/9// CORNELL UNIVERSITY LIBRARY 3 1924 073 537 254 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924073537254 A TEEATISB ON PLANE TEIGONOMETBY CAMBRIDGE UNIVERSITY PRESS ILonftan: FETTEE LANE, E.G. C. F. CLAY, Manager ffiliinlmtBl): 100, PRINCES STREET asniin: A. ASHER AND CO. Eeipjis: ^ A. BROCKHAUS 0tto Bork: G. P. PUTNAM'S SONS JSomtia!) anti CalctittB: MACMILLAN AND CO., Ltd. All rights reserved A TREATISE ON PLANE TEIGONOMETEY BY E. W. HOBSON, Sc.D., LL.D., F.E.S., SADLEIRIAN PROFESSOR OF PURE MATHEMATICS, AND FELLOW OF CHRIST'S COLLESB, CAMBRIDGE THIRD EDITION Cambridge : at the University Press 1911 First Edition 1891 Second Edition 1897 Third Edition {revised and enlarged) 1911 PEEFACE TO THE THIED EDITION T HAVE taken the opportunity afforded by the need for a new edition to subject the whole work to a careful revision, and to introduce a considerable amount of new matter. In Chapter i I have inserted a theory of the lengths of circular arcs, and of the areas of circular sectors, based upon arithmetic definitions of their measures. Much of that part of the work which deals with Analytical Trigonometry has been re-written. Proofs of the transcendency of the numbers e and tt have been introduced into Chapter XV. It is hoped that the proof there given of the impossibility of "squaring the circle" will prove of interest to many readers to whom a detailed discussion of this very interesting result of modem Analysis has hitherto not been readily accessible. E. W. HOBSON. Christ's College, Cambridge, October, 1911. PEEFACE TO THE FIEST EDITION TN the present treatise, I have given an account, from the modern point of view, of the theory of the circular functions, and also of such applications of these functions as have been usually included in works on Plane Trigonometry. It is hoped that the work will assist in informing and training students of Mathematics who are intending to proceed considerably further in the study of Analysis, and that, in view of the fulness with which the more elementary parts of the subject have been treated, the book will also be found useful by those whose range of reading is to be more limited. The definitions given in Chapter iii, of the circular functions, were employed by De Morgan in his suggestive work on Double Algebra and Trigonometry, and appear to me to be those fi-om which the fundamental properties of the functions may be most easily deduced in such a way that the proofs may be quite general, in that they apply to angles of all magnitudes. It will be seen that this method of treatment exhibits the formulae for the sine and cosine of the sum of two angles, in the simplest light, merely as the expression of the fact that the projection of the hypothenuse of a right-angled triangle on any straight line in its plane is equal to the sum of the projections of the sides on the same line. The theorems given in Chapter vii have usually been deferred until a later stage, but as they are merely algebraical consequences of the addition theorems, there seemed to be no reason why they should be postponed. PREFACE Vll A strict proof of the expansions of the sine and cosine of an angle in powers of the circular measure has been given in Chapter viii ; this is a case in which, in many of the text books in use, the passage from a finite series to an infinite one is made without any adequate investigation of the value of the remainder after a finite number of terms, simplicity being thus attained at the expense of rigour. It may perhaps be thought that, at this stage, I might have proceeded to obtain the infinite product formulae for the sine and cosine, and thus have rounded off the theory of the functions of a real angle ; for convenience of arrangement, however, and in order that the geometrical appli- cations might not be too long deferred, the investigation of these formulae has been postponed until Chapter xvii. As an account of the theory of logarithms of numbers is given in all works on Algebra, it seemed unnecessary to repeat it here ; I have consequently assumed that the student possesses a know- ledge of the nature and properties of logarithms, sufficient for practical application to the solution of triangles by means of logarithmic tables. In Chapter xii, I have deliberately omitted to give any account of the so-called Modern Geometry of the triangle, as it would have been impossible to find space for anything like a complete account of the numerous properties which have been recently discovered; moreover many of the theorems would be more appropriate to a treatise on Geometry than to one on Trigonometry. The second part of the book, which may be supposed to commence at Chapter xiii, contains an exposition of the first principles of the theory of complex quantities ; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Vlll PREFACE Gauss, and others, had placed the use of imaginary quantities and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis- In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity. I owe a deep debt of gratitude to Mr W. B. Allcock, Fellow of Emmanuel College, and to Mr J. Greaves, Fellow of Christ's College, for their great kindness in reading all the proofs; their many suggestions and corrections have been an invaluable aid to me. I have also to express my thanks to Mr H. G. Dawson, Fellow of Christ's College, who has undertaken the laborious task of verifying the examples. My acknowledgments are due to Messrs A. and C. Black, who have most kindly placed at my disposal the article "Trigonometry" which I wrote for the Encyclopcedia Britannica. During the preparation of the work, I have consulted a large number of memoirs and treatises, especially German and French ones. In cases where an investigation which appeared to be private property has been given, I have indicated the source. I need hardly say that I shall be very grateful for any corrections or suggestions which I may receive from teachers or students who use j;he work. E. W. HOBSON. Christ's College, Cambridge, March, 1891. CONTENTS CHAPTER I. THE MEASUREMENT OF ANGULAR MAGNITUDE. ARTS. 1. Introduction 2 — 3. The generation of an angle of any magnitude 4. The numerical measurement of angles 5 — 10. The circular measurement of angles 11. The length of a circular arc . 12. The area of a sector of a circle Examples on Chapter I PAGES 1 1—3 3—4 4—6 7—10 10 10—11 CHAPTER II. THE MEASUREMENT OF LINES. PROJECTIONS. 13 — 16. The measurement of lines 17. Projections 12—13 13—14 CHAPTER III. THE CIRCULAR FUNCTIONS. 18 — 21. Definitions of the circular functions 22 — 24. Relations between the circular functions 25. Bange of values of the circular functions 26 — 29. Properties of the circular functions 30. Periodicity of the circular functions 31. Changes in the sign and magnitude of the circular functions 32. Graphical representation of the circular functions 33. Angles with one circular function the same 34. Determination of the circular functions of certain angles 35 — 38. The inverse circular functions Examples on Chapter III 15—18 19—20 20 21—24 24 24—26 26—28 28—29 29—32 32—33 33—35 CONTENTS CHAPTER IV. THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES. AETS. 39—43. 44—45. 46. 47. 48. 49. 50. 51. 52. 53. 54. The addition and subtraction formulae for the sine and cosine 36 — 41 Formulae for the addition or subtraction of two sines or two cosines ........ 41 — 44 Addition and subtraction formulae for the tangent and cotangent 44 — 45 Various formulae 45 — 47 Addition formulae for three angles .... 47- — 48 Addition formulae for any number of angles . . 48 — 49 Expression for a product of sines or of cosines as the sum of sines or cosines 50 — 52 Formulae for the circular functions of multiple angles 52 — 53 Expressions for the powers of a sine or cosine as sines or cosines of multiple angles 53 — 54 Relations between inverse functions .... 54 — 55 Geometrical proofs of formulae 55 — 57 Examples on Chapter IV 58—62 CHAPTER V. THE CIRCULAR FUNCTIONS OF SUBMULTIPLE ANGLES. 55—63. Dimidiary formulae 63—69 64. The circular functions of one-third of a given angle . 70 — 72 65 — 66. Determination of the circular functions of certain angles 72 — 75 Examples on Chapter V 75 — 77 CHAPTER VI. VARIOUS THEOREMS. 67. Introduction 68. Identities and transformations 69. The solution of equations 70. EUminations 71. Relations between roots of equations 72. Maxima and minima. Inequalities 73. Porismatic systems of equations . 74 — 77. The summation of series Examples on Chapter VI 78 78—82 82—83 84—85 85—87 87—89 89—90 90—94 94—103 CONTENTS XI CHAPTER VII. EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES. ARTS. 78—79. 80—83. 84. 85. 86—91. Series in descending powers of the sine or cosine Series in ascending powers of the sine or cosine The circular functions of submultiple angles Symmetrical functions of the roots of equations Factorization Examples on Chapter VII . PAGES i 104- -106 106- -109 109- -110 110- -114 114- -120 120- -123 CHAPTER VIII. RELATIONS BETWEEN THE CIRCULAR FUNCTIONS AND THE CIRCULAR MEASURE OF AN ANGLE. 92— 9.5. 96. 97—98. 