit of sin --7--, when a is constant and n is increased indefinitely, n n is 1; Wherefore all the terms of the above series which is the value of ( cos - ) vanish in the limit, except the first, .'. 1 = limit of ( cos - | , when a is constant and n is increased indefinitely. •■• °°^''=i-o+i:2^4- «• By a similar substitution in 113, (3), Sma = a-3-^3 + 3-2;|-^- (2). If a be less than ^ tt (or a right angle), the series (1) and (2) here obtained are immediately convergent. Note. To anive at these expressions — ^ lias been sup- posed to become 1 when is written for 0, which is the case only when the angle is referred to the circular measure. Hence a, w^hich is equal to nO, is also referred to that measure in the series (1) and (2). 74 PLANE TEIGONOMETEY. Cor. If a be an angle so small that a^ and higher powers of a may be neglected when compared with unity, equation (1) becomes Cosa = l. If a^, a^ be retained, but higher powers of a be neglected, (1) and (2) give, Ct tt Sina = a-7r; Cosa = l- — ; b 2 approximations which are often practically very useful. 115. If €^~^ and e'^"^"^ be expanded in terms of Oj — \ and — OJ — 1 in the same manner as ^ is expanded in terms of x in the series^ e' = l + x + ^^ + ^-^+„. (App. I. 18. Cor.) then, (1) Cos^ = i.(/^'^+e-*^^). (2) Sin^ = -i_.(e«^^l-e-«^l). 1 e2^^^-l (3) Tan^=-74^.^— 7= . V-1 ,2eV-l 1 For. e^-^ = U.V-l-,^-,4:,^--l.,:^...- and e-^-'- ^ .. ^ 0^ - O'^ >~^ 0^ '=^-^'^^^-r:2+i72T3'^+i.2.3.4-- •Ve^^^--^-^ = 2(l-^.,,^^-...) = 2cos.. (lU), and e^^l-6-^^-"l = 2Vri(^^_^_|_^ + ...) = 2V^sin^; .-. Cos^ = i(e^^^l+e-«^~l) (1). And, Sin^=-J=(e«V3i_,-eV3i) ^2). Also, Tan^= = . , : — _— — ^ _- ; cos^ ^_1 gSV-i^g-eV-i and multiplying the numerator and denominator by e^^~\ I e20V^_. "^"''=7^-^^ <^)- PLANE TRIGONOMETRY. 75 Series for determining the Yalue of tt. 116. Gregorie's Series. To 'prove that ^ = tan^-^tan'^ +itaii*(9- ... ; where $ lies between — ^tt and + ^tt. Let 6=mr + dQ; where ^o is any angle between -^tt and +^7r, and n any positive or negative integer. Then Cos ^ = cos (mtt + ^o) = ( - 1)" cos 9^ . Now e''^-l = cos7r+^/31 sin7r=-l; .-. (- l)" = €"'^^~i ; .-. cos6' = £"^^~lcos5o; And from (1) and (2) of Art. 115, e^^~^ = cos d + J~l sin ^ = cos ^(l + »/^ tan 5) = e .Tin-V-l cos Oqj 1 + J -Hand .'. le^^^-^ = l«e"'^^^ + l,cosS'2 + 54-^6+'") Sin (a + /3 + ... + X)=cosacos^...cosX(>S'i->S^3 + ^g-...); .-. Tan(a + /3+...+X)= ^^^^"-'^+- + ^> ='^ ^-'^^ + '^^-- . If there be n of the angles a, /S,...X, it maybe shewn as in (113. Cor.) that i-i Tan(a+^+... + X)= ^^~'^^+^^-- + ^"^^'"^'^---\ n even, _ ^1-^3 + ^5-. ..+(-l)i("-i)^. ■l-'52 + ^4-... + (-l)i'"-i)Vi , n odd. 120. If Sin p = sin P . sin (z + p), required to expand p in terms of Sin P and of the Sines of z and its multiples. Hymers' Astronomy, Art. 247. Since Sin p = — — = (ePV^- e-i'^/^) by (115), the equation becomes by substituting such values of Sinjs and Sin (z+^), \ -I _ — —^ (eP V-i _ e-P^^=^j = gin p — Z^^^ (e(3+P)V-i _ e-(2+p)V-i)j and, by multiplying both sides of the equation by 2j^l . ePV^^ esp V=i - 1 - sin P (e {«+2p) V^ _ ^-z V^) = sin Pe^ V^ . esp V^- sin Pg-^ V=i ; PLANE TRIGONOMETRY. 79 .-. e^PV^i (1 - sin Pe" V~ ) = 1 _ sin Pe"' ^^^ ; g2pV^l. 1 - sin Pe-^ ^ ~^ l-sin/^e"V~i ' .-. 2pJ-l = h{l- siu Pe-2 V~i) - 1, (1 - sin Pe^^ = - sin Pe-^ ^-l sin2 Pe-s^ V^^ - i sin3 Fe-^' ^^- ... + sin Pe^^-i+isia2Pe2W~i4.i sin3 Pe^^ v^^+ .. Appendix i. 14. iii. .-. p = 8inP — J=r(€^^^-e-^^~) + 4sin2p _ J,^ (eaW^- e-s^ v^) + ... 2V-1' 2V-I = sinPsin2 + ^ sin2Psin22+^ sin^P sin 3z+ ... Cor. The number of seconds contained in the angle p, which is expressed by the circular measure, is (107) « sin P . , sin^P . ^ , ein^P . „ -r^, = -. — 7-r, sin 2 + i ■ .,, sm 2z + ^ -, .. „ sm 82 + ... jm 1 sin 1 ' ^ sin 1 '' sin 1" sin P . sin2 P . ^ sin3 P . 01 =— — — , 8mz+-i — -77 sin 22 + -^ — -77smd2 + ... sin 1' sm 2" sm 3 ' 121. In tlie same manner if tan l'=n. tan I, and tlie tangents be expressed in terms of ^^-1 and l'J-\ by the formula of (115), it may be proved that, \-n r = l-m sin 21 + Im^ sin il - Im^ sin 6^ + . . . where m = ^ . Cor. The number of seconds in V, is (107) sm 1" sm 1 sm 1'' sin 1'' I m sin 21 m^ sin 4? ^' ^iLir'~"sinl" "'"sin 2" "*'* [The angle I, which is the observed latitude of a place, is read oS in degrees, minutes, and seconds from the instrument by which the observation is made ; therefore to find the degrees, minutes, and seconds in V, the only computa- tion necessary is to determine the number of seconds in the latter part of the series, viz. m . -, 77l2 . --:— TT, sm2?+-^-p-,sm4Z- ...] sm 1" sm 2 80 PLANE TRIGONOMETRY. 122. To expand -z ,. in a, series of the Cosines of and 1 — e cos d '' -^ its multiples ; e being less than 1. , By hypothesis, e is less than 1 ; and since (1 - 6)2, or 1 - 25 + W, being a square, is necessarily positive, 1 + 6^ is greater than 2h. Let .'. e=. .'. l-e2 = l_ 1 + 62 462 (1+62)5 ~ Vi + &V ' •*' T+h^~"^ ' Whence, 62=l--A^-l:Vl-^ 1 + ^^ l + N/l-e2 l + Jl-e'^' l + Ji-e"^' .-. 6= ', 1 + ^1-^2 Let 2 cos ^=ic + -. 1 Then ;; = =(l + 5S) z. 1-ccos^ . 6 / 1\ ^^ ''•(l-6x){a;-6)- 1- i+6n X X — . X A B Assume -^ — ^i-^--, y. = - — - — l- (1 - 6a!;) (a; - 6) 1 - 6a; x-b* .'. x=^A{x-'b) + B{l-lx). "Whence, by putting x successively equal to ^ and 6, 4 = A., 5=: " 1-62' 1-62* Wherefore, 1 ^ 1+62 / 1 , 5 \ 1 - e cos ^ "" 1 - 62 ■ \ 1 - 6a; a - 6/ 1 6 1 V -bx x' 6 J 1 + 62 l-62-(l j__ X' l + lx + h^x^ + ¥x^ + ...^ - ,2 /'■«--rox-t-y-x--rw ^ -t-...\ ^ x x^ x^ ' PLANE TEIGOXOMETRY. 81 1 + 62 1-62 1 . {l + 26cos5 + 262cos2^+...} {l+2icos^ + 262cos2^+...}. g . Since e is less than 1, h, or , , is small, and therefore this series l + Ji-e'' converges rapidly. 1-e- CoR. The equation to an ellipse is r = a- -, the focus being the 1-ecos^ pole and 6 being measured from that vertex which is the further from the focus ; , e .-. r = aJl-e- {1 + 26 cos 5 + 262 cos 2^+ ...}; where 6 = :; n=^ . ^ 1 + A^l-r^ 123. The approximations of 114, Cor. to the values of the sine and the cosine of a very small angle, may often be applied to determine the magnitudes of Astronomical Corrections. Ex. 1 . If Sin (oj — y) = sin w . cos u, where y is very small, required an a2)proximate valine of y. Here, Sin w cos y - cos w sin y = sin w cos u ; .-. (1-^2/2) siim,._y cos w = sin w cos «; .• . y cos 0) + ^ y^ sin w = sin w - sin w cos u = sin w (1 - cos !<) = 2 sin w sin^ i u ; .-. y{l + ^y tan w} =2 tan wsin2^M; 2tana)sin2iw „, . „, ,1 , i i. ,_i .•. ?/=— — - — ; ^ = 2 tan wsm^^it {1 + iwtan w} ^ = 2 tan wsin2iit {1-iytan w+ ...}. By neglecting the second and all the succeeding terms of the expansion, as being small when compared with 1, a first approximation [y^] to the value of y is obtained, 2/^ = 2 tan w sin2 i u ; And by putting for y in the second member of the equation this its first ap- proximate value, a second approximation (^g) is obtained ; S. T. 6 82 PLANE TRIGONOMETRY. 2/^ = 2 tan oi sin^ | w {1 - tan w sin^ ^u tan w} = 2 tan w sin^ Jm {1 - tan^ oj sin^ i^}. The number of seconds contained in y.^, is (107) sm 1 \sm 1 , tan3 w . . , \ hu- - — 777 sm^ hu ) ', ^ sm 1" / and tlie two terms of the expression having been separately determined by means of logarithms, the number of seconds in y.^ is known. Hymers'' Astronomy, Art. 176. Ex. 2. In the same manner it may be shewn from. Cos {z + y) = sin n . siii z . cos m + cos z . cos ?^, where y and n are very small, that The number of seconds in y is y n cos m n^ , . « sin 1'' sin 1'' sin 2' nearly. The terms of this expression are, as in Ex. 1, determined separately by means of logarithms. Hymers^ Astronomy, xirt. 161. 124. The expressions of Arts. 115, 110, have been employed in Arts. 120, 122, to expand certain quantities in the form of a series. The operation can be reversed, as in the following in- stances. To find the sum of the series, (1) Sin a + sin 2a + sin 3a+... +sm7?c». (2) Cosa + cos 2a+cos 3(X-|- ... -fcosijot. ) 1 (1) Let 2>y-lsina = a; — ; .'. 2^~^mi2a = x'-\\ (110). 2 >/ - 1 sin 3a = x^ — r 2^/^si sm na = a3"- PLANE TRIGONOMETRY. 83 representing the sum of the series (1) by S, „ , , 1 X"-^ + 05-1 _ cc" - 1 1 £^ a" ' X-\ X ' I ~ x-1 X \ x"+V V xy 2cosi(2n + l)rt-2cosia Ji_l 2 J -Is'm ha i«. « 1 f 1 1 /.-. ■!< , Sin ^na . ... ,. .*. S=i^—. — i — . {cos^a-cosi 2n + l)a} = -^-^:^ — .smi(n + l)a. 2 sin ia - ' sin ^a ''^ ' (2) Similarly, 2 (cos a + cos 2a + cos 3a + ... + cos na) V g a ^ \ / _ sin^(2n + l) a- sin|a 1 1 "" sin J a a;* .•. the series (2) = —r-^z — .cosi(« + l)a. ^ ' sm ^ a 125. The formulre of Chapters II. and 111. may sometimes be employed for the like purpose. To sum the Series, (1) Sma + sin(a + 6) + sin(a + 2&)+ ... + sin {a + (ii- 1) 5}, (2) Cosa + cos(a4-^) +cos(a + 26) + ... f cos{a + (/i- 1) 6}. G— 2 84 PLANE TRIGONOMETEY. (1) Cos {a - J6) - cos {a + J6) = 2 sin J& sin a, . . . (51, 4). so cos {a + J5) — cos {a + §6) = 2 sin J6 sin {a + 6), cos {a + fS) - cos {a + -15) =2 sin J6 sin {a + 26), cos{,, = . ^, , . cos {a + ^(n-l) b}. sin J6 ^ ^ ^ ' ' Cor. "Writing « for 6 in these results, the results of (124) are immediately obtained. CHAPTER VI. ON THE SOLUTION OF EQUATIONS AND THE RESOLUTION OF CERTAIN EXPRESSIONS INTO FACTORS. 126. To solve a Quadratic Equation. I. Let x2 + 23x - 2 = be the equation. Let ^ = tan2^ (1) ; _ /- l=Fsec(?_ /- cos^tI ~"^'^- tau^ --V2. ^^Q ' „ cos^-1 -2sin2i^ , ,- Now — -. — -— = -r-^ — T— — =-r-: = - tail 4 6. &me 2 sm }, d co?,\d * and co8^ + l 2 cosH^ sin d 2 sin 4 ^ cos 4 9 tan \ d If therefore a^ and x^ be the two roots of the equation, — r x,= V2tan|^, (2); a^2 = t^^ ' (3). From (1), Z tan =li, 2 + ^ . lio2 - l^^p + 10, which determines 6, (2), lioaa = i.lio2 + ^tani0-lO, (3), lio{-a;2) = i.lio2-^tan^^ + 10. II. If the equation \)Q x'^-px-q — 0, the roots are the same as those of the preceding equation with their signs changed ; for the product of the roots with their signs changed ( - q) remains the sam-e, and the sum of the roots with their signs changed (the coefficient of the second term) changes its sign, but continues of the same magnitude. {Wood's Algebra, 8th Ed., Art. 281.) 86 PLANE TRIGONOMETRY. III. Let the equation hex^-px + q = 0; tlien If -| be less than 1, assume -4=sin2^ n\ Then, cc^ and ajg heing the roots of the equation, x^ = ^p{l + GOse)=pcos,^ie (2). X2 = ^p{l-cose)=psm^^e (3). From(l), Zsin^=lio2+i.lio2-lioi> + 10, (2), lioa^i = lioi) + 2i cos 1^-20, (3), lioa;2=lioP + Si sin 1^-20. IV. If the equation he x^+px + q = 0, the roots are the same as those of the last equation with their signs changed. {Wood's Algebra, Art. 281.) In the last two cases, if ^ be > 1, let -|-=sec2^; p^ p^ then x=Ip {1^ J - {^eo^ d -1)} =Ip {l:i^i2iQ.eJ^^}^ and both roots of the equation are impossible. These solutions may be employed in preference to the common method of solution when ^ and q are very large numbers. Ex. Eequired the roots of x^ + 36542x - 3469-1 = 0. By case (i.) itan^=lio2 + i .lio3469-l-lio365-42 + 10. By the tables, lio2= -3010300 I . lio 3469-1 = J X 3-5402168 = 1-7701084 10 12-0711384 Subtract lio365-42= 2-5627923 9-5083461 = i tan 17«, 52', nearly ; ••• lioaa=i.lio3469-l + Ztan80, 56'-10. i.lio 3469-1 = 1-7701084 Z tan 80, 56' =9-1964302 10-9665386 Subtract 10 •9665386 =lio 9 -2584, nearly. PLANE TRIGONOMETRY. 87 Again, Ik, (-a;2) = i.lio 3469-1 -X tan 80, 56' + 10. 10 + 1.I10 3469-1 = 11-7701084 Z tan 80, 56'= 9-1964302 2-5736782 =lio 374-69 nearly. Hence the approximate values of the roots of the proposed equation are 9-2584 and -374-69. After ccj was found, x^ might have been more easily determined from the equation - {3^ + x^=SQ5-^2. {Wood's Algebra, Art. 271 ) 127. To solve a Cubic Equation. Let the equation when transformed, if necessary, to another which wants the second term {Wood's Algebra, Art. 284), be x^-qx-r — ^)-, if a;=-, this /J equation becomes y"^ - qn-y - ni^ = 0. Now Cos^ 0-4^ cos 0- J cos 30 = 0, (55, Cor.) And this equation is identical with the equation y'^- qn-y-rn'^ = 0, if (1) Cos

= rn'^; and .-. Cos30 = 4rn3 = ^^-Z . Let a be the least value of 30 which satisfies the equation (3) ; then one value of y is cos 0, or cos | a. Also since, (103), Cos a = cos (2m7r±a), the two other values of y are con- tained among the values of Cos -J {2mir±a). Now m, being an integer, must be of one of the forms dp, 3p±l, and Cos ' ' ^ = cos (22)7ri^a) = cos^a. (103). ^ 2(3j)±l)7r±a /„ 27r±a\ 27r=l=a Cos -^^ — ^-^ = cos ( 2pir i — - — 1 = cos — - — , 88 PLANE TRIGONOMETRY. Wherefore the three values of y are Cos^a; Cosi(27r + a); Cos^(27r-a); V and the three values of x, or - , are n 2Y^|.cosia; 2 ,y^| . cos i (^tt + a) ; 2 y^ ^ . cosi(27r-a). 128. The solution of a cubic equation given in the last Article applies to those cases only where all the roots are possible : i. e. to the irreducible case of CardarCs Rule. \Wood's Algebra, Art. 331.] Since Cos a is necessarily less than 1, r /27 r /J" r^ f^ ''' 2\ ^^ ' 2^V27' 4^27' which is the irreducible case of Cardan's Rule. 129. To resolve the equatio7i x^ — 1 = into its quadratic factors, n being a positive integer. Here x^^ = 1. Now (cos d + J^^ sin ^)2« = cos 2nd + J^sm. 2nd, and if 6 — — , m and n being positive integers, I cos h x/ - 1 sin — I = cos 2mir + J -Ism 2nnr \ n n J ^ = 1, since sin2»i7r = 0; = X2« mir I — r • '^t .'. a; = cos — + J-lsm n ^ n Now m, being an integer, must be of the form p.2n + r, where ^ is or any positive integer, and r is or any positive integer less than 2n. (p.2n + r)ir / — =- . {p.2n + r)'jr Wherefore, x = cos ^ — + */ - 1 sm ^ n It' :COS =.cos- + v/-lsin-, by (103). 71 n PLANE TEIGOXOMETRY. 89 And for r writing 0, 1, 2,... 2n - 1, successively in n (1) If r = 0, a;«l, rtr I — - . rir x = cos — -\-^J -Ism — ; (2) r = l, x=cos- + sj -1 sin- , (3) r = 2, a; = cos — + J' - 1 sia — , 71 ?i (n + 1) r = n, z = -l, /n IN o o 2n-2 /^ . 2»-2 (2n-l) r = 2n-2, a; = cos - 7r + x/-lsm tt ^ cos --^/^ sin-, (103). n n r. . 2n-l /— - . 2n-l (2n) r = 2n-l, a; = cos Tr + ^-lsm tt IT / . TT = cos - - W - 1 sin - . n ^ n Now the equations (1) and (n + 1) give the quadratic factor {x-l){x + l) = x'^-l] the equations (2) and (2n) give ( a; - cos --v,/^ sin ^11 a; -cos - + v/- 1 sia - ) =jt2_2a;cos- +1; \ n ^ nj\ n ^ nj n and so on ; Wherefore, x^^'-l^ix'-l} x2-2iccos- + lj ... |a;2-2a;cos^^7r + l j . 130. To resolve the equation x^ + 1 = into its quadratic faxitors, n being a positive integer. Now (cos d + J^l sin e)-« = cos 2nd + J^l sin 2nd. And making 2nd = (2j» + 1) TT, and proceeding as in the last Article, 2m+l i—zr . 2w + l x = cos-^^7r+V-lsin-^^7r. 90 PLANE TEIGONOMETRY. Also, assuming m =p . 2n + r, where p is or any positive integer, and r is or any positive integer less than 2n, as before, 05 = cos ^ /-— . 2r + l 2/1 ^ 2n by means of which it may be proved that £c2n + l = |x2-2a;cos|^+lj \x^-2xcos ^ + ll... { , „ 2n-l J ...jar - 2a;cos — - — tt + IK 131. To resolve the equation x^^"*"' — 1 = into its quadratie factors, n being a positive integer. As in the last two Articles, (cos d + V -^ sin ^)-"+i = cos {2n + 1)6+ V "^ sin {2n + 1)6 = 1, if (2/1 + 1) ^ = 2M7r, or ^=|^, 2/?i ; — - . 2m .'. a; = cosT -T+fJ-lsin jTT+J-l 2/1 + 1 ^ 271+1 and making m=p . (2n + l) +r, this becomes 2r7r , / — L . 2r7r . £c = cos7^ + J -1 sm^r r . 2/1+1 ^ 2/1 + 1 Whence a;-^»+i-l={x-l} x2-2xcos-^ + l| ! x^ - 2x cos -?^, + 1 j . ' 'i 2n + l ) i 2/1 + 1 ) 132. To resolve the equation x^'*'^^ + 1 = into its /actors, n being a positive integer. Here a^'*+i = - 1, (cos 6 + V'^Tsin ^)2'H-i = cos (2?i + 1) ^ + J~^ sin {2n + 1) ^, 2772, + 1 and let (2?i + 1) ^ = (2//i + 1) tt, or ^ = ^7—7^'^ J * Here if r=0, a; = l, and one factor is x-1; the values of x corre- sponding to r = l and r=2n form one quadratic factor; those corresponding to r = 2 and r = 2w-l form another; and so on. PLAXE TRIGONOMETRY. 91 . •. (cos d + V^sin ^)2«+i = - 1 = x2"^'» ; 2m + 1 / — 1 . 2m + 1 „ — ^ IT + J -Ism- -: 2n + 1 ^ 2?i + 1 and makiug m—]).{2n + l)+r, this becomes 2r + l /— , . 2r+l 05 = cos TT + V - 1 sin ^ -. IT. 2n + l ^ 2?i + l "VYhence, as before, ^ ^ ( 2^1 + 1 J < a;' -2a; cos ^ r tt 4 1 v.* ( 2?i+ 1 j 133. To resolve Sin ^ and Cos into factors. The values of 6 which satisfy the equation Sin^ = are 0, ±7r, db27r, ±37r, ±n7r; n beiug any integer whatever. Assuming then that the series which expresses the vahie of Sin ^ is divisible by 6, d-ir, d-^ir, 6 — 2ir, d + 2w, &c., let Sin^ = a.^ ((9-7r) (^ + 7r)(^-27r) (^ + 27r)..., where a is some constant quantity whose value is to be determined. /. Sin^=±a.^(7r-^)(7r + ^) (27r- ^j (27r + ^) -^a..^.2V-.BV xo(l-f,)(l-^,) Now if 6 become 0, " — - becomes 1, (lOi), and this equation becomes 6 I=±a.7r2.227r2.327r2.... * Here the values of x corresponding to the values and 2n of r form one quadratic factor; those corresponding to the values 1 and 2n-lform another; and so on. When r is w the value of a; is - 1, and «+ 1 is therefore a factor. 92 PLANE TRIGONOMETRY. ••— <-S)(-.-£"0(-s£.) ■•■••«• Again ; the values of d which satisfy the equation Cos ^ = are ± - , ± -^ , Bit 2n + l . . . , , ^ ± — ,... ± — - — TT, ; n being any integer whatever. Hence, making an assumption like that in the former case, Cos^==.6.^-,.^.... (^l-_^j^l-_j ...... Now if ^=0, this equation becomes 1=± 6. — .-—- ... Cos*=(l-_rj (l-p-;j (^1-^,J (2). CoE. If d = \ir, the expression for Sin B becomes, TT 22-1 42-1 62-1 82-1 22 • 42 • 62 ^TT (2-l)(2 + l) (4-l)(4 + l) (6-l)(6 + l) 2 • 22 • 42 • 62 _22 42_ 62 82 **' '^~ • 1.3*3.5 'STT'TTQ"" This is Wallis's Theorem for the determination of tt; in which the successive factors become more and more nearly equal to 1. APPENDIX I. ON THE LOGARITHMS OF NUMBERS, AND THE CONSTRUCTION OP THE LOGARITHMIC TABLES OF NUMBERS. 1. Def. If 71= a', X is called the Logarithm of the Number n to the Base a ; or the Logarithm of a Number to a given Base is that power to which the base rnust be raised to give the -number. If a logarithmic formula be generally true whatever may be the value of the base, the logarithms of the quantities involved will be written thus, log m, log n ; but if the logarithms are cal- culated to some particular base, (as 10 for instance), they will be written thus, log^j^m, log^^n ; or thus, \^^m, \^^n. If 7i = a', and, while a remains the same, successive values be given to n, and the corresponding values of x be registered, the tables so formed are called " Tables of a System of Logoirithms to the Base a." It will hereafter (10) be shewn that a system of logarithms calculated to the base 10 is attended with peculiar advantages. 2. Since if n = a', x = \a,n ; therefore, in all cases, n = a' = a' . Cor. 1. If aj = 1, n becomes = a, and .'. La = 1. If ic = 0, a' (or n) becomes 1 ; and . *. 1, 1 = 0. Cor. 2. If a be the base of any system of logarithms, Since m = a * , and tz = a , 1 1 .-. m^''"* = a, and 71 ^"^^ = a, 1 j^ . •. m " = w* ; and m " =71" j a tnie result, whatever be the value of a. Wherefore, w'°s " = rt}"^ "*. 94 PLANE TRIGONOMETRY. 3. Required from tables of logarithms calculated to a given base, as e, to form tables of logarithms to any other base, as 10. By (2), 71 = ^""', 71=10^^°^; 10 = e^*^^ and equating the indices of e, leTO l£»=lel0.1ioW; and lio"' = -,— tf^ • le^- Hence the logarithm's of any number n in two systems calcu- lated to any bases, as 10 and e, are connected by a constant multi- plier , i viz. _-y- j which is called the " modulus ; " and therefore from tables of logarithms calculated to a base €, tables m.ay be formed to the base 10. Cor, By writing a for 10 in this proof, L'^ = = — . L'^ ; or, i^a the modulus connecting the logarithms of a number in the two systems whose bases are a and e, is , — , 4. The logarithm of the product of any number of factors is equal to the sum of the logarithms of the several factors. For m.^.r...=J"'\a^'''\aK.. = a^«"*^^«^-^^«^+-^ But m.n.r...—a'^^^^^^"; .-. \a,{m.n .r...)~\am + \an-\r\ff^- ... or, log (?7i. ?i.r...)=log??i+logw + log?*+ .... Hence if there be found in the tables the number whose logarithm is the sum of the logarithms of the several factors, the j)roduct of those factors is obtained. 5. The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. For a^V"/ = - = -r— = a ; n ^\an I TIX \ f 7Yh\ .*. L(-j=Lw-Lw; or, log(-j=Iogm-logw. PLANE TRIGONOMETRY. 95 Hence if there be found in the tables the number whose logarithm is the logarithm of the dividend minus the logarithm of the divisor, the quotient is obtained. 6. 2%e logarithm of the c^ i^ower of any nuniber is equal to c times the logarithms of the number ; — c being either whole or fractional. For a"^ '^Tnv^—\a ) =a " ; .•• la{m') = c.\a.m; or, log (m"^) = c . log w. Also, al«('«0 = 7n'=(ala'»)'=a' ; 11 i 1 •'. !»('«') = -■ Lw; or, log {ill') = ~. log m. C Hence if any number be given, the c"' power of it is that number in the tables whose logarithm is c times the logarithm of the given number ; and the c^ root of it is that number in the tables whose locjarithm is -th of the logarithm of the eiven number. 7. From the last three Articles it appears that the operations of multiplication, division, involution, and evolution cau be per- formed with greater facility by means of tables of logarithms than by the common arithmetical methods, particularly when the numbers are hirge. Tlie easier arithmetical operations of addition and subtraction cannot be j)ei'formed by logarithms. 8. In the common (or Briggs') system of logarithms, tJie base of which is 10, the logarithms of 10° x N and — — may be deter- mined from the logarithm ofN. For l.„(10''x.Y) = 1^^10" + l^„iV^, by (4) = n.l,„10 + l^,^,... (6) = n + l,,N- (2, Cor. 1). And l.o@-lo^^-L10", (5). = 1,0^-^. 96 PLANE TRIGONOMETRY. Thus from the pages of logarithms printed at the end of this Appendix, 1^,6 = 0-7781513; .-.1^,60 =1,„10 + 1^,6 =1+0-7781513 = 1-7781513, lio600=l,„(10^) + \fi= 2 + 0-7781513 = 2-7781513, L-6 =1.6 -UO =0-7781513-1, l,„-006 = 1,„6-1^,(10^)= 0-7781513_- 3. The last two logarithms are written thus, 1-7781513, 3-7781513. Def. The integral part of the logarithm is called the " Cha- racteristic " of the logarithm of the number ; the decimal part is called the " Mantissa ^""^ of the significant digits of which the number is composed. Thus, in 1^,600 = 2-7781513, 2 is the Characteristic of the Logarithm of the Kumber 600, -7781513 is the Mantissa of the significant digit 6. 9. In the common system, to determine the Characteristic of the Logarithm of any given Number. If a number be between 1 and 10, its log. is between and 1; .-. the characteristic is 0, 10 and 100, land 2; .• 1. lOOandlOOO, 2and3; .♦ 2. 10""^ and 10" w-landw; .• n-\. Hence the Characteristic of the logarithm of a number of n integi-al places (and which therefore lies between 10""^ and 10"), is n-1, or is less by one than the number of integral places of figures in the number. Agara, if the number be between 1 and r^ , its log. is between and - 1; .". the characteristic is 1» ^^^^4' -land -2; .- 2, lO^a^^lO^T' -(»i-l)and-w; .• n. * "Mantissa;" a handful thrown in over and above the exact weight; an overplus. PLANE TRIGONOMETRY. 97 Hence, the Characteristic of the logarithm of a decimal fraction having w - 1 cyphers after the decimal point, is n. Generally therefore, the Characteristic of the Logarithm of any Number is the number of its digits minus one; where if the number be a decimal frac- tion, the cyphers which follow the decimal point are alone counted, and are reckoned negatively. Wherefore, conversely, if logarithms be given having characteristics 1, 2, 3...1, 2, 3... there are in the integral parts of the numbers to which these logarithms belong 2, 3, 4 0, -1, -2... digits respectively. Thus the logarithms of 245 and 25400 have for Characteristics 2 and 4, and the Characteristics of the Logarithms of 2-54, 2o*4, 0*254, -000254 are 0, 1, 1,4. The Mantissa given in the tables for Ijo36o2 is -5625308. .'. lio 3652 = 3-5625308, l^o 36-52 = 1-5625308, lio 365200 = 5 -5625308, l,o '3652 =1-5625308, 1,0-003652 = 3-5625308. 10. The advantages of Briggs^ system of logarithms. From the last Article it appears, that if the Base of the system be 10, it is requisite to register the MuntisscB only in the tables ; because the Charac- teristics can be determined by counting the digits in the integral part of the numbers whose logarithms are required. This omission of the characteristics renders the common tables less bulky than those calculated to any other base. Also, from (8) it appears that in this system the Mantissa of N is also the N Mantissa of 10" x N, and of — ^ , — where n is any integer : this circumstance renders the common tables more comjyrehensive than if any other base were taken; for if any other base were used, the Mantissa of N would not be the N Mantissa of 10" . A^, or of y— . In the same way it might be shewn that if the system of arithmetic in common use were duodecimal instead of being decimal, tables calculated to the base 12 would possess the same advantages which have been here she\\'n to belong to the tables in common use. 11. The tables of logarithms in common use register, some to five, and others to severi, places of decimals, the mantissae for numbers from 1 to 100000. Two pages of logarithms are printed at the end of this Appendix, in which the mantissae are calculated to seven places of decimals. The hne at the top of the second of the pages begins with the number 3650 (or 36500), and its mantissa -5622929 is placed opposite to it. And because the mantisss of all numbers from 36500 to 36559, (comprised in the first six lines of the S. T. 7 98 PLANE TEIGONOMETRT. page), have the same initial three figures, viz. -562, these three figures are registered, once for all, opposite to the number 3650, and the la^t four figm-es of each succeeding mantissa are placed under the number to which they belong. Thus the first line of the page is Num. 1 2 3 4 5 6 7 8 9 3650 5622929 3048 3167 3286 3405 3524 3642 3761 3880 3999 Whence is obtained, Mantissa of 36500= -562 2929 36501 = -562 3048 36502 = -562 3167 36503 = -562 3286, and so on. In the same manner, the next line gives the mantissae of numbers from 36510 to 36519 inclusive. 12. Since a change in the value of the third figure may not take place at the beginning of one of the horizontal lines, whenever a mantissa is sought from the tables, care must be taken to get the right initial figures. Thus (see p. Ill), the mantissa of 36643 is '5639910, and the last four figures of the mantissa of 36644 are put down as 0029. Now if the mantissae of these two numbers had the same thhd figure, the mantissa of 36644 would be less than that of 36643, which cannot possibly be. A change in the value of the third figm-e does, in point of fact, take place here ; and the mantissa of 36644 (as do those of the numbers immediately following 36644) begins with -564, and not with '563. Similar changes of the third figure of the mantissa occur at the numbers 36729, 36813, 36898, 36983, and are indicated by printing in a smaller type the fourth figm-e of the mantissae of those numbers. The construction and use of the small tables in the last column of page 111 will be explained hereafter. 