100. Theorems Euler's product The limits of certain expressions .... Series for the sine and cosine of an angle in powers of its circular measure The relation between trigonometrical and alge- braical identities .... Examples on Chapter VIII .... 124—127 127—129 130—131 131—134 135 135—138 CHAPTER IX. TRIGONOMETRICAL TABLES. 101. Introduction 102—105. Calculation of tables of natural circular functions 106. The verification of numerical values 107. Tables of tangents and secants 108. Calculation by series 109. Logarithmic tables 110—111. Description and use of trigonometrical tables 112—114. The principle of proportional parts 116_117. Adaptation of formulae to logarithmic calculation 139 139—143 143 143 144—145 145 145—147 147—152 152—154 xu CONTENTS CHAPTER X. RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE. ARTS. 118—124. 125. 126. 127—128. 129. Theorems The area of a triangle Variations in the sides and angles of a triangle . Relations between the sides and angles of polygons The area of a polygon Examples on Chapter X PAGES 155—159 159 160—161 161—162 162—163 164—166 CHAPTER XI. THE SOLUTION OF TRIANGLES. 130. Introduction 131 — 133. The solution of right-angled triangles . 134 — 140. The solution of oblique-angled triangles 141 — 144. The solution of polygons 145 — 149. Heights and distances . . . . Examples on Chapter XI . . . 167 167—169 169—175 176—178 178—182 182—189 CHAPTER XII. PROPERTIES OF TIUANGLES AND QUADRILATERALS. 150. Introduction 151. The circumscribed circle of a triangle 152 — 154. The inscribed and escribed circles of a 155. The medians 156. The bisectors of the angles 157. The pedal triangle .... 158. The distances between special points 159. Expressions for the area of a triangle 160 — 163. Various properties of triangles 164 — 167. Properties of quadrilaterals 168. Properties of regular polygons 169. Examples Examples on Chapter XII 190 190—191 191—195 195—196 196—197 197—198 198—201 201 201—203 203—208 208—209 209—213 213—223 CONTENTS XUl CHAPTER XIII. COMPLEX NUMBERS. ABTS. 170. Introduction 171 — 174. The geometrical representation of a complex number 175 — 177. The addition of complex numbers . 178. The multiplication of complex numbers 179. Division of one complex number by another 180 — 185. The powers of complex numbers 186—187. De Moivre's theorem 188. Factorization . 189. Properties of the circle . 190. Examples .... Examples on Chapter XIII PAGES 224 224—227 227—229 229—231 231—232 232—236 237—239 239—240 241 241—242 243—245 CHAPTER XIV. THE THEORY OF INFINITE SERIES. 191. Introduction 192 — 196. The convergence of real series 197. The convergence of complex series 198. Continuous functions .... 199 — 201. Uniform convergence .... 202. The geometrical series .... 203 — 208. Series of ascending integral powers 209. Convergence of the product of two series 210. The convergence of double series . 211—212. The Binomial theorem .... 213 — 217. The circular functions of multiple angles 218 — 219. Expansion of the circular measure of an angle in powers of its sine .... 220 — 222. Expression of powers of sines and cosines in sines and cosines of multiple angles . 246 246—251 251—252 253 253—257 257—258 258—265 265 266—268 268—272 272—279 279—280 280—283 XIV CONTENTS CHAPTER XV. THE EXPONENTIAL FUNCTION. LOGARITHMS. AliTS. 223 — 227. The exponential series 228. Expansions of the circular functions 229 — 230('). The exponential values of the circular functions 231 — 232. Periodicity of the exponential and circular func- tions ....... 233 — 237. Analytical definition of the circular functions 238—239. Natural logarithms . 240 — 244. The general exponential function 245. Logarithms to any base . 246 — 248. Generalized logarithms 249 — 250. The logarithmic series 251. Gregory's series .... 251(')_251P). The quadrature of the circle . 252 — 254. The approximate quadrature of the 255. Trigonometrical identities . 256 — 257. The summation of series . Examples on Chapter XV circle PAGES 284—287 288 288—290 291 291—296 296—297 297—300 300 300—301 302—304 304^305 305—310 310—311 311—312 312—315 315—321 CHAPTER XVI. THE HYPERBOLIC FUNCTIONS. 258. Introduction 259. Relations between the hyperbolic functions 260—261. The addition formulae .... 262. Formulae for multiples and submultiples 263 — 265. Series for hyperbolic functions . 266. Periodicity of the hyperbolic functions 267 — 270. Area of a sector of a rectangular hyperbola 271. Expressions for the circular functions of com- plex arguments 272 — 274. The inverse circular functions of complex argu ments 275 — 276. The inverse hyperbolic functions 277. The solution of cubic equations 278. Table of the Gudermannian function Examples on Chapter XVI 322 322—323 323—324 324 324—325 326 326—331 331 331—333 333—334 335—336 336 337 CONTENTS XV CHAPTER XVII. INFINITE PRODUCTS. AKTS. PAGES 279 — 281. The convergence of infinite products . . . 338 — 342 282 — 292. Expressions for the sine and cosine as infinite pro- ducts 343 — 354 292 W. Representation of the exponential function by an infinite product 354 — 355 293 — 295. Series for the tangent, cotangent, secant, and co- secant 355 — 359 296 — 299. Expansion of the tangent, cotangent, secant, and cosecant in powers of the argument . . 360 — 364 300. Series for the logarithmic sine and cosine . . 365 — 367 301. Examples 367—369 Examples on Chapter XVII 369—373 CHAPTER XVIII. CONTINUED FRACTIONS. 302—303. Proof of the irrationality of tt . . . . 374—375 304. Transformation of the quotient of two hypergeo- metric series 375 305. Euler's transformation 376 Examples on Chapter XVIII .... 376—377 Miscellaneous Examples 378 — 383 CHAPTER I. THE MEASUREMENT OF ANGULAR MAGNITUDE. 1. The primary object of the science of Plane Trigonometry is to develope a method of solving plane triangles. A plane triangle has three sides and three angles, and supposing the magnitudes of any three of these six parts to be given, one at least of the three given parts being a side, it is possible, under certain limitations, to determine the magnitudes of the remaining three parts; this is called solving the triangle. We shall find that in order to attain this primary object of the science, it will be necessary to introduce certain functions of an angular magnitude ; and Plane Trigonometry, in the extended sense, will be under- stood to include the investigation of all the properties of these so-called circular functions and their application in analytical and geometrical investigations not connected with the solution of triangles. The generation of an angle of any magnitude. 2. The angles considered in Euclidean Geometry are all less than two right angles, but for the purposes of Trigonometry it is necessary to extend the conception of angular magnitude so as to include angles of all magnitudes, positive and negative. Let OA be a fixed straight line, and let a straight line OP, initially coinci- dent with OA, turn round the point in the counter-clockwise direction, then as it turns, it generates the angle AOP; when OP reaches the position OA', it has generated an angle equal to two right angles, and we may suppose it to go on turning in the same H. T. 1 2 THE MEASUREMENT OF ANGULAR MAGNITUDE direction until it is again coincident with OA ; it has then turned through four right angles; we may then suppose OP to go on turning in the same direction, and in fact, to make any number of complete turns round 0; each time it makes a complete revolution it describes four right angles, and if it stop in any position OP, it will, have generated an angle which may be of any absolute magnitude, according to the position of P. We shall make the convention that an angle so described is positive, and that the angle described when OP turns in the opposite or clockwise direction is negative. This convention is of course perfectly arbitrary, we might, if we pleased, have taken the clockwise direction for the positive one. In accordance with our convention then, whenever OP makes a complete counter- clockwise revolution, it has turned through four right angles reckoned positive, and whenever it makes a complete clockwise revolution, it has turned through four right angles taken negatively. As an illustration of the generation of angles of any magnitude, we may consider the angle generated by the large hand of a clock. Each hour, this hand turns through foui- right angles, and preserves no record of the number of turns it has made ; this, however, is done by the small hand, which only turns through one-twelfth of four right angles in the hour, and thus enables us to measure the angle turned through by the large hand in any time less than twelve hours. In order that the a,pgles generated by the large hand may be positive, and that the initial position may agree with that in our figure, we must suppose the hands to revolve in the opposite direction to that in which they actually revolve in a clock, and to coincide at three o'clock instead of at twelve o'clock. THE MEASUREMENT OF ANGULAR MAGNITUDE 3 3. Supposing OP in the figure to be the final position of the turning line, the angle it has described in turning from the position OA to the position OP may be any one of an infinite number of positive and negative angles, according to the number and direction of the complete revolutions the turning line has made, and any two of these angles differ by a positive or negative multiple of four right angles. We shall call all these angles bounded by the two lines OA, OP coterminal angles, and denote them by (OA, OP); the arithmetically smallest of the angles (OA, OP) is the Euclidean angle AOP, and all the others are got by adding positive or negative multiples of four right angles to the algebraical value of this. The numerical measurement of angles. 4. Having now explained what is meant by an angle of any positive or negative magnitude, the next step to be made, as regards the measurement of angles, is to fix upon a system for their numerical measurement. In order to do this, we must decide upon a unit angle, which may be any arbitrarily chosen angle of fixed magnitude ; then all other angles will be measured numerically by the ratios they bear to this unit angle. The natural unit to take would.be the right angle, but as the angles of ordinary size would then be denoted by fractions less than unity, it is more convenient to take a smaller angle as the unit. The one in ordinary use is the degree, which is one ninetieth part of a right angle. In order to avoid having to use fractions of a degree, the degree is subdivided into sixty parts called minutes, and the minute into sixty parts called seconds. Angles smaller than a second are denoted as decimals of a second, the third, which would be the sixtieth part of a second, not being used. An angle of d degrees is denoted by d°, an angle of m minutes by m', and an angle of n seconds by n", thus an angle d° m n" means an angle containing d degrees + m minutes + n seconds, and is equal to qK + qq qq + q q qq gQ of a right angle. This system of numerical measurement of angles is called the sexagesimal system. For example, the angle 23° 14' 56"-4 23 14 56-4. f ■ 1,. 1 denotes 1 1- ^ — ^^~-„^ oi a right anele. aenoieb go ^ 90 . 60 90 . 60 . 60 ^ ^ 1—2 4 THE MEASUREMENT OF ANGULAR MAGNITUDE It has been proposed to use the decimal system of measurement of angles. In this system the right angle is divided into a hundred grades, the grade into a hundred minutes, and the minute into a hundred seconds ; an angle of g grades, m minutes and n seconds is then written g^ ni «". For example, the angle 13^ 97' 4"-2 is equal to 13-97042 of a right angle. This system has however never come into use, principally becavise it would be inconvenient in turning time into grades of longitude, unless the day were divided differently than it is at present. The day might, if the system of grades were adopted, be divided into forty hours instead of twenty-four, and the hour into one hundred minutes, thus involving an alteration in the chronometers; one of our present hours of time corresponds to a difference of 50/3 grades of longitude, which being fractional is inconvenient. It is an interesting fact that the division of four right angles into 360 parts was used by the Babylonians ; there has been a good deal of speculation as to the reason for their choice of this number of subdivisions. The circular measurement of angles. 5. Although, for all purely practical purposes, the sexagesimal system of numerical measurement of angles is universally used, for theoretical purposes it is more convenient to take a different unit angle. In any circle of centre 0, suppose .45 to be an arc whose length is equal to the radius of the circle ; we shall shew that the angle AOB is of constant magnitude independent of the particular circle used; this angle is called the Radian or unit of circular measure, and the magnitude of any other angle is expressed by the ratio which it bears to this unit angle, this ratio being called the circular msasure of the angle. THE MEASUREMENT OF ANGULAR MAGNITUDE 5 6. In order to shew that the Radian is a fixed angle, we shall assume the following two theorems : (a) In the same circle, the lengths of different arcs are to one another in the same ratio as the angles which those arcs subtend at the centre of the circle. (6) The length of the whole circumference of a circle bears to the diameter a ratio which is the same for all circles. The theorem (a) is contained in Euclid, Book vi. Prop. 33, and we shall give a proof of the theorem (b) at the end of the present Chapter. From (a) it follows that arc^.B ZAOB circumference of the circle 4 right angles ' Since the arc AB is equal to the radius of the circle, the first of these ratios is, according to (b), the same in all circles, conse- quently the angle AOB is of constant magnitude independent of the particular circle used. 7. It will be shewn hereafter that the ratio of the circum- ference of a circle to its diameter is an irrational number ; that is, we are unable to give any integers m and n such that m/n is exactly equal to the ratio. We shall, in a later Chapter, give an account of the various methods which have been employed to calculate approximately the value of this ratio, which is usually denoted by tt. At present it is sufficient to say that ir can only be obtained in the form of an infinite non-recurring decimal, and that its value to the first twenty places of decimals is 3-14159265358979323846. For many purposes it will be sufficient to use the approximate value 22 • 355 3-14159. The ratios — = 3-i4285'7, -^5 = 3-1415929... may be used as approxi- mate values of w, since they agree with the correct value of w to two and six places of decimals respectively. 8. We have shewn that the radian is to four right angles in the ratio of the radius to the circumference of a circle ; the 2 radian is therefore - x a right angle ; remembering then that TT a right angle is 90°, and using the approximate value of tt, 3-1415927, we obtain for the approximate value of the radian 6 THE MEASUREMENT OF ANGULAR MAGNITUDE in degrees, 57°-2957796, or reducing the decimal of a degree to minutes and seconds, 57° 17'44"'81. The value of the radian has been calculated by Glaisher to 41 places of decimals of a second i. The value of l/n has been obtained to 140 places of decimals 2. 9. The circular measure of a right angle is ^tt, and that of two right angles is nr ; and we can now find the circular measure of an angle given in degrees, or vice versa ; if d be the number of degrees in an angle of which the circular measure is 6, we have d - = -— ■ , for each of these ratios expresses the ratio of the given IT 180 angle to two right angles ; thus j^ d is the circular measure of 180 an angle of d degrees, and 9 is the number of degrees in an angle whose circular measure is 6; if an angle is given in degrees, minutes, and seconds, as d° m n", its circular measure is (d + m/60 + n/3600) tt/ISO. The circular measure of 1° is -01745329..., of 1' is -0002908882..., and that of 1" is -000004848137 10. The circular measure of the angle AOP, subtended at the arc JLP centre of a circle by the arc AP, is equal to — ^. ;; — ; — ,- . for ■^ radius oi circle ' ^,. . . , ^ arc^P ^.AOP this ratio is equal to ^-^ or TT^n- ^ axe AB zAOB The arc AP may be greater than the whole circumference and may be measured either positively, or negatively, according to the direction in which it is measured from the starting point A, so that the circular measure of an angle of any magnitude is the measure of the arc which subtends the angle, divided by the radius of the circle. The length of an arc of a circle of radius r is rd, where 6 is the circular measure of the angle the arc subtends at the centre of the circle. The whole circumference of the circle is therefore 27rr. ' On the calculation of the value of the theoretical unit angle to a great number of places. Quarterly Journal, Vol. iv. ^ See Grunert's Archiv, Vol. i., 1841. THE MEASUREMENT OF ANGULAR MAGNITUDE 7 The length of a circular arc. 11. It has been assumed above that the length of ar circular arc is a definite conception, and that it is capable of numerical measurement; this matter will be now investigated. The primary notion of length is that of a linear interval, or finite portion of a straight line ; and the notion of the length of an arc of a curve, for example of a circular arc, must be regarded as derivative. That a given finite portion of a straight line has a length which can be represented by a definite rational or irrational number, dependent upon an assumed unit of length, will be here taken for granted. In order to define the length of a circular arc AB, we proceed as follows : Let a number of points of division A^, A^, ... ^,i_i of the arc J. 5 be assigned, and consider the unclosed polygon AA-,Ai...A^-^B\ the sum of the lengths of the sides AA-^ + A-^A^-V ... +^„_i£ of this polygon has a definite numerical value pi. Next let a new polygon AA^Ai ... An'-^B, where n' >n, be inscribed in the arc AB, the greatest side of this polygon being less than the greatest side of AA^A^ ...B; let the sum of the sides of this new unclosed polygon be p^. Proceeding further by successive subdivision of the arc AB, we obtain a sequence of inscribed unclosed polygons of which the lengths are denoted by the numbers p^, p^, ...pn, ■•• of a sequence which may be continued indefinitely. In case the number pn has a definite limit I, independent of the mode of the 8 THE MEASUREMENT OF ANGULAR MAGNITUDE successive sub-divisions of the arc AB, that mode being subject only to the condition that the greatest side of the unclosed polygon corresponding to p„ becomes indefinitely small as n i& indefinitely increased, then the arc AB is said to have the length I. In order to shew that a circular arc has a length, it is necessary to shew that this limit I exists, and this we proceed to do. It is clear from the definition that, if ABG be an arc, then if AB, EG have definite lengths, so also has AG; and that the length of ABG is the sum of the lengths of the arcs AB, BG. It will therefore be sufiicient to shew that an arc which is less than a semicircle has a definite length. In the first place we consider a particular sequence of polygons such that the corners of each polygon are also corners of all the subsequent polygons of the sequence. Denoting by Pj, Pa, ...P„, ... the lengths of these unclosed polygons, it can be shewn that P, AA^. Using this and the similar inequalities AiB + Se + ...