13. Examples. (1) To multiple/ 23 by 16. Art. 4. By the tables, p. 110, Mantissa of 23 is -3617278, Mantissa of 16 is -2041200 ; .-. li, 23 = 1-3617278 lio 16 = 1-2041200 2-5658478 And, p. Ill, the significant digits corresponding to the mantissa '5658478 are 3680; .-. lio368-0 or I^q 368 = 2-5358478; and /. 358 is the product sought. PLANE TRIGONOMETRY. 99 (2) To multiphj -0172 hy -00214. lio •0172 = 2-2355284 lio-002U = 3 -330^38, 5-5659422, =lio -000036808 J p. Ill; Wherefore '000036808 is the product sought. (3) To divide 3G72 hj 51000. Art. 5. 1^0 3672 = 3-5649027 p. 111. lio 51000 = 4-7075702 p. 110. 2-8573325 =lio-072; \Vherefore '072 is the quotient sought. (4) To find the values q/*(15-4)^, and (650)'. Ai-t. 6. liol5-4 = 1-1875207 p. 110. 3 3 502562 1=1^0 3652-8, nearly, p. 111. .-. 3652-3 is the approximate cuhe of 15-4. Again, 1^0650 = 2-8129134, p. 110. .-. ?:.lio650 = -5625826 =lio 3-6524, nearly, p. 111. .-. 3-6524 is the approximate fifth root of 650. (5) To find the values of (•085)^ and o/ (-000005)1 2-9294189 = lio -085 4 5-7176756 and the mantissa is that of 522006, nearly. Therefore -0000522006 is the number sought, nearly Again, lio'000065 = 5-8129134 = 6 + 1-8129134; .-. 3)5-8129134 2 -6043044 = l^^ -040207, nearly. Note. In dividing it is to be remembered that 5-8... = 6 + 1-8... 7—2 100 PLANE TRIGONOMETRY. 14. To expand l^(l + x) in a series ascending hy powers of X. Let p denote tlie logaritlim of (1 + x) to the base a, so that \+x = a^. Then(l +£cr=a'"^ = {! + (« -l)}"^. And expanding both sides of this equation by the binomial theorem, m—\ , 111— \ 7)1 — 2 „ 1 + mx + rri . — ■ — . ocr + 7n . — ^ — . — ^ — . x + = 1 -^mj) {a— 1) + mp . ~ — . {a— ly + ; and arranging according to powers of m, \-^{x-\x^ + \:f? - ...) m + Prtf + Qnf + ... = l^{{a-\)-l{a-\Y-¥l{a-lY-...}pm + Fm' + ... This relation being true for all values of m, the coefficients of like powers of m on each side of the equation will be equal ; wherefore {{a-\)-l{a-iy-^^{a-lf-...}p = x-kx' + k^'-...l •'•^''^^"^^^'"^''"(a-l)-l(a-l/+|(a-l)^-... ^'-^ Let € be the quantity which when substituted for a makes the denominator of this fraction equal to unity ; then le (1 + cc) = cc - J-x^ + -g-cc^ — . . ., (ii-) and.-. a^{a-l)-l{a-lY + l{a-lf- (iii.) And hence \{l +x) =^ {x -Ix"" + lx^ - ...) (iv.) Cor. The modulus =-— tt of the common system of loorarithms LIO ' ^ may be calculated thus ; by taking for granted the formulae (ix.) and (vi.) which are proved hereafter. PLANE TRIGONOMETRY. 101 By(ix.), 1^5 = le4 + 2JQA^+|.^Q^+..^3:p^+... j ^,„a 11 11 ^J (vi.), i U 1 1 1 ^ ^ (3 ^3 '33 +5 '35 + ,1111 ^ + 3-9+5-iP+- 11111 + -H — . — +-.— +., ^9 3 ya^5 95^ = 2-30258509... Whence is obtained, —J-^ = 0-434294819, 15. Some rapidly-converging formulae for the calculation of logarithms will next be investi^jated. But le - = lei -leX = - hx 1 X x-1 ifx-iy , , II. Again; le (l + 2)=2- ^ +^- ...; and h (l-s) ^23 .•.l<(l + 2)-le(l-z), orle^ = 2 12 ^ 1 V ... 1 + 2 iC — 1 And if :r— =ic, and .-.2= -, this becomes, 1-2 a: + l leX = 2. lx-1 l/ic-l\3 l/rr-l\5 Iic + l'3\a; + iy '5 ViC'+ 1, If X be a little greater than 1, this series converges very rapidly. (^i.) 102 PLANE TRIGONOMETRY. j_ _i III. Again, hx = h (^"')'" = m . le x™ ^ X 1 = m{{x>''-l)-l{x'"^-iy+... }, by (ii.) 1^ and, (x being greater than 1,) by assuming m of sufficient magnitude aj^* may be made to differ from 1 by any definite quantity, however small the quantity may be ; in which case the succeeding terms of the series may be neglected as being of inconsiderable magnitude with respect to the first, and 1 leX = m{x'^^-l) (vii.) 16. Having given Lx, to find l^ {k + z) ; z being small when compared with x. Expanding h (l + |) by (tI.). [since r^l{^ = ^^ » le(.^.)=leX + 2|2-^-+|(^J+... I (Viii.) CoH. If3=l;l.(l + x) = le:« + 2J2^ + |.^-^34.^^^ | which is useful in computing U [l + x) from lea;, particularly when x is large. 17. Having given the Napierian Logarithms of fivo successive numbers, x- 1 and x, to find that of the nuinber next following . = 2leX-le(^-l)+le(l-^,). /IN r. V'^V^ 1 "I Expanding [I- ^^jhj (vi.), |^smce ^ j-^ ="2^^^J ' PLANE TRIGOXOMETRY. 103 18. To expand d^ in a series ascending hy poiuers of x ; i.e. to expand the number in a series ascending hy powers of the Logarithm. = \ +x{a-l) +x . — rt— • (^*- 1)' + ic . — ^ . — ^ .{a-lf + ... let the coefficient of x {which h (a - \) ~ h(a — If + I. {a — \f - . . .} be represented by p^, and let the coefficients of a;^ x^, ... be repre- sented by p^, p^, ... ; then a' = 1 +p^x +pjx' + p.^x^ + . . . . •. rt' = 1 + p^z + p^z^ + p.^z^ + ... Kow a'''^*' = a^ . a', or 1 + p^ {x + z)-ifp, {x + z)- ... +p^^ {x -\-zY -^ ... = {\ +p^x + p^x'+ ... ^px" + ...} X (1 +p^z+p/+ ... +p^z''+ ...I and by equatLng the coefficients of the terms involving xz, x'z, JU Zm • • • JU 'V* • • ■ -Pi^l'rPi' ■'■P, = '-4 ?>.' ^P,=P,-Ps, -'-p.^i-f 1.2.3.4 l^owp^ = (a-l)-i{a-iy + ^{a-iy-...=U; by (iii.) Cor. If the base be e, (xi) becomes 104 PLANE TRIGONOMETRY. 19. To find the value of c, the base of the N'apierian system of logarithms. If a; = l, and the base be e, the series for a* in the last Article become? 1 -. 1 1 1 1.2^1.2.3 1.2.3... n Now, 1 + 1 + J^ =2-5 = 4666666666 = -0416666666 = -0083333338 = -0018888888 = -0001984126 = -0000248015 = -0000027557 = -0000002755 1.2.3 1 1.2.3.4 1 1 . 2 .'3.4.5 1 1. 2.3.4.5.6 1 1.2.3.4.5.6.7 1 1.2.3.4.5.6.7.8 1 1.2.3 8.9 1 1.2. 3.4 9.10 2-7182818 This result gives the correct value of e, the base of the Napierian system, so far as the figures are put down. By taking more terms and a greater number of figures in each, the value of e might be determined to any degree of accuracy required. 20. On the construction of the common tables of logarithms. From one of the series (v), (vi), (vii), the Napierian logarithms of low prime numbers may be found. The logarithm of a high number which is not a prime may be determined from the logarithms of its factors, by resolving the number into powers of its prime factors ; Thus, Ie288=le25.32=le25+le32 = 5.l62 + 2.1e3. And the expressions (viii), (ix), (x), will greatly facilitate the computation of the logarithms of high numbers. PLANE TKIGOXOMETKY. 105 The Napierian logarithms having been determined, the tables to base 10 are deduced from them by multiplying each by .—yttj "which is equal to •434294819... le 10 Arts. 3 and 14, Cor. 4. Some of the artifices employed in computing the tables may be found in Sharpe's "Method of making Logarithms'' prefixed to Sherwin's Tables. 21. In the common tables are registered the mantissse of the logarithms of numbers of five places of figures, these mantissae being computed to seven places of decimals. At the side of each page is placed a " Table of proportional parts," by which, as it will be shewn, the manti.ssa may be found of the logarithm of a number containing six or seven places of figures ; and conversely, if a logarithm be given whose mantissa is not contained exactly in the tables, the number corresponding to it may be determined, by means of these additional tables, to six or seven places of figures. 22. On the construction and use of tlie Tables of Proportional Parts. Let m-y and m^ be the mantisste of two consecutive integi-al numbers, n and w+ 1, which contain five digits each ; and let m be the mantissa of the number n+ ^ , which contains six digits, the last of which (a) is after the decimal point. Now, since n and n + ^T^ have the same number of integral places, their logarithms have the same characteristic ; .-. m-Wi = lio^«+^j-lioW = lio— ^. Art. 5. =lio^l+j^J which, by expanding 1^ f 1 + r^ j by (iv.) and neglecting the succeeding a terms as being small compared with the first term, becomes r^. j^ttt: » ueaily. 11 Similarly, W2-Wi = lio(« + l) -lio « =j-i;Q- - ; .-. w-mi = (???2-Wi). — . Whence m may be found, if m^, m^ , a, be given ; or a may be determined, when Wj , m^ , m, are known. Should n+ 1 be a nmnber of the form 10', where ^ is an integer, the above proof will hold, if the characteristic and the mantissae of l^, {n + 1) be taken to be J) - 1 and 1 respectively. 106 PLANE TRIGONOMETRY. 23. By the Tables, p. Ill, Mantissa of 36633 = ^2= -5638725 36632 = Wi= -5638606 .-. m2-77ii= -0000119 And in tlie expression m ~ 7n-^^ = {m.2 - m^) -j: , writing for a the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 successively, the former of these tables is obtained. a m.2 - m^ 1 •0000119 X ^ or -00000119 •0000012 nearly. 2 •00000238 •0000024 8 •00000357 -0000036 4 -00000476 •0000048 5 -00000595 •0000060 6 -00000714 •0000071 7 -00000833 •0000083 8 '00000952 •0000095 9 •00001071 •0000107 119 1 12 2 24 3 36 4 48 5 60 6 71 7 83 8 95 9 107 Now the "difference" put down in the Tables for numbers near 86600 is 119, and the "Table of Proportional Parts" is the latter of the Tables above. It appears then, that the significant digits only of the whole difference, and of the differences corresponding to the several digits, are inserted in the table of proportional parts. Hence the following Rule for constructing Tables of Proportional Parts is evident : Of the significant part of the whole difference point off the last digit as a decimal ; (this is the same thing as multiplying the whole difference by t,— , the whole difference beinig: treated as an integer). Multiply the resulting number by 1, 2, 3,... 9 succes- sively, and the whole numbers thus obtained (the last digit in the integral part being increased by unity where the decimal part is not less than •S) are the significant parts of the differences for the digits respectively. PLANE TRIGONOMETRY. 107 Thus, let the significant part of the whole difference be 156. 15-6x6= 93-6= 9-4 nearly. ... X 7= 109-2= 109 ... X 8= 124-8= 125 ... X 9= 140-4= 140. 15*6 X 1 = 15*6 = 16 nearly. ... X 2= 31-2= 31 ... X 3= 46-8= 47 ... X 4= 62-4= 62 ... x5= 78-0= 78 If the digit be given, the difference is immediately known from such a table, or vice versa. To avoid the necessity of per- forming the operation of subtraction in any particular case in order to find the whole difference, there is a line in the tables marked at the top with " Diff ", in which the difference is placed opposite to that logarithm at which such difierence begins. To know what the difference therefore is in any particular case, it is merely requisite to take the number in this line next above the logarithm in question. Ex. 1. To find the Number whose Logarithm is 3-5677766. By the Tables, p. Ill, the mantissa next below the given mantissa is that of li(j 36963, and the whole difference put down is 117. Mantissa of the given Logarithm = ?» =-5677766 Mantissa of liy869G3 =7?ii = •5077672 m - m^ = 94 By the Table of Proportional Parts to "Diff." 117, the difference 94 corre- sponds to the digit 8; therefore the significant part of the Number sought is 369638. Also, since the given Logarithm has 3 for its characteristic, the Number requu-ed is 3696-38. Ex. 2. Eequired the Logarithm q/" 367 "654. By p. Ill, lio 367-650 = 2-5654346 And "Diff." being 118, Part for 4= 47 .-. lio367 -654 = 2-5654393 24. To find the Mantissa of the Logarithm of a Numhe/r which has seven places of digits. Let m^ and m^ be the mantissas of n and n + 1, two successive integers of five digits each : let M be that of n + y^r + -r^ , which has the same number 108 PLANE TRIGONOMETRY. of integral places as n and w + 1, and has also one digit (a) in the place of the tenths, and another (&) in that of the hundredthg. Since a and 6 are digits after the decimal point, tlie numbers n and n + — + YKK have the same number of integral places, their logarithms have the same characteristic, and therefore the difference of their mantissas is the same as the difference of their logarithms ; a b neglecting the terms of the series after the first. Similarly, since n and n + 1 are integers of the same number of digits, W2-Wi = lio(n + l)-lioW = lio^l + -j = PQ. -; Now the first part of the expression, {m^ - mj) — , is the quantity to be added to ?% for the ^rst additional digit a ; Art. 22. And (m^-mj) j^ is what ■would have been added, had b been the^rsf additional digit, instead of being the second; wherefore the quantity which is to be added for b when it is the second additional digit, namely, — . \ (wig-mi) r^[ , is the tenth part of what would have been added, had b been the Jirst, instead of being the second, additional figure. The following Examples will explain what has been said. Ex. 1. To find\JQU-2^^, By the tables, p. Ill, l^o 3684 -2 = 3 -5663432, and "Diff." being 118, the part for a first additional digit 8 is 94 ; and for a first additional digit 6 the ... 71 part is 71, and therefore the pai-t when 6 is a second additional digit is j^r , or 7'1. The operation is thus performed; lio ,3684-2 = 3-5663432 Part for 8 94 Part for 6 71 .-. lio 3684 -286 = 3 -5663533 PLANE TRIGONOMETRY. 109 Ex. 2. To find the Number whose Logarithm is 2'5656560, 2-56o6560 Now 2-56o6471 = lio367-83, p. 111. 89 83=Part for digit 7; "Diff." being 118. 6 =Part for a second digit 5. .-. 2-5G56560 = lio367-8375, and therefore 367"8375 is the number sought. 25. On the adaptation offormulce to logarithmic computation. After a table of logaiithms has once beeu constructed, the labour of certain arithmetical operations can be materially dimi- nished, while at the same time the chance of committing errors is lessened. But by referring to Articles 4, 5, 6 of this Appendix, it will appear that the sole arithmetical operations which can be performed by logarithms, are those of Multiplication, Division, Involution, and Evolution. Before, therefore, the value of an expression can be calculated by means of logarithms, the expression must be put into such a form that no other arithmetic operations than these have to be performed. Such an arrangement of an expression is called the adaptation of it to logarithmic computation. Thus if a, h, c, the three sides of a triangle, be given, logarithms cannot be directly applied to determine the value of cos A from the equation Cos^ = . ; but if from this equation the formula, Cos 44 = /'li^^ , where S=Ua + b + c), be deduced, logarithms can be immediately applied to determine the value of Cos U. For a, b, c being given, S and S-a are easily determined; and these being known, Cos ^A is determined from the equation UGosU = k{\io[S{S-a)]-l,,ic}=h{\ioS + \nAS-a}-\,,b-\,,c}. The two accompanying pages of logarithms are taken from Babbage's Tables, the most correct and the best arranged, per- haps, of any which have been published. The columns have been omitted which in those Tables are given to determine the number of seconds in an angle containing a given number of degrees, minutes, and seconds, and conversely. 110 PLANE TRTGONOMETEY. 1 0000000 51 7075702 101 0043214 151 1789769 201 3031961 2 3010300 52 7160033 102 0086002 152 1818436 202 3053514 8 4771213 53 7242759 103 0128372 153 1846914 203 3074960 4 6020600 54 7323938 104 0170333 154 1875207 204 3096302 5 6989700 55 7403627 105 0211893 155 1903317 205 3117539 6 7781513 56 7481880 106 0253059 156 1931246 206 3138672 7 8450980 57 7558749 107 0293838 157 1958997 207 3159703 8 9030900 58 7634280 108 0334238 158 1986571 208 3180633 9 9542425 59 7708520 109 0374265 159 2013971 209 3201463 10 0000000 60 7781513 110 0413927 160 2041200 210 3222193 1] 0413927 61 7853298 111 0453230 161 2068259 211 3242825 12 0791812 62 7923917 112 0492180 162 2095150 212 3263359 13 1139434 63 7993405 113 0530784 163 2121876 213 3283796 14 1461280 64 8061800 114 0569049 164 2148438 214 3304138 15 1760913 65 8129134 115 0606978 165 2174839 215 3324385 16 2041200 66 8195439 116 0644580 166 2201081 216 3344538 17 2304489 67 8260748 117 0681859 167 2227165 217 3364597 18 2552725 68 8325089 118 0718820 168 2253093 218 3384565 19 2787536 69 8388491 119 0755470 169 2278867 219 3404441 20 3010300 70 8450980 120 0791812 170 2304489 220 3424227 21 3222193 71 8512583 121 0827854 171 2329961 221 3443923 22 3424227 72 8573325 122 0863598 172 2355284 222 3463530 23 3617278 73 8633229 123 0899051 173 2380461 223 3483049 24 3802112 74 8692317 124 0934217 174 2405492 224 3502480 25 3979400 75 8750613 125 0969100 175 2430380 225 3521825 26 4149733 76 8808136 126 1003705 176 2455127 226 3541084 27 4313638 77 8-64907 127 1038037 177 2479733 227 3560259 28 4471580 78 8920946 128 1072100 178 2504200 228 3579348 29 462::i980 79 8976271 129 1105897 179 2528530 229 3598355 SO 4771213 SO 9030900 130 1139434 180 2552725 230 3617278 3L 4913617 81 9084850 131 1172713 181 2576786 231 3636120 32 5051500 82 9138139 l;^2 1205739 182 2600714 232 3654880 33 5185139 83 91911781 133 1238516 183 2624511 233 3673559 34 5314789 84 9242793 134 1271048 184 2648178 234 3692159 35 544U680 85 9294189 135 1303338 185 2671717 235 3710679 36 5563025 86 9344985 136 1335389 186 2695129 236 3729120 37 5682017 87 9o95l93 137 1367206 187 2718416 237 3747483 38 5797836 88 9444827 138 1398791 188 2741578 238 3765770 39 5910646 89 9493900 139 1430148 189 2764618 239 3783979 40 6020600 90 9542425 140 1461280 190 2787536 240 3802112 41 6127839 91 9590414 141 1492191 191 28 10334 241 3820170 42 ()2:i2493 92 9637878 142 1522883 192 2833012 242 3838154 43 63346S5 93 96^4829 143 1553360 193 2855573 243 3856063 44 6434527 94 9731279 144 1583625 194 2878017 244 3873898 4> 6532125 95 9777236 145 1613680 195 2900346 245 3891661 46 6627578 96 9822712 146 1643529 196 2922561 246 3909351 47 6720979 97 9867717 147 1673173 197 2944662 247 3926970 48 6812412 98 9912261 148 1702617 198 2966052 248 3! (445 17 49 6901961 99 9956352 149 1731863 199 2988531 249 39611-93 60 6989700 luO 0000000 15U 1760913 200 3010300 250 3979400 PLANE TRIGONOMETRY. Ill Log. 562. N. 3G 5. Num. 12 3 4 1 5 6 7 8 9 r>iff. 3G50 5622929 3048 3167 3286 3405 3524 3642 3761 3880 3999 119 119 1 4118 4237 4356 4475 4594 4713 4832 4951 5070 5189 1| 12 2 53U8 5427 5.546 5664 5783 5902 6021 6140 6259 6378 •2\ 24 3 dti 3 6497 6616 6734 6853 6972 7091 7210 7329 7448 7567 4 43 5 60 4 7685 7804 7923 8042 8161 8280 8398 8517 6636 8755 H 71 7 K3 5 8874 8993 9111 9230 9349 9468 9587 9705 9824 9943 K 95 ! 107 6 5630062 0181 0299 0418 0537 0656 0775 0893 1012 1131 " 7 1250 1368 1487 1606 1725 1843 1962 12081 2200 2318 8 2437 2556 2674 2793 2912 3031 3149 3268 3387 3505 9 3624 3743 3861 3980 4099 4218 4336 4455 4574 4692 3660 4811 4930 5048 5167 5285 5404 5523 5641 5760 5879 1 5997 6116 6235 6353 6472 6590 6709 6828 6946 7065 2 7183 7302 7421 7539 7658 7776 7895 8013 8132 8251 3 8369 8488 86(J6 8725 8843 8962 9081 9199 9318 9436 4 9555 9673 9792 9910 0029 ol47 o266 0384 o503 o621 5 5640740 0858 0977 1095 1214 ia32 1451 1569 1688 1806 118 118 6 1925 2043 2162 2280 2398 2517 2635 2754 2872 2991 1 12 ■J 24 3 a5 7 3109 3228 3346 3464 3583 3701 3820 3938 4056 4175 8 4293 4412 4530 4648 4767 4885 5004 5122 5240 5359 4 47 5 59 9 5477 5595 5714 5832 5951 6069 6187 6306 6424 1 6542 6 71 7 H3 3670 6661 6779 6897 7016 7134 7252 7371 7489 7607 7726 H 94 it 106 1 7H44 79()2 8080 8199 8317 8435 8554 8672 8790 ■ 8908 2 9027 9145 9263 9382 9500 9618 9736 9^55 9973 i 0091 3 5650209 0328 0446 0564 06S2 0800 0919 1037 1155 1273 4 1392 1510 1628 1746 1864 1983 2101 2219 2;337 2455 5 2573 2692 2810 2928 3046 3164 3282 3401 3519 3637 6 3755 3873 3991 4109 4228 4346 4464 4582 4700 4818 7 4936 5054 5173 5291 5409 5527 5645 5763 5881 5999 8 6117 6235 6353 6471 6590 6708 6826 6944 7062 7180 9 7298 7416 7534 7652 7770 7888 8006 8124 8242 8360 3C80 8478 8596 8714 8832 8950 9068 9186 9304 9422 9540 1 9658 9776 9894 0012 0130 o248 0366 o484 o602 o720 2 5660838 0956 1074 1192 1310 1428 1545 1663 1781 1899 3 2017 2135 2253 2371 2489 2607 2725 2843 2960 3078 4 3196 3314 3432 3550 3668 3786 3903 4021 4139 4257 5 4375 4493 4611 4728 4846 4964 5082 5200 5318 5435 G 5553 5671 5789 5907 6025 6142 6200 6378 6496 6614 < 6731 6849 6967 7085 7203 7320 7438 7556 7674 7791 8 7909 8027 8145 8262 8380 8498 8616 8733 8851 8969 9 9087 9204 9322 9440 9557 9675 9793 9911 o028 0146 3690 5670264 0381 0499 0617 0734 0852 0970 1087 1205 1323 1 1440 1558 1676 1793 1911 2029 2146 2264 2382 2499 2 2617 2735 2852 2970 3087 3205 3323 3440 3558 3675 3 3793 3911 4028 4146 4263 4381 4499 4616 4734 4851 4 4969 5086 5204 5322 5439 5557 5674 5792 5909 6027 117 117 5 6144 6262 6379 6497 6615 6732 6850 6967 7085 7202 1 12 2 23 6 7320 7437 7555 7t)72 7790 7907 8025 8142 8260 8377 3 35 4 47 7 8495 8612 8729 8847 8964 9082 9199 9317 9434 9552 5 59 6 70 8 9669 9787 9904 0021 0139 o256 o374 o49l o608 o726 7 82 9 5680843 0961 1078 1196 1313 1430 1548 1665 1782 1900 8 94 9 105 1 2 3 4 5 6 7 8 9 APPENDIX II. ON THE CONSTRUCTION AND USE OF TABLES OF GONIOMETRIC EATIOS. 1. If it be required to find the value of a trigonometrical formula in which the sines, cosines, tangents, secants, &c., of given angles enter, much labour will be avoided if the values of these quantities be determined once for all, and registered in Tables. In very small angles the sines and tangents are exceedingly- small quantities, and if they be expressed as decimal fractions, two or three cyphers will follow the decimal point before the sififnificant diofits are arrived at. Now in order to avoid the inconvenience of printing these cyphers, the real values of all Goniometrical Ratios are multiplied by 10,000 (or the decimal point is moved four places to the right) before they are regis- tered in the tables ; and the tables so formed are called, " Tables of natural sines, cosines, &c,*" * The tables of Goniometric Katios are sometimes said to be "calculated to a radius of 10,000." To explain the meaning of this expression. If with C as centre, and radius CA, an arc ^^ be described, and BN, ^T be J- to CA, the lines NB, CN, AT, CT, AN respectively, are sometimes de- fined to be the sine, cosine, tangent, secant, and versed sine, of the angle ACB to the radius CA, — or the sine, cosine, tangent, secant, and versed sine, of the arc AB. The expression "to the radius CA" is necessary to these definitions, because the Unes NB, A T. . .depend on the magnitude of CA as well as upon that of the Single ACB; in fact, for a given value of the angle ACB, those lines vary directly as CA. Now PLANE TRIGONOMETRY. 113 2. To find the sine and cosine of 10". Let 6 be the Circular Measure of an angle of 10". ^. d__ 10 1_ TT 180 X 60 X 60 ~ 6^800 * .-. ^ = -000048481368110. But Sin ^ > ^ - :i d\ Art. 105. .*., a fortiori, Sin 10">^-i (-00005)3 > -000048481368110 - -000000000000032 > -000048481368078. Also, Sin 10" <^, or, -000048481368110. Wherefore a near approximation to the value of Sin 10" is obtained by taking the first twelve places which these two quantities have in common, and therefore Sin 10"= -000048481368 very nearly. By substituting this value of Sin 10" in the formula Cos 10" = \/{l - sin^ IC'I . there is obtained Cos 10"= -9999999988248. Now by the definition of the sine which has been adopted in this treatise, Sin ACB = ^; .'. 10,000 X sin ^C5 = iV^.'^^;^. But 10,000 X sin 4 CB is the tabular, or natural, sine of A BO; . '. tab. sin ACB = NB . - '^ = .V5, if CA = 10,000. L/A Wherefore the tab. sine of ACB expresses the magnitude of the line N^>, (the Sine of ACB to the radius CA), the magnitude of CA being represented by 10,000. Similarly, the tab. cosine, tab. tangent, &c., of ACB, express the magr>i. tudes of the Hues CN, AT, &c., the magnitude of CA being 10,000. S. T. 8 114 PLANE TRIGONOMETRY. 3. The Sine and Cosine of 10" being hnown, the Sines of all angles between \0" and 90° may be calculated. ^va[A + B) = 2B\nA .cos B-^in[A- B) ; and writing n . 10'' for Z A, and 10" for Z B, Sin (n + 1) 10" = 2 sin nlO". cos 10" - sin {n - 1) 10". Now 2 cos 10" = 1-9999999976496 = 2- -0000000023504 = 2 - fc suppose ; . •. Sin {n + 1)10"= 2 sin nlO"--Jc. sin nlO" - sin {n - 1) 10" = {sin n . 10" - sin {n - 1) 10"} + sia wlO" - ft . sin nlO". And by writing successively for n the numbers 1, 2, 3... Sin 20" = (sin 10" - sin 0") + sin 10" - ft . sin 10", Sin 30"= (sin 20" - sin 10") + sin 20" - ;t . sin 20", Sin 40"= (sin 30" - sin 20") + sin 30" - ^fc . sin 30". This method is not very laborious. In the last line, sin 30" and (sin 30"- sin 20") are known from the two lines preceding, and the chief labour is in multiplying sin 30" by k. The Sines of angles up to 60" having been successively cal- culated by the above method, those of angles between 60° and 90° may be thus determined. Sin (600 + ^) - sin (60^ -A) =2 cos 60^ . sin ^ = sin A ; .-. Sin (600 + ^) = sin ^+ sin (600-^). So that if A be made to increase by 10" at a time from 0" up to 60", this last formula, by addition merely, will give the sines of angles from 60" to 90". 4. The Sines of angles up to 90° having been determined, their Cosines are also known. For Cos 4= sin (900-^). Thus Cos 25« = sin (90° - 25") = sin 650 ; Cos 720 = sin (900 - 720) = sin IS"^ ; &c. 5. The Tangents, Cotangents, Secants, ^nd Cosecants can be determined from the Sines and Cosines. For TanA= ^, CoiA = -. — -, Sec4 = r, GosecA^—. — ,. cos A Bm A cos A sm A PLANE TRIGONOMETRY. 115 6. Since, Sin ^ = cos (90°-^), Tan^ - cot (90" -^), Sec^ = cosec {90° -A), the Sines, Tangents, and Secants of angles from 45" to 90" are re3j)ectively the same as the Cosines, Cotangents, and Cosecants of angles from 45" to 0". Wherefore it is unnecessary to cany the tables further than to the angle 45". Thus Cos 72", 20'= sin (90" - 72", 20') = sin 1 7", 40', Sin 72", 20' = cos 1 7", 40', Tan 72", 20' = cot 1 7", 40', Sec72", 20' = cosecl7", 40'. At the bottom of the page containing the Sines, cfec. of angles from 17" to 18" is placed the angle 72", and the column which at the top of the page is marked to indicate the Sines of angles from 17" to 18", is marked at the bottom to shew the Cosines of angles from 72" to 73"; and so fur the other Goniometric Eatios. See page 117. 7. FormulcB of Verification. Since the Goniometrical Ratios are determined succes'aive/t/ one from another, one error will affect every successive result As checks against the possibility of errors, several formula3 (of verijication as they are called) are used to examine the accuracy of the results ; and the values registered in the tables are presumed to be correct if they satisfy these formulae. The following are the principal formulae of verification. (1) Sin A = I [^1 + sin 2.4 -Jl- siu 2 J } ) ^^^ ^^ (2) Cos A = k{Jl + sm 2A+Jl- siu 2^ } / A being an angle less than 45". A + 1. n...7oo_V'^-l Again, Cos360 = ^j^; Cos720 = ^-j— ; Art. 58, (3), (1). .'