+tcA > A^A^, we have by addition, and remembering that yA^, A^S, ... are all less than e/2r, pn+ e> A A 1 + .^1^2 + . . . + Ar-iB >l — e, there- fore pn> I— 2e, provided n = n. Next consider a polygon AAi'AiAg ... B, of the sequence whose lengths are Pj, Pj, ..., so far advanced that the greatest side is less than the least side of AoL^y ... kB, and also less than e/2s, where s is the number of sides in this latter polygon ; as before we see that Pn (11). tan (m + i TT + a) = + cot a J The angle tt - a is called the supplement of the angle a, and the angle ^tt— a is called the complement of a. We have shewn that the sine of an angle is equal to the sine of the supplementary angle, and the cosine of an angle is equal, with opposite sign, to the cosine of its supplement ; also that the sine of an angle is equal to the cosine of its complement, and the cosine of an angle is equal to the sine of its complement. 24 THE CIRCULAR FUNCTIONS The formulae (6) to (11) enable us to find the circular functions of an angle, when we know the values of the circular functions of that angle between zero and ^tt, which differs from the given angle by a multiple of ^tt, or also when we know the circular functions of the complement of this latter angle. Periodicity of the circular functions. 30. When a function /(») of a variable has the property f(^sc) =f{x + k) for every value of x, k being a constant, the function /(«) is called periodic; if moreover the quantity k is the least constant for which the function has this property, then k is called the period of the function. It follows at once that i{f(x) = f(x + k), then f(x)=f{x + nk), where n is any positive or negative integer; if then we know the values of the function for all values of a: lying between two values of x which differ by k, we know the values of the function for all other values of x, the function having values which are a mere repetition of its values in the interval for which they are given. The property (6), of sin a and cos a, shews that these functions are periodic functions of a, the period being 27r, or if the angle is measured in degrees, sin A and cos A are periodic functions of A, the period being 360°. The property (7) shews that these functions are such that their values, for values of the angle differing by half the complete period, are equal with opposite sign. The property (8) shews that the tangent is periodic, the complete period being tt, half the period of the sine and cosine. Obviously the period of the secant or of the cosecant is 27r, and that of the cotangent is tt. It will be hereafter seen that the circular functions derive their importance in analysis principally from their possession of this property of periodicity. Changes in the sign and magnitude of the circular functions. 31. We shall now trace the changes in the magnitude and sign of the circular functions of an angle, as the angle increases from zero to four right angles. (1) To trace the changes in the value of the sine of an angle, THE CIRCULAR FUNCTIONS 25 we must observe the changes in magnitude and sign of the projection ON, in the figure of Art. 18. When the angle A is zero, ON is zero, and as A increases up to 90°, ON is positive and increases until when A is 90°, ON is equal to OP, thus sin A is positive and increases from to 1. As A increases from 90° to 180°, ON is positive and diminishes until when A is 180° it is again zero, therefore sin A is positive and decreases from 1 to 0. As A increases from 180° to 270°, ON is negative and increases numerically, until when A is 270°, ON=-OP, hence sin 4 is negative and changes fi-om to - 1. As ^ increases from 270° to 360°, ON is negative and diminishes numerically, until when A is 360° it is again zero, thus sin A is negative and changes from - 1 to 0. (2) In the case of the cosine, we must observe the changes in magnitude and sign of the projection OM. We find that as A increases from 0° to 90°, cos A is positive and diminishes from 1 to 0; as -4 increases from 90° to 180°, cos J. is negative and changes from to — 1 ; as 4 increases from 180° to 270°, cos A is negative and changes from — 1 to 0; and as A increases from 270° to 360°, cos A is positive and increases from to 1. (3) To trace the changes in the tangent of an angle, we must consider the ratio of ON to OM ; when the angle is zero, this ratio is zero, and is positive and increasing as the angle increases from 0° to 90°; when the angle is 90°, the projection OM is zero, and ON is unity, hence tan 90° = oo ; a.s A increases from 90° to 180°, the tangent is negative and changes from — oo to 0. As, A increases from 180° to 270°, tan A is positive, since ON and OM are both negative, and it increases until it again becomes infinite when A = 270°. As A increases from 270° to 360°, the tangent is negative and changes from — oo to 0. It will be observed that tan A changes from + oo to — oo in passing through the value 90°, and from — oo to + oo in passing through 270°; to explain this, it is only necessary to remark that as a variable x changes sign by passing through the value zero, its reciprocal Ijx changes sign in passing through the value oo . (4) The changes in the values of the cosecant, secant, and cotangent of A may be deduced from the above, if we remember that they are the reciprocals of the sine, cosine, and tangent, respectively. Their values for ^=0°, 90°, 180°, 270°, 360° are 26 THE CIRCULAR FUNCTIONS given in the following table, which also includes the results obtained above for the sine, cosine, and tangent. sin cos tan cot sec cosec 0" 0°-90° + 1 + + + C0 + 1 + + ac + 90° 90°-180° 180° 180°-270° 270° 270°-360° 360° 1 + -1 - - -1 - + 1 ±00 - + + 00 - - + =c + - + 00 + 00 - -1 - + 00 + 1 1 + + C0 - -1 - + 00 Graphical representation of the circular functions. 32. In order to obtain a graphical representation of the changes in value of the circular functions, we shall suppose that the circular measure x of an angle is represented by taking a length X measured along a fixed straight line, according to any fixed scale, from a fixed point, and that the numerical value of the function to be represented is the length of a corresponding ordinate drawn perpendicularly to the given straight line, through the extremity of the length x ; the function is represented graphically by the curve traced out by the extremity of this ordinate. This curve is called the graph of the function. The first of the three figures opposite gives the graphs of sin x and of cos«. If is the origin from which the length x is measured along the fixed straight line OX, and OA = ir, OB = 27r, 00' = i7r, O'C = 1, the curve OPAP'B is such that any ordinate represents roughly the value of sin a; corresponding to any value of X between and 2-ir. If 0' is taken as origin, and O'B' = 27r, the curve G'PP'jy represents the value of cos x for values of x between and 27r ; this follows from the relation cos x = sin (^tt + x). Beyond OB, the curve OPP'B will be repeated indefinitely on both sides of the origin 0. The second figure represents, in a similar manner, the values of tan x and cot x, being the origin for tan x, and 0' for cot x ; the ordinates through 0', A', B' are asymptotes of the curve, where the functions change sign by passing-through an infinite value. The third figure represents the values of sec x THE CIRCULAR FUNCTIONS 27 and coseca;, being the origin for cosec x, and 0' for sec a;; the ordinates a,t 0, A, B are asymptotes of this curve. 0' B X 28 THE CIRCULAR FUNCTIONS Example. Draw graphs of the following functions (1) sinx + cosx. (2) cos {n sin x). cos {w cos x). (3) tanx+secx. (4) sin(ircosx)/eos(7rsinx). (5) sin^x-2cosx. (6) siii (Jtt + Jn- cos x). Angles with one circular function the same. 33. We shall now find expressions for all the angles which have one of their circular functions the same. (1) If in the figure, AOP is a given angle, and PP^ is drawn parallel to OA, the angles {OA, OP) and {OA, OPi) are the only angles which have their sine the same as that of AOP, for they are the only angles for which the projection of the radius on OB is equal to ON; these angles are 2mr +'« and 2mr + tt — a, where a is the circular measure of AOP, and n is any integer; they are both included in the expression m7r + (—!)'"«, where m is any positive or negative integer ; this is therefore the expression for all the angles whose sine is the same as that of a. (2) Next draw PP^ parallel to OB, then the angles (OA, OP) and (OA, OP.^) are the only angles which have the same cosine as a, for they are the only angles for which the projection of OP on OA is equal to OM; they are both included in the formula 2m7r + a, where m is any positive or negative integer. THE CIRCULAR FUNCTIONS 29 (3) If PO is produced to P3, the angles {OA, OP), (OA, OP,) are the only ones which have the same tangent as a ; these angles are respectively 2«.7r + a and 2»i7r + tt + a, and are therefore both included in the formula mir + a, where m is any positive or negative integer. (4) Since angles which have the same cosecant have also the same sine, we see that rmr + (—!)"'« includes all the angles whose cosecant is the same as that of a ; also 2rmr ± a includes all angles whose secant is the same as that of a, and ottt + a includes all angles whose cotangent is the same as that of «. In every case zero is iacluded as one value of m or n. Determination of the circular functions of certain angles. 34. The values of the circular functions of a few important angles can be obtained by simple geometrical means. (1) The angle 45" or ^tt is an acute angle in a right-angled isosceles triangle, the sine and cosine of this angle are therefore obviously equal to one another ; and since the sum of their squares is unity, each of them is equal to 1/V2 ; the tangent of the angle is therefore unity. (2) Each of the angles of an equilateral triangle is 60° or Jtt. Let ABC be such a triangle ; draw AB perpendicular to BG, then the cosine of the angle -^ is -j^, and this is equal to ^ ; the sine of the same angle is Vl-i = |\/3. The complement of 60° 30 THE CIRCULAK FUNCTIONS is 30° or ^TT, hence we have cos 30° = | V3, and sin 30° = f We have also tan 60° = -v/3, and tan 30° = l/\/3. (3) Draw AE bisecting the angle DAB, then the angle DAE is 15° or ■^TT. We have by Euclid, Book vi. Prop. in. DE_DA_^ ,„ therefore and thence ^^^ or DA From this we obtain sin 15° = EB DE DB AB V3 ^ 2 + V3 ' V3 tan 15° is equal to ,„ ,„ . .„. or 2 — \/3 V6-\/2 cos 15° = \/6 + V2 4 ' 4 ■ We can, from these values, obtain the sine, cosine, and tangent of 75° or ^TT, the complementary angle. If we proceeded in the same way, bisecting the angle DAE, we should obtain the tangent of 7° 30' or ^TT, and we might continue the process so as to obtain IT the tangent of all angles of the form , where ^ is a positive integer, but we shall hereafter obtain formulae by which the functions of these angles may be successively calculated, thus obviating the necessity of continuing the geometrical process. By a similar geometrical method we might obtain the circular functions of the angles of the form 7r/2*'. (4) Let ABC be a triangle in which each of the base angles is double of the vertical angle A ; the base angles are each 72°, or A THE CIRCULAR FUNCTIONS 31 f-TT, and the vertical angle is 36°, or ^tt. If AB is divided at D so that AB . BD = AD^, then it is shewn in Euclid, Book iv. Prop. x. that AD = DG=GB. Draw AE perpendicular to BC. Denoting the ratio of AB to ^S by x, we have \—x = x^, and solving this quadratic, we find a; = ^(+ VS - 1); we must take the positive AD — root, hence ^-g = ^ (VS — 1), thus BC cos72° = sinl8° = iJ| = i(V5-l); from this we obtain sin 72° = cos 18° = ^ VlO + 2 V5. AG Also cos 36° = ^ -j-j^ , since DAG is an isosceles triangle, therefore cos 36° = J (V5 + 1), hence sin 36° = J VlO^^YvS. Since 54° is the complement of 36°, we have therefore the values of sin 54° and cos 54°. In the following table the values we have obtained are collected for reference. The functions in the first line refer to the angles in the first column, and the functions in the last line to the angles in the last column. sine cosine tangent cotangent iV7r = 15° 4 \/6 + V2 4 2-V3 2+V3 A7r = 75° \/5-l 4 V10 + 2V5 4 T^T=18° 1V25-10V5 V5+2V5 fT = 72° ln = ZQ° i W3 1 V3 V3 J7r = 60° Vs + i 4 VlO -2^5 4 ^^ = 36° \/5 - 2 V5 iV25 + 10V5 ftT = 54° J7r = 45° 1 x/2 1 V2 1 1 ^77 = 45° cosine sine cotangent tangent We can find at once the circular functions of any angle which differs from any one of those in the table by a multiple of a right angle, by employing the formulae (6) to (11). 32 THE CIRCOLAR FUNCTIONS Example. Find the sine and cosine of 120°, and of — 576°. We have 120° = 90° + 30°, hence sml20° = cos30°=iV3, cos 120° = -sin 30°= -^. Again - 576°= - (3 . 180° + 36°), therefore sin(-576°)=sin( + 180°-36°) = sin36°, also cos - 576° = cos (180° - 36°) = - cos 36°. The inverse circular functions. 35. If y is a function /(a) of x, then x may also be regarded as a function of y ; this function of y is called the inverse function of /(«), and is usually denoted by f~^ {y) ; thus x =f~^ (y). If /(«) is a periodic function, of period k, so that/(a;)=/(a; +m^), where m is any positive or negative integer, the function f~^ (y) will have an infinite number of values given by a; + mk, where x is any one value of /"' (y) ; such a function of y is called multiple- valued, since it has not a single value for each value of the variable y. We see therefore that, corresponding to a periodic function f(x) = y, there is a multiple-valued inverse function f~'(y) which has an infinite number of values for any one valioe of y, these values differing by multiples of the period of f (x). 36. If there are two or more values of x, lying between and k, for which /(«) has equal values, the multiplicity of values of f-^ (y) is still further increased, since it will have each of the values of x for which f{x) = y, and the infinite series of values obtained by adding multiples of k to each of these. For example, suppose that there are two values Xj, x^, each lying between and k, for which /(«) = ^, then the inverse function /^^ (y) has the two sets of values x^ + mk, x^ + nk. 37. In the case of the circular function sin x = y, the values ■ of the inverse function sin~'y are mr +(— l)"a!i, where «, is any value of X for which sin a;, = y ; in this case the complete period of sin X is 27r, and there are two values of x, say Xi and ■n- — a^, lying between and 2^, for which sin a; = y ; thus the values of sin^'y are the two series of values w.27r+a^ and ?i.27r + tt — ajj, both included in n7r+(— !)"«,. In a similar manner, we see that the values of cos~' y are in- cluded in 2M7r + x, where cos x = y. The periods of the functions tana;, cot a; are tt, only half EXAMPLES. CHAPTER III 33 those of sin x and cos x, and there is only one value of x between and IT for which tan x or cot x has any given value ; thus tan"' y has the values nir + x^, and cot-^j/ the values 71-^ + x^, where x-^ is that value of x between and tt, such that tan x-^ or cot ^i is equal toy. 38. The numerically smallest quantity x which has the same sign as y, and is such that sin a; = t/, is called the Principal Value of sin-^2/; a similar definition applies to the principal values of tan~'y, cot"'^2'> cosec"'^/. The numerically smallest positive value of x which is such that cos a; = y is called the Principal Value of cos~^ y ; a similar definition applies to sec~iy. Thus the principal values of swr^y, tan^^y, cot-'^y, cosec^^/ lie between the values +^7r, and the principal values of cos'^y, sec~^y lie between and tt. In some works, the principal values of sin~' y, cos^^ y, tan~^ y are denoted by Sin~^ y, Cos~' y, Tan~' y ; the general values are then given by sin~^2/=)i7r+(— l)"Sin~^t/,cos""'2/=2w7r+Cos~'2/,tan~'2/=W7r+Tan~^2/;. we shall however not use this notation. It must be remembered that in many equations connecting these inverse functions it is necessary to suppose that the functions have their principal values, or at all events that the choice of values is restricted. For example, in such an equation as sin~' y + cos~^ y = ^ir, the choice of values of the inverse functions is restricted. It should moreover be noticed that the functions cos~^ y, sin~^ y have only been defined for values of y lying between + I ; beyond those limits of y, the functions have no meaning, so far as they have been at present defined. The student should draw, as an exercise, graphs of the various inverse circular functions. In Continental works, the notation arc sin x, arc cos x, arc tan x is used for sin"^ x, cos~^ x, tan"^ x. EXAMPLES ON CHAPTER III. 1. Prove the identities (i) tan^(l-cotM) + cot^(l-tanM)=0, (ii) (sin^ + sec4)2 + (cos.4 + cosec^)2=(l + seo4cosec4)2. 2 The sine of an angle is — = ^ ; find the other circular functions. ° m^ + n' H. T. 3 34 » EXAMPLES. CHAPTER III 3. If ta,nA+8mA—m, ta,a A - ain A = n, prove that m^ — m^ = 4 ^/mn. 4. Having given -^ — 5=p, 5~?' ^P*^ ^^^ ^ ^^^ ^^^ ^' 5. If ^ = V2, ^^=V3, ^nd^andA smB ^ ' tan 5 6. If cosA= tan 5, cos £ = tan C, cos C= tan 4 , prove that sin 4 = sin 5 = sin C= 2 sin 18°. 7. Solve the equations : (i) sinfl+2cos5=l, .... C0Sa_3 (iii) V3 coseo^ fl = 4 cot 6. 8. Solve the equations : COS (2.j?+y)=sin(.» — 2^)"! COS (a; + 2y)=sin (Zx—y)) ' 9. Find a general expression for 6, when sin2fl=sin*a, and also when sinfl= -003^ = 1/^2. 