. Sin{36o + ^)-sin(360-^)=2cos360. sin ^ = ?^^^^ . sin J , l and Sin (720 + ^) _ si^ ^720 -A) =2 cos 720 . siu A =- ^'^~^ . sin ^ ; I .'. by subtraction, Sin (360 + A) + sin (72o -A)- sin ^36° - 4) - sm (72o + ^) = sin ^. 8—2 116 , PLANE TRIGONOMETRY. (3) _ .-. Sin ^ + sin (360-4) + sin (720+.d)=sin (36o + ^) + sin (72«- J), which is Euler^s formula. Again, Sin 540=^^^^; Sin 180:=^^^~- ; Art. 58, (3), (1), and 2 sin 54°. cos ^ =sin(54<' + ^) + sin (540- J)) 2 sin ISO. cos ^ = sin (18«+ 4) + sin (.180-^)) * (4) .-. Cos^, or Sin (900-^), = sin (540 + ^) + sin {54P-A) - sin (180 + ^) - sin (IS^ - A), which is Legendre's formula. This formula might have been proved by writing 90" — J. for A in Euler's formula. Ex. To exemplify the use of these formulae. By making ^ = 13° in Legendre's formula; Cos 130= sin 67° + sin 41'-' - sin 310 - sin 50. Now the tables give for the quantities in the second member of the equa- tion 9205 -049 + 6560-590 -5150-381 -871-557, which =9743-701, the quantity given by the tables as the Cosine of 13**. Since, therefore, these quantities satisfy the relation which ought to exist between the Sines of 67^', 41", 31", 5", aad the Cosine of IB", it may be con- cluded, without much chance of error, that the values of these Goniometric Ratios are correctly given by the tables. 8, The values of the Goniometric Ratios having "been thus calculated, multiplied by 10,000, verified, and registered in tables, are called " Tables of Natural Sines, Cosines,. Tangents, Cotangents, Secants, and Cosecants." To find the real Gonio- metric Katios from these tables, the tabular numbers have to be dividtd by 10,000; that is, the decimal point has to be re- moved four places to the left. 9. The logarithmic sines, cosines, tangents, &c. of angles will next be treated of, by which, rather than by the natural goniometric ratios, mathematical calculations are most frequently made. A page from Sher win's Logarithmic Tables, calculated for angles which difier from one another by 1', is here subjoined, the Natural cosines, tangents, secants, and cosecants being omit- ted. The column following that of the N. sines, which is marked ^'^ N.D. 1"," will be explained in the next Appendix. 17 Degrees. 117 2965-416 2y;aji94 2!)8 734 ;JO<)4-284 3007-058 3009-832 :J0I2()(K) 3015 ;m» 3018-163 3O20-926 .•<023-699 3026-471 3029-244 3032 016 3034-788 ;jo;{7 5")9 3040 331 3043-102 3045-872 3048643 3051-413 ;}054 183 3056-953 3059-723 3062-492 30(i5-261 30f;8-030 3070 7y8 3073-566 3076 334 3079 102 3081 -8f)9 3084-(i3() 3087 -403 3090-170 9-46-93.53 9-466.i483 9 467l7.iO 9-4675848 9-467f>960 9-46«4(k;9 9-46!{8173 9-4/J92273 9 46963()9 9-4700461 9-4704)48 .9 4708631 94712710 9 4716785 9-47-208-)6 9 472 92-2 !>-47:--8:»85 9 473:«)43 9 4737097 9-4741146 2-4745192 9-47492;<4 9-4753271 9 4757304 9-4761334 9 47(i.'>359 9-47693.'!(i 9 4773;«x; 9-4777409 9-4781418 9 4785423 9 47HJM23 9 4793420 94797412 9-4801401 9 480.V»»5 9 48(0366 9 -48 1 3342 9 4817315 9 4821 2a3 9-4H2.-)-J48 9 4829208 9-483316.5 9-4837117 94841066 9-484.5010 9-484895 1 9 4850888 9-4856820 9-4860749 9-4864674 9 48<)8595 9 4872512 9-4876426 9-4880335 9-4884240 9 4888142 9-4892040 9-489'>.934 9-4899824 o.j^- 68 833 68 766 68-683 68 633 68-533 68-483 6*i 401) m 2m 68-200 68-116 68 0.50 f)7 9a3 67 916 67-850 67 766 (.7716 67 6",'{.3 67-.5()6 07 483 f7 4.^3 67 -■■«»«; (.7-283 (7 216 67-16() (17 (m (7 016 66 9.33 (U] 8}t.3 66 816 667.50 (Ki cm («6I6 (;6-.533 66-483 66-400 (!(i-,3.5(t (K3 266 66 216 66133 66-083 (i6-(K»0 65-9.50 ()5 8()6 65-816 65 733 65 683 65 616 65.533 65-483 65-416 65-350 (J5-283 6.5-233 65-150 65-083 65-003 64-f»6'i 64-900 64-83.3 Co-tecants. 10-5.340647 10-5.336517 10-53M2.391 10 5.328270 10-5324152 10-.5.320040 10-5315931 10-5311827 105307727 10-53U36J1 10 .52995.39 10-5295452 10-.522 ! 9 5(l.'J.54.59 i 9.5039822 9. 5( (44 1 82 9-3048338 95052891 95057240 9.5061586 9-.5(K».5928 9 5070267 9-.5074602 9-5(789:33 9.5083261 9 5087586 95091907 9-5096224 9-5100539 95104849 95109156 9 5113460 9 51177h-0 1 SfC. 75 283 75 2(»0 75 150 75 (183 75 0(XJ 74 9.50 74 9(KI 74 816 74 750 74-700 74 6'i3 74 ^('yi 74 616 74450 74 366 74 316 74 2(;o 74-ia3 74 \-x\ 74 050 74(116 74 0(^) T\ 883 73 816 73416 73 700 7nii(; 7.3 58.-J 73.5(H) 73 450 7.3 3a3 7.3 ;{.•« 73 ^m , 73 2(H) I 73130 73 083 73 016 72!K;() 72 9(H) 72 850 72 7a3 7271 6 72 6«)6 72-600 72-550 72 483 72 433 72 34 10 0201448 100201842! 10 (»-2o22:i6 10 0_'0:.'();(1 lo().'o;i027 10 0203422 10 020,3818 ' I0(t.'04215 10020-1612 10 0-J05 9 97!H5!73 9 97!-6378 ' 9-^796182 9 '795785 9 f793.388 9 9794!)91 9 9794593 9 9794195 9 J79.379<) 9 979^398 9 !79i:'»!i8 9 9792599 (^nnr ! 9 9792198 ^ ^*^ 9 9791798 9979i;«7 9 97f)09!M) 6 683 6(>83 «7J«,9 97;»0594 6716 6 716 6716 6 733 6733 6 750 6-750 6 750 6-7(i6 6 766 ' 9-r7.90I92 9-J78:789 9! 789386 9-!788 i83 9 9788579 9-9788175 9 9787770 9 -.9787365 9-97H6!;60 9-9786554 6-783 6-783 9 9786148 9 97a^741 fi7H.3 9 978.'^334 fi-80n 9 ■''7«4927 «8J«j9 9784519 10-0215889 R.o.fi. 9-9784111 100216298 fl.o ft 9-978.702 10-02167(7 fi.oqo 9-97a3293 10-0217117 I fi.oT«i 9 9782883 100217526 «.„![! I 9-.9782474 100217937 r ^^ I 9 9782063 APPENDIX III. ON THE LOGARITHMIC TABLES OF GOXIOMETRIC RATIOS. 1. When the Sines, Cosines, &c., of angles have been de- termined, their logarithms may be found from the tables of the logarithms of numbers. There are, however, methods by which the logarithms of the Goniometric Ratios can be found inde- pendently. 2. To find Ijg sin ^, sin B not being given. Sm^ = ^.(l-^j . ( 1-2^0) . I l-7:;r^J Art. 133. By makine; ^ = — . — , and taking the logarithms of both sides of this '^ ?i 2 equation, I and lio f 1 - ^^^^ j , ho f 1 - ^2) ' ^^- ^^ing expanded by (ii), p. 100 j , fm 7r\ , ['l^n^-m^ = \mA --o +1 in«"2y'"^'in 2-7i2 1 10" V 4^^ ■*" ^ • 4%5 "*" ^ • 46,i6 + m° 1 f m"^ ^ Tff^ " iTio ' \^^ ■*■■ ^ ' e-^H^ ^ '^ • 66^6 1 / w? J m* J m^ -&c. Now lio ( f • I) = lio "^ ^ ho •^ - ho 'i - ho 2, and ho^2^=lio{(2« + m).(2n-7n)}-lio(2%i^) = ho (2/z + >/i) + ho (2?i - w) - 2 {ho2 + lio « } ; PLANE TKIGONOMETRY. 119 •'• ^10 ^"^ (!?•?)" ^10 ^ + ^10 (2« + "?) + lio (2n - ra) + l^o tt - 3 {l.^ 7i +1^0 ' — A 1 1- LIO + -^ ^^14+64 + .34+- j^ ^. n 1 2 \ '■ ' I- + &c. By giving m and n different values, the logarithmic sines of all angles may be found by this formula. 3. Ill like manner from the series 3'-7r-/ V o'-TT- Cos^=f 1- the following formula is obtained; liocosf— . ^j=liol'* + "^) + lio('i-w)-21ion leiO 1 1 1 3- 5- /- + , /I 1 1 + .. , /I 1 1 + &c. .Art. 133, ?(•* ). . 4. The logarithms of the Sines and Cosines having l)een thus determined, those of Tan^-"^, Sec^=-i^, cos 6 cos u may be severally found. Cot^^'';'^^, Cosec^ = ^-, sin 6 sm $ ' 5. Since all Sines and Cosines are, generally, less than 1, their logarithms to base 10 are negative. In order to avoid tho inconveniencs of printing negative characteristics, the logarithms to base of 10 of all Goniometrical Ratios are increased by 10, and the resulting numbers being registered are called "Tables of Logarithmic Sines, Cosines," &c. 120 PLANE TRIGONOMETRY. Hence if any one of these tabular logarithmic quantities be given, by subtracting 10 from it the real logarithm of the go- niometric ratio may be obtained. These tabular logarithmic quantities will be indicated by the letter L; thus the tabular logarithmic sine of A, or 10 + 1^^, sin -4, T/ill be written L sin A. 6. The common Logarithmic Tables of Goniometric Ratios are calculated for angles which differ from one another by one minute. If, beside degi-ees and minutes, the angle contain some seconds, its tabular logarithmic function may, with certain ex- ceptions, be found on the principle proved in the next Article. 7. The increments of tabular logarithmic sines, &c., of angles vary, except in certain cases, as the increment of the angle. Let the angle A receive the increments a", and 60", successively. sin [A + a") - sin A sin A Then Sin {A + a") = sin ^4 j 1 + = sin^ . Iih -. — r— -[ > unless ^ = 90"^ nearly; Art. 59, Cor. ( sm ^ ) . •. lio sin {A + a") = lio sin ^ + l^o (1 + cot A sin a") ; . •• {10 + lio sin {A + a")} - {10 + Ijo sin A) =\^q (1 + cot A sin a") ; .'. L sin (^ + a") - L sin ^ = -1^ . {cot A sin a" - \ cot^ A sin^ a" + ...} App. i. (14) (iv.) = . cot^ sin a"; by neglecting the higher powers of cot ^ . sin a"; which may be done unless A = 2n . 90^ nearly. And writing 60" for a" in this equation, it becomes L sin (^ + 60") - L sin ^ = -— . cot A sin 60" ; . L sin (^ + a") - L sin ^ ^ sin a'[ _ a ^^^ ^^^ ^^^ •■ L sin (4 + 60")-!. sin ^ sin60" 60* • > • And in an exactly similar manner it may be shewn that for the Cosine, Tangent, &c., of an angle, the increment of the tabular logarithm varies as the increment of the angle, except in those cases mentioned in the Corollaries to Arts. 60, 61, 62. PLANE TRIGONOMETRY. 1*21 8. To explain the meaning and use of the columns of differ- ences for one second (Diff. 1"), which are placed after the columns of logarithmic sines, tangents, d:c. If a" become V\ the equation arrived at in the last Article becomes, L sin U + 1") - L sin ^ = {L (sin ^ + 60"^ - L sin ^} . ^ , which is the difference for L sin A corresponding to one second. Now, if this quantity be computed and registered, L (sin A + a") - L sin A may be determined, when a is given, by merely multiplying this registered difference by a; and when L sin {A +a") is given, a may be found by dividing L sin [A + a") - L sin A by this difference. Por L sin [A + a") - 1, sin J = {L sin {A + 60") - L sin A} . -r Lsinf.-<+GO")-LsinJ .a, and a = GO L sin {A + a") - L sin A -- . {L sin (.4 + GO") - L sin ^1} Thus, L sin 17^ 1' = 9-4663483 Lsinl7<' =9-4659353 .-. L sin 17^ I'-L sin 17^= -0004130 Now -^— = 68-833, which is the quantity put down in the Tables as the difference for one second to L sines of angles between 17° and 17*^, 1'. The significant part of the difference is considered as a whole number, or the real difference is multiplied by 10^, in order to avoid the necessity of printing the three or four cyphers which in nearly every case precede tlie significant part of the difference*. 9. Ex.1. ro^/?«rfLsinl7^ 14', 12". i:sinl7^14', =9-4716785 Now Diff. for 1" = 67*850 .-.Diff. for 12" = 814-200= 814-2 L sin 17", 14', 12"= 9-4717599. * The column of differences for the natural sines, &c. of angles are computed after a manner similar to this, and the differences themselves are all multipUed by 1000, to avoid the necessity of printing the cyphers imme- diately following the decimal point. 122 PLANE TRIGONOMETRT. Ex.2. IJ L sin A = 9-4685537, rcgtm-6tZ A. I. sin ^=^9-4685537 L sin 17°, 6'= 9-4684069 Diff.= 1468 Now Biff, for 1" is in this case 68-400, and ^^^^ =21-46; .-. ^=17^ 6', 2r-46. Ex.3. If L cos A = 9-9784328, reguireiZ A. lu this case, because the increase of the angle is attended by the decrease of the L cosine, Art. 60, Cor. 2, the given L cosine must be subtracted from that in the tables which is next greater than it. Now L cos 17", 15' = 9-9784519 LcosA =9-9784328 Diff. = 191 191 Now Diff. for 1" in this case is 6-800, and -rr~^- =28*088 = 28-09 nearly : jj J J 6-aoo '' and the angle required is 17**, 54', 28"-09. Ex. 4. Required the L cosine of 72^, 5', 8". By the Tables, p. 117, L cos 720, 5' =9-4880335 , and Diff. for 1" = 65-15; .-. Difi. for 8"= 521-2 .-. L cos 720, 5'^ 8" = 9-4879814 the difference for the additional seconds being in this case subtracted from M L cos 720, 5'. Art. 60. Note. It may here be observed, that the difference for additional seconds must be added for L sines, L tangents, and L secants, Arts. 59, 61, 62 ; and subtracted for L cosines, Ait. 60, L cotangents, and L cosecants. 10. To shew that the same colwmns of '■^ Differences for 1"" serve for L sin A and L cosec A, for L cos A and L sec A, and for L tan A and L cot A. For Sin^ = -; .*. Lo sin J. = - L,, cosec J, cosec 4 ^ ^^ .-. Lsin^, =10 + 1^0 sin ^, =20- (10 4-lio cosec ^) = 20-1/ cosec ^, Shnilarly, L sin {A + 1") = 20 - X cosec {A + 1") ; .-. D sin ^A + 1") - L sin J. = - {I, cosec {A + 1") - L cosec A\» PLANE TRIGONOMETRY. 123 Hence a column of "differences for 1"" is printed between the column of logarithmic sines and cosecants; serving to the former as a column of incre- ments for 1", and to the latter as a column of decrements for 1", In like manner it may be shewn that L cos (.4 + 1") -Lcos, A=-\L sec [A + 1") - L sec A}, L tan [A + 1") - L tan ^ = - {L cot {A + 1") - L cot J }. Wherefore the columns of cosines and secants have the same differences for 1", as also have the tangents and cotangents: and it is to be observed that these ditierences serve respectively as increments to the secants (Art. 61), and to the tangents (Art. 62), and as decrements to the cosines (Art. 60), and to the cotangents. 11. Before the increment of the tabular logarithm of a Go- niometrical Ratio can be determined from the small given incle- ment of the angle, or conversely, these two conditions must be fulfilled ; I. The logarithmic increment must in that particular case vary as the increment of the angle ; II. The increment of the logarithm must not be an exceed- ingly small quantity. Thus, if it were required to determine L sin 89", 40', 3", from Tables in which L sines were registered for all angles from 0° to 90** which differed from one another by 1', it would be found that, L sin 890, 41', = 9-9999934 L sin 890, 40', = 9-9999927 .-. Difference for 60"= 7 Wherefore, if even the first of these conditions held for the L sines of angles about 89", 40' in magnitude (which it does not, Art. 59, Cor.), yet a differ- 60" ence of 1 in the L sine would produce a difference of —=- , or ^" nearly, in the angle; and therefore any increment of 89", 40' which was not greater than 8 ', would produce no change at all in the first seven figures following the decimal point of L sin 89^, 40'. 124 PLANE TRIGONOMETRY. 12. To determine the degree of accuracy to which additional seconds may he calculated in a given case hy assuming that the in- crease of the angle is proportional to the increase of some logarithmic function of the angle. Let n be the number of seconds by which a difference of unity is produced in a certain logarithmic function of a given angle, and let I be the difference for 60" ; — the differences being considered in both cases as whole numbers ; — - TV. ™ 1 1 ^'^ The quantity -y- therefore gives the number of additional seconds 1/ corresponding to the least possible increase of the given logarith- mic function of the angle, and is consequently the measure of the degree of accuracy to which small increments of the angle may be calculated on the principle that the increments of the angle vary as the increments of the logarithm of some particular Goniometric function of it. 13. It has been observed, App. iii. 5, that the real logarithm of a Goniometrical quantity is obtained by subtracting 10 from the tabular logarithm. It is therefore necessary To establish a general rule for supplying the tens when the tabular logarithms of goniometrical quantities are used. Let Cos"ui =a. sin"* 5 . tan^C be any trigonometrical formula adapted to logarithmic computation, App. i. 25. Then n . 1^,, co^ A =\^a + m .\^^y!ii B ■¥ p . \^ tan G ; .-. w.(10-^lJ„cos^)-?^. 10 = ljoa + m. (10 + l,„sin ^) - m. 10 + p .(10+l^„tan(7)-;) . 10; .-. n . Lco^A= \^a + m. LsinB + p . L tan C +{n-(m + p)] . 10. And here n, m, p may be whole or fractional. "Whence the Rule ; Add to the second member of the equation as many tens as the number of times the tabular logarithms of t/ie goniometrical ratios have been taken in the former member of tJie equation exceeds the number of times they have been taken in the latter member. PLANE TRIGOXOMETRY. 125 Ex.1. Tan5 ^ = cos i? . cos^ C \ .-. 5 . L tan ^ =L cos ^ + 2 . L cos C 4- {5 - (1 + 2)} . 10 = L cos J5 + 2 . L cos (7 + 20. Ex. 2. Tan2 A . sm^ B = %. sec* C, o 2 . L tan ^ + 6 . L sin 5 = lioa - lir/^ + i . ^ sec C+ {(2 + 6) - 4} . 10 = lio«-lio^ + "='^ • -^ s^c C+40. Ex.3. Tan^^ = 2-^^^^^; cos^ C 10 .-. ^ ^ tan 4 = l3o2 + - L Sin i? - - L COS C + w - f - - - J 5 2 = lio2 + -Lsm 5-^ Lcos C- J . 10. [Had both sides of this equation been raised to the sixth power, the frac- tional indices would have di^iU)i)eared, and the value of tan A would have been practically determined luuch more easily.] 14. Lastly, the methods will be explained by which small angles are determined from their L sines, ajid conversely. When an angle receives a small increment, the Differential Calculus affords the means of determining with facility the con- sequent increase of the Goniometric Katio. Required the increase of L sin $ arising from receivi'ng a siiialL itbcrement 6B. By Taylor's Theorem, L sin {d + d9)^L sin O + deL sin ^ . SO + d^eL sin d . \ ■ J+... Now L sin ^ = 10 + lio sin ^; , -r ■ ^ 1 co^ ^ 1 . '. deL sni 6 = ,--T-^ • , = — ;— . cot d, lelO auxd lelO ' whence, d L sin d — - —~ . cosec- d ; le J-0 .: I. sin {e + be) -Lsmd= -— . cot ^ . 5^ - — — . co^ec- 6 . ^-^ + ... 126 PLANE TRIGONOMETRY. 1 5. Now if ^ be very small, cosec is very largftj and there- fore (unless B6 be exceedingly small also) the second difference, 1 (?if))^ . cosec^ . \ — ~ , is of such a maomitude that it cannot be IJO 1.2' neglected in comparison with the first difference, =— — . cot . hd. In this case, therefore, since the increment of L sin 6 does not vary as B9, the simple power of the increment of 0, the quantity to be added to L sin for a small increment of cannot be obtained by a simple proportion, but will have to be determined approximately by the tedious process of computing the first two terms of the series wliich gives the value of the increment of X sin 6. There are three methods of escaping this inconvenience. 16. The first method is to construct tables for the first few degrees to intervals of a second, instead of to intervals of a minute as the tables in ordinary use are constructed. Here Bd is less than one second, and L sin [d + dd) may be roiighly com- puted to decimal parts of a second by neglecting the second term of the series obtained in the last Article but one, in which case, Lsin(^ + 5^)-Lsin^ = — —.cot^. 5^; le -L^ Therefore, for any particular value of 0, The increment of L sin ^ oc 5^, and Increment of L sinO for 56'' 88 Increment of L sin d for 60" 60 * 17. Second method. By the following formulfe, which are given by Maskelyne in his introduction to Taylor's Logarithms, a small angle may be determined very accurately to decimal parts of a second from its L siae and conversely, by the aid of tables of L sines and L cosines which are calculated to every second for a few degrees, Sin^ = = d- 9' -2.3+- ■e- 2 3 nearly. Cos^ = = 1- -1.2^-- -.1- - nearly. Sin^ 1p ftnrtt. -.1- flin 2.3 ' inc n". i ^1. 6^ 2 T n : j nearly, = d cos^ . ,,, , or 6 = n. sin 1"; sin 1 and .-. Sin 5 = n. sin 1" . cos^ 0. PLANE TRIGONOMETRY. 127 "Writing n" for ^, and taking the logarithms, L sin n" = l^g n + L sin 1" + \L cos n" - ^ . 10 ; .-. L sinn" = lioW + jL sin 1"-^. (10 -L cosn") (i.)» and Ij^n = L sin n'' + i (10 - L cos n") - L sin 1" (ii.) Def. The quantity 10 - L cos n" is called " the arithmetic complement" of L cos n'. 18. It is to be remarked, that in using these formulae to detennine L sin n" when n is given, or conversely, an approximate, value of L cos n" may be taken from the tables and written in the second member of the equation.-^ without sensibly affecting the result, because the variation of L cos n" is exceedingly small when n is small, as may easily be shewn*. The whole matter will be rendered more clear by an example. Ex. If L sin n" = 7*3217783, required n. Taylor's tables give, L sin 7', 12' L sin 7', 13' 7-3210583 I L cos 7', 12" 7-3220G21 Z cos 7', 13" 9-9999990 9-9999990 Therefore the angle is 7', 12" nearly ; and 7', 12" is the approximate value of the angle which must be taken in the second member of (ii.) of the last Article. Now, by (ii.), L sin n" = 7-321 7783 M10-I-cos7',12")|_ or 1 X -0000010, S - "^"^^^^ 7-3217766 Lsin 1" = 4-6855719 .-. lion = 2-6362037 = lio 432-717; .*. 71 = 432-717; or the angle required is 7', 12"-717. * Cos^ = l-^^+ ^ ., 3^^ = l--nearly; .-. liocos^, orLcos^-10, =-^^ -. |- +-|( j +... ; 1 6'^ .'., by the Differential Calculus, de (L cos ^ - 10) = - .-— ir {d+ '^ + ■..), a le 10 ^ very small quantity, if 6 be an angle of a few minutes only. 128 PLANE TRIGONOMETRY. 19. In like manner a formula may be established for finding L tan 'Th' from ?i, and conversely. cos^ ^ ^ sin^ 2.3 , Tan 6 = = — — -r- nearly ; 1-2 1 _ _^ Tl-^'^^ tan ^ 2.3 V 2 7 / ^'A^l .. Sr- = 1 - T7 = (cos ^) 5 ; 2 2 tan n" , ,. ^ n . sm 1 ' — 1 -no. r cilT 1"_ ^^„^,„ , 3 3 2 2 i .-. L tan?i"=ljo^ + -^ s^^l"" q --^ cos7i"+-. 10; " 1 2 •. L tan n"=lion+L sinl"+ -(10-Lcos n") (ui.) 2 And lion = L tan7i"-Lsin l"-o (lO-Lcosn") (iv.) 20. Delamhre's Tables. The third method alluded to, (15, p. 126,) is to construct tables as far as an angle of one degree, which . , sin ^ , , , , , . , . . sin ^ t - i>'\ e give Ijp —7T- , (or tables which give 1^^ — ^ + i/ sm 1 ), tor every second. Such tables are printed in no collection, perhaps, except those of Callet; they may be easily constructed in the following man- ner : Let 6 be an angle of n seconds ; .*. d=n . sin 1". „, , sin 6 . sin n" , . „ , , • i « Then lio -y- = lio ,,,gi^i^. =lio ^^^ ^ " lio^ " lio sm 1 = L sin n" -l-^^n-L sin 1" ; .-. Ijo^^^ — + IiSinr' = Lsinn"-lio^. u Similarly, if 9 be an angle of n minutes, sin 6 T Ijo —— + L sin r = L sm n - l^Qti. TLAXE TRIGONOMETRY. 129 21. To determine, the Sine of a given small angle, or con- versely, from Delambre's Tables. r,- ^ sin 6 ^ sin 6 • , ,/ Since Sin d - --— . 6 = -^— . n . sm 1 ; sin^ T Lsind=[ lio —5- + i sm 1 + I^q n ; ^ • „ /, sin ^ T • 1/A 1 ,-\ .-. Lsmn"=Mjo-^+I'SmlM + lio« (O- And lio» = i'Sinn"- (lio — ^ + I'Siul" ) (ii). The most convenient tables are evidently those which give ( lio "V" + L sin 1" J for every second. 22. Ex. 1. To determine Lsiii n" h>/ Delambre's Tables. Since, as shewn below*, 1^^ ^ + Z sin 1" increases very slowly as 6 iucreases, the value of L sin n" is obtained without sensible error by taking for 1^^, — ^ — 1- L sin 1" the quantity which is given in the tables for the angle containing that number of seconds which is the nearest integer to the given number (n) and adding lj,?i to it. Thus ; If 11 = 546'2o, required L sin 546"-25. By Taylor's Tables, L sin 546", or L sm 9', 6", is 74227670; and lio 546 = 2-7371926. Therefore, when 6 is an angle of 546", lio ^^ + ^ sin 1" = L sin n" - l^o "• Art. 20 : p. 128. = 7-4227670 -2 -7371926 = 4-6855744, For —- =1-^-^ + 3-^-3-^3-. ..=l-g nearly; sin5_ J_ ^^2 ^ /^2\2 .-., by the Differential Calculus, d^l^Q ^^ ="riO' (3 "^ 6^+ ••') a very small quantity, if be an angle of a few minutes only. S. T. 9 130 PLANE TRIGONOMETRY. the quantity corresponding to the angle 546" which would be given in the tables of L,, — -— + 1, sin 1". u . : By (i) of the last Article, I- sin 564" -25 = 4 -6855744 + lio 564-25 =4-6855744 + 2-7373914 = 7-4229658. Ex. 2. To determine n when L sin n" is given. By referring to the tables of L sines in common use the in- tegral number of seconds is found which is contained in the angle ^vhose L siue is next below the proposed quantity : suppose this number to be m. Then substituting in (ii) of the last Article, the sin 9 T which corresponds to the angle m", a near approximation to the value of n is obtained. quantity given in the tables for that value of 1^^ — ^ h L sin 1" Thus, 1/ L sin n" = 7 '4230612, required n. By Taylor's Tables, L sin 546" = 7*4227670, and L sin 547"= 7-4235617 ; .-. the value of m in this case is 546. Now, as in the last example, the value of l-^^ — ^ + ^ ^-^ ^" ^°^ ^^ angle 564" is 4-6855744. Therefore by (ii) of the last Article, lion = 7-4230612-4-6855744 = 2-7374868=lio564-37; and . ', 546" "37 is the angle sought. APPENDIX IV. THE GENERAL PROOF OF THE FORMULAE Sin(^±i?) = Sm^ .Cos^±Cos.4 . Sin i/, Cos(J «fc B) = Cos ^ . Cos ^ =F Sin ^ . Sin JB. On the Theory of Projections. 1. Let X'OX be an indefinite straight line, and PQ a finite straight line from which PJ/, QN' are drawn perpendicular to X'OX ; the length MN intercepted on X'OX between these perpendiculars is termed the Orthogonal Projection* (or sira])ly the "projection") oi PQ on X'OX, and X'OX is termed the Line of Projection. Let lengths on X'OX be reckoned positive if measured in the direction OX, and negative in the opposite direction OX' ; and let ihe sign of the projection of any line on X'OX be determined in accordance with this convention. Then in the above figure the projection of PQ (being MX, measured from 31 to X) will be positive ; while the projection of QP (being measured from X to M ) will be negative. 2. From this explanation of projections it will be seen that the projection of a line on a line parallel to it is equal to the line itself, — that the projection of a line on a line perpendicular to it is zero, — and that the projections of two equal and parallel lines, taken in the same direction, are equal in magnitude and of the same sign. * An orthogonal projection is made by means of straight lines that &xe per- pendicular to the line of projection. There are other modes of projecting a line on another line, as for instance, by means of lines drawn from a fixed point through the extremities of the first hne. 9—2 132 PLANE TRIGONOMETEY. 3. If two 'points he joined hy any series of straight lines, the algebraical mm of the projections of the lines taken in order from one point to the other is constant. Thus, let AB, BC, CD, BE, EF be any series of straight lines com- mencing at A and terminating at F. Their projections are MN, NP, FQ, QR, RS, of which the algebraical sum = M^ + NF-QP+QR-SR =r MS, the projection of AF. JV Q. This is a proposition of very frequent application ; and it may be remarked that it will be equally true if the straight lines do not all lie in the same plane. It is sometimes convenient to express this result by saying that the sum of the projections on any straight line of the sides taken in order, of a closed polygon, is zero. 4. If FR (Fig. to Art. 1) be drawn parallel to X'OX, it will be seen that UN = FR = FQ . cos QFR ; and therefore the length of the projection of a line is obtained by multiplying the length of the line by the cosine of its inclination to the line of projec- tion. The same will be true with regard to sign as well as magnitude, if the following conventions be made. 5. From a fixed point in X'OX draw a line parallel to FQ, the line to be projected, and in the same direction as FQ ; and let the angle which this line makes with the positive portion {OX) of the line X'OX be considered as the in- clination of FQ to the line of pro- jection; then will the sigTi of the projection be the same as that of the cosine of the inclination. Thus the inclination of FQ to X'OX is XOT, au angle whose terminal line lies in the first quadrant, and whose cosine therefore is positive, as is also MN the projection of FQ. But the incli- nation of QF is XOT', an angle whose terminal line lies in the J PLANE TEIGONOMETRT. 