10. Find the general values of the limits between which A lies, when ain^A is greater than cos^A. 11. Find the general value of 6, when 9 sec*fl=16. 12. If tan(7rcot6) = cot(7r tanfl), then tan5=i{2?i + l±\/4?iH4m-15}, where n is any integer which does not lie between 1 and — 2. 13. Give geometrical constructions for dividing a given angle into two parts, so that (1) the sines, (2) the tangents of the two parts may be in a given ratio. 14. Construct the angle whose tangent is 3 - ^J2. 15. Divide a given angle into two parts the sum of whose cosines may be a given quantity c. What are the greatest and least values c can have ? 16. If M„=cos"fl+sin''5, prove that 2«j - 3% + 1 = 0, &iiio—l5us + lOug-l=0. 17. Two circles of radii a, b touch each other externally ; 6 is the angle contained by the common tangents to these circles, prove that . . 4(a~b)/^ab sm5 = — ^- ~ — . (a + 6)2 EXAMPLES. CHAPTER III 35 18. A pyramid has for base a square of side a; its vertex lies on a line through the middle point of the base, perpendicular to it, and at a distance A from it ; prove that the angle a between two lateral faces is given by 19. Two planes intersect at right angjes in a line AB, and a third plane cuts them in lines AD, AC; if the angles DAB, CAB be denoted by a, /3 respectively, prove that the angle BA makes with the plane CAD is tan a tan /3 tan" V'tan^a + tau'''/3 20. Shew that, if OD be the diagonal of a rectangular parallelepiped, the cosines of the angles between OD and the diagonals of the face of which OA, OB are adjacent sides are respectively AB OA^~om OD ^"° OD.AB ■ 21. Two circles, the sum of whose radii is a, are placed in the same plane, with their centres at a distance 2fls, and an endless string, quite stretched, partly surrounds the circles, and crosses itself between them. Shew that the length of the string is {^n + ^^JZ)a. 22. Prove that 3tan-ismcot ^=(-2X0) ■ 23. Illustrate graphically the change in sign and magnitude of the func- tions 3 sin .»+ 4 cos x, e' sin as, and sin f -^ sin a; j for all values of x. Shew that the equation 2^=(2?i+l) ir vers x, where m is a positive integer, has 2n + 3 real roots and no more, roughly indicating their localities. 3—2 CHAPTEE IV. THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES. The addition and subtraction forTnulae for the sine and cosine. 39. We shall now find expressions for the circular functions of the sum and of the difference of two angles, in terms of the circular functions of those angles. Suppose an angle AOB of any magnitude A, positive or negative, to be generated by a straight line revolving round from the initial position OA, our usual convention being made as to the sign of the angle, and suppose further that an angle BOG of any magnitude B is described by a line revolving from the initial position OB; then the angle AOG is equal to A+B. In 00 take a point P, and draw PN perpendicular to OB. According to the convention in Art. 15, the straight line ON is positive or negative according as it is in OB, or in OB produced ; , also NP is positive when it is on the positive side of OB, revolving counter-clockwise, and negative when on the other side. The positive direction of the straight line on which NP lies makes an angle ^+90° with OA. We have ON = OP cos B, and NP = OP sin B; for ON and NP are the projections of OP on OB and on the line which makes an angle A + 90° with OA . In fig. (1), each of the angles A, B is positive and less than 90° ; in fig. (2), the angle A lies between 90° and 180°, and the angle B also lies between 90° and 180° ; in fig. (3), the angle A lies between 180° and 270°, and the angle B is negative and lies between - 90° and - 180°. In figs. (1) and (2), NP is of positive length, and in fig. (8), NP is of negative length, since, in the last THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES 37 38 THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES case, PN is the direction of a line making an angle A + 90° with OA. By the fiindamental theorem in projections, given in Art. 17, the projection of OP on OA is equal to the sum of the projections of OiVandi\^PonOJ,or OP cos (A+B) = ON cos A+NP cos (A + 90°) = OP cos ^ cos B + OP sin B cos (A + 90°), therefore cos {A+B) = cosAcosB — sirxAsmB (1). If, instead of projecting the sides of the triangle ONP on OA, we project them on a line making an angle + 90° with OA, we have OP sin (A+B)= ON sin A + NP sin (A + 90°) = OP sin AcosB+ OP sin {A + 90°) sin B, therefore sin {A+B) = sin A cos B + cos .4 sin 5 (2). The formulae (1) and (2) have thus been proved for angles of all magnitudes, both positive and negative. The student should draw the figure, for various magnitudes of the angles A and B, in order to convince himself of the generality of the proof. If we change B into —B,m each of the formulae (1) and (2), we have cos (A—B)= cos A cos (— B) — sin A sin (— B) and sin (A—B) = sin A cos (— B) — cos A sin (— B), hence cos{A-B) = cosAcosB + smAsmB (3), and sin(A—B) = sinAcosB — cosAsmB (4). These formulae (3) and (4) would of course be obtained directly, by describing the angle B in the figure in the negative direction, so that the angle POA would be equal to A— B. 40. The formulae (1), (2), and (3), (4), are called the addition and subtraction formulae respectively; either of the formulae (1) and (2) may be at once deduced from the other; in (1) write A + 90° for A, we have then cos (90° + ^ + B) = cos (90° + A) cos B - sin (90° + A) sin B or — sin {A +B) = — sin AcosB — cos As\nB; and changing the signs on both sides of this equation, we have the formula (2) ; in the same way, by writing A + 90° for A in (2), we should obtain (1). It appears then that all these four fundamental formulae are really contained in any one of them. THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES 39 41. The proof of the addition and subtraction formulae, given by Cauohy, is as follows ;— With as centre describe a circle, and let the radii OP, OQ make angles A, B, respectively, with OA ; join P§, and draw PM, QN per- pendicular to OA, and QR parallel to OA, then we have PQ^=QB?-irRP^ = {0N- OMf+{PM- QNf = 0^2 {(cog s - cos ^)2+{sin A - sin Bf} = 20A^(l-cosA cos B-ainA sin B). Let FS be drawn perpendicular to the diameter §§', then PQ^=QS, qq = 20A{0A-0S) = 2042{i_cos(4-2J)}, therefore cos(j4 — 5)=cos^ cos 5+ sin 4 sin 5 (3). The other formulae may then be deduced ; (1) by changing B into —B, (2) by changing B into 90° -5, (4) by changing B into 90° + 5. 42. Besides the two proofs which we have given of the fundamental addition and subtraction formulae, both of which are perfectly general, various other proofs have been given, some of which are in the first instance only applicable to angled between a limited range of values, and require extension in the cases of angles whose magnitudes are beyond that range. We shall make this extension in the case in which the formulae have been first proved for values of A and B between 0° and 90°. Whatever A and B are, it is always possible to find angles A' and B', lying 40 THE CIKCULAR FCNCTIOXS OF TWO OR MORE ANGLES between 0° and 90°, such that A=m.90'' + A', B = n.90° + B', where m and n are positive or negative integers ; we have then cos {A+B) = cos (»ir+^ 90° + A' + B') ; (1) if m and n are both even, we have ' m+n COS (^ + 5) = (- 1) 2 cos{A' + B') m+n = (-1) ^ (cos 4' COS 5' - sin ^' sin jB'), now cos A={—\)^ COS A', sin A={—\)'^ sin -d', with similar formulae for B, hence cos {A+B) = cos A cos 5 — sin ^ sin B ; (2) if m and n are both odd, we have cosil=(-l)"^cos(90° + 4') = (-l) ' sin J.', 7M— 1 7ft — 1 sin4=(-l)^~sin(90°+^')=(-l) ^ cos 4', with similar formulae for B; hence as before we obtain, by sub- stituting the values of cos A', cos B', sin A', sin B, the formula for cos {A +B); (3) if m is odd and n is even, cos {A+B) = (- 1) 2 cos (90° + A' + F) 9ft+»+l = (-1)'"^^ sin (^' + 5') m+«+l = (- 1) 2 (sin ^ ' cos B + cos A' sin £'). m+1 m now cos ^ = (— 1) '^ sin A\ cos B = (~ 1)^ cos 5', m-l » sin JL =(-l)"2"cos^', sin5 = (-l)2sinB'; hence, substituting as before, we have the formula for cos (A + B). The other formulae may be extended in the same manner. 43. The form in which the addition formulae were known in the Greek Trigonometry' is Ptolemy's theorem given in Euclid, Bk. vi. Prop, d; this theorem is, that if A BCD be a quadrilateral in- scribed in a circle, AB.CD + AD.BG = AG.BD. Any chord AB is the sine of half the angle which AB subtends at the centre of the circle, the diameter of the circle being taken as unity, and 1 See the Article "Ptolemy" in the Encyclopaedia Britannica, ninth Edition. THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES 41 this half angle is the angle subtended by the arc AB at the cir- cumference. We shall shew that the formulae for sin (a + /8) and cos (a + /3) are contained in Ptolemy's theorem. (1) Let BI) be a diameter of the circle, and ABB = a, BBG = ^; then ABD = ^ir-a, DBG = ^ir - /3, AC = sm(a + ^), AB = sin a, CZ) = cos/S; thus the theorem is equivalent to the formula ■ / r.x • • ' sm {a. + p) = sin a cos p + cos a sin (8. (2) Let CD be a diameter of the circle, and BCD = a,AGD = /3, thus AB = sin (a — /3), and the theorem is equivalent to sin (a — yS) + sin /8 cos a = cos /S sin a. (3) Let BD be a diameter of the circle, and ADB = a, GBD = ^, then ADG = \iT-irOL-^, thus ^C=cos(a-/3), and the theorem is equivalent to cos (a — ;S) = cos a cos /3 + sin a sin /3. (4) Let CD be a diameter of the circle, and BGD = a, AI)G = /3; then BCA=a + ^-^7r, AB = - cos (a + /3), and the theorem is equivalent to — cos (a + j8) + cos a. cos /3 = sin a sin 0. Example. Em/ploy Ptolemy's theorem to prove the following theorems : sin a sin — 7) + sin /3 sin {y-a) + sin y dn (a - 0) = 0, sin (a+/3) sin {^■^y)=sin a sin y+ sin ^ sin (0+^+7). Formulae for the addition or subtraction of two sines or two cosines. 