133 third quadrant, and whose cosine therefore is negative, which is also the case with i\'i/, the projec- tion of QP. Again, in the annexed figure, where OT, OT' lie in the fouiih and second quadrants respective- ly, the same will be seen to be true. 6. The line to be projected has not in the preceding Articles been considered as afiected by any sign*; but \i it lie on a line on which the positive and negative directions are assigned, the inclination of the line to be projected must be taken to be the inclination of the j^ositive side of the line on which it lies to the positive side of the line of projection. Thus, if PQ be considered as negative, (that is, if PQ lie on a line on which lengths measured in the same direction as from P towards Q are taken as negative), its inclination to the line of projection must be considered as XOT' in both the figures to Art. 5 ; and the cosine being in both cases negative, PQ x cosine of the inclination will be positive as before. 7. From the preceding considerations, a perfectly General Proof may be derived of the formulje which give the values of Sin {A ± jB) and Cos (^ d= -5) in terms of the Sines and Cosines of the simple angles. Saj)pose a pair of rectangular axes xOx, y'Oy to be originally coincident with the axes X'OX, Y'OY, having their positive sides Ox^ Oy coincident with the })0si- tive sides OX, OY ; and then suppose them to revolve through any angle A, in the positive or negative direction of revolution according as J. is a positive or negative angle. Thus, in the figure, Ox and Oy have revolved from OX, OZ through an angle * A distinction has been made throughout between the lines PQ and QP, but we have not before supposed that distinction to be expressed by means of the algebraical signs + and - . 134 PLANE TRIGONOMETRY. in tlie positive direction greater than two right angles and less than three, or tlirough an angle in the negative direction greater than one right angle and less than two. Also, suppose a line OP, initially coincident with Ox, to revolve through any angle B, measured from Ox in a positive or negative direction according as B is positive or negative. Thus, in the figure, OP has revolved through a positive angle xOP that is greater than one right angle and less than two, or through a negative angle greater than two right angles and less than three. Let ON, NP be the projections of OP on x'Ox, y'Oy ; .♦. 0N= OP . cos B, and NP = OP . cos (B - 90°) = OP . sin 5 ; as well with regard to sign as to magnitude (Art.- 5). I^ow projecting OP, and also the lines ON, NP on X'OX, the projection of the first will be equal to the algebraic sum of the projections of the other two (Art. 3). But the angle between OP and OX (the positive part of X'OX) is A +B, .'. the projection of OP on X'OX = OP . cos {A + B), Also, by Art. 6, the projection of ON on X'OX = ON. cos ^ = OP . cos ^ . cos ^ ; and the projection of NP on X'OX = NP . cos (^ + 90°) = - OP . sin ^ . sin B. Therefore, OP . cos {A+B) = OP. cos J . cos P - OP . sin ^ . sin P, or Cos (A +B) =cobA . cos P — sin ^ . sin P j for all values of A and B, positive or negative. 8. The formula for Sin (A + B) may be found in like manner, by projecting on Y'OY, but it may be derived from the formula just proved by writing in it 90° + ^ for A, when Cos (90° + ^ + P) = cos (90° + ^) . cos P - sin (90° + ^) . sin P, .-. - sin (^ + P) = - sin ^ . cos P - cos ^ . sin P, .-. Sin(il + P)= sin ^4 . cos P + cos ^ . sin P. PLANE TPJGOXOMETRY. 135 9. The demonstration given above is due to Professor De Morgan : see Chap. iii. of his " Trigonometry and Double Alge- bra." And it ought to be remarked in his words, that *4t can be convincing only to those who enable themselves to understand in the most general sense the preliminary theorems. Any want of such mastery over the universal character of theorems in projection will follow the student through all his course, particu- larly in the higher Geometry and in Mechanics." 10. Examples. (1) Shew, by projecting the sides of a regular polygon on any line, that Co.se + cosf« + ?^V +cos(g + ^^"~^^" | = 0, Sine + Hinfe + ^^y +sin{fl+?i!Lrih')=o, whatever be the value of 6, n being an integer. (2) If X, 2/, and x\ y be adjacent sides of two parallelograms described on the same diagonal, shew that y^ , . ^^ , . -^ y . sin i/x = x . sin x'x + y' . sin y'x, X . sin xy = X . sin x'y + y' . sin y'y ; where yx denotes the angle which y makes with x, and so foi the other expressions x'x, (fee. [These are the general formulae for the transformation of co-ordinates in Analytical Geometry.] (3) If the inclinations of the sides a, 6, c of a triangle, taken in order, to a given line be a, /?, y respectively, prove that a . cos a + b . cos /S + c . cos 7 = 0, and that a . cos (f3 + y) + b . cos (y + a) + c . cos (a + ^) = 0. EXAMPLES. 1. Prove tliat 45", 15', 20" = 50^, 28\ 39" -50 ; 10", 15', 37" = IP, 40; 3"-09 j 18", 10', 48" = 20^, 20^ ; -^= 115^, 47\ II. The Complements of 17", 36', 43"; 29", 27', 6"-82 ; and 216", 45'; are 72", 23', 17"; 60", 32', 53"-68 ; and- 126", 45'. III. The Supplements of 37", 4', 3" ; 115", 13', 24"-66 ; and 226",14',17"; are 142", 55', 57"; 64",46',35"-34; and -(46", 14', 17"). IV. 1. If Cot^=|, find tlie values of Sin^, Cos J, Cosec^, Yersin^, and Sec^l. 2. What angles have the same sine as 320" 1 Ans. The form 2m. 180"+^..,see Art. 24 (1),... gives 320", 680", 1040". ..for the values 0, 1, 2 ...of m, and -40", -400", - 760".., for the values - 1, ~ 2, - 3, ... of m. The form (2m + 1) 180° -^, see Art. 24 (4), gives - 140", 220", 580".. for the values 0, 1, 2, ...of m; and-500", -860", -1220"... forthe values-1, -2, -3,... ofm. 3. "What are the angles that have their tangents of the same magnitude as that of - 110", but afiected with a different sign? Ans. 110", 290", 470", 650",... -70", - 250", -430", ... The angles are comprised under the form (see Art. 25) m. 180"+ 110°, where m is or any positive or negative integer. V. Prove the formulae, 1. Sec^ A cosec^ A = sec^ A + cosec^ A. 2. Cot' A cos' A = cot" A - cos' A. o n^o A ^o* ^ ^ Tr • ^ sec ^ - 1 6. Cos A =—rp. Tg-rr . 4. Yersiu A = -. — . ^(1 + cotM) sec^ 4. Sin ^ cos ^ = -. r . tan A 4- cot A PLANE TRIGONOMETRY. 1S7 YI. 1. If Tan-"^ + 4sm-^-i = 6; A = 60\ 2. If m = tan A + sin A, and w = tan A — sin ^ ; Cos^ = 3. If m sin ^ = n cos A ; Sin A=^ m + It, n J' 4. Tfl = (''!^V+(oos^cosCr; SinC = *^i. \sin BJ ^ tan -S 5. If Cos a; = ^—;^,, and Cos (QO^-o?) = ^?^ ; sin 6 sm 6 then Cos'' A + cos^ ^ + cos^ (7=1. VII. Prove the following formulae ; 1. Tan A + cot A ^2 cosec 2 A, 2. Cot ^ - tan ^ = 2 cot 2 A. ^ , sin 2yl , /1+sin^ 1 + tan -M 3. Tan A = -^ — r . 4. ' 1 + cos 2^1 * * V 1 - sin ^ 1 - tan J^I ' 5. Cosec 2 A + cot 2^1 = cot A. G. 2 cosec 2 A = sec A cosec A . ^ sin ^ ... ^ Versin A ^ o. , 7. ^j — r = cotJ^. 8. . ,-,0^0 — j-, = tan'p. 1 - cos A ^ versm ( 1 80 - J) ^ 9. 8 cot 2 A cosec^ 2^1 = cot A cosec^ ^ — tan A sec^ .4. 10. Yersin (180" -A) = 2 vers J (180° + A) vers J (180" - A). cot ^ + tan ^ 1 o rr o 1 i 2 sin ^ - sin 2^ 11. »ec lA = -. . 1 2. Ian" }y A = ~- — -, cot J. - tan A "2 sm A + sm 2 A 13. Cosec 24 = 2. cos 24 + sin 24 (cos ^ — sin ^1) — (cos ZA — sin 3-4) ' 1 4. Cos" A = (cos \A - sin ^Ay + 2 cos J^ sin 1^ (cos \A - sin J^ y. 15. ^(1 + sin ^) = 1 + 2 sin J^ . ^(1 - sin ^A). Shew also that the radicals have their proper signs in this equation if A be between - 90° and 180'. 138 PLANE TEIGONOMETRY. 16. Cot A + cot 2A + cot 41 = . , , . (2 + 2 cos 2.4 + 3 cos 4:1). SlU 4:A ^ .^ -n XT. . Sec (2^''-^^l) - 1 tan (2"'^'A) 17. Prove that \.-,,„,/ ., =- — ^.,„-i .: ; sec (2^"1) - 1 tan (2-" ^A) and thence shew that (sec 24 - 1) (sec 2^1 - 1) (sec 2'^ - 1) ... to y;, factors (sec A-1) (sec 2^ A - iy(sec 2* J. - 1) ... to t^ factors = cotlltan(2'"-M). 18. Cosec 21 + cos 41 = cot 1 — cosec 41. 19. Sin 31 sin^ 1 + cos 31 cos^ 1 = cos^ 2 A. g.^ Sin nA sec (7^ + ^) 1 — sec (7i — |) 1 _ cos 2nA + cos 1 4 sin ^1 ' and thence shew that Sin 1 sin 2A sin 31 /, . x o-i 7 + n T + ^1 :< + "• (**^ '^ terms) cos 2 A + cos 1 cos 41 + cos 1 cos bA + cos ^ sin ^nA . sin |- (ti + 1 ) 1 ~ sin 1 . cos J (2w +1)1* VIII. Prove that, whatever be the values of the angles, 1. Cos 21 + cos 2^ = 2 cos (1 + £) cos (1 - £). 2. 1 + cos 21 cos 25 = 2 (sin' 1 sin' B + cos' 1 cos' 5). 3. Cos' (1 + 5) - sin' 1 = cos 5 cos (21 +5). Sin (1-5) sin (5- (7) sin ((7-1) _Q sin 1 sin 5 sin 5 sin C sin C sin 1 5. Cos (1 + B) sin (1 - 5) + cos (5 + C) sin (5 - C) + cos(C + i))sin((7-i))+cos(Z> + l)sin(Z)-l) = 0. 6. Cos (1 + B) sin5- cos (1 + (7) sin C = sin (1 + B) cos 5 - sin (1 + (7) cos C. PLANE TRIGONOMETRY. 139 7. Sin(^ + ^)cos^-sm(.4 + C)cosC'=:sm(^-C)cos(^ + ^+C). 8. Sin {A +B-2C) cosB -sin {A + C-2£)cosO = sin (B - C) {cos {B + C - A) + cos {A + C-£) + cos (A + B-C)}. 9. Sin ^ sin (^ - C) + sin^sin (G - A) + sin C'sin (^ -^) = ; and Cos Asm(B -C) + cos ^ sin (C - ^ ) + cos C sin (^ - ^) = 0. 10. Cos 2A + cos 2B + cos 20 =cos{B+C)cos{B-C)+cos{0+A)cos{C-A) + co3{A+B)cos{A-B); and Sin 2A + sin 2^ + sin 20 =^sm{B+C)cos{B-0) + sm(C+A)co3{0-A) + sm(A+B)cos{A-B). 11. I{2S = A+B + 0, 4cosA cos ^ cos 6' = cos 2 (S-A) + cos2{S-B) + cos2{S-0) + cos2S; and 4 sin A sin B sin = sin 2 {S - A) + sm 2 {S-B) + sin 2{S-C)-shi 2S. 12. Sin ^ sin (.1 - i?) + sin Csin (A - 0) = cos {B - 0) cos 2 {S-A)- cos A. 13. (SinM + sin«i? + sin*C){sin^(J-^)4-sin^(^-C) + sin^((7-.4)} = {cos^-cos(^-C)cos2(,S'-^)}''4-{cos^-cos(C-^)cos2(aS-5)}- + {cos - cos {A - B) cos 2 {S- C)Y ; wliere 2^= :! + ^ + (7. Sin(^-^) sin{B-C) sm( C-A) sin (7 sin A sin i? sin {A - B) sin (^ - 0) sin ((7 - ^) sin A sin ^ sin C ~ /Tan ^ tan ^\ /tan ^ tan (7\ /tan tan ^' \tan^ tanZ>/ \tan (7 tunBj \tanJ. tan (7/ _2 sin 2 (^ - ^ ) + sin 2 ((7 - ^) + sin 2 (^ - (7 ) sin 2A sin 2i? sin 20 16. The formula in tlie last Example is also equal to 8 sin (J - B) sin (B~0) sin (0- A) sin 2 A sin 2^ sin 2C 140 PLANE TRIGONOMETRY. IX. Prove that 2. Sec (45" + A) sec (45'' - ^) = 2 sec 2yl . 3. 2 sec ^ = tan (45° + i^) + cot (45° + ^A). 2 cos 2A-1 2 cos 2A + 1 4. Tan (30" + A) tan (30° - A) = 5. Sin (60° + A)- sin (60° - A) --= sin ^. 6. ^(1 - cos ^) =gi^ (450 _i^)^^Qg ^450^1^^- 7. Cos ^ + cos (120° - ^) + cos (120° + ^) = 0. 8. Cos' ^ + cos- (60° -A) + cos' (60° + ^) = f . 9. CosM + cos' (72° + A)+ cos^ (2 . 72° + A) + cos" (3 . 72° + i) + cos"(4.72° + ^)=y. X. Find the numerical values of Sin 15°, Sin 9°, Cos 12°, Versinl5°, Tan 22°, 12', Sin 150°, Cos 135°, Sin 3°, Sin 75°, Sec 225° j and prove that 1. Tan 50° + cot 50° =2 sec 10°. ^ 2. 2 cos ^n = J^ + \/{2 + &c. + ^(2 + 2 cos A)} ; where n is a positive integer, and the symbol ^ is repeated n times, each time affecting all the quantities which follow it. XI. Determine A in the following equations : L Sin ^ = sin 2^. 2. Tan 2^ = 3 tan yi . 3. Tan A = cosec 2 A. 4. Cos A = tan A. 5. Tan' 2^ = 3 tan 2X 6. Tan^ + 3cot^ = 4. PLANE TRIGONOMETRY. 141 7. Tan ^ tan 2.1+ cot ^ = 2. 8. 2 sin' 3^ + sin- 6^1 = 2. 9. Tan ^ + cot 2.4 = sin ^ (] + tan A tan ^A). 10. Cos nA + cos {n — 2) A = cos A. 11. If Sin A + sin 2 A + sin 3^ = 0, the values of A are 7i . 120", (2/1 + 1) 180", or (2?i + 1) 90", where ?i is or any integer. 12. If Sin74-sin^ = sin3^; A = n . 60' or {6n^l) . 15' ; n being any integer. 1 13. If Cos 3/1 + sin 3.1 =-^ )A=n, 120'' + 15''± 20" ; wbeing any integer. 14. If Tan (45" - i.l) + cot (45"- J-y1) = 4; .4 = (6n ± 1) . 60" ; n being any integer. 15. If Sin5^= IG sin^ ^; O^rinr, or imr^-, m being any integer. XII. Determine x from the following equations : 1 . Sin (a — x) = cos (a + x). 2 Sin (x + a) + cos (a; + a) = sin (x — a) + cos (x — a). 3 Sin a + sin (x-a)-h sin (2a; + a) = sin (x + a) + sin {2x — a). 4. Tan a tan x = tan" (a + x) — tan" (a — x) ; — to find Cos x. 5. m versin x = oi versin (a — x). m tan (a ~ x) n tan x 6. cos^ X COS" (a — £c) * A 7is. Tan (a — 2a;) — . tan a. ' n -\- ni ^ m / V / \ 1 — 2 COS 2a , „ „ 7. Tan (a + a;) tan (a - x) = . Ans. x = 30 . ^ ^ ^ ^ 1 + 2 COS 2a 8. n see" x tan (a — x)= on sec (a - x) tan x. Ans Tan 2a; = ^r- . m + n cos 2a 142 PLANE TRIGONOMETRY. 9. Sin 03 = sin a sin (5 + £c). Ans. Tana; = :i ; — '■ — ^—71. 1 — sin a cos (i 10. Tan X = cos a tan ^. Shew that tan^lasin'2/S Tan {/3-x) = 1 + tan^ Ja cos 2^ * 11. ^ = ^'^ /°^f^"> FindTanc.. ?^ sm p cos (a — x) 12. If 2 sin (x — y) = 1, and Sin {x — y)= cos (a? + ?/) ; then X = 45°, and 3/ = 1 5". 13. If the cosines of ^ — a, <^, and <^ + a be in harmonical progression, then cos ^= J 2 . cos ^a. 'sin a. cos ^ 14. If Tan£c = -; — -p, —. sm (5 + cos a then Tan Jcc = tan J a tan (^^tt - J j8). Sin (^ — ^) cos a cos (^ + a) sin /? sin (+cot (7=cot ^ cot-Scot C. 3. Tan ^ + tan JB + tan C = tan A tan ^ tan C + secA sec i5 sec C. PLANE TRIGONOMETRY. 143 XIV. If A+B + C=180'', 1. 4 sin ^ sin -S sin 6' = sin 2^ + sin 2^ + sin 2(7. 2. Sin' ^A+ sin' -^B + sur^C + 2 sin ^A sin J^ sin 1(7= 1. XV. Prove that, 1. Tan-' J + tan-' J = 45°. 2. Tan-' } + 2 tan"^ J = 45'1 3. Cot-'f + cot-'^ = 135°. 4. Sin-'-|_ + cot-'3 = 45''. 5. If ^ = tan-' -7^ , and ^ = tan-'-7rr=; J'd V15 tlien Sin (> + ^) = sin 60" cos 36^ /S+ /2 6. If ^ = cos-' J I and + e=-- G0\ 7. If Cot x = 7i cot (a - a;) ; then 2x = a- sin-' ( . sin a j . 8. If Versin"' (l+x)- vei-sin"' {l-x)= taa-' 2ji-x' ; then a; = ± 1, or — ^% 9. If Versin"' — versin"' — = versin"' (1 — &) ; a a X /2b then - = ± / z — y a V 1 + 6 10. If Sin"' 2x - sui-'x J'd = sin"' a;, then x = 0, or =t |. 11. If Sin[2cos-'{cot(2tan-'x)}] = 0, thenic=±l, or l=^j2, or-l^V^. 12. Given Tan"' J + 2 tan-'i + tan"'! + tan''a; = Jtt, shew 144 PLANE TRIGONOMETEY. 13. Prove that Tan"' (1 tan 2A) + tan"' (cot A) + tan"' (cot^ A) = 0. 14. Shew that Tan (2 tan~^ a) = 2 tan (tan~^ a + tan"^ a^). 2 15. If SinT^ X + smT^ Ix = ^TT, x= 7= ^ ^ 5 + 2J2' 16. Shew that, (a being ^^-rr), 04. -ir+ 4/1 \ + 1 01 _i tana + cos ^ 2 tan {tan^(l.-a).tanl^} = cos j^^^^,-^ • 17. The sum of anj number of angles c,. _i 2ab . _, 2a'6' bm -2 — ^, sm '-73 — 772? ••• or + lr^ a^ + 6 ^ may be expressed in the form sin"^ — 5 r where m and n are rational functions of ah, a'h', ... 18. Prove that (2 ) Cot~^ <- +J/i(?z+l)a> = tan~^ J (?z + 1) a — tan"' ^na ; and thence sum the series cot"^ (- + a) + Cot"V - +3aj + cot"V - + 6a j +cot~^/-+ 10a j + ... (to n terms). Ans. The Sum = cot"' ( — + — - — . a ) . \na An J 19. Shew that J a' cosec^ (J tan"' ?) + i^^ sec^ T Jtan"' - j = (a + 5) (ct^ + lf\ PLANE TRIGONOMETRY. 145 XVI. 1. If m = tan 6 — sin 6, and n = tan $ + sin ; fiud the relation between m and ?i. 2. If Tan^ = -, then acos2^ + 6sin2^ = a. 3. If (ic-a)cosS + csin(^-S) = 0, (y — b) cos S + c cos (I- S) = ; then (a; — a) sin 1 + [y — h) cos ^ + c = 0. 4. If a sin' ^ + a' cos' e = h, a sin' 6' + a cos' 6' = h', and a tan' = a' tan' ^'; then aa = bb'. 7)1" 1 5. If Tan ^ = tan' J <^, and Cos> = -V— ; then m = ~ j — - . (cosa + sin^ 0)^ cosw-e /iTe ^ , 6. If cos V = -z : then Tan i v= . / . tan A- u. i—e cos u - V 1 — e 7. If a sec' ^ - 6 cos ^ = 2a, and 6 cos' ^ - a sec ^ = 26, then a'' = 6'. 8. If a tan ^ + 6 tan B = {a + b) tan J (.4 + ^), then a cos /? = 6 cos^. 9. If Cos a = cos /3 cos <^ = cos ^' cos <^', and Sin a = 2 sin |- ^ sin J \ then Tan J a = tan ^ y8 tan J )8', 10. If Sin^sin(a-^) =sin/?sin(a + ^), shew that CotyS - cot ^ = cot (a + ^) + cot (a - )8). 11. If ^ be eliminated from the equations (a + 6) tan (^ - <^) = {a-b) tan (0 + = 6'. S. T. 10 146 PLANE TRIGONOMETRY. 1 2. If Cos 6 = 7t , cos a = t^, , and 7^7 = > , then coH /8 cos (3 tan 6^ tan a Tan ^ j3 = tan |^ a tan ^ a'. 13. If (a + h) sin ^ + (ot - 6) cos ^ = J {a' + 6^), and a sin^ + hcos^e = J3ab, then ^ = 2^j3, or (^\+^ + l=.0. 14. If a sin a + 6 sin/? + c siny = 0, and a cos a 4- 6 cos /8 + c cos y — 0, then a : b : c :: sin {/B — y) : sin (y - a) : sin (a - ;8). 15. If ^ and i be eliminated from the equations /sin^d) cos^s]ab Ijub 18. In a triangle in which {a-b) is small compared with c, shew that {A — B), expressed in seconds, _ 9 a-b sin B fa - bY sin 25 •sini"^l^j-inn^'"^'''y- 19. A side a and the opposite angle -4 of a triangle remaining constant, shew that the correspondent variations of the other sides are connected by the relation Sb . sec B + clc. sec (7=0. XIX. 1. The sides of a triangle are 13, 12. 5 ; find its area. 2. Given the perimeter, area, and an aiigle, find the side opposite to the given angle. 150 PLANE TRIGONOMETRY. 3. Given B, a, and the area, to determine the triangle. 4. Given the vertical angle, the perpendicular on the base, and the rectangle under the segments of the base, to determine the triangle. 5. The length of a road, in which the ascent is 1 foot in five, from the foot of a hill to the top is a mile and two-thirds. What will be the length of a zigzag road in which the ascent is 1 foot in 12] 6. The area of a triangle is four-thirds of that of the triangle whose sides are equal to the three lines which join the angles of the first triangle with the bisections of the sides respectively opposite. 7. If in the three edges of a cube which meet in a point 0, points A, B, C be taken which are distant a, b, c from 0, the area of the triangle formed by joining A, B, G is ^ (a^b^ + b'c' + a'c')k 8. The sides of a triangle are as n—1, n, n+1, where n is very large ; find the angles, and determine the difference of each from 60". 9. An object 6 feet high, standing at the top of a tower, subtends an angle Tan~^ 'OlS at a station which is 100 yards from the base in a horizontal line. Find the height of the tower. Ans. ±1^^30009 — 1 yards. Explain the meaning of the double result. 10. Two objects, A and B, were observed from a ship to be at the same instant in a line inclined at an angle 15*^ to the east of its course, which was at the time due north. The ship's course was then altered, and after sailing 5 miles in a N.W. direction, the same objects were observed to bear E. and N.E. respectively. Required the distance of A from B. Ans. 5(3- ^3) miles. 11. A vessel observed another a" from the north, sailing in a direction parallel to its own. In p hours its bearing was ^°, and in q hours afterwards -/ from the north. To what point of the compass were the vessels sailing? Ans. If 6 be the PLANE TRIGONOMETRY". 151 inclination to the north of the course of the vessels, the equation for determining it is sin {0 — y) p sin (/? — a) sin(^-^ ^q' sin (y - (i) ' 12. The shadows of two vertical walls, which are at right angles to each other, and are a and a^ feet in height, are ob- served, when the Sun is due south, to be 6 and b^ feet in breadth. Shew that if a be the Sun's altitude above the horizon, and ^ be the inclination of the first wall to the meridian; Cota= /(K+K). Cot/i^'^' 13. A boy wiis flying a kite at noon when the wind was blowing a from the south, and the angular distance of the kite's shadow from tlie north was ft. The wind suddenly changed to a^ from the south, and the shadow to jB^ from the north, and the kite was raised as much above :|^ tt as it had before been below that elevation. Shew that 6 being the angular elevation of the Sun, and ^ TT — <^ that of the kite at first. sin (a - (B) sin (a^ - /3 ) ' Sin (a^ - pj Sin p 14. A privateer observes the direction in which a trader leaves a harbour at a known distance. Having given the rates of sailing of the two vessels, shew how to compute the angle at which the privateer must pursue the trader, so that she may just have prepared for action at the time she comes within, gun-shot. 15. There are two towns lying N. and S. of each other, and distant 2^ miles. Their angles of depression as observed from a balloon are 45° and 60", and after the balloon has pro- ceeded horizontally 6 miles in a S.E. direction, the angles of depression of each are one-half of what they were before. Shew that the height of the balloon was 3 miles very nearly. 152 PLANE TRIGONOMETRY. 16. The sides a, b, c of a triangle are 4, 5, 6, find the angle B ; having given — lj^2 = -3010300. Zcos27°, 53'= 9-9464040. 1,^5 = -6989700. Zoos 27°, 54'= 9-9463371. 17. Given 1,^3 = -4771213, andZtan57'', 19', 11"=10-1928032, nearly. Shew that if one angle be 60", and the two sides contain- ing it are as 19 to 1, the other two angles are 117", 19', 11", and 2", 40', 49". 1 8. Two sides of a triangle are as 9 to 7, and the included angle is 64°, 12'; determine the degrees, minutes, and seconds in the other angles. Given 1^,2 = -3010300, L tan 57", 54' = 10-2025255. Ztanir, 16' = 9-2993216. Ztanll", 17'= 9-2999804. 19. A man ascends a mountain by a direct course, the incli- nation of his path to the horizon being at first a, and afterwards changing suddenly to (3, which continues to the summit. Given that the mountain is a feet high, and the angle of depression of the starting-point as observed from the summit is y, find the length of the ascent. cos{l(a4-^)-y} ^ siu y cos i (/8 — a) ' Ans. a . 20. At noon a person on a cliff h feet above the sea-level, observes the altitude of a cloud in the plane of the meridian to be a, and the angle of depression of its shadow on the water to be ft. If y be the Sun's altitude at the time of observation, the height of . - . 1,7 sin y sin (a + /?) the cloud above the sea-level = n . -: — -^ — : — -, r . sm p sin (y + a) 21. To determine the distances between three objects A, B, C a person places himself in a line with A, B and then walks at right angles to AB until he is in the same lines with A, G and with B, C respectively, measuring the distance he walks over; he also observes the angular bearings at these points ; shew how the distances between the points A, By G can be found. PLANE TRIGONOMETRY. 153 22. A person wishing to know his distance from an in- accessible object finds three points in a horizontal plane at which the angular elevation of the summit of the object is the same. How may the requii-ed distance be determined ] 23. The altitude of a balloon, at noon, is observed at three places A, B, C to be 45°, 45°, 60° respectively, A and £ being respectively west and north of C ; find the height of the balloon, and the position of its shadow. Ans. If AC = a, £G=b, the , the ambiguity height required = ^.^ ^1 -^ |l - [-j^) | of sign corresponding to the cases where the balloon is east of £0 and south of AC, and where it is west of £C and north of AC. 24. The lengths BC and CD in the straight line BBF sub- tend equal angles at a point A. If i ACE = 9, i ADE = (^, BC = a, AC = b, shew that Tan ^ . (b cos 6 + a cos 2^) = b sin + a sin 20. 25. Having given AB and the angles subtended hj AB and AD Sit Cy and hj AB and BC at D, find the distance CD. Ans. If AC, BD meet in E ; LACB--=a, /.ADB=a; /,ACD = (3, I BDC = ft' ; AB=a, DC = x... this equation may be obtained, x^{mr + m'^ + ^mm! cos {ft + ft')] = a sin (ft + ft') sin (ft + ft' - a) sin (ft + ft' - a), where m = sin a sin ft' sin (ft + ft' — a), and 7?>/= sin a sin ft sin (ft + ft' — a). 26. If the sides J56', CA, AB, or the sides produced, of a triangle ABC be intersected by a straight line in the points D, E, F respectively, shew that AE y, BF x CD = DB x EC ^ FA, and that the product of the areas of the triangles EAF. FBD, DCE is (DE xEFxFD.sinD sin ^ sin i^)' - 8 sin ^ sin J? sinC. 154 PLANE TRIGONOMETRY. 27. Aa, Bh, Cc are straight lines making angles witli the sides AB, BG, CA of the triangle ABC each equal to a, and they intersect in A', B\ C ] shew that the triangle A'B'C is similar to the triangle ABCy and that the ratio of two homologous sides is sin*^ +sin^^ + sin*C . = cos a ^r—. — -—. — p . ^ . sin a. 2 sm A smB sin 6 XX. 1. The radii of the circles described within and about a triangle whose sides are a, &, c being r and R, prove that 2Rr = 5 — : and r = 4i? sin i^ sin A -5 sin J (7. a + h + c' 2 2^ 2. The product of the radii of the four circles which touch three lines intersecting each other = the square of the area of the triangle which the three lines include. 3. Three circles whose radii are a, h, c, touch each other ex- ternally; prove that the tangents at the points of contact meet in a point, whose distance from any one of the points of contact is abc \i a + b + c 4. The areas of all triangles described about a given circle are as their perimeters. 5. If a, a be homologous sides of similar triangles described about and within a ciicle, a = ia sin ^ A sin | B sin ^ C. 6. Express, in terms of the sides, the diagonals of a quadri- lateral whose opposite angles are supplementary; also shew that the diagonals are to one another as the sines of the angles which they respectively subtend. Ans. If the sides AB, BO, CD, DA of the quadrilateral be represented by a, b, c, d; ~\ ab-hcd J ' 1 bc + ad ) ' 7. If E be the radius of the circle which circumscribes the triangle ABC, then a cos ^ -I- & cos ^ + c cos (7 = iB sin A sin B sin 0, PLAXE TRIGONOMETBY. 155 8. Three circles touch one another externally, shew that the area of the triangle formed by joining their centers is equal to the square root of the product of the sum and product of their radii 9. If r„, r^,, r, be the radii of the circles inscribed between the sides containing the angles A, £, C respectively of a triangle and its inscribed cii'cle (radius r), shew that J fan + Jr^r, + Jr^r, = r. 10. If h, h, I be the diameters of the greatest cii'cles touching each a side of the triangle ABC and its circumscribing circle (radius R), prove that 2hkl = S'B, where 2S^a + b+ c. 11. and being the centers of the circles described about and within a triangle, and li the radius of the former, then, if ii*^ , 7/^ , li^ be the radii of the circles which circumscribe the triangles BOC, CO A, A OB respectively, and r^, r^, r, the radii of the circles which circumscribe the triangles BoC, Co A, AoB respec- tively, + ^ + -^ = -^T^ , and * * ' R, R, R, R^ ' abc a + 6 + c * 12. The perpendiculars from the angles of a triangle on the opposite sides meet in a point P ; prove that PA + PB+PC=2{R+r), R and r being the radii of the circles described about and within the triangle. 13. Perpendiculars can be drawn from one point within a triangle, and from only one point, to the three sides, so as to divide the triangle into three quadrilaterals which can have circles inscribed within them. Prove this ; and shew that, r^, r^, r , r being the radii of the circles described within the quadrilaterals and the triangle, \r^ t] \i\ r) Vg r) \r^ r) \r^ r) \r^ r) " r" ' 156 PLANE TRIGONOMETRY. XX T. 1. In a regular polygon of n sides, of wMoh a side is 2a, ^ + r = a cot ^r— . In 2. The square of a side of a regular pentagon inscribed in a circle = the square of a side of the inscribed hexagon, together with the square of a side of the inscribed decagon. 3. The area of a circle and that of a regular polygon in- scribed in it being given, find the number of the sides of the polygon, and the angles they make one with another. 4. The area of a regular polygon that circumscribes a circle is to that of the inscribed polygon of the same number of sides as 4 to 3. Shew that the polygon is a hexagon. 5. The perimeter of a regular polygon circumscribing a circle is double that of the inscribed polygon of the same number of sides. Find what that number is. 6. One of the angular points of a regular polygon of n sides is joined with each of the other angular points, and circles are inscribed in the triangles of which the sides are bases. If R be the radius of the circle circumscribing the polygon, shew that the sum of their radii is 2i? [X—n sin^ -^ ] . XXII. Adapt to logarithmic computation ; 1 / — 7T , I A o \/« + ^ Ja-h 1. Ja + o + Ja~o. 2. ~ r-+~ ;— . a— b a+o In the following questions $ is required. 3. Sin (^ + a) = m sin 0. 4. Cos (a + ^) = cos a sin d + sm/S. 5. Cos mO + cos nO + cos pO =1. ^ ^ ^ na cos A — h sin B . . x. ^ 6. ian^ = -. : — -; where A, B. C, are the anmes nia cos A + c sm (J ° of a triangle, and a, b, c the sides. PLANE TRIGONOMETRY. 157 7. Sin 0J{1 + tan-^ tan'^) + cos ^ ^(1 - tan^ A . tan' B) = tan A + tan B. 8. Adapt to logarithmic calculation, u = a^ J (or + x~). Ans, If x = a tan 6, u = ~- with the sim + : and ' cos ^ ° ' u = :^ — -with the sign — . cos^ ^ 9. Also adapt, m = a; ± J(cc^ — of), where x < a. Ans. If a; = a sin 6*, u = a J 2 sin (^ =•= ^ tt). 10. Also adapt, u = x^ J{2ax — x^), where x < 2a. Ans. If cos ^ = / ^ , u = 2 J 2 . a cos 9 cos (0 ^f ^tt). 11. Adapt to logarithmic computation the formula (see Example 25 in Class XIX), y ^-{nf + ?n- + 2mm'cos(/3 + ^')Y, where ?7i = sin a sin j8' sin {(3 + /3' — a), and m' = sin a sin fi sin (^ + ^' — a). -4 ns. Supposing ni < m, take m' Cos tan | (/? + ^'), then y = 2m cos" ^ ^ . cos -J (/3 + yS') . sec ^. ANGULAR UNITS. XXIII. 1. Find the cii'cular measure of (1) 15^ .30'. (2) r. (3) 15 grades. Ans. (1) -270526... (2) -000290888... (3) -235644... 2. Find the number of degrees, minutes, and seconds in the angles whose circular measures are (1) |; (2) ^; (3) 1-5708. Ans. 85°, 56', 37"-U; 16°, 21', 49"-09; 90°-00018. 158 PLANE TRIGONOMETRY. 3. Wliat equation connects the angle, arc, and radius, when 180 are the angular unit is an angle of 1*"? Ans. Angle = . — -r^— . 4. If the angular unit be n times the angle which subtends an arc equal to the radius, what is the equation that connects the ancjle, arc, and radius ? Ans. Angle = — . — ^. — . " ' * ^ n radius 5. If the angle be represented by - . — -. — , the angular unit is two right angles. 2 6. If when the angular unit is ^^ of a right angle the measure of an angle be 3, find the ordinary circular measure of the same angle. Ans. g tt. 7. If six times the ordinary circular measure of an angle be unity, what will be its circular measure when the unit of measurement is one-third of a right angle 1 Ans. — . TT 8. Find the number of units of circular measure in an angle of 20 grades; also the number of units in the same angle when the angular unit is an angle of 30". Ans. j^ j ^^^ f • 9. Compare the units of angular measure when is the measure of an angle with reference to one unit, and mO the measure of an angle n times as great with reference to the other unit. Ans. The magnitude of the first unit : that of the second :: m : n. XXIV. Prove that 1. § (8 sin :| 6 - sin I ^) = ^ ; nearly. 2. Yersin^ = -l{€i^>/^-€-2«V^}. 3. If from an arc AJB of a circle of radius r whose center is 0, AC be cut ofi'=rsin^O^; then sector 00^ = segment ^C^. PLANE TRIGONOMETRY. 159 4. The tops of two vertical rods on the Earth's surface, each of which is 10 feet high, cease to be visible from each other when 8 miles distant. Prove that the Earth's radius is nearly 4224: miles. 5. The top of a mountain may be just seen from a ship at sea which is a miles otf, and at a distance of h miles the angular elevation of the mountain above the horizon is 0. Prove from 2 12 these data that the Earth's radius is — ^r^— , cot $ nearly, and 26 "^ determine the height of the mountain. 6. If Sin {a + ax) = m tan {b + (3x) for all values of x, then, so long as ax and I3x are small quantities, a cot a— 23 cosec 26 , X = b — yry^ 5-? , nearly. a- + 2)8' sec' 6 ' -^ _ ^. y cos ^ - my sin B + n/x ((S - a) y 7. Given x = ^ j-^ ^ ^^-^ -i tan<^ = -^; , ysin(a — fi) _ \ // ? n m = cota; „ = ^'^^-A__; A = (^-„)^(y +^-); shew that, if h be small compared with y, a: x sin 2 nearly. 8. A level stratum of clouds is one mile above the earth's surface ; prove that the distance in miles of the edge of a cloud whose angular altitude is a is very nearly -: — . ( 1 ^r — ) , '' -^ "^ sma \ 2/i J' unless a be too small ; n being the number of miles in the earth's radius. 9. Altitudes of the same heavenly body are observed from the deck of a ship and from the top of the mast, the height of which from the deck is known ; find the dip of the horizon, and the true altitude. Aiis. If h be the height of the mast, a and a' the observed altitudes, a the dip, B the radius of the earth, h cosec (a' — a) . , , i . i • . i ^-T,' . .,,, — nearly : and the true altitude h cosec {a' — a) = a- K' sin 1" 160 PLANE TRIGONOMETRY. XXY. Solve tlie equations ; 1. 03^+1=0. 2. x^-6x-4: = 0. [a; = - 2, or 1 ± ^3.] 3. cc^- 147£C- 343 = 0. [One root is 13-1556964: ; given that Cos 20" =-9396926.] XXYI. Logarithms : 1. The logarithm of x to the base a is m ; find x. Ans. a'". 2. Find the number whose log to the base 8 is 4. Ans. 4096. 3. To what base is 8 the log of 6561 ? Ajis. 3. 4. To what base is 0*6 the log of 49 1 Ans. 343. 5. To what base is - 5 the log of 32768 ? Ans. i. 6. li n = log^p, what are the numbers of which m + n, n-m, 2m -n are respectively the logarithms 1 Ans. pa"" ; -^ ; — • 7. If m and n be the logs of p to the bases a and a' respec- tively, what is the relation between m and n 1 Ans. m = nr. 8. Given the log of a number to the base a, find the bases of the systems (1) in which its log is half that magnitude, and (2) in which it is thrice as great. Ans. a^ ', l/a, 9. The log of a number to one base is the same as that of its reciprocal to another base ; find the relation of the bases to each other. An^. The one is the reciprocal of the other. ■ 10. Shew that (1) Log{3a-'6*(4-a;')-^} = log 3 - 2 log a + 4 log 6 - log (2 - £c) - log (2 + x). (2) Log (6 + £c) -« = —.{n\oga + q log (6+ a:;) -^ log (a-x)-p)\og {a + x)]. PLANE TRIGONOMETRY. IGl (3) Log fl-^^ = \og{x + l) + log (cc - 1) - 2 log X. 11. If log X + log 6 = log 1 2 - log 2 — log X, then x' =^1. 1 2. From 1,, 2, and 1^, 3, find 1,, 72. 13. From 1,„U, and.l,„16, find U2,l,„5, 1,,7, 1,,^-^. l+N 1-N ,;^2 ;\r4 ir^ u. i.{(i^3-) ^ .(1-ivO M = 0+371 + ^ + - 1 5. If log x = log a — log b, shew that log [a + b) = log « + log ( 1 + - j , and log (a — h) = log a — log . \ X/ X — x. 1 IG. lfy=€^~^^'\ and ;^ = €^"'«^; then a;= €^~^^^. -irr Ti' a R r. .1 /? log m +' 8 log 71 1 7. If m<^^ - ^ - ?i>''^'+^ = 0, then x = S-^ r-^— • a log ni — y log ?i 1 8. If a' + 6^ = c, and a' -V = d, then ^^log^(cW) ,^^^^ log|(c^^ loga ' -^ log 6 19. Prove that log (tan 2"^) = 2 log (2 sin 2" 6) - log (2 sin 2"^' 6) ; and thence shew that i log (tan 2"0) + -^Aog (tan 2"^) + . . . (to 9i terras) = log (2 sin 2^) -^. log (2 sin2''-''^). S.T. 11 SPHEKICAL TEIGONOMETEY. CHAPTER I. ON CERTAIN PROPERTIES OF SPHERICAL TRIANGLES. 1. Def. a Sphere is a solid bounded by a surface of which every point is equally distant from a fixed point which is called the center of the sphere. Def. The straight line joining the center with any point in the siu'face is called the radius of the sphere. 2. Every section of a sphere made hy a plane is a circle. Let ABCD be a sphere of which the center is ; AFCG the curve in which a plane cutting the sphere intersects its surface ; OE a perpendicular from upon the cutting plane. Join E with F any point in AFCG, and join FO. Then since OE is perpendicular to the cutting plane, it is perpendicular to EF, a line meeting it in that plane ; .-. EF^J{OF^-OE^), a constant quantity. Now ^ is a fixed point in the cutting plane, and F is any point in the curve AFC. Therefore AFC is a circle whose center is E and radius is EF. Euclid, i. Def. 15. CoR. If the cutting plane pass through the center of the sphere, EG vanishes, and EF becomes equal to OF^ the radius of the sphere. SPHERICAL TEIGOXOMETRY. 163 3. Def. The circle in wliich a sphere is cut by a plane is called a great, or a small circle, according as the cutting plane passes, or does not pass, through the center of the sphere. Cor. Since the radius of every gi'eat circle is equal to the radius of the sphere (2, Cor.), all gi-eat circles of the sphere must be equal. Note. Unless the contrary be expressly mentioned, when hereafter an arc of a sphere is spoken of, an arc of a great cii'cle is meant. 4. Def. Spherical Trigonometry investigates the rela- tions subsisting between the angles of the plane faces which form a solid angle, and the angles at which the plane faces themselves are inclined to one another. For the sake of convenience the angular point is made the center of a sphere, which, being cut by the plane faces containing the solid angle, presents on its surface a figure of which the sides are arcs of gi-eat circles. Let ABC be a triangle of this kind, whose sides ^^, BC, CA are formed by the intersectic^n of the planes A OB, BOO, CO A, with the surface of a sphere of which is the center. The angle of any face, as AOB, is — —^ , PI. Trig. Chap. vi. ; and the angle contained between any two faces (as BOA and CO A), is the angle contained between the lines AD and AJE which are drawn in the planes BOA and COA at riglit angles to theii' intersection OA. Euclid, xi. Def. 6. The lines AD, AE, being at right angles to the radius OA, and lying in the planes of the arcs AB and AC, are tangents to those arcs. 5. After certain properties of spherical triangles have been proved, it will not be requisite, in pursuing further investigations, to represent in the figures attached to the Propositions the center of the sphere on which the triangles are described. It must not, however, be forgotten, that when the words "the angle ^^40" or "the angle J." occur, the angle meant is that of the inclination of the two planes which pass through the center of the sphere and the arcs A B and AC', and that this is the angle contained between two lines drawn from any point in AO at right angles to it, and respectively lying in the planes AOB and AOC. Also, whenever the expression "the sine oi AB" occurs, the sine is meant of the angle which the arc AB subtends at the center of the circle of which it is a portion. If this circle be a great circle, its center is also the center of the sphere. 11—2 1G4 SPHERICAL TRIGONOMETRY. 6. In the following pages the angle BAC will commonly be indicated by the letter A, and the angle subtended by BC, the side of the triangle opjDosite to z BAG, will be indicated by a. The other parts of the triangle ACB will be represented in like manner. 7. Def. If OE (fig. Art. 2), which is perpendicular to the plane AFC, be jDroduced both ways to meet the surface of the sphere in B and D, these points are respectively called the nearer and the further poles of the circle AFG j and the straight line BOD is called the axis of the circle AFG. 8. The pole of a circle is equally distant from every point in its circumference. (Fig. Art. 2.) Join B with F any point in AFGG. Then, BEF being a right angle, BF^ = BE' + EF' = BE^ + {OF' - OE'), a constant quan- tity. And F being any point in AFGG, B is equally distant from every point in the circumference of that circle. SimHarly, DF' = BE' + EF' = BE' + {OF' ~ 0E'\ a constant quantity. Hence D is equally distant from every point in the circumference of A EGG. Again ; let BNF be an arc of a great circle passing through B and F, any point in AFG. Then since the chord BF is the same for every point in the arc AFG, and since in equal circles equal circumferences are subtended by equal straight lines (Euclid, III. xxix.), the arc BNF is constant ; and also, because the radii of all great circles are equal, the angle BOF subtended at the center of the sphere is constant. Hence it appears that every point in the circumference of a circle of a sphere is equally distant from the pole of the circle, whether the distance be measured by the length of a straight line joining the point with the pole, or by the arc of a great circle connecting the same points, or by the angle which such an arc subtends at the center of the sphere. 9. Since BO is at right angles to the plane AFG, every plane through BO is at right angles to that plane. Hence, the angle between any circle whatever and a great circle passing through its pole is a right angle. SPHERICAL TRIGOXOMETRY. 165 10. If FB he a quadrant drawn at right angles to the great circle AFC from any point F in it, B is the pole of AFC. Join B and F wifh the center of the sphere, and in the plane of AFC dra\y OA at right angles to FO. Then, AG being perpendicular to OF, and (since FB is a quadrant) BO being perpendicular to the same line, therefore z BOA — L between the planes AOF aud BOF, whose intersection is OF \ an angle which is by hypothesis a right angle. And BO, being at right angles to OA and OF, is at right angles to the plane AOF in which they lie; and therefore B is the pole oi AF(J. Cor. If B be the pole of AFC, AO and FO, any two lines in the plane CFAG, are at right angles to BO, and AOF is the iuclination of AOB and FOB ; .-. Art. 5, lAF=iABF. Def. Great circles which pass through the pole of a great circle are called secondaries to that cii'cle. 11. If from a point on the surface of a sphere there can he draiun two arcs {7iot jmrts of the same circle) lohich ar^e at right angles to a given circle, that point is the pole of the circle. (Fig. Art. 10.) Let B be the point, BA and BF arcs (not parts of the same circle) through A and F. points in the circle AFC ; and let BA and BF be at right angles to AF. Then since BA and BF are at right angles to AF, and are not parts of the same circle, the planes BOA and BOF must intersect, and their intersection BO is at right angles to AOF, Eucl. xi. 19. Therefore B is the pole of AFC. [The proof holds whether AFC be a great or a small circle.] Cor. Hence also it appears that the intersection of two arcs drawn at right angles to a given circle through any two points in itj is the pole of that circle. 166 SPHERICAL TRIGONOMETRY. 12. The avgle subtended at the center of the sphere hy the arc of a great circle which joins the poles of two great circles, is the inclination of the planes of those circles. Let the given circles be FD and FE intersecting in F, A and B their respec- tive poles, and ABDE the circle through A and B. Now ^0 is perpendicular to OF^ which is a line in the plane DOF, And BO is perpendicular to OF, a line in the plane EOF ; .'. OF is perpendicular to the plane A OB ; and therefore to OD and OF, which are lines in that plane j .-. DOE is the angle of inclination of the planes FOB, FOE. And AOB = AOD-BOD = ^0'-BOD = BOE-BOD = DOE. CoE. Hence also it appears that l AB = l BE = z DEE. Arts. 5, 4. 13. Two great circles are bisected by their intersections. (Fig. Art. 10.) Let BAB and BFB be arcs of two great circles intersecting each other in B and B. Join BB. Then since BA i) is a great circle, the center of the sphere is a point in its plane ; similarly, it is a point in the plane of BFB ; therefore the center of the sphere is a point (0) which is in the intersection of these planes, that is, in the line joining B and B. Therefore BAB and BFB are semicii^cles, having BOB for their common diameter, and the two gi'eat circles are bisected by the points B and B. 14. Any side of a s^jherical triangle is less than a semi- circle, and any angle is less than two right angles. Since Euclid takes two right angles as the limit of a plane angle, this is also his limit for the angle of any plane face of a solid SPHERICAL TRIGONOMETRY. 167 angle. Hence in a spherical triangle no side can be equal to a semicircle. Thus, if ACB intersect AIJBF in the points A and B, and CD be any other arc, the triangle connecting the points B, C, J) is not the figure formed by BG, CD, JJAFB (of which the side DAFB is greater than a semicircle), but the figure formed by BC, CD, DEB. If a side, as A DEB, become a semicircle, then the arcs which join A and B with any other point C, are portions of the semi- circle BCA, and the arcs joining the points A, B, C cease to form a triangle. Wherefore the side of a triangle is less than a semicircle. Cor. 1. Hence it follows that an angle of a triangle must be less than two right angles. For if possible let BFADC be a triangle having the angle BCD greater than two right angles. Produce BC to A ; then BFA is a semicircle, and BEAD is greater than a semicircle ; which is impossible. If the angle BCD become equal to two right angles, BC and CD become parts of the same circle, and the figure ceases to be a triangle. Wherefore the angle of a triangle is less than 'two right angles. Cor. 2. Hence also it follows, that if the great circle be completed of which any side JJ5 of a tri- angle ABC forms a part, the triangle lies within it. For if possible let C lie without the circle, and let CA cut it in i) ; then since DA is a semicircle (13), a side CDA of a triangle is greater than a semicrrcle ; which is impossible. 15. IfVhe the pole of a great circle BAG and of a small circle bac which are cut by the great circles PaA and PbB, then Arc ab ^. ^^ -r t-tc = bm ra. Arc AB 168 Spherical trigonometry. Let be the center of tlie sphere, and OP, which is at right angles to the planes of both circles, cut the plane of the small circle in D. Join AO, BO, aD, hD. Since these lines lie in planes which are perpendicular to OP, each of them is perpen- dicular to OP. Join aO. Then aD, being perpendicular to PO, is parallel to AO, ?l line which lies in the same plane with itself. Similarly, hD is parallel to -50 ; .-. I aDh = ^.AOB. Eucl. xi. 10, arc ah _ arc AB ^ lD'^'~AO~ ' that is, Arc ah aD aD . „-. . „ . •. -, r^ = -r^ = —7^ = sin POa, or sin Pa, Arc AB AO aO Cor. 1. Since POA = 90", sin POa = cos aOA ; Arc ah ^ . , • '. -^ r^ = cos aOA, or cos aA. Arc AB Cor. 2. li AB be any circle of which P is the pole, it may be proved in like manner that Arc ah sin Pa AvG AB ~ ^m.PA ' 16. Def. !^f the angular points A, B, of a spherical tri- angle ABC, Fig. Art. 17, be the poles of the three great circles D£J, EF, FD respectively, the triangle ABG is called with respect to DEF the Primitive Triangle, and DEF is called with respect to ABG the Polar Triangle. 17. The angular points of the polar triangle are the poles of the sides of the ^primitive triangle. Join AE and BE, by arcs of two great circles. Then, since A is the pole of ED, AE^^ir, Art. 10. And, since B is the pole of EF, BE =^7r. Therefore the great circle of which E is the pole passes through A and B, or E is the pole of AB. Similarly D and F are the poles oi AC and BG respectively. SPHERICAL TRIGONOMETRY. 169 18. The Polar Triangle is called also the Supplemental Tri- angle, from the following property it possesses. The sides and angles of the polar triangle are the supplements of the angles and sides respectively of the primitive triangle. (Fig. Ai-t. 17.) For z ^ = HG, Art. 10, Cor., =EG-EH = EG- {ED - DH) = EG + ER-EB ; but EG and EH are quadrants, or each subtends an angle Jtt, .-. lA = tt-ED. Similarly, lB^tt-EF, and z C = tt - FD. Again, AB = AG-BG = AG-{IG-BI) = AG + BI-IG = lTr + ^7r-E, since E = IG. (10, Cor. ) = 7r-E. Similarly, BC^tt- F, and AC = 7r-E. 19. If a general equation he established hetiueen the sides and the angles of a spherical triangle, a true result is obtained if in the equation the supjdements of the sides and of the angles respectively be written for the angles and sides which enter into the equation. For if the equation be proved for any triangle whatever, the sides a', b', c', and the angles A', B', C, of the j^olar triangle may respectively be substituted in it, in the place of the sides a, b, c, and the angles A^ B, C, of the primitive triangle. And in the equation as it then stands, j^utting for the sides and angles of the polar triangle their equivalents drawn from the primitive triangle, viz. IT — A = a'y TT — B — b', TT — C = c', -rr-a^A', Tr-b=B', 7r-C=C\ a true result is obtained, which differs from the original equation in having the supplements of the sides and of the angles respec- tively written for the angles and the sides of the triangle. 170 SPHERICAL TRIGONOMETRY. Ex. If A, B, C be the angles, and a, h, c be the sides, of any triangle, it will hereafter be proved (29) that _, , cos a — cos h cos c Cos A = -. — z — ; , sin sm c in which the cosine of an angle of a spherical triangle is expressed in terms of the sides. Therefore Cos A' = cos a' — cos h' cos c' sin b' sin c' Now A' = 7r-a, a' = 77- A, h' = tt — B, c' = tt - (7 ; .-. Cos (^ - g) = ""^ <" -^\ - "°^^r ;^> °°^/" - ^> ; ' sm (tt - i^) sio (tt - (7) Cos a = cos A + cos ^ cos G \ in which the cosine of a side of a sin jB sin (7 triangle is expressed in terms of the angles. 20. Any two sides of a spherical triangle are together greater than the third; and the sum of the three sides is less than the circumference of a great circle. For, Eucl. XI. 20, any two of the angles AOB, BOG, GO A, arc AB arc BG arc GA^ or AO AO AO which form the soKd angle at 0, are together greater than the third. Therefore any two of the arcs AB, BG, GA are together greater than the third. Also, Eucl. XI. 21, the three angles forming the solid angle at are less than four right angles ; arc AB arc BG arc GA ■+ r^T— + rT^-<27r. AO AO AO ,'. Arc AB + arc BG + arc GA <^Tr , AO, which is the circum- ference of a circle whose radius i^ AO. SPHERICAL TRIGONOMETRY. 171 Cor. 1. Since the sum of the plane angles which contain any Kolid angle is less than four right angles, Eucl. xi. 21, it follows from the same mode of proof, that the sum of the sides of any polygonal figure which is described on a sphere, and whose sides are arcs of great circles, is less than the circumference of a great circle. [The polygonal figure, however, must be such that the angle between any two adjacent sides is less than two right angles, and also that its area is con- tained on the surface of a hemisphere : for by EucHd, the angle between two planes is less than two right angles, and in the proof of the proposition re- ferred to, it is supposed that all the plane faces which contain the sohd angle may be cut by one plane; which cannot be the case unless all the edges of the solid angle lie within the same hemisphere.] Cor. 2. Also, let ABC BE A be a five-sided fiscure described on a sphere, and let it be divided into triangles by the arcs AC, AD. (A figure is easily drawn.) Then^i? + i?(7>yl(7; .-. AB + BG + CD>AC-¥CD^AD, for AC + CD > AD ; .: AB + BC + CD + DE > AD + DE > AE. And the same method of proof being applicable to a polygon of any number of sides, it follows that the sum of all the sides but one of a polygon described ou a sphere is greater than the re- maining side. Cor. 3. If a and h be two sides of a spherical triangle, since each is less than it, .'.a + b<:27r, .•. J (a + 6)<7r. And a - 6 < TT, .: ^ (a — b) < |-7r. So, I (A + B)<7r, and ^ {A - B) < Jtt. Cor. 4. a', b', c, being the sides of the polar triangle, a' + b'^c; .:{Tr-A) + {7r-B)>{7r-C); ,'. 7r> A + B- C ; oi; A + B-C <7r. 172 SPHERICAL TRIGONOMETRY. 21. The sum of the angles of a spherical triangle is greater than two right angles, and less than six. Let A, B, C be the angles, and a, h, c the sides, of a triangle ; A', £', C, and a\ b', c', the angles and sides of its polar tri- angle. Now27r> a'+¥ + c\ (20) >{7r-A)+{7r-£) + (7r-C)>37r-{A+B + C); ,\ A+B + G^TT. Also, since each, of the angles A, B, G is less than tt, .-. ^ + ^ + C<37r. 22. The angles at the base of an isosceles triangle are equal to each other. Suppose ABG to be an isosceles tri- angle, having AB — AG, and therefore / AdB=iAOG. From D, any point in OA, draw DG ^ perpendicular to the plane BGG, — and therefore at right angles to every line it meets in that plane ; and from G draw GE and GF perpendicular to OB and OCi join BB, BF, GO. Then OB' = OG' - GE' = {OB' - DG') - {ED' - BG') = OB' - EB' ; .*. BE is perpendicular to OE, and .*. z BEG = inclination of the planes BOG and BOA, = i B. Similarly, BF is perpendicular to OG, and L DFG = l C, Now DE=OB. sin AOB - OB . sin AOG = BF. And EG' = BE'-BG' = BF''-BG' = FG'. SPHERICAL TRIGONOMETRr. 173 Hence, since GE, ED are equal to GF, FD, each to eaoh, and GD is common, .-. L BEG = L DFG, or lB=lC. 23. Conversely, if z ^= z C, or z DEG = DFG, it may be shewn from the same figure that AB = AC', that is, the angles at the base being equal, the sides opposite to them are equal. Cor. Hence also it follows that every equilateral triangle is also equiangular ; and conversely, that every equiangular triangle is also equilateral. 24. Of the two sides ivhich are opposite to tiuo unequal angles in a triangle, that is the greater which is opposite to the angle which is the greater. Let z ABC be greater than z BAG. Then AC >BC. Make z ABD = z BAD ; .-. DA = DB. And AC=CD + DA = CD + DB; But CD + DBiB> BC (20) ; .: AC > BC. 25. Conversely, it may easily be shewn that of two angles in a triangle which are opposite to unequal sides, that is the greater which is opposite to the side which is the greater. CoR. Hence A — B and a — b have the same sign. 26. Article 24 has been proved after the manner of Euclid, I. 19. The following propositions may be enunciated for spheri- cal triangles in the terms used for plane triangles, and may be Droved in nearly the same words. Euclid, i. Props. 4. 8. 24. 25. •26. (fee. 174 SPHERICAL TRIGONOMETRY. 27. To recapitulate the more important properties of Spheri- cal Triangles which have been proved in this Chapter. Ai-t. U. 14. 20. 24. 25, Cor. 20. 21. 20, Cor. 3. 20, Cor. 3. 20, Cor. 4. 1. A side must be less than a semicircle. 2. An angle must be less than two right angles. 3. Any two sides are together greater than the third. 4. The greater side is opposite to the greater angle j and conversely. 