44. We obtain at once from the addition and subtraction formulae ■ , . n\ • / ^ t,\ c, ■ a d sm (A+B) + sin (A-B) = 2 sin A cos B, sin (A+B)- sin {A-B) = 2 cos ^ sin 5, cos (A+B) + cos (J. — 5)= 2 cos A cos 5, cos (il - 5) - cos {A+B) = 2sinA sin 5, let u4-|-5=C, A-B = B; we obtain then, since A=^{C + D), B = \{C-D), the formulae sin(7 + sini)=2sini(G + i))cosi(C'-I>) (5), sinC-sinD = 2cosi(C + i))sinJ(Cf--D) (6), cos + cosi) = 2 cos HC + -0) cos k{G-D) (7), cosD-cos(7 = 2sinHC + ^)sinHC'--0) (»)■ 42 THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES These important formulae (5), (6), (7), (8) are the expressions for the sum or difiference of the sines or of the cosines of two angles as products of two circular functions; they may be ex- pressed in words as follows : The sum of the sines of two angles is equal to twice the product of the sine of half the sum and the cosine of half the difference of the angles. The difference of the sines of two angles is equal to twice the product of the cosine of half the sum and the sine of half the difference of the angles. The sum of the cosines of two angles is equal to twice the product of the cosine of half the sum and the cosine of half the difference of the angles. The difference of the cosines of two angles is equal to twice the product of the sine of half the sum and the sine of half the reversed difference of the angles. 45. These formulae may be proved geometrically by the method of projections. Let BOA = G, GOA=D, and let OB =00; draw ON per- pendicular to BC, then N is the middle point of BG, also N-OA = i(G + I)), NOB = NOG = \{G-D). The sum of the projections of OB, OG, on OA, is equal to the sum of the projections of ON, NB, ON, NG, on OA, and, since the projections of NB and NG are equal with opposite sign, this is equal to twice the projection of OiV^; therefore OB cos G + OG COS B = 20N cos ^ (C + B), and since ON = OB cos ^ (C - D), THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES 43 we have the formula cosC+cosi) = 2cosi(C + i))cos|(G-D) (7). If instead of projecting on OA we project on a straight line perpendicular to OA, we have 0£sin C+ 0(7sinD = 20Fsini((7 + i)), hence sin(7 + sinD = 2sin^(a + D)cos J(G-i)) (5). Also the projection of OG on OA is equal to the projection of OB, together with twice the projection of BN, or 00 cos D = 05 cos C + WN sin i (C + D), hence cosi)-cos(7= 2sini(0+Z))sini(0-D) (8), and if we project on the line perpendicular to OA, we have 00 sin Z> = 05 sin (7 - 25iVcos i (0 + D) or sinO-sinD = 2sini((7-D)cos^(C' + i)) (6). A curious method of multiplying numbers, by means of tables of sines, was in use for about a century before the invention of logarithms. This method depended on a use of the formula sin 4 sin jB = ^ {cos (4 -5) -cos (^ + 5)} ; the angles A and B, whose sines, omitting the decimal point, are equal to the numbers to be multiplied, can be found from a table of sines, and then cos {A-\-B), cos {A — B) can be found from the same table ; half the difterence of these last gives the required product. This method was called irpoa-6a\ cot A cot 5+1 ,,„. cot{A-B)= — -^ ^ (12). ' cot ii - cot ^ ^ ' The formulae (9), (10), (11), (12) are the addition and sub- traction formulae for the tangent and cotangent. Various formulae. 47. The following formulae may be deduced from the for- mulae which we have obtained for two angles, and are frequently useful in effecting transformations. The student should verify each of them. sin (A + B) sin {A-B) = sin^ A - sin^ B = cos^ B - cos'^ A . . .(13), cos (A + B) cos (A-B) = cos^ A - sin^ B = cos°- B - sin^ ^ . . .(14), sin (A + B) cos {A — B) = sin J. cos ^ + sin 5 cos 5 (15), cos {A + B) sin {A - B) = sm A cos A — sm B aos B (16), sin {A+B) _ tan A + tan B _ si'n{ . ,, ^s sin {A - B) " tan ^- tan £ ..^...-..^^^.....(10, cos (A + B) 1 - tan A tan B _ S>X(i/::,4iP:) /,o\ cos {A -B) ~ 1 + tan 4 tan B "' '"7 .' ',K'*-'s>wi 'i _ sin (A + B) ,., ., tan^ +tan5 = \ " ^ (19). cos A cos B From the formulae for the addition and subtraction of two sines or cosines we obtain at once sin A + smB t an \{A + B) sinX^=^sin]B'~tan|(^-ii) ^ '' sin A + sin ^ , i / ^ , d\ /oi \ cos .4 + COS jD sin J. ± sin .B 4. i / j t n\ c9-)\ =-^^ ^ =coti(^ + B) (2^), COS .B — cos -A cos ^+ cos ^ ^ ^^^ i (^ + iJ) cot \{A-B).. .(23). cos iJ — cos A 46 THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES Examples. (1) Prove the identity 1 — cos^ A — cos^ B — cos^ C + 2 cos A cos B cos C = 4sm^(A+B + C)«mi(-A + B + C)siMi(A-B + C)smi(A+B-C). The expression on the left-hand side may be written - cos2 A - cos (B+ G) cos {B - C) + cos A {cos {B+ C) +cos {B - C)}, which is equal to {cos A — cos {B + C)} {cos (B — C) — cos A} ; then, splitting each of these factors into two factors, we obtain the expression on the right-hand side. If ±.4+5+Cisa multiple of 2ir, then 1 — cos^ A — cos^ B — cos^ C+2 cos A cos B cos C is zero ; this result is sometimes useful. (2) Ffove that 1 - cos'' A — cos'' B — co^ C — 2 cos A cos B cos C = -4cos^(A-t-B-l-C)cosi(-A-f-B-l-C)cosH-A--B-t-C)cosi(A-|-B-C). This may be deduced from (1), or proved independently. (3) Prove that i/ A + B -t- C = »nr, sira 2 A -(- Sim 2B -H sm 20 = ( — 1 )""" 1 4 SMi A si» B s m C . We have sin 24 -1-sin 25-1- sin 2C=2 sin A cos y1 -}-2 sin (mtt — 4) cos {B—C) = 2 sin -4 {( - 1)» cos (5-f C) - ( - 1)" cos (.8 - CO} =( — l)"-! 4 sin A sin 5 sin C. (4) Prove that, under the same supposition as in Ex. (3), 1 ■'rcos 2A-I-C0S 2B-)- cos 2C = ( — 1)" 4 cos A cos B cos C. Prove the identities (5) stra 3A =4 sin A dn (60° -F A) sm (60° - A). (6) cos 3 A = 4 cos A cos (60° -H A) cos (60° - A). (7) si»iA-|-simB-hsiwC-si>i(A-f-B-HC) = 4 sire ^ (B -t- C) sira i (C -h A) «m J (A-f- B). (8) cos A -I- cos 5 -H cos C-f- cos (A H-B -1-0) = 4 cos ^ (B + 0) cos ^ (0 -1- A) cos i ( A -t- B). (9) 2 sim 2 A sin^ (B -t- 0) - sin 2 A sin 2B sire 20 = 2 sm (B -I- 0) »i» (0 -)- A) S191 ( A -H B). (10) 2 cos 2A cos'' (B -t- 0) - cos 2 A cos 2B cos 20 = 2 cos (B -H 0) cos (0 -f A) cos ( A -t- B). (11) 2sim2 As!ira(B-|-0 — A) — 2smA«iraBsimO = Sim (B -I- - A) «m (0 -I- A - B) SIM ( A -I- B - 0). (12) 2cos2Aco«(B-f-C-A)-2eosAco«BcosO = cos (B -I- - A) cos (0 -I- A - B) cos ( A -I- B - 0). THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES 47 (9) and (10) correspond to the algebraical identity 2 2a(6 + c)2-8ci6c = 2(6 + c)(o + a)(cH-6); (11) and (12) to the identity 2a2(6+c-a)-2a6c = (6 + c-a)(c + a-6)(a + 6-c), Addition formulae for three angles. 48. From the addition formulae (1) and (2) we may deduce formulae for the circular functions of the sum of three angles in terms of functions of those angles; we have sm(A+B+G) = sin (A + B) cos C + cos {A + B) sin G = (sva. Acqs B + cos Asia. B)cosG ■\- (cos 4 cos 5 — sin J. sin £) sin (7, and cx>^{A+B + G) = cos {A + B) cos C- sin {A + B) sin G = (cos A cos B — sinA sin B) cos C— (sin^ cos 5 + cos ^ sin 5) sin (7, hence we have sm{A+B + G) = sva.A cos -B cos + sin -B cos (7 cos J. + sin (7 cos ^ cos 5 - sin 4 sin 5 sin G (24), cos{A+B+G) = cos.4 cos5cos G—cosA sin £ sin (7 — cos 5 sin (7 sin .4 — cos (7 sin -4 sin 5 (25). The formulae (24), (25) may be written in the form sin{A+B+G) = cos A cos B cos G (tan A + tan B + tan C — tan A tan B tan G), cos{A+B-\-G) = cos^ cos £ cos 0(1 — tan £ tan (7— tan (7 tan 4. —tan .4 tan 5); hence by division we have the formula ta.n{A+B + G) tan A + tan B + tan G — tan A tan B tan C 1 — tan B tan G — tan G tan A — tan A tan B We might obtain in a similar manner the formula cot{A+B + G) cot A cot B cot G— cot A — cot B — cot (7 .(26). cot 5 cot (7 + cot (7cot ^ + cot J. cot £ - 1 .(27). 48 THE CIECULAE FUNCTIONS OF TWO OR MORE ANGLES Examples. (1 ) Prove that tan (45° + A) - tan (45° - A)= 2 tan 2A. (2) Prove tliut if A + B + C = tott, tanA. + tan B+tan C — tore A tan B tanG = ; andif A + B + C = (2to + 1)|, taJiB toJiC + tonC tore A+toM AtojiB = l ; and state the corresponding theorems for the cotangents. Addition forimulae for any number of angles. 49. It is obvious that we might now obtain formulae for the circular functions of the sum of four angles, then of five angles, and so on ; we shall prove by induction that the formulae for the sine and the cosine of the sum of n angles .4,, A2...An are sin{A^ + A^-\- ... + An) = S^- S^ + S,- (28), cos(4i+A + .-.