5. A — B and a — h are of the same sign. 6. « + 6 + c < 27r. 7. u4 + ^ + (7 > TT, and < Stt. 8. a + 6 < 27r, a—h<.-K. 9. A^B<1iT, 10. A+B-C<-TT. A-B<7r, The following properties will also be proved hereafter : 34. 39. 11. \{A^ B) and J (a + 6) are "of the same affection;" i. e. they are both > ^tt, or both < A TT. 12. It ABC be a right-angled triangle, C being the right angle, then A and a (as also B and b) are of the same affection. CHAPTER II. FORMULA COXXECTIXG THE SIDES AND ANGLES OF A SPHERICAL TRIANGLE. 28. The Sines of the Sides of a Spherical Triangle are as the Sines of the Angles to luliich they are respectivelij opposite. Let ABC be the triangle, tlie cen- ter of the sphere. From JJ, any point in OA, draw DG perpendicular to the plane BOC, and from G draw GE and GF perpendicular to OB and OG. Join DE, O. JJF. Then a line through G parallel to OB would be perpendicular to DG and GEj and .*. to the plane DGE : wherefore, Euc. XI. 8, OE is perpendicular to the plane BEG^ and BE and EG ; and .-. z BEG =^ i B. SimUarly, l BFG = jlG. Now BE.BmBEG = BG=BF,BmDFG; .-. OB. sin AOB . sin BEG = OB. sin AOC . sin BFG ; . •. Sin AB . sin 5 = sin ^ C . sin C ; Sin B sin G ''' SmAG^siu^AB' Similarly it may be proved that Sin^ Sin^ Sin^C Sin^C* Wherefore Sin A _ sin B sin G Sin BG " &iD.AC " sin^i^ ,(i.) 176 SPHERICAL TRIGONOMETRY. The Proof will be found to hold whether G fall within the angle BOG or without it. For if, for instance, G fall without OB^ the angle DEG will be the supplement of the angle B, and sin DEG = sin B. Also if E he in. BO produced, the proof will still be found to hold. 29. To express the Cosine of an Angle of a Triangle in terms of the Cosines and Sines of the Sides. From any point D in OA draw DE and JDE,- in the planes A OB and AOC, at right angles to AO. Therefore z EBF is the incli- nation of those planes to each other, that is, the angle A. Join EF. Then, from the tri- angles FOE and FEE, 0F'+ OE' -20F.0E. cos FOE = FE ' = FE' + EE'-2FE . DE . cos FEE ; .-. 20F . OE . cos FOE = {OF' - FE') + (OE' - DE') + 2FD . DE . cos FDE = 20D' + 2FD . DE . cos FDE ; „, ^„^ od od fd de ^^^ ••• ^''^^^=of'oe'-of-oe''''^^^'^ or, Cos a= cos b cos c + sin h sin c cos A (ii.) ,„, ^ . cos a — cos b cos c ^ Whence Cos A = -. — ^ — : .^ sm sm c * In the proof DE and DF are supposed to cut OB and 0,C, which will be the case only if AB and ^C be less than quadrants. The result arrived at is notwithstanding general, as may be thus shewn. [The figures may easily be drawn from the descriptions.] 1st. If ^C be greater than 90*^, and AB less than 90°, produce CA and CB until they meet in C". Then AC'B is a triangle in which AC, AB are each less than 90*^, and therefore by the proof in the text, Cos C'B--= cos A C. cos AB + shiA C. sm AB . cos BAG' or — cos ^(7=- cos ^C. cos 45+ sin 4(7. sin 45. (-cos 54(7); .'. cos 5(7=cos4(7. cos 45 + sin 4(7. sin 45 . cos 54(7. 2nd. SPHERICAL TRIGONOMETRY. 177 Cor. By writing tt - a for A, &c. (19), Cos A + cos B cos C = Sin B sin C cos a (iii.) ,,,, ^ cos A + cos B cos C vV hence, Cos a = — jr-~. — . sm M sui 30. To shew that Cos a sin b = sin a cos b cos C 4- sin c cos A. Produce the side CA to F, making AP a. quadi-ant. Then the formula (ii.) applied to the triangles FCB and PAB gires ^ Cos BP = cos (90° + b) cos a + sin (90° + h) sin a cos C = - sin b cos a + cos 6 sin a cos C, and Cos BP = — sin c cos A ; Equating these values of cos BP, Cos a sin 6 - sin a cos 6 cos (7 + siu c cos yl (iv.) 2nd. If AC and AB be both greater than 90'', produce AC and ^5 until they meet in A'. Then A'BC is a triangle in which 4^ and AC are each less than 90*>. .•. Cos £C = cos ^'5 . cos ^'C+sin^'^ . sin^'C. cos ^', = cos (1800 - AB) . cos (ISO^ -AC) + sin (180" -AB).sm (1800 - AC) . cos A = cosAB. cos ^C+ sin 45. siu ^C. cos ^. 3rd. Let one of the sides, as AB, be 90<>. From B draw BD perpendicular to AC ov AC produced. Then the angles at D are right angles ; and fi-om the A BCD, Cos BC= cos DB . cos DC+ sin JOB . sin DC . cos i), where cosDB = cosA; cos DC=cos {90°^ AC) = sm AC \ coaD—0, and the equation becomes cos a = cos A . sin h, which is the form (ii) assumes when = 90". 4th. If both the sides AB, AC be quadrants, cos A =cos BC -cosa ; and this is the form which (ii. ) assumes when b and c each become 90*^. S. T. 12 178 SPHERICAL TRIGONOMETRY, Cor. 1. By (iv.), Cos a sin & = siiK^^ cos bcosC + siii c cos J, cos a . , , • ^ sin c . .-. -. . sui = cos cos C + —. — . cos A sm a sin a sin C . = cos COS 6 + -T — -, . cos A : sin^ .*. Cot a sin h = cos i cos C + sin C cot A (v.) Cor. 2. By writing Tr-a for A, &c. (19), the formula (iv.) becomes, Sin C cos a = cos ^ sin B + sin ^ cos JB cos c (vi.) 31. To find the values of Cos J A, Sin J A, Tan ^A, and Sin A in terms q/* a, b, c. By (ii.), Sin 6 sin c cos A = cos a — cos h cos c ; .*. sin h sin c (1 + cos -4) = cos a — (cos h cos c — sin h sin c) ; .*. sin 6 sin c 2 cos' ^A = cos a — cos (b + c) = 2 sin J (a + 6 + c) sin J (6 4- c - a). And if >S= J (a + 6 + c), S - a = ^ {a + h + c) -a = ^{h + c - a), „ . , sin aS' sin (aS' — «) .♦. COS^ i ^ = r-T— ^^ ; ^ sm 6 sm c /sinAS'sin(AS'-^ sin (>S' - a) ' ^"^'^ SPHERICAL TRIGOXOMETRY. 179 And Sin A = 2 sin J A cos J A 2 " sin 6 sin c • '^^^"'^ ^S' sin (^ - a) sin {S - h) sin (>y - c)} (x.) The positive signs of the square roots are taken in these cases, because ^^-4 is necessarily less than ^tt. 32. To find the values of Cos ^sl, Sin J a, Tan J a, and Sin a in terms of A, B, C. _, ,... V r, cos A + cos 5 cos C By (m.), Cos a = ^ — j^—. — -^ : •^ ^ " sm i> sin C ' o 2 1 n cos .4 + cos B cohO + sin ^ sin C . *. 2 cos i a = 1 + cos a = : — - — ■. — -. sui Jj sm C __ cosA+co3{B-C) _ 2 COB h{A +C - B) cos ^ (A + B -C) sin B ain C sin B sin C And if aS" = 1 (^1 + i? -f- C), then S' - B=h (A + C -B\ &c. ; ^ , /cob (S'-B) COB (S'-C) . .. .-. Cos Ja= / ^ — : — ; . \^ ^ (xi.) "= V sin i? sin C ^ ^ . . o • " 1 , , cos A + cos ^ cos C Again, 2 sm" A a = 1 - cos a = 1 - - sin ^ sin 6' _ • cos A + cos (B + C) sin i> sin C wTi. a- 1 /- cos aS^ cos (.S^ - ^) , ._ Whence, Sm Ja= / ^ — ^-^^S^^ (xii.) ^ V sin^smC ^ ^ Ai rr. 1 sin^rt / - cos /S" cos (*S" - yl ) . ... , Also, Tanja = ^= / —-, ^, ,^, Va (xm.) 2 cos Ja V cos (aS - B) cos (/S - 6') ^ ^ And Sin a = 2 sin ^a cos ^a 12—2 180 SPHERICAL TRIGONOMETRY. Note. Since the angles of a spherical triangle are together greater than two right angles and less than six, S' is greater than one right angle and less than three, and its cosine is there- fore a negative quantity. Also since (27) A + £ — C <7r, there- fore cos ^ (A + B — C), or cos (>S" — C), is a positive quantity ; and in like manner cos {jS' — B) and cos (/S' — A) are positive. Where- fore the last three formulae are not imjwssible quantities, as at first sight they appear to be, but real quantities. 83. To prove Napier's Analogies, By (iii.), Cos A + cos B cos G = cos a sin B sin C ; so Cos B + cos A cos G = cos h sin A sin G. By addition, (cos ^ + cos ^) (1 + cos G) = (cos a sin ^ + cos 6 sin ^) sin (7 ; .-. (cos A + cos B) 2 cos^ J G — (cos a sin A . —. 1- cos b sin A) 2 sin ^ C cos |- (7 ; sm CI/ . '. cos A + cos B — — — . (cos a sin h + cos h sin a) tan J (7 sm.a sin A . sin (a + h) tan ^ C. sm a sin ^\ . . /, sin &"' Again, Sin ^ + sin ^ = sin ^ ( 1 + - — 7 ) =:^in ^ ( 1 -f- ) * \ s,uiAJ \ sma/ sin A , . . T\ = ■—. . (sm a + sm 0) : sm ct ' sin A + sin ^ sin a + sin ^ . 1 /^ cos J. + COS B sm (a + 6) ^ 2 sin J (^ + ^) cos J (^ - ^ 2 sin J (a + &) cos J (S^ sin (S - 6) felQ o^ {-a. + Jj) = / ; 1 — ; ; ; ysin /S' sin (S — a). sin (^ — a) sin (S—c) sin 6 sin c . sin a sin c 'sin (S—h) ^ sin (aS'— a)) /sin /S'sin (aS' - c) sin c sm c ) sy sin a sm b ^ 2 sin {^ - 1 (a + b)} cos H^-^) ^^Q^^Q sine * ^ = — ^-S ^.cos^C, since*S-i- ((^+-6) -ic : cos Jc ^ ' ^ \ / ^ ^ .-. cos Jc sin J (^ + ^) = cos I C cos J (a - b). And the formnlse (xx.), (xxi.), (xxiii.) can be proved in like manner. There will be no ambiguity respecting the algebraical signs in these formulae, if it be borne in mind that if A he >= <.B, then a is > = < 6, and that ^ (A + B) and J (a + 6) are of the like affection. CHAPTER III. ON THE SOLUTION OF RIGHT-ANGLED TRIANGLES. 36. KrGHT-ANGLED triangles may in all cases (with an ex- ception which will be pointed out, Art, 40,) be solved by means of the following formulte, when, besides the right angle, two other quantities are given out of the three sides and the two remaining angles. If A, B, C he the angles of any spherical triangle, and a, b, c the sides respectively opposite to them, it has been proved that (ii.) Cos c = cos a cos b + sin a sin b cos C (a) (iii.) Cose sin ^ sin ^ = cos C + cos A cosB (6) Cos a sin B sin C = cos A + cos B cos C (c) (i.) Sin a sin C = sin c sin -4 (d) (v.) Cot a sin b = cos b cos C + sin C cot A (e) Cot c sin a = cos a cos ^ + sin 5 cot C {/) By making C = 90", there will be obtained, From (a), Cos c = cos a cos b. (b), Cos c — cot A cot B. (c), Cos A = cos a sm B. Cos B = cos b sin A. From (/), Cot ^ = cot c tan a. Cos A = cot c tan 6. From (J), Sin a = sin J^ sin c. Sin 5 = sin B sin c. (e), Sin b = tan a cot A. Sin a = tan b cot -S. 37. These results are comprised under the following for- mulae, which the Student will find it necessary to keep in his memory. 184 SPHERICAL TRIGONOMETRY. (1). Cos hyp = product of cosines of sides. (2). Cos hyp = product of cotangents of angles. (3). Sin side = sin opposite angle x sin hyp. (4), Tan side = tan hyp x cos included angle. (5). Tan side = tan opposite angle x sin the other side. (6). Cos angle= cos opposite side x sin the other angle. 38. An artificial method of remembering tliese formula3 is by Napier's Rules. The formulae of the last Article are comprised under two Kules, which take their name fi-cm Napier, who first gave them. The right angle being left out of consideration, the two Sides which include the right angle, and the Complements of the Hypothenuse and of the other Angles, are called the Circular paints of the triangle. Any one of these being fixed upon as the middle part (M), the two circular parts next to it and immediately joining it are called the adjacent parts {A^, A^), and the other two parts are called the opposite parts {0,. 0,). Thus in the triangle ABC whose right angle is C ; If M be rhe adjacent Parts are The opposite Parts are a, (one of the sides including the right angle) ^TT-A, (the complement of an angle) Jtt - c, (the complement of the hypothenuse) iTT-B, b; b, Jtt-c; ^TT-A, ^TT-B; \ir-A, ^ir~c. Itt-B, a. a, h. And Napier's E,ules are, Sin l^=product of the Tangents of the Adjacent par^5= tan A^ .tan A^,. Sin M.=product of the Cosines of the Opposite parts =cos Oj . cos 0^\ with which the formulse of the last Article will, on trial, be found to agree. SPHERICAL TRIGONOMETRY. 185 39. To shew that in a triangle ABC in which C is a 7ight angle, A and a are of the like affection, as are also B and b. By37,(5).Si.5 = ^. Now since h is less than tt, sin b is positive ; therefore tan a and tan A must be of the same sign. And because tt is the limit both of a and of A, these angles must be both greater or both less than a right angle ; that is, A and a must be of the like affection. Similarly, from Sin a = „ it appears that B and b are of tan Jj the like affection. 40. If, in a right-angled triangle, an Angle and the Side opposite to it be the only quantities given, the triangle cannot be determined. For if the circles AB and AG intersect again in A\ and (7 be a A<'^ ) ^X. right angle, it is evident that ACB and A'CB have the angles TT A, A' equal, and CB, the side opposite to these angles, is the same in both triangles. It is therefore ambiguous whether ABG or A'BC be the triangle sought. This ambiguity will also be found to exist, if it be attempted to determine the triangle by 37, (5). For it cannot be deter- mined from the equation Sin ^ C = tan (7^ cot ^ whether the angle AC is to be taken, or its supplement A'C. 41. The solutions of the other cases of a right-angled triangle from two given parts are not ambiguous, if attention be paid to these two principles ; (1) The greater side is opposite to the greater angle, (2) An angle and the side opposite to it are of the like affection. 186 SPHERICAL TRIGONOMETRY. [For example : Let c and A be given, to find a, B,l. Now Sin a = sin ^ sine ; and since a and A are of the like affection (39), the greater or lesser angle which satisfies this equation is to be taken for a, according as A is greater or less than ^ir. Again, Cos c=cot ^ cot 5 ; .'. Tan S = cot ^ secc. And ^ is < 90°, or > go**, according as the second member of the equation is positive or negative ; that is, as A and c are of like or unlike affection. Again, Cos c = cos a, cos 6 ; .-. Cos & = cos c sec a. And 6 is < 90°, or > 90°, according as the second member of the equation is positive or negative ; that is, as a and c are of like or unlike affection.] 42. In selecting a formula, attention must be paid to the principles laid down in Appendix ii. to PI. Trig. The following formulae may be used with advantage, when the side or angle required is small, or nearly equal to one right angle, or to two. Cos c = cot 4 cot 5; whence c cannot be accurately determined, if it be either a very small angle or nearly equal to two right angles. , . , ^ ■, cos A cos B Now 2 sin2ic = l-cosc = l-cot^ eot^ = l--^ — -. — -. — ^ ', ^ sm A sm B g;„,. /-C0S(4+i)) J»=V- So, Co&\c=J\{^+c>osc) = j^- 2 sin 4 sin B cos [A - B) 2 sinj.sin5* . ,. . -I <• n cos 4 sin(|7r-^) In like manner there is obtamed from Cos «= . „ = — . p — r _ , /sin {1 { B-A)-\-\ir](iOB{\{B^A)-\ ir} . Cosia=^ ^^-^ ., „. , /sin{4 (g + ^)-i7r}cos{4(^-^) + |7r } Sin4a = ^ ^^ • Since (21), A-\-B^-G>Tr, .-. if C=iT, A + B>kTr, and since (20, Cor. 4), A + B-G When c and il are given, if a be nearly a right angle it cannot be ac- curately determined from its sine. In this case Cot ^ = cose tan ^4 deter- mines B, and a may be found from the formula for Sin ^ a, or from that for Cos i a, (32). 43. Def. a triangle is called Quadrantal, if any one of its sides be a quadrant. A Quadrantal Triangle maij he solved by applying to its Polar Triangle the formulae employed for solviog a Right-angled Triangle. A Collection of Examples for practice is added at the end of this Treatise. CHAPTER IV. ON THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES 44. Let the three Sides be given, (a, b, c.) The Angles may be determined from one of the formulae (vii.), (viii.), (ix.), (x.). 45. Let the three A ngles he given, (A, B, C.) The Sides may be determined from one of the formulae (xi.), (xii.), (xiii.), (xiv.). 46. Let two Sides and the included Angle he given. (a,C,bO By Napier's first and second Analogies, (xv.) and (xvi.), J (^ + B), and J (>4 — B), are determined ; A^h{A + B) + l{A-B),\ ^ ^ are known. B = l{A^B)-l{A-B\ And A and B being known, c is found from sin (7 „ Sin c = sin a . sin J. * The easiest practical method of solving this case is by letting faU from A a perpendicular (AD) on BO or BO produced either way, and then de- termining the right-angled triangles ABD and ACD. Supposing D hes be- tween B and C, then Tan CD = cos C tan h, which gives CD. And . •. DB, =a- CD, is known. COS 5 Also Cos c = cos DB cos AD = cos DB . T^rv, . cos CD SPHERICAL TRIGONOMETRY. 189 47. To determine c independently of A and B, hy forms adapted to logarithmic computation. Cos c = cos a cos h + sin a sin 5 cos C = cos 6 (cos a + sin a tan h cos (7). Let 6 be an angle such, that Tan = tan b cos (7 (1). Then Cos c = cos b ( cos a + sin a . sm \ cos Oy = - — -7,. cos (a - 6) (2). cost/ ^ ^ ^ From (1), -Z^ tan ^ = Z tan 6 + Z cos C — 10 ; which gives 6. (2), L cos c = Z cos b + L cos (a — 6) —L cos 0; which gives c. [On comparing this solution with that given in the foot-note to Ai-t. 46, it will be found to be identically the same, if CD be represented by ^.] 48. Let two Angles and the included Side he given. (A, c, B.) From Napier's third and fourth Analogies, (xvii.), (xviii.), rr 1 / 7 X cos i (^ - 5) Tan h(a + b) = -. f-p- — ■„( . tan * c, ^ ^ ^ cos h {A + JB) 2 » Tan l(a-b)= !^ 'i\\~ d • tan i c, "^ ^ ' sm |(^ + Z) 2 ' \{a-^b) and |- (a — 6) are determined ; And a and 6 being known, C is found from •. ^ . sine bm C = sin ^ . -.- - . sina 190 SPHERICAL TRIGONOMETRY. 49. To find C independently of a and b, hy forms adapted to logarithmic computation. Cos C — cos c sin A sin B — cos A cos B; .'. 2 cos* J (7 -1 = (1 - 2 sin^ J c) sin J. sin ^ - cos A cos 5 = — cos {A+ B) — 2 sin^ |^ c sin ^ sin. 5 ; .-. 2 cos^ J (7 = 1 - cos (J. + ^) - 2 sin^ |- c sin ^ sin 5 ; .♦. cos^ \G = sin^ \{A + B) - sin^ J c sin ^ sin B. Now Sift^ I" c sin A sin ^ is necessarily positive, and less than unity ; wherefore there is an angle 9 such that Sin^ 6 = sin^ J c sin ^ sin ^ (1) ; .-. Cos' 1 (7 = sin' i{A+B)- sin' 6 = sin {1 {A + B)+ 0} sin {1 (A + B)- 0}. PI. Trig. Art. 54 (2). From (1), Xsin^ = J (Zsin^ + Z sin^) 4-Zsin J c- 10 ; which gives 6. Prom (2), L cos^C= l[Zsin {i(^ + ^) +^} +^ sin {1(^ + Bj-O}] ; which gives C. 50. Let two Angles and a Side opposite to one of them be given. (A, B, a.) . sin^ ISm = sm a . — — j : sin^ from (XV.), Tan J (7 = |2^M . cot J (^ + 5) , from (xvii.), Tan i c = ^~)r-, J, . tan ^ (a + &). ^ ^' "^ cos J (.4-^) - ^ ' And h having been determined from the fii*st of these equations, C and c may be found from the other two. SPHERICAL TRIGONOMETRY. 191 51. To determine C and c independently o/b, hy forms adapted to logarithmic computation. (A, B, a.) Cos A = cos a sin B sin C - cos B cos C = cos B (cos a tan B sin (7 — cos (7) . Let ^ be an angle such that Cot = cos a tan B (1) ; 'cos 6 Cos A = cos 5 ( - — - . sin C - cos C ] \sin J = -r-^ . BUl{C-0) {2). sin c* ' From (1), Z cot ^ = Z cos a + Z tan Z - 10 ; which gives 0. (2), Z sin (C - (9) = Z cos ^ + Z sin ^ - Z cos B; which gives G — 6, and thence C. Again, from (v.), Sin B cot A = cot a sin c — cos c cos B = cot a (sin c - cos Z Um a cos c), and, if , , 1 — ; = sin B cot ^1 tan a . cos (fi tan ^ cos B tan a ' sin(c-<^) tanZ ... Qp } L = f'2), sin = cos^ tan b (1), a- / ,\ cos a sin <^ ,^. there is got, bm(c + d») = 7 — {■^)y ° \ -r/ cos 6 and from these two equations <^ and c can be found*. * If d=lTr-d', tan e'=^^— - , and 6' and {G-d') are the segments of the ^ cos 6 angle G made by a perpendicular let fall from C on ^lU. Also if (p — ^Tr- TT. If therefore b be < ^ tt, a must he > ^ tt. In this case, there- fore, a may be determined, and the triangle may be solved. But if 6 be > -^ TT, the condition a + 6 > tt affords no means of determining whether a be > or < ^ tt. II. If ^+^ be < TT, then a + b is <7r; and a is therefore < i TT if 6 be > I TT, but cannot be determined if 6 be < -| tt. III. If A+B = 7r; then, (by xv.), « + 6 = tt, and therefore a = Tr — b. Whence it appeal's that, (A, B, b being given), (1), when A + B>7r, and b <^'jr ; then a is greater than |- tt, (2), A + B ^tt; then a is less than | tt, (3), ^ + ^= tt; then a = 7r — 6 ; and in no other case can a be found, and the triangle determined from these data. S. T. 13 194 SPHERICAL TRIGONOMETEY. (2) Given two Sides, and an Angle opposite to one of them, to determine when the remaining parts of the triangle may be determined, (a, b, B.) > > > If « + 6 = TT ; then, as before, A + B = 7r, and . '. A = tt - B. < . < < Whence it is collected, as in the last proposition, that (1), when a + 5 > TT, and ^ < J tt ; then A is greater than J tt, (2), a + b<7r, and B^^-rr; then ^ is less than | tt, (3), a + b = 7r ; then A = '7r- B ; and in no other case can A be determined. 55. By letting fall a perpendicular from any angle upon the side opposite, an oblique-angled triangle may be divided into two right-angled triangles, which in most cases may be solved by Napier's Rules %^TLth not less facility than by the methods just given. The very same ambiguities, however, will arise (40) when this construction is used for the determination of the triangle as have been pointed out in (54). 56. Peob. To find the radius of the small circle described about a given triangle in terms of the Angles of the triangle. Let A BC be the triangle ; bisect CA and ^ CB in D and E, and draw from those points at riffht anejles to AC and to GB arcs inter- secting in P. Join PA, PB, and PC. Then, from the right-angled triangles PCD and PAD, Cos PC = cos PD cos DC = cos PD cos DA = cos PA ; .'. PA = PC, Similarly, PB = PC. Therefore P is the pole of the circumscribing circle. SPHERICAL TRIGONOMETRY. 195 Now Cos PBE = cot PB tan BE = cot FB tan J a, . •. Cot FA, = cot /*i?, = cos FBB cot J a ; and since FAC, FCB, FBA are isosceles triangles, .-. 2 L FAC + 2 I FAB + 2 i FBE = A + B + C = 2S' ; . •. z P^^ = S'-{i FAC + z P.l^^) = .S" - A. Also, by (xiiL), Cotia= / — ^ ^, j-^, -r-^ ; ' -^ ^ ^ ^ V -COSaS cos(.S -^) ^ , r^ ^ r>, /cos (^' - ^) cos {S' - B) cos (*S" - C) ,'. Cot y ^ = / — — ^ , . V - COS aS 57. Prob. To determine the radius of the circumscribing circle in terms of the Sides of the triancjle. As in (5G), Cot FA = cot J a cos FBE ; And as before, Cos PBE = cos (,S"- A) = cos | {(2? + C) - A] = cos -J (B + C) cos J yi + sin J (^ + C) sin ^ J[ sini^cosi-^ , , ., > , ,, \) 1 / \ 1 / • \ = ^ J — - — . {cos I [0 + c) + cos ^ (6 + c)\, by (xx.) and (xix.), sin J. , , , = ,-- . cos A cos i c : cosja ^ .■.TanP^ = -^"= ./^^ A cos P^^ cos 1^ 6 cos -^ c sin J. 2 sin I a sin ^ h sin |^ c , by (X,). J {sin S sin (>y - «) sin [S — h) sin [S - c)] 196 SPHERICAL TRIGONOMETRY. 58. Prob. To jmd the radius of the circle inscribed in a given triangle in terms of the Sides of the triangle. Let ABC be the triangle ; bisect z A and l C by AP and CP, arcs of great circles meeting in P] and from P (kaw the arcs PD, PE, PF perpen- dicular to the sides. Then it may be proved that PE = PD = PF \ and therefore P is the pole of the inscribed circle. Also it may be shewn that CE = S-(AF-\-F£) = S — c, and thence that Tan P^ = sin C^ tan PC^ = sin (aS'- c) tan 1 a /sin (S — a) sin (aS' — b) sin (S— c) ~ V sin/S' * 59. Prob. To determine the radius of the inscribed circle in terms of the Angles of the triangle. As in (58), Tan PE = smGE. tan J (7, and ^uiCE = sm{S-c)=Bmi{{a + b)-C} = sin J (a + 6) cos J c - cos J (a + 6) sin J c; Whence, by means of Gauss' Theorem, (xxii.) and (xx.}, it may be proved that Cot PE= 2 cos J ^ cos J P cos J C ^{-cosa^^cos(>S" - A) cos {S' - B) cos (.S" - C)} ' CHAPTER V. ox THE AREAS OF SPHERICAL TRIANGLES, AND THE SOLUTION OF TRIANGLES WHOSE SIDES ARE SMALL COMPAUED WITH THE RADIUS OF THE SPHERE. 60. Def. The poi-tion of the surface of a sphere which is contained within two ijreat semicircles is called a lane. 61. To find the Area of a Lune. If ACBDA, ADBEA, AEBFA be lunes each having the same angle at A, any one may be placed on another so as to coincide, and therefore be equal with it. Thus if the angle CAD be repeated any number of times, the area ACBDA will be repeated the same number of times. Wherefore the Area of a Lune varies as its angle. Area of the lune whose angle is ^" C D Area of the sphere (whose angle is 360") 360 ' And Area of a sphere = A-rrr', if r = radius of the sphere j (See Hymers' Integral Calculus.) .*. Area of the Lune wliose angle is ^''= -^^^^^ . iirr^ ooO 180* STrr . 198 SPHERICAL TRIGONOMETRY. 62. To find the Area of a Spherical Triangle. Let ABC be a triangle upon a liemi- /^^ sphere ABDEGA (14, Cor. 2); and let / AC, BG be produced until they meet again a;^ -ll!.- iii F, which is a point on the side of the [ ><^' sphere turned from the spectator. \ ..••'' "]^ Then since CDF— a semicircle = yiCZ), .-. DF = AC; Similarly, FE=GB; and i DFE = l ACB ; .'. A DFE — A AGB in every respect, Now 2 (= area of a ABC) = surface of hemisphere - BE DC - AG EC - DCE = 27rr' - (lune AHBA - 2) - (lune BGECB - 2) -(hme CDFEC~1) = ,fs.«-', i£i;° = A° + £'' + C"-180': Def. The quantity A' + B' + C -180', by which the sum of the degrees in the angles of the spherical triangle exceeds 180", | is called the Spherical Excess of the triangle. The SjAerical Excess is generally written thus, E=A+B + C-\8(}'\ Cor. 1. Hence for all triangles described on the same sphere, '^ oz A^ -\- B'^ + C^ — 180'', and on this account the Sj^herical Excess has been taken as the measure of the surface of a triangle. CoE. 2. To find the Area of a Spherical Polygon. Divide the Polygon into as many triangles as it has sides, by means of arcs of great circles drawn from each of the angular points to any point within the polygon. Let n be the number of the sides of the polygon. vr^ Then Area = area of the n triangles = r— jr x (number of degrees in the angles of the triangles - n . 180) — i^c\ ' {nuiiaber of degrees in the angles of the polygon - (w - 2) 180}. 180 SPHERICAL TRIGOXOMETRY. 199 63. Cagnoli's Theorem. To shew that if E he the Splterical Excess, then Sin -i- E = v^l^^^^ ^' ^in (S - a) sin (S- b) sin (S - c)} ^ ^ 2 cos ^ a cos ^ b cos ^ c Sin ^E=Bmi{A+B+C- 180'>) = sin {^- (.4 + ^) - i (180*' - C)} = sm^{A+B)sm^C-cos-^ {A + B) cos I C, COS -^ C^ And by (xix.), Sin i M + 5)= — ^-f— .co3i(a-6), *' ' " ^ cos ^ c (XX.), Cos ^ (.4 + i?j = — ^^ . cos i (a + 6). ^ ^ cos ^ C " ^. , ^ , ,/ ,^ ,, l^. siniCcosiC siniasin?>/j . Smi^={cosMa-i)-cosi{a + i)}.— ;-^^-^=-^^^^^.siuC sin Tasini 6 2 „ • « • /o n • /t i\ • /o m = — ; — =^ . ~. -. — r . \/{sin S&iaiS- a) sin (5 - 6) sm (S - c)} cos i c sin a sin ^ V{sin g sin (.9 - a) sin {S - h) sin (5 - f)} ~ 2 cos i a cos ^ 6 cos ^ c C4<. Llhuillier's Theorem. Io 5^ei^ that Tan i ^ = ^[tan ;^ S tan 1 (^' - a) tan \{S-b) tan -^ (.S' - c)}. By PI. Trig. (51), Sin H^ + B) - sin ^ (ISO^ - C) ' &\n\{A+B + C- ISO") Cos ^ (^ + i?) + cos h, (ibo'J - C) cos {[A + B + C- 180*') = tau I E ; ^ ,^ sini(^ + 5)-cosiC cosi(a-6)-cosic cos^C, , . , ,, .-. Tan|E= — p7- ^- ^^= — TT TT h- ■ " -. by (xix.)and(xx.) * cos^{A+B)+sin^C cos^(a + 6) + cosic sm^C -^ ^ ' ^ ' _sin| (c + a - 6) sin I (c + 6 - a) / sin <§ sin (uted Spherical Excess (which, for all prac- tical purposes, may be supposed to be the real Spherical Excess), is -23". Hence it appears that the whole error of observation, viz. real Spherical Excess — apparent Spherical Excess, is -23" — (— 1"), or l"-23, which the observer must add to the three observed angles, A^, B^, C^, in such proportions as his judgment may direct. (See the next Article.) 70. To skew how the observed angles of a Spherical Triangle whose sides are small compared with the radiibs of the sphere may he best freed from the errors of observation. SPHERICAL TRIGOXOMETRY. 203 Let A, B, C\ be the real angles of the triangle, -4 J, B^, C'j, the observed angles, a, /?, y, the eiTors made in observing A, B, Cj respectively. So that a + p + y = {A - a;) + {B - b;) + {c - c y, = {A + B + C- 180") -(A^+B^ + C^- ISO*^) = real Spherical Excess — computed Spherical Excess. Now if a value of the Spherical Excess which differs very slightly from the real value, be found by the method of the last Article, the above equation will give the sum of the errors of observation which have been made. The distribution, how ex er, of this sum — (that is, the determining what part of the whole en-or is to be assigned to each angle individually) — must evidently be left to the judgment of the observer, who, from knowing the state of the atmosphere at the times of the observations, may be able to form an opinion how far his optical observations can be depended on, and may then assign to each of the observed angles such a portion of the whole error as he thinks will be the most likely to lead to a correct solution of the triangle. [If a + /34-7 (the sum of the errors of observation) be found by means of ((57), and thence the several angles A, B, C be determined by the arbitrary assignment of the several parts of this whole error to each of the observed angles A^, B^, Cj, the sum of the errors of observation may be sui)posed to be got rid of. Yet it is highly probable that the judgment of the observer has not been absolutehj correct in this arbitrary assignment of the parts of the whole error. The following theorem will point out what relation the sides of the triangle ought to bear to each other, in order that the small quantities by which the corrected angles differ from the real angles of the triangle may have the least possible effect in producing errors when the other two sides of the triangle have to be determined from a measured side and the corrected angles. ] 71. Having given the Corrected Angles and one Side of a Spherical Ty^iangle whose sides are small compared with the radius of the sphere, required the relation which the Sides of the Triangle ought to bear to each other in order that the other sides may be determined from these data with the least probable amount of error. 204! SPHERICAL TRIGONOMETRY. Let A, B, C he the real angles of the triangle, and x, y, z the sides respectively opposite to them ; a, ^', y the errors of the corrected angles ; therefore A + a', £ + (3', C + y' are the corrected angles. Then, since the Spherical Excess (and therefore the sum of the real angles of the triangle) is supposed to be known exactly, the sum of the corrected angles is known exactly, and therefore the sum of the separate errors a, jS', y must vanish ; .-. a' + /S' + y' = 0, or a' = -(/?' + /)• [Throughout this investigation, powers of a', /S', 7' of any order above the first will be neglected.] ISTow, considering the spherical triangle t;0 be very nearly a plane triangle, sm(C + y) sm(G + y') ~x . — X. sin (^ + a') • sin {A - (^' + y')\ sin G cos y + cos C sin y sin A cos {j^ + y) - cos A sin (/?' + y) sin (7 1 + y' cot (7 , ~ — 7 • 1 77v T\ — T~A > nearly, sm^ 1 - (/3 + 7 ) cot ^ ' -^ ' = a;.-^^.{l+/(cot^ + cotC) + j8'cot^} sin G ( , sin G sin (A + C) ^, cos A sin C") sm J. (^ ' sm A sin J. sm C sm J. J And Sin B = sin (7r — B)= sin (^ + C) nearly ; Therefore the error in the value of «, ( or ;s — a? . - — , ] , is sin A, X ( , sin B _, cos A sin G) ,^ , t' sm J. '^ sm^J. j ^ ' SPHERICAL TRIGONOMETRY. 205 Similarly the error in the value of ?/ is {^, sin C , cos^isin^) .^, ^ sin^^ ' sin- J. J ^ ' The question is now reduced to determine what are the values of A^ B, and C which make both the quantities (1) and (2) the least possible. Since a' + /5' + / = 0, two of the three quantities a, /3', y must be of the same sign and one of the contrary sign. Where- fore the probability is that any particular two, as ^' and y', are of different signs. If then (3' and y be of different signs, the expres- sions (1) and (2) are diminished in magnitude by giving Cos A a positive value *j that is, by supposing A to be less than a right angle. And if it is further supposed, — that the errors ^ and y', though diflferent in sign, are yet nearly equal in magnitude, — it is clear that (1) and (2) satisfy this hypothesis if ^ be nearly equal to C. This conclusion is therefore arrived at, — that there is the great- est probability of a small spherical triangle having been correctly solved from thi-ee observed angles and a measured side, if the angle opposite to the kno^vn side be less than a right angle and the other two sides be nearly equal. And these conditions will be best fulfilled for a series of triangles if each triangle be nearly equilateral. 72. [It may be as well to recount the gratuitous suppositions made in the last Article. I. The Spherical Excess has been accurately determined. II. The Errors ^' and 7' are of different signs. III. These errors are of the same magnitude. * It is evidently more advantageous in determining a series of triangles from one another, that the errors should be inconsiderable and equally dif- fused through all, than that any one calculated side, by differing much from its real value, should affect with considerable errors all the triangles succes- sively determined from it. If, /3' and 7' being of different signs, cos A become negative, the errors (1) and (2) are increased, and considerable inaccuracies might so be introduced into the calculations. 206 SPHERICAL TRIGONOMETRY. Now for any particular triangle it is very probable tbat some of these suppositions may not be true. It may happen that none of them may be correct. In very few cases indeed will they all be fulfilled. "With respect to the Spherical Excess, it may generally be supposed to be known accurately. With respect to the signs of j8' and 7', since two of the quantities a\ j8', 7', are of the same, and the third is of the contrary sign, the probability of j3' and 7' being of different signs is twice as great as the probability that they are of the name sign. And if they be of different signs, then (1) and (2) of (71) shew that the errors will not be so great if Cos A be positive, as if it be negative. This consideration renders it advisable that each of the angles of the triangle should be less than a right angle ; which will be the case if the triangle be nearly equilateral. But if the third hypothesis be admitted, since a vanishes in this case, the angle A is supposed (which is highly improbable) to have been determined with mathematical exactness. Besides, it is very unlikely that an observer after making an error + /3' in observing an angle, should make an error - ^ (for 7' = — ^') in observing, with the same instrument, an angle (7 which is nearly of the same magnitude as B ; and again, using stiU the same instru- ment, and observing a third angle A, (nearly equal to B or to C), that he should make no error at all.] 73. Legendre's Theorei^i. If each of the angles of a Spherical Triangle whose sides are small when co7npared with the radius of the sphere he diminished hy one third of the Spherical Excess, the triangle may he solved as a Plane Tri- angle, whose sides are equal to the sides of the Spherical Triangle, and whose angles are these reduced angles. Let X, y, z be the lengths of the sides respectively opposite to the angles A, B, C o£ a. small spherical triangle, and A', B', C the angles of that plane triangle [A'B'C) whose sides are x, y, z. Then Sin ^ + sin ^' = —. — ^ . sin jB + - . sin B' sm y X = - . (sin B + sin B') nearly ; if ,'. 2 smi(A + A') cos i(^ -A')=-. 2 sin J {B + B') cos J {B - B') ; .'. cos i{A-A') = ^. 1^ . cos 1 (^ - B'), nearly ; .-. ooHi{A-A')^cosi{B-B'), or A-A' = B-B'. Similarly A-A' = C-C'. SPHERICAL TRIGONOMETRY. 207 Now B = A+B+C -ISO", and O^^' + ^ + C^'-lSO"; .-. I!={A-A') + {B-B') + {C-C') = S(A-A')', .'. IE = A-A', or =B-B', or =C-C\ Hence if BC (x) be measured, and one third of the computed Spherical Excess be subtracted from each of the observed angles of the triangle A, B, C, the other two sides [y and z) of that triangle can be determined by solving the plane triangle A'B'C whose angles are A —~B, B — ^ E, C — ~ B, and whose sides arc X, y, z. Note. It may here be remarked that the determination of ^ is a matter of considerable impoi-tance when triangles have to be solved whose sides are small compared with the radius of the sphere on which they are desciibed. The observed angles have to be corrected by means of it (70), and this Article yhews that it is employed in Legendre's approximate method of solution. How it may be determined from three measured or observed parts of the triangle is given in (08). 74. To find the angle contained between the chords of two spherical arcs luliich subtend given angles at the center of the sphere, the angle between the arcs themselves being also given. Let AB and AC he the arcs, the center of the sphere. Let the straight lines AO, AB, AC meet the surface of a sphere which is described with center A and any radius AD, in the points D, E, F respectively. Then the angle EDF is the inclination of the planes BxiO and CAO, i.e. the angle contained between the arcs AB and AC, or i A. A ^ ^ Arc^^ Ave AC ^ Now lCA0 = i(7r-^A0C) = i{7r-b). So LBA0 = {{7r-c}. And from the triangle EDF, Cos EF= cos BF cos DE + sin BF sin BE cos EDF ; Or Cos EAF= sin i 6 sin i c + cos i 6 cos J c cos ^ (1 ). 208 SPHEEICAL TRIGONOMETRY. 75. Two formulae will now be deduced from (74), wMcli are convenient for determining the angle EAF practically. Let LEAF=A-e', Then Cos (J. - ^) - cos ^ = sin ^ 5 sin J c + (cos J h cos J c - 1) cos ^ ; And since, PL Trig. (54), ISin I 6 sin J c = sin^ J (6 + c) - sin^ \(b- c), I Cos ^ 6 cos J c = cos^ J (6 + c) - sin^ i (^ - c) = l-sin^J(6 + c)-sin4(6-c); .-. 2sin(^-l^)sinJ^ = {sin4(5 + c)-sin4 {b - c)} - {sin^ J (5 + c) + sin* i (^ - c)} cos A = {1 - cos J } sin* :|(6 + c) - {1 + cos ^} sia* i (6 - c). Eor an approximation to the value of 6, take Sin ^6= ^0, and Sin (A-^0) = sm.A j then, since 1 - cos j1 , - . , 1 + cos ^ . . — ; — : — =tani^J., and — ; — -. — = cotl^, sm^ ^ smJ. ^ ' the above equation becomes 6 = tan J A sin* ^i; (& + c) - cot J ^ sin* l{b- c). The number of seconds in this angle is = — — =-r, sm 1 = ^^^ . tan J ^ sin* J (& + c) - ^^j^ . cot J^ sin*i (6 - c) (1); and by determining the two terms of the second member of this equation separately by means of tables of logarithms, the number of seconds in ^ is obtained, and the angle A — 6, contained by the straight lines BA and CA, may then be found. CoK. If y, z be the measured lengths of the arcs AC, AB, h^ c . j/ =^z y ^z sm -; — = sin —. — = —. — , nearly. 4 4r 4r ' *^ SPHERICAL TRIGONOMETRY. 209 .*. the number of seconds in 9 rr'{(^7*"'' i ^ - (V)' <=°* i ^ } ' ""^'^^ (->• If then the radius of the Earth and the sides of the spheri- cal triangle be approximately known, the chordal triangle can Ije determined ; since its sides can be found by the expression, (Jhord = 2r . sin (^ — j- j , and its angles can be determined by one of the last two Articles. 7G. Another method is to solve the triangle by the rules laid down for the solution of s])herical triangles whose sides are not small in comparison with the radius of the sphere. In this case the logarithms of the sines and tangents of the sides, whicli are very small, must be found by the methods pointed out in Appendix iii. to PI. Trig. S. T. 14 CHAPTER YI. ox GEODETIC MEASUREMENTS. 77. The object of Geodetic Measurements is to obtain a cor- rect representation of a part of the Earth's suiface whicli is too large in extent to be considered as lying in one plane. A horizontal line of considerable length is first measured, ■which is called a base. Next an object is fixed upon, so situated that it forms with the extremities of the base a triangle which is nearly equilateral (Arts. 71, 72). The angles of this triangle are then measured by a Theodolite, (an instrument described hereafter in Art. 84,) and the remaining parts of the triangle computed by some one of these three methods : 1. By the common processes of Spherical Trigonometry, Chap. IV. 2. By the Chorda! Triangle (74), (75). 3. By diminishing each of the observed angles by one third of the Spherical Excess, and then treating the figure as a plane triangle '^ (78.) 78. The use of Geodetic Measurements. The form and position of the polygon being thus determined, it may be represented on paper according to any projection of the sphere (See Hymers Astronomy, Appendix i.), and a map of a * The first of these methods was preferred by Delambre, the second was employed in the English Survey, and the third in the French Survey. SPHERICAL TRIGONOMETRY. 211 countiy obtained. If the object in view be to determine the fionire and dimensions of the Earth with great accuracy, the length of an arc mn, which passes through two given points of the polygon, may be found by calcula- tion. If these points, m and n, be situated on the same meridian, and the difference of the zenith distances of the same fixed star when on the meridian be noted 1 y obser- vers at in and n, the Earth's radius of cur- vature at the middle point of the calculated arc may be approximately found. From arcs thus measured in different latitudes the figure of the Earth has been determined with great exactness. {^Ilyiners Astronoin ij, 2nd Edit. Chap. ii. Arts. 123 — 142.)* Next, one of the computed sides is taken as the base of ano- ther triangle, whose angles are observed and sides computed ; and thus, by a series of triangles, the figure and dimensions of a polygonal area on the Earth's surface are determined. The form of the Earth is, in fact, spheroidal, but it is so nearly spherical, that each triangle may individually be supposed, without ap})reci- able error, to be described on the same sphere. If the triangu- lation be carried over a very extensive tract of country, it will become necessary to take into consideration the alteration which a change of latitude produces in the Earth's radius, {llymerti Astronomy, Art. 125.) 79. Base of Verification. A side of the last triangle of the series, after being computed in this manner, is carefullv measured. The deijree of exactness with which the measured coincides with the computed length tests the accuracy of the survey. Any considerable error is easily de- tected in the course of the calculations, in the following mannei'. (Fig. Art. 78.) If the value of 6, as calculated from the original * The student will find this part of the subject treated clearly and con- cisely in the Articles on "Trigonometry" and "The Figure of the Earth," written by the Astronomer Royal for the Encyclopedia Metropolitana. From Section 179 of the former of these Articles, 71 of this treatise has been taken. For a more particular description of the details of Geodetic operations any of the pubHshed accounts of the EngHsh Survey may be consulted. 14—2 212 SPHERICAL TRIGONOMETRY. base a through the triangles A, B, C, D, E, be found to agree closely with its value as computed from the sara.e base through a different set of triangles, as A, B', C, E, it may be presumed to have been accurately determined*. 80. Corrections of Measurementi^. — There are several causes productive of error in Geodetic Measurements. (1) The refraction of light is affected by the perpetual variations of the atmosphere in density and temperature, and thus the observed angles cannot always be relied on. (2) It is almost impossible to find a line perfectly straight and perfectly horizontal, to measure for a base. (3) Neither can a portion of the Earth be found which is altoge- ther free from inequalities of surface. To get rid of the errors thus introduced several corrections are used. Some of these, which do not depend on experiment but are capable of mathema tical investigation, will now be given. (1) Jf there he a slij^ht rise in the line of the base, to determine the reduction to the horizon. Let CB be horizontal, CE the line of the base, ED the sniall rise, — which is obtained by levelling with a spirit- level (83). Describe a circle with center C and radius GE. Then BD=^GB-CD = CE - CD - the " Reduction to the horizon." A C D B ^^ ^^ = AB=WE=2CE^^''^^''^^"' the formula made use of in the English Survey of 1784. * In the English Trigonometrical Survey of 1784 and succeeding years, the original base on Hounslow Heath was by admeasurement 27404 2 feet ; and the base of verification on Sahsbury Plain was, as measured, 36574-4 feet, and as computed through three different series of triangles, 36574*3, 36574-6, and 36574-9; any one of which is an approximation sufficiently near for all practical purposes. t If BD = x, CE=a, DE — h, x—— — = —^ ; and neglectmg powers of X above the first, x — -^^ nearly. SPHERICAL TRIGONOMETRY. 213 (2) Let the vuasured base, instead of being one straight line, consist of two straight lines, a and b, enclosing an angle -rr - 6, where 6 is very small. Required the correction to find c. c^ = a' +b'+ lab cos 6 = a' + b' + 2ab(l- | 0'), nearly, ^ (a 4- 6)- ' 1 - I (ci + W c = {a^b)\\ ll_ ab ..) ^ ' 2 (a + 6) ,e\, nearly ; .-. Correction = (« + 6) — c =— - «& ^^ 2(a + 6) Or if ^ be an angle containing n\ where n is very small, 1 ab (n sin V'f abn' sin 1 ~ 2(rt+6) ~ the Correction, in seconds, sini"' 2(rt+6) 2(« + 6) ' [In practice it is seldom necessary to apply this correction.] (3) From observations made at a poi/d D lohich is at a small known distance from a signal C, required to find the angle which A and B subtend at C. Let the angles BDA and ADC be observed at D. The dis- tances CB and CA are known approximately from the base AB and the observed angles BAG, ABC, Then L BDA + l DBE = exterior angle BE A = l BCA -¥ lCAD, .: The reduction, \'iz. i BCA - z BDA, = I DBE -L CAD = sin I DBE - sin z CAD, nearly, ^^ . sin BDC - ^f? . sin ADC CB CA _ /sin BDC sin ADC ) -^^\~CB CA~]'' CD (sin BDC sin ADC) , , . -, c = — — TTf i — 7Tn — TTi — tj when expressed m seconds. sm 1 ( CB CA j ^ 2U SPHERICAL TRIGONOMETRY. (4) The angles of elevation or depression^ being small quan- tities, of two objects having been observed, and also the angle the objects subtend at a certain 2^oint, to find the horizontal angle they subtend. (Encycl. Metrop.) Let Z be the Observer's zenith, GDRO his horizon, A and B the two objects ; l BOA -=6; l DOC, its reduction to the horizon, —Q-vx) CA--^h, DB^h'. From the a AZB, ^ ^^^ cos ^ - cos ZA cos ZB Cos CZB = sin ZA sin ZB cos B — sin h sin h' ^ cos h cos h' Making ^inh-h, Cos7i=l— JA^, tkc, and neglecting powers of the ansfles above the second, this becomes Qo^CZD = cos B — hh' l-l{1f+h'') = (cos 6 - hh') {1 + 1 {h' + h'% nearly, = cos ^ - hh' + 1 {h' + h") cos 6. But Cos CZD = cos {B + x) = cos ^ cos £c - sin 6 sin x = cos B — X sin ^, nearly. And from these two equations, hh' , ,j2 ,, ox cos (9 sm B "^ ^ ^smB Kow if ^.= 7^+7/, and q=h-h' ; then7Ji'=l(/-0, h' + N'^l{p' ^q'); \jf-(f . c o. cos^^ a; = i{/ sm 1 cos B sm 1 + cos ^ -^ sin^ ^ ' sin^ = i{/tani6-^'coti^}. SPHERICAL TEIGONOMETRY. 215 In seconds, this correction = -4-r77 •{{p sin V'Y tan i ^ - (q sin V'f cot J 6}, where /;', ff are the seconds in the angles i^ and ^, = I sin 1"{ /^ tan J (9 - ^'' cot \ B]. Note. This reduction is not required when a Theodolite is used. INSTRUMENTS USED IN SUPwVEYING. 81. Def. The Vernier is a contrivance for sub Uvidiiiij equal graduations that have been made on a straight line or a circle. A Jl M K AB\s> 2, portion of a straight line, or a circle, which is divided by straight lines at right angles to it into any number of equal ])arts. CD^ the Yernier, is another scale, which slides along AB when ^^ is a straight line, and revolves round the center of AB when it is a circular arc. If it be required to subdivide each of the divisions of AB into n equal ])arts, take QM = n — \ of these parts, and divide QM into n equal parts by straight lines at right angles to it ; if a be the length of a division of AB^ the magnitude of each of these 72 — 1 parts will be . a. * n Let R^ the r"" division from Q, coincide with a division on AB\ then PQ ^ PR - QR =^ra- r . ^^ . a = r . - , and the 7t n length of PQ is known. Ex. If each of the original divisions were an inch, and nine inches were divided into 10 parts on the "Vernier, then, supposing that the extremity of a line AQ which it was wanted to measure came to Q^ and that the inches marked at P were p, if the third division of the Vernier coincided with a division on the scale, the length required would be (^ + ttt ) inches. 21 G SPHERICAL TRIGONOMETRY. 82. The Spirit Level is a glass tube BABE of circular bore, which is ground into the form of a circular arc of very large radius — sometimes 800 feet. It is then nearly filled with, some fluid, and the ends are closed. If the instrument be placed in a vertical plane, and the extremities D and E rest on a horizontal surface, the ^^ ^^ bubble {AB) of air left in the tube will be at the highest part of it ; and if E be gradually raised the bubble will continu.ally keep moving towards E''\ If there be a plane of an instrument (such as a Theodolite) which it is necessary to bring into a horizontal position, it is pro- vided with, two levels, as nearly equal to one another in every respect as possible, which, are placed at right angles to each other and permanently attached to the plane. The instrument-maker marks the positions of the bubbles when the plane is horizontal, and therefore if the bubbles occupy these positions on any occa- sion, the plane to which, the levels are attached must then be horizontal. If the plane be inclined at any angle to the vertical and the positions of the bubbles be noted, then if at a second observation they occupy the same positions, the plane will iiave the same inclination to the vertical which it had before. S3. To level between two points. For the purpose of finding the altitude of one point above another point, a spirit-level is attached to a telescope, and so * The grinding the bore of a Spirit-level is done with a plug of metal covered with emery. The elasticity of the glass, assisted probably by that of the metal of the plug, enables the workman, by means of -pressure on the outside, to wear away any particular portion of the interior surface he chooses. If, after the grinding is finished, the bubble be found to move through equal lengths of the tube for equal increments of incHnation, and to be always of the same length, the bore of the tube must be uniform, and the form of the tube a truly circular arc. If I be the length through which the bubble moves in consequence of an increase of n" in the inclination {n being a small quantity), the radius of the circular arc = — -. — -rr = - x 206265. n sm 1 n Ether is the best fluid with which spirit-levels can be fiUed ; because in ether the bubble is found to come into a state of rest in the shortest time after a sudden displacement. SPHERICAL TRIGOXOMETRY. 217 adjusted that the optical axis of the telescope is horizontal when the bubble is at the middle of the level. Let there be two staffs set up verti- cally, A£, CD ; and when the instru- ment is at i? and its axis is horizontal, let the point G be seen on the cross wires ; and when the instru- ment is placed at D with its axis horizontal let E be seen on the cross wires ; then, by measuring AB and CD, EF, the altitude of E aboA'^e the horizontal line AF passing through A is found. The operation can be repeated as often as it may be necessary. *■ 84. The Theodolite. This is an instrument for measuring the angles of elevation of objects above a horizontal plane; and also the horizontal angle which two objects subtend at the obser- ver's eye. The accompanying figure is an elevation (or projection on a vertical plane by lines perpendicular to the plane) of a Theodolite. The two circles / and K fit close to each other, the lower one having attached to it three horizontal bars, making aughis of 120° with each other, on which the instrument rests. The feet of the instrument are thi-ee screws, J/, J/, M, working in these bars; and by moving the screws in one direction or the other, the planes of the circles / and A", which are parallel to each other, can be brought into a horizontal position. The upper circle / is graduated, and revolves with the utmost possible nicety upon the lower circle by means of an axis attached to the upper circle, and working within the collar L which forms a part of the frame of the instrument. The circle K has two Verniers engraved upon it, and in the best instruments the divisions are read off by microscopes attached to the Verniers. To the revolving circle 1, which is called "the limb" of the Theodolite, two levels at right angles to each other are attached. \t;. 218 SPHERICAL TRIGONOMETRY. To this circle also two stands are fixed diametrically opposite to each other (one of which, DOE, is here represented) support- ing an axis parallel to the revohdng circle /. To this axis a circle ABC is permanently attached at right angles to it; and a telescope {GH) is fastened to the circle, with its line of collimation perpen- dicular to the axis of the cii-cle. The circle and attached telescoj)e revolve along; with the axis in such a manner that the whole can turn completely round without touching "the limb" of the instru- ment. The rim BB of the circle is graduated ; and there are two Verniers, A and G, unconnected with the axis, but carried by a support which is seen to enter the cii'cle / (but not the circle A') ati^. 85. To explain how the Theodolite is used. The two cii'cles / and K are first rendered horizontal by lengHi- enino; or shortenin.o' the screw-feet. The limb is then turned round upon the lower circle until the plane of the vertical cii'cle passes through the object whose altitude it is required to find, and the axis carrying the vertical circle is made to revolve until the observer, on looking through the telescope, perceives the object on the cross wires with which the telescope is furnished. The g]*aduations at the points of the rim of the vertical circle which are opposite to the beginning of the scales engraved on the Vernier plates are then read off. Next, without touching the axis to which the telescope is attached, the limb is made to revolve through 180°. The plane of the vertical circle again passes through the object, but the direc- tion of the telescope {A'B') is now as much de- pressed below the horizon as it was before {AB) b elevated above it. Let the vertical circle and its axis be turned in the direction A'AB' until the object is again seen on the cross wires of the tele- scope ; the telescope A'B' is therefore now brought into the same direction (AB) that it had at the first observation. The graduations are again read off at the Ver- niers, and the number of degrees marked on the arcs AA', BB' of the instrument, (which are the arcs that have passed between the two Verniers), is four times the zenith distance of the object. [For let he the real center of the graduated circle ; then twice the zenith distance is BPB', oxAPA'; jom BA', AB' \ SPHERICAL TRIGONOMETRY. 219 .-. 4xzenith disiance = £PB' + APA' = {PAB' + FB'A) + {PBA' + PA'B) = BAB' + A'B'A +ABA' + B'A'B = 2BAB' + 2ABA' = BOB' + AOA'\ EucHd, in. 20 = sum of the differences of readings at the Verniers at the two observations. It appears, therefore, that any errors arising from the axis of rotation of the circle not coinciding exactly v:ith the center of its graduation are wholly avoided by using the instrument in this manner. {Hymers' Astronomy, Ait. 117.)] To observe the liGrizontal angle between two objects, the limb of the Theodolite is made horizontal, and the angle is noted (bj four readings off, as in the last case) through which the limb revolves to bring the V'vo objects successively on the cross wires of the telescope. 8G. The Repeating Circle was the instrument used in the French Surveys for observing the angles. The observations, how- ever, which are made by it, are of questionable value for this reason, that although the errors of imperfect gi'aduation (which are but slight, if the instrument be a good one) may possibly be destroyed, as they j^'^'ohably are, by repeating the observations and taking the mean of them, yet, for anything that can be known to the contrary, the errors arising from the inability of the ob- server to distinguish the position of a point with perfect exactness, may have been accumulating all the while. Also, from the construction of the Repeating Circle, it is scarcely possible to avoid errors arising from the instahllity of the instrument. CHAPTER VII. ON (l.) THE SMALL CORRESPONDING VARIATIONS OF THE PARTS OF A SPHERICAL TRIANGLE j AND (ll.) THE CONNEXION EXISTING BETWEEN SOME FORMULA IN SPHERICAL TRIGONOMETRY AND ANALOGOUS FORMULA IN PLANE TRIGONOMETRY. I. The process of Differentiation can be applied to deter- mine tlie errors introduced in determining the other parts of a Spherical Triangle from three given parts, when one of the given parts is affected by a small known error. 87. If C and c remain constant, the corresponding small variations 3a and Ih of the sides a and b are connected by the equation, 8b . cos A + 8a . cos B = 0. Considering G and c constant, and differentiating with respect to a, the formula Cos G sin a sin. b = cos c — cos a cos b, Cos G {cos a sin b + dj) . cos b sin a} = sin a cos b + dj) . sin b cos a ; .-. Cos G {Sa . cos asiab + Sb. cosb sin a} = Sa . sin a cos 5 + 86 . sin 6 cos a, nearly ; Ba and 86 being small corresponding increments of a and b ; . •. = Sa. {sin a cos b — cos G sin b cos a} + Sb . {sin b cos a — cos G sin a cos 6} = 8a . cos £ sin c + 86 . cos A sin c j by (iv.) .*. = 8(X . cos.5 + 86. cos^. This method is always applicable. A small spherical triangle, however, is frequently treated as a plane triangle after the follow- ing manner; particularly in establishing formulae for calculating the corrections used in Astronomical investigations. SPHERICAL TRIGONOMETRY. 221 88. If c, tlie side opposite the right angle in a inght- angled spherical triangle, receive a small known increment, to determine the corresponding increments of the sides enclosing the right angle. Let AB = c, BD = hc, lBA^ = which is true ^ tein i3 sm b for a Spherical Triangle, to deduce the analogous formula in the case of a Plane Triangle. Let a, h' be the lengths of the arcs subtending the angles A and B ', r the radius of the s^Dhere. . a' a' 1 a'3 1 a'a Sin^ ^^^^7 F "273-7? + - a' ^-273-7^+- -^^'^""Sini? ./ T 1 b'\ -h'\ 1 6- And if a' and 5' be indefinitely small compared with the radius of the sphere, the formula becomes -^^^ — 77 = — ; a property of Plane triangles. 90. And, in like manner as in the last Article, from theformulce _, ^ cos a — cos h cos c , „ , , . ^. sin h(a — h) , -, r^ Cos A = : — ^—. , and Tan A- (^ -J5) = -. — f ) , ; . cot A C, sm 6 sm c ^ ^ ^ sni^{a + b) '^ there m,ay he deduced these analogous formulcz of Plane Trigo- nometry, r^ -, , . -r.s a —a ...^ i/^ , -^ c —a Tan 1 ( J. - ^) = , ,, . cot i G, and Cos A = ^-r^-f . ^ ^ ^ a +b "^ zhc 91. Prob. If two arcs of great circles intersect each other in a small circle, the product of the tangents of the semi-segments of the one is equal to that of the tangents of the semi-segments of the other. Let A B, CD be the arcs of great chcles intersecting in a point E within the small circle whose pole is F ; FEPGr an arc of the great circle through E and P ; PQ perpendicular to AB. Then AB is bisected in Q, and APQ,, BPQ, are tri- angles equal in every respect. Also PA^PB=PF=PG. SPHERICAL TRIGONOMETRY. 223 By (37), Cos PE= cos EQ cos PQ, and Cos PA =go^AQ cos PQ ; Cos EQ, - cos AQ _ cos PE - cos PA Qo^ EQ + Gos AQ~ coiPE + QOs PA ' .-. Tan ^ (^(2+ fQ) tan \ {AQ-EQ)=tan i {PA+PE) tan^ (P^ - PE), or Tan i BE tan J- £'J = tan ^ {?^ tan 4 EF = tan 4 C'-Ctan ^ ED, in like manner. Cor. If a', b' represent the lengths of the segments of the arc AE, and c', d' represent the lengths of the segments of the arc CD ; and these quan- tities be indefinitely small compared with the radius of the sphere, this formula becomes, in the case where the radius of the smaU circle vanishes with respect to the radius of the sphere, a'b' = c'd' ; which agrees with Euclid, III. 85. CHAPTER YIII. ox THE REGULAR SOLIDS. 92. Defs. (1) A Polyhedron is a solid bouncled bj plane rectilinear figures. If the bounding surface be composed of any similar and equal regular rectilinear figiu-es, the polyhedron is called a Regular Polyhedron. (2) A Tetrahedron is bounded by four equal and equilatenil triangles. (3) A Hexahedron, or Cube, is bounded by six equal squares. (4) An Octahedron is bounded by eight equal and equilateral triangles. (5) A Dodecahedron is bounded by twelve equal and equi- lateral pentagons. (6) An Icosahedron is bounded by twenty equal and equi- lateral triangles. It will be proved hereafter that no more Eegular Polyhedrons exist than these five. 93. In any regular Polyhedron, if F = number of Faces, S = number of Solid Angles, E = number of Edges, m = number of Sides in each Face; then 2E = mF, an^ S + F = E + 2. SPHERICAL TRIGONOMETRY. 225 (1) Since every edge is made by two sides, the whole num- ber of sides in the polyhedron is 2E, and this = number of faces x number of sides in a face, (2) Take any point within the polyhedron as the center of a sphere whose radius is r, and join it with each of the angular points of the polyhedron. Let the points in w^hich these lines meet the surface of the sphere be joined by arcs of great circles ; the sur- face of the sphere will then be di\'ided into as many polygons as tlie polyhedron has faces, and the Area of one of these polygons Ttr^ Tnumber of degrees in tlie angles of the polygonal 180* ( - (number of sides of polygon - 2) . ISO J ' (62, Cor. 2.) .-. Area of all these polygons 2 ^number of degi'ees in the angles of all the polygon sj = T— - . \ on the sphere /• ( - (number of all the sides - 2F) . 180 ) ..^^ AS ZC>0-(E-F).^mf=2irr\{S-E + F). loO But Area of all the polygonal areas = area of the sphere = iirr^ ; .'. S-E+F^2, and S+F = E + 2. [These results are evidently true whether the Polyhedron be Regular or Irregular.] 94. The Slim of all the Plane Angles which form the Solid Angles of a Regular Polyhedron = (S — 2) . 36()'\ For the Sum of the Plane Angles = sum of all the Interior Angles of each face. 'O' = F. {m - 2) . 180' Eucl. i. 32, Cor. 1. = 2(^-i^).180'') ^ ,^^. = (^-2). 3600 jby(93;. S. T. 15 226 SPHERICAL TKIGONOMETRY. 95. To prove that in a Regular Polyhedron 2(m + n)-mn' 2(m+iij— mn' 2(m+n)-mn* where m and n are respectively the number of Sides in every Face and the number of Plane Angles in every Solid- Angle. Since every face lias m plane angles, .-. number of the Plane Angles wMcli form all tlie Solid Angles = niF, and . '. = Sn. Hence, (93,) Sn = mF = 2F ; and since S+F = F+2, .:2 = S^-F-E = s(l+~-t)', \ m 2/ ,.S= ^ ; E=„. ^""' ; ^= *" 2 (m + oi) — mn ' 2 {in + n) — mn ' 2 {rn + n) — mn ' 96. There can be but five Regular Polyhedivns. In any regular polyhedron, m, 7i, S, E, F must each be a positive integer. In order that the values of S, E, F obtained in the last Article may be positive, 2 (m 4- n) must be greater than mn ; and that each of them may be integral, 4m, 2mn, and 4rt miist be severally divisible by 2 (m + n) — mn. Now if 2 (??i + n) be greater than inn, 111 111 -+->^, or->^--; m n z m z n but n cannot be less than 3, .'. — cannot be so small as ^r — ^ , or - : m z 3 o Therefore, since m must be an integer, and cannot be less than 3, it can only be 3, 4, or 5. Similarly, since - > ^ , and m cannot be less than 3 : ^ n 2 m' ' .'. the values of n can only be 3, 4, and 5. SPHERICAL TRIGONOMETRY. 227 It will be found, on trial, that the only values of m and n which satisfy all the required conditions are the following. Each regular solid takes its name from the number of its plane faces. m. 3 n. s. E. 6 F. Name of the Regular Solid. 3 4 4 Tetrahedron. (Regular Pyramid.) 4 3 8 12 6 Hexahedron. (Cube ) 3 4 6 12 8 Octahedron. 5 3 20 30 12 Dodecahedron. 3 5 12 30 20 Icosahedron. Cor. If 2 {m + n) = mii, S, E, and F become infinite quanti- ties, and the solid itself becomes a sphere. 97. // 1 he the inclination of two contiguous faces of a Regular Polyhedi^on, then. Sin i I = cos 1800 sin 18U m •) or = cos -p . 90° sin % . 900 XL. Let C and E be the centers of the circles inscribed in two adjacent faces whose com- mon edge is AB\ bisect AB in D, and join -4, B, and D with the points C and E \ CD and ED are manifestly perpendicular to AB^ and .-. L CDE^I. In the plane CDE draw CO and EG at right angles to CD and ED respectively ; let these lines meet in 0; join OA, OB, OD. About as center describe a spherical surface which is cut by the planes AOD, DOCy CO A in ad, dc, ca. Now since AB is perpendicular to CD and to ED, it is perpendicular to the plane CDE, and therefore the plane AOB, in which AB lies, is perpendicular to the p]ane COE; angle ; z ado is a right 15—2 228 SPHERICAL TRIGONOMETKY. And, L cad — \' 360" 4- number of edges whicli meet in a solid angle 1 Q AO = 1 SO*' -4- number of plane angles which meet in a solid angle = . n Also L acd (\ L which each side of a plane face subtends | 1 360" 180" (at the center of the circle inscribed in the face j 2' m jii ' Now by (37), Cos z cad = cos dc sin z acd, ^ 180" ^ . 180" .'. (Jos =cosacsin : n in And Cos dc = cos DOC = cos J COE = cos |(180° - CDE) = siiD J /; 1800 .-. Sini/- cos 71 2 " . 1800* sm m '\LE 9E Again, Since 7Z = ^ . . . (95), and ^ = '^r. • • . (93), cos (-.90" .-. Sinl/= 1£_ sm(^^.90^ 98. To find the Eadius of the Sphere luhich may he inscribed in a Regular Polyhedron. (Fig. Art. 97.) 00 = OE, (= r), is this radius. Let AB = 2a, 180° Then CD = AD. cot ACD = a, cot ACD = a . cot ; til And r = CD.tsinCDO = CD. tan -ll = a. tan l / cot ^-— . til 99. To find the Radius of the Sphere described about a Regular Polyhedron. (Fig. Art. 97.) OA = OB, (= R), is this radius. SPHERICAL TRIGONOMETHY. 229 And r—U. cos aG = R . cot acd cot cad = R . cot cot - — - : m n 180" 1 SO" 1 80" .*. R — r . tan tan — a . tan ^r I tan , by (98). 100. If a Hexahedron and an Octahedron he described about a given spJtere, the sphere described about those Polyhe- drons will be the same ; and conversely. Let R and r. R' and r', be the radii of tlie inscribed and circumscribed spheres for a hexahedron and an octahedron re- spectively. 72 , 180" , 180" R' Ihen - = tan — ^r— .tan — -— = —. . r 6 4 r "Wherefore, if R' be equal to R, r' is equal to r ; or if ?•' be equal to r, R' is equal to R. That is, if a hexahedron and an octahedron be described about the same si)here, the spheres circumscribing them will also be the same ; and conversely. In like manner it may be proved that if a dodecahedron and an icosahedron be described in a given sphere, they will have the same circumscribing sphere; and conversely. 101. To find the Volume of a regidar Polyhedron. From in fig. Art. 97, draw OA, OB, &c. to all the angles of the polyhedron. The solid will thus be divided into F pyramids, whose common altitude is r, and common base, being the area AB of a face, — m. -— . CD = in.a. CD = m. a.r . coth I ', (^8). .'. Whole Volume = ^ . F .vi. a .r^ . cot ^ I 1 80" = |.i^.m.tanl/.cot' . «^ by (98). Cor. Therefore in similar Polyhedrons, Volume a «'. 239 SPHERICAL TRIGONOMETRY. 102. In a Parallelopiped, given the three Edges luJiich meet, and the Angles between them, to find the Altitude^ Sur- face, and Volume of the solid. Let AO, BO, CO be tlie three edges meeting in the point ; and let z BOG=A, L COA=B, l A0B=C; OA = a, OB = b, OC^c; S = i{A + B + C). CD perpendicular to the plane AOB. Join OD, BB, DA. About describe a spherical surface, and let it be cut by the planes AOB, BOC, CO A, DOC in ah, he, ca, dc. Then, the angles at d being right angles, Sin dc = sin ac sin cad, or Siu DOC = sin B sin cad ; And, by (x.), Sin cad= ■ ^. ^ •. Jlsiu S sin (^S^- A) sin (.S' -B)sm(S-C)}: bin B sm 6 ^ ' .-. i)(7 = c. sini)(9C' = ^^. V{sm>S'sin(.S'-^)sin(>S'-^)sin(>S'-(7)}. The Surface = 2 {be sin A + ac sin B + ah sin C}. rYolume of rectangular parallelopiped on the \ same base, and of the same altitude. = area of base x altitude. Euclid, xi. 31. = 2abc ^{sin S sin (.S'- A) sin (>S'-- B) sin {S- C)}. 103. The same things being given, to determine the Dia- gonal which passes through the Solid angle of the Parallelo- 2nped. Let D be the diagonal required; and now suppose OcZT), in the fig. Art. 102, to be in the direction of the diagonal of the face AOB. Let this diagonal = d. SPHERICAL TRIGONOMETRY. 281 Then d^ = a^^h'' + 2ab cos C, and B^ = e^d^- 2c^cos (180" - BOC) = c'' + d' + 2cdco^D0G = 0^ +¥ + c" -\- 2cd cos DOG + 2ab cos C. Now Cos DOC = cos cd = cos ac cos a). 14. It A = a, and 5 = 6, then C=l80'-c. IX SPHERICAL TRIANGLES NOT RIGHT-ANGLED. 15. If each of the three sides be quadrants, and a, /3, y be the distances of a point within the triangle from the angular points ; Cos^ a + cos^ ^ + cos^ y = 1. 16. If d be the length of the arc which bisects G and is terminated by the opposite side. Tan d sin (a + b) = 2 sin a sin b cos ^ G. 17. If a and 6 be nearly equal, a = 1 (a + 6) + tan ^{a + b) tan ^{A- B) cot J (J. + 5), very nearly. 18. If one angle of a triangle, plane or spherical, be equal to the sum of the other two angles, the greatest side is double of the distance of its middle point from the opposite angle. 19. If be any point in which arcs of great circles drawn through the angular jjoints of ABG intersect, then Sin A sin B sin G sin a sin b sin c sin 10 sin 50 sin (70 , , .^ . r>/^/7 i. T>r\ • nr\ a = , ^ — ^-^ . {cot AG sm BOG + cot BO sin GO A sin a sm o sm c + cot 00 sin 105}. SPHERICAL TRIGONOMETRY. 235 20. Find the locus of the vertices of all right-angled spheri- cal triangles which have the same hypothenuse ; and from the equation prove that the locus is a circle when the radius of the sphere is infinite. 21. Divide, by drawing an arc from an angle to the side opposite, a given triangle into two others whose areas are in a given ratio. 22. The sides of a spherical triangle are each 111°, 28'; find its angles, and shew that its area = ^ surface of the sphere. {Ill", 28' = 2 X (55", W), and Tanl(55^ ii') = j2}. 23. If be a right angle, E (the Sjiherical Excess) = 2 tan"' (tan J a tan J b). Ako Sin 1 £ = ^'°^°r^^ , and Cos i E = <=£i^ » «°^ cos ^ C - COS J c * 24. HE be the Spherical Excess, _,,,_, cot i a cot i 6 + cos C Qoi^E^ ^, , or = sin C 1 + cos a + cos h + cos c CosiJ^ = In seconds, \ E = 2 J {sin S'^m {S - a) sin (S - h) sin {S -c))' 1 + cos a + cos h + cos c 4 cos J a cos J h cos J c ' tan |- h tan ^ c sin yl tan^ i 5 tan^ i c sin 2^ + sin i" sin 2' tan^ ^ h tan^ ^ c sin ?>A 25. If P be the Perimeter and E the Spherical Excess of the triangle ABC, then 2 sin I P sin J A sin J 5 sin | G = {sin J J- sin (^ - 1 J') sin (^ - 1 JE') sin (C - 1 ^) }^. 26. Determine the area of a spherical triangle from the data a, b, C ; and shew that if a and b be constant, and also a + b he less than tt, the area admits of a maximum value. 236 SPHERICAL TRIGONOMETRY. 27. The angles of a spherical triangle, of which the area is 7 rr-rTTT^, wherc r is the radius of the sphere, form an arithmetic 1 'Ji progression of which the common difference is 45°. Find them. 28. If the sides AC, BG of a triangle be produced to D and E, points such that Tan \AC. tan ^ BG — tan ^ DG . tan ^ EG, and DE be joined by an arc of a great circle, the triangles ABG, GDE are of equal area. 29. If ^ be the surface of a spherical triangle whose angles are each 120°, and S' that of its polar triangle, Tani/S' : Tan|>S' = 6^2 + ^3 : 2 J2 - J3. 30. If a be one of the n sides of a regular spherical polygon, its surface (/S) may be found froni the equation ^. TT— h ^ TT ^ Cos — = COS - sec * a, n n 31. The Spherical Excess of a triangle is T'-S. Find its area, the radius of the Earth being taken to be 7757 miles. 32. If the three sides of a spherical triangle measured on the Earth's surface be 12, 16, and 18 miles, find the Spherical Excess. 33. A plane triangle whose sides are a, b, c, is placed in a sphere of radius r. Prove that the angle between the arcs of the great circles of which a and b are the chords is a right angle, ii2rj{a' + b'-c') = ab. 34. If P be the pole of the small circle circumscribing a triangle ABG, prove that z AFB is double of the angle between the chords oi AG and BG. 35; The middle points of the sides AB and AG of b. triangle are D and E respectively, and P is the pole of DE ; shew that Z.BPG is double of z DPE. 36. If an arc of a great circle be bisected, its segments will subtend equal angles at any point on the great circle of which its middle point is the pole. 37. A lune is formed by two great circles which intersect at a right angle; prove that from any point in one of the circles two arcs of great circles can be drawn to the other cutting it at equal angles, and find the least value of these angles. [The points are equidistant from the extremities of the lune.] SPHERICAL TRIGONOMETBY. 237 38. ABO and A'B'C are equal triangles; prove that arcs drawn at the middle points of the arcs of the great circles AA' , BB', CC, and at right angles to those lines, meet in a point at which AA\ BB', CO', subtend equal angles. What limitation is there to this proposition] [The triangles ABC, A'B'C must be such that if placed one upon the other they would coincide.] 39. There is a great circle ABC, and AA', BB', OC are arcs at right angles to it, which are reckoned positive on one side and negative on the other : prove that the condition of A', B', C lying in a great circle is Tan A A' sin BC + tan BB' sin CA + tan CC sinAB = 0. 40. Two quadrants {OA, OB) of great circles include a right angle ; a great circle meets them in 0, D respectively, and through P, any point in it, arcs of great circles APY, BPX are drawn meeting OB, OA respectively in Y, X ; if OX — 0, 0Y=<1>, 00 = a, On=.(S, shew that Tan 6 U\n cfi tan a tun (i 41. Two great circles, inclined at an angle w, intersect at 0, NN' and MM' are equal arcs on the two circles respectively, and JVJ/ and N'JI' are arcs perpendicular to OMM'; if NM=h, N'M = 8', shew that Cos 8 cos 8' = cos w ; also that Cos J/J/ = ^1'^^'''''^' , Cos XX' = sin OX . sin OiV . ( 1 + cos a>). 1 — cos W 42. P is the pole of a small circle; S^S^S^.-.S^^ a series of points in this circle equidistant from one another; if Z be any other point and ZS^, ZS^... be joined, shew that the sum of the cosines of Z>>\, Z^S^,... = n cos PZ . con S, where 8 is the radius of the small circle. 43. Three small circles are inscribed in a spherical triangle whose anofles are each 120", in such a manner that each circle touches each of the other circles and also two sides of the triangle ; prove that the radius of each circle is 30°, and that the centers of the cii'cles coincide with the angular points of the polar triangle. 44. Three small circles, whose radii are p^, p^, p^, touch one another in P, Q, R. If A, B, be the degrees in the angles of a spherical triangle formed by joining their centers, prove that Ai-ea PQR = {A cos p^ + B cos p^ + C cos p^ - 180) r^, r being the radius of the sphere. 238 SPHERICAL TRIGONOMETEY. 45. If in a triangle, R, r be the radii of the small circum- scribing and inscribed cii'cles, and r^, r^, r^ the radii of the circles touching one side of the triangle and the other two sides produced, prove that Cot r^ + cot r^ + cot r^ — cot r = 2 tan R, and shew from this result that the corresponding property in a plane triangle is 1111 — + —+—=-. rt« A» n* fv* 1 2 3 46. Determine the points in the sides of a triangle at which tangents being drawn, they will meet two and two and form a triangle. If J.', B\ C be the angles of this triangle, prove that Tan ^A' = cos (aS' - a) tan |- A. 47. Prove that the square of the area of the triangle formed by the tangents as in the last question is equal to r^{tan(*S'-a)+taD (/S'-6) + tan (/S'-c)}tan (aS'-ct) tan(AS'-5) tan (/S'-c), r being the radius of the sphere. 48. If the vertical angle of a triangle be equal to the sum of the angles at the base, the locus of the vertex, while the base remains fixed, will be the small circle described with the middle point of the base as pole and the base as diameter. 49. ABCD is a quadrilateral whose sides are arcs of great circles, E and F the middle pointg oi AC and BD ; prove that Cos AB + cos BG + cos CD + cos DA = 4: cos J AC. cos J BD . cos FF. 50. If ABCD be a spherical quadrilateral, P the intersection of AB and JDC, Q that of AD and BC, R that of ^C and BD, shew that Sin AB . sin CD . cos P - sin AD . sin BC . cos Q = cos AD . cos BC - cos AB . cos CD — ± sin AC . sin BD . cos R. 51. If a, /3, y be the angles which three diameters of a sphere of radius a make with one another, prove that the volume of the parallelopiped formed by tangent planes at their extremities, (2(r being = a + /5 + y), is ^ ;^/{sin o- sin (o- — a) sin (a — jS) sin (cr - y)} * SPHERICAL TRIGONOMETRY. 239 52. If i be the inclination of a plane to tlie horizon, and a the inclination of a line in it to the intersection of the plane with the horizontal plane, the inclination of the line to the horizon will be found from the equation. Sin = sin i sin a. 53. The shadow of a cloud is observed to fall upon a spot at a known distance on the side of a hill. Given the altitudes and the azimuths of the cloud and shadow, and the azimuth of the Sun, find the distance of the cloud. Atis. If a^, a^, ttg be the azimutlis of the Sun, cloud, and shadow respectively, and a,, a^ the altitudes of the cloud and shadow, d the known distance, the distance of the cloud is cosa, . sin (ttg- ttj) ' cosctj . sin(a2 - a J ' 54. If Z be the zenith, K the pole of the limb (which is not exactly horizontal) of a theodolite, and S be an object whose azi- muth is observed, the error is knoM'n from the equation, z SKZ- L SZK= KZ. cot .S^. sin SZK. 55. If Z be the zenith, K the pole of the circle of a theo- dolite, which is not exactly vertical, and KZ be produced to Q till KQ is a quadrant, then if S be an object whose zenith distance is to be observed, the error of observation is known frojn the equation, HQ - SZ= cot SQ . sin \ QZ. 56. Given two sides, a and h, of a triangle, spherical or jtlane, and the included angle C, to find the variation produced in A corresponding to a small given variation in C. 57. If a solid be bounded by plane figures, of which some have an odd and some have an even number of sides, shew that there must be an even number of those faces which have an odd number of sides. 58. In a triangle G and c remain constant, and a, h receive small increments Set, hb respectively ; shew that ha 86 ^ , sin (7 + ~77i 2 • 2 i\ — ^') where n = J (I —n° sin^a) J{\-n^ sin^ b) sin c ' 59. A solid is formed of an equal number of faces bounded by 3, 4, and 5 sides; find the least number of faces in such a solid, and the numbers of its edges and solid angles. Ans. F=6, ^----12, ^-8. 240 SPHERICAL TREGONOMETRY. 60. Every solid in wMch four or more edges meet in each solid angle must have at least eight triangular faces ; and those in which live or more meet in each angle must have at least twenty- triangular faces. Also no solid can be formed so that not less than six edges meet in each solid angle. 61. No solid is entirely composed of faces all of which have more than five sides ; and if there be neither quadrilateral nor pentagonal faces, there must be more than lour triangular faces, unless the faces be all triangles. 62. The base of a pyramid is an equilateral hexagon whose alternate angles are equal and adjacent angles are unequal. Given the angles between any face of the pyramid and two adjacent faces, find the angle between a normal to any face and the axis of the pyramid. Aiis. The angle required is equal to that which any plane face makes with the base. If a, a be the given angles, and be the angle required, then Sin = 2 cos ^ (a' + a) . cos ^ (a— a) . sec 0, where is known from the equation Tan cfi = -p . tan ^ (a' + a) . tan | (a - a). 63. If r, R be the radii of the spheres described within and about a regular tetrahedron, r', R the radii of the spheres to which the edges, and one face and the planes of the three others produced, are resj^ectively tangents, prove that r' = J^, and R' = j2^. 64. Given the six edges of a triangular pyramid, to find its volume. 65. Of all triangular pyramids of given volume, the regular teti-ahedron has the least surface. 66. A cube is turned round one of its diagonals through 180"; shew that 2J2 is the tangent of the angle at which any one of the faces is inclined to its original position. 67. The diagonal of a cube is produced until the length of the part produced is equal to the diagonal, and from the ex- treme point as pole the cube is projected on a plane perpen- dicular to the diagonal. Shew that the projection will be an equilateral hexagon, in which the alternate angles are equal and the adjacent angles are unequal, their sines being in the ratio of 8 to 5. CAMBEIBGE : PllINTED BY C. J. CLAY, SI. A. AND SONS, AT THE UNIVEKSIXY PKESS, \ ♦*'i ^flNCflUffifsci 3 9031 01550245 3 Tri^"- ■^ m >eJmuM fail jf *«ATH, DEPT, BOSTON COLLEGE LIBRARY UNIVERSITY HEIGHTS CHESTNUT HILL. MASS. Books may be kept for two weeks and may be renewed for the same period, unless reserved. Two cents a day is charged for each book kept overtime. If you cannot find what you want, ask the Librarian who will be glad to help you. The borrower is responsible for books drawn on his card and for all fines accruing on the same. ■,;:.;:-;. f;v::;^;VV/'..*/-:^;vv:;-?'' ' \: ^: -A-.. ,...■■•1 •■*/»■■ ■<; :.