+^«) = -80-/82 + .84- (29), where Sr denotes the sum of the products of the sines of r of the angles and the cosines of the remaining n — r angles, the r angles being chosen from the n angles in every possible way, thus /So = cos ^1 cos A^... cos An /Si = sin.4iCos-(l2 ... cos-4„+ cos.4isin-4aCos J.3 ... cos An + .... The formulae (28), (29) agree with the formulae (1), (2), and (24), (25), for the cases w = 2, w = 3 ; assuming the formulae to hold for n angles, we shall shew that they hold for n+\ angles ; we have sin (^1 + J.2 + . . . + 4» + An+i) = sin (^1 + . . . + ^„) cos An+i + cos (4i + . . . + ^ „) sin An+^ = cos An+1 (/Si - /Ss + /Sfj . . .) + sin An+i (/So -8^ + 8^..), now let 8,.' denote the sum of the products of the sines of r of the angles A^, A^-.-An+i, and of the cosines of the remaining w + 1 — r- angles, the r angles being chosen from the n + 1 in every possible way, then we have Si = /Si cos An+i + So sin An+i , for in 8iCosAn+i there is in each term the sine of one of the angles A^, A^... An, and in each term of /Sosin-4,i_,,i there is only sin^„+i. THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES 49 Similarly S,' = S3 cos An+i + Si sin An+i Sf = Ss cos An+i + St sin An+i hence sin (^i + . . . + A^+i) = Si -Sa' + S^' .... We may similarly shew that cos(^, + ... + A„+i) = S„' - (SV + (S; ..., thus if the formulae (28), (29) hold for n angles, they also hold for n + 1; and they have been shewn to hold for re = 2, 3, hence they are true generally. These formulae may be written in the form sin (Ai + Ai+ ... +.4„) = cos^iCos^2... cos An {ti-ta + ts... ), cos {Ai + A^ + ... + An) = cos Ai cos A^... cos Anil — t^ + ti...), where t,. denotes the sum of the products of tan Ai, tan A^... tan An, taken r together ; hence by division we have Un{A, + A, + ... + An) = ^^^^^^ (30), which is the formula for the tangent of the sum of n angles, in terms of the tangents of those angles. The formula (30) may also be proved independently. Assuming it to hold for n angles, we shall prove that it holds for ?i + 1 ; we have -ta n(^i+.l2+ ... +^„)+tan^„.n {h-h+h- ...) + ta,nAn + iO--h+k+ ■■• ) ~'(l-<2 + <4-...)-tan^„ + i(^ tan A ± tan B (1) To prove the formulae tan(.i±^)= ^_^^^^^^^^ . £ C 56 THE CIRCULAE FUNCTIONS OF TWO OE MOEE ANGLES Let AB, CD be two chords of a circle at right angles, and let the angles AD£!, BDEhe denoted by 4 and B; since AE. EB^CE. ED, we have AE+EB ED AE±EB AB , AE EB~ ED+EG~ BF' whence ED' ED tan A + tan B =\&q{A±B). 1 + tan A tan B (2) To prove the formulae sin 2^ = 2 sin il cos A, cos 24 = cos^ A - sin^ A . Let AOA' be the diameter of a circle, and let FAA'=A, then POA'=iA ; draw PN perpendicular to AA'. PN Then sin 24=^ , now PN. AA'=2AAPA'=AP. PA', X,. ^ . „, AP.A'P AA'^BinAcosA ... . therefore sm 24 = -p^^ — j-j-, = 7775 — ttt = 2 sin 4 cos A, also cos 24 = :=ry; = OP. AA' OP. AA' ON Am-A'N^ AP'^-A'P^ AA'^ =008^4— sin^ 4. OP 2.AA'.0P (3) To prove the formulae sin 34 = 3 sin 4 - 4sin'4, cos 34 =4 cos' 4 — 3 cos 4. Let CAB=AGB=A ; let AB meet the tangent at G to the circle round the triangle ABG in E; draw BD perpendicular to CE. THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES 57 The angle BED is. ZA, or 180° - ZA . Now AE £!.ACE _AC^ BE~ hBGE" BO^ = 4 cos^ A ; therefore hence and A n 9^=4oosM-l=3-4sinM; . „ , BD BD AB „ . . . ■ ■, . sm34=-g^=^.^=.3sm4-4sm3^, „._-DE_DC _EC^_DG BC _AC cos 3^-+^^-^^ Qg-BCBE AB = cos 4 (4 oos2 4-l)-2cos4=4 cos^ A-ZooaA. The proofs in (1) and (3) were given by Mr Hart in the Messenger of Mathematics, Vol. iv. Examples. Prove geometrically the formulae o A 1 — co«2A (1) tarflA.=^-, XT. ^ ' l + co«2A (2) tan (45° + A) - tan (45° - A) = 2 tan 2 A. (3) dn A «■» B = si»i2 J (A + B) - MJi^ i (A - B). (4) sin^ a + dn^ |3 = dn^ (a + j3) - 2 sin a dn ficos(a + ^). , m , m — n jr (5) toJJ-i tan~^ — ; — = t. ^ ' n m + n 4 (6) oos^A+aos^'B + cos^C+2cosAcos'BcosG=l, where A + 'B+G = 180°. (7) sinA+sinB-dnC = 4sin^Asinj^'BcosiC, where A + B + C = 180°. (8) cotd=cosec2d+cot2d. (9) cos36°-dnl8° = i. 58 EXAMPLES. CHAPTER IV EXAMPLES ON CHAPTER IV. Prove the identities in Examples 1 — 15: 1. cosM+oos2(120°+J) + cos2(120°-J)=f. 2. (cos^+sin4)* + (cos^— sin J)*=3 — cos4il. 3. sin 3A sin' A +oos 3^ cos' A =oos' 2A. 4. 4 cos' j1 sin 3^ + 4 sin' ^1 cos 3.4 = 3 sin 4 J . 5. sin' A +sin' (120° + .4) - sin' (120° -.4)= - 1 sin 3.4. _ sin 4 + sin 3A + sin 5 A + sin 74 , 6. 1 TT-i p-; ^^. = tan 4.4. cos A + cos SA + cos 5 A + cos TA 7. 16 cos5 4 - cos 54 =5 cos 4 (1 + 2 cos 24). 8. cosec {m+n)a; cosec mx oosec no; — cot {m+n) x cot »ia;cot tisc =cot ma; +cot nx— cot {m + n)x. 9. 2 cos A (cos 35 - cos 3C) = 4 (cos 5 - cos G) (cos C— cos A ) (cos 4 — cos B) (cos 4 + cos B + cos C). 10. 2sin4(sin2 5+sin2C)sin(5-C) =sin(5-C)sin(C-4)sin(4-£)sin(4+5+C). 11. tan(4 + 60°)tan(4-60°)+tan4tan(4+60°)+tan(4-e0°)tan4 = -3. 12. cot (4+60°) cot (4 - 60°) +oot 4 cot (4 + 60°) +cot (4 - 60°) cot 4 = - 3. cos 34 cos 64 cos 94 cos 184 cos 4 cos 24 cos 34 cos 64 = 2 {cos 24 - cos 44 +oos 64 — cos 124}. 14 2 An{B+C+D-A) *■ 8in(4-fi)sin(4-C)sin(4-2>) 15. cos 44 . cos 45 sin 4 sin (4 - JB) sin (4 - C) ^ sin Bsay{B-C) sin {B-A) COS 4t/ + -. — -p^-. — ryi — TT-- — 77; — ^v = 8sin(4 + 5+C) + cosec 4 cosec B cosec C. sm Csm (C- 4) sin (C-5) li A+B+C=ir, prove the relations in Examples 1 6 — 27 : 16. 2 tan 4 cot 5 cot C=S tan 4 -22 cot 4. 17. 2 cot 4 = cot 4 cot B cot C+cosec 4 cosec B cosec C. 18. 2 sin (5-C)cos'4 = -sin (5- C) sin (C-4) sin (4 -iS). 19. S(sin5+sinC)(co8(7+cos4) (cos4 + cos5) =(sin 5 4- sin C) (sin C+sin 4) (sin 4 + sin B). 20. 2 sin 4 cos (4 - B) cos (4 — (7) =3 sin 4 sin 5 sin C+sin24 sin 25 sin 2(7. 21. 2 sin 25 sin 2C= 4 {sin^ 4 sin^ 5 sin* C+ cos^ 4 cos* 5 cos* C + cos 4 cos B cos C}. EXAMPLES. CHAPTER IV 59 22. S cos 2A (tan B - tan C) = - 2 sin {B -C)sm{G-A) sin {A - B) sec A sec 5 sec C. 23. 2 cos"'' .4 (sin 25 + sin 2C) = 2 sin 4 sin B sin C. 24. S cos il sin 34 = {2 sin 24} {f + 2 cos 24 }. 25. (sin A +sin J5+sin C) ( - sin 4 + sin £+sin C) (sin 4 - sin 5+sin C) (sin 4 + sin fi - sin C) = 4 sin^ 4 sin^ B sin^ C. 26. sin^ 4 cot 4 1 sin2£ cot 5 1 sin^ C cot C 1 =0. 27. Scosec5cosec Csec(5-(7) = sec {B - C) sec (C- 4) sec (4 - 5) (3 + 8 cos 4 cos 5 cos C). 28. Prove that, if a+04-y=i7r, sin^ a+sin^/S+sin^ y + 2 sin a sin /3 sin y = 1. 29. Prove that 1 + 2 cos (jTT + fl) 1 +2 cos (^ JT - e) 2 cos fl- 1 ■ 30. Prove that sin2 (5 + a) + sin2 (5 + ;3) - 2 cos (a - /3) sin (5 + a) sin (^ + (3) is independent of d. 31. If tan /3=-:j — ^ „;„a_ i shew that tan(a-^) = (l-»i)tana. 32. If tan ^= 1— Jisin^a sin a sin 6 iu i J. /I sm a sin d) , prove that tan 5= ; 2:-. C0S) — (sin fl+sin d>) sin (5+rf>). 2cos(5-<^)-l ^ ^ :r/ \ -r/ t/ 35. If 6 and (^ satisfy the equation sin 5 + sin (^ = Vs (cos (^ — cos 6), then will sin 3fl + sin 3(/) = 0. 36. Prove that tan 70° = tan 20° + 2 tan 40° + 4 tan 10°. 37. If cos* a sin* a , ,, cos*/3 , sin*j3 , r+^^-B=lj then — :7r-+:^-5^=l• cos2/3 sin^/S" ' cos" a ' sm" a 38. If cos(4+5)sin(C+Z>) = cos(4-5)sin(0-Z)), then cot4cot jB cote = cot i). 39. If a+/3 + y = in-, then (cos a+sin a) (cos /3 + sin /3) (cos y + sin y) =2 (cos a cos /3 cos y + sin a sin ;3 sin y). 60 EXAMPLES. CHAPTER IV 40. If A+B+C=ir and cos 4= cos 5 cos (7, then will cot B cot C=^, 41. If 4sin2asm2j3sin2y+sm*a+sin*/3+sin*y — 2sm^/3sm*y - 2 sin^ y sin^ a — 2 sin^ a sin^ ^ = 0, shew that a±^±y is a multiple of w. 42 If tan(a+/3-y) ^tany tan(a— ;3+y) tan3' prove that sin 2a + sin 2/3 + sin 2y = 0. 43. If seoa = seCj3secy+tanj3tany, prove that secj3=secyseca+tanytana and secy=secasecj3+tanatan|3. -„ sin^flcosd) — cos^flsind) sin^ A cos d — cos'' rf) sin 5 ,^ ,. 44. If ^'^ ^= — ■^ -rj — ^ =cos(d + d>), cose tan a cos(ptan|3 ^ sin^ a cos fl — cos^ a sin a sin^Scosa-cos^/Ssina , . _, then =-r — 3 = ^z — -r^ = cos (« +/3)- cos a tan fl cos p tan 9 45. If A, B, Ohe positive angles such that A+B+G=QO°, prove that sec A sec B sec Ch- 22 tan B tan (7=2. 46. If cos (g+^) cos (g+y) + 1 _ cos (fl+y) cos {6+a.) + l _ cos {6+a) cos {6+p) + l cos(j3+y) ~ cos(y + a) ~ cos(a+/3) ' prove that . cosec (/3 - a) cosec (y - a) + cosec (y - j3) cosec (a — /3) +cosec (a — y) cosec (^ - y) = I. 47. Having given sin*5+sin*, yj/ must be equal to mTr+^w, mr-^w, or else one of them and also the sum of the other two must be multiples of w. 50. If "^"(^-y^ cos [6- 2a) + ''^^cos (fl- 2/3) cos a cos/3 prove that cos 6 = cos a cos /3 cos y. 51. If a, /3, y, 8 be any four angles and 2(r=a+0+y+8, then cos a cos /3 cos y cos 8 + sin a sin ^ sin y sin 8 = cos (o- — a) cos (o- - ^) cos (o- — y) cos (cr — 8) + sin (o- - a) sin (o- — /3) sin (o- - y) sin ( \ + Vl + tan^ a/ cosHa = ifl+ — ]; ^ ^V +Vl + tan»a/' hence sinia= ^^^(l- ^-^ji=_), cos^c , ^.1 i 1 +Vl+tan='a-l and consequently tan ^ a = 7 ; each of these formulae contains ambiguities. We leave to the student the discussion of these ambiguities, which should be made as in the previous cases. THE CIRCULAR FUNCTIONS OF SUBMULTIPLE ANGLES 69 It should be noticed that the values of tan |a are the roots of the quadratic equation in tan^a, 2 tan i a tan a = z — - — fi— > l-tani'^a obtained by replacing A by ^a, in the formula (41) of the last Chapter. 63. The functions sin a, cos a, tan a can be expressed without ambiguity in terms of tan^a; for all the angles which have the same tangent as ^a are included in the formula n7r + |^a, and 2 (wTT + ^a) or Imr + a are angles which have all their circular functions the same as those of a. To find the expressions, we have 2 sin Aa cos 4a 2 tan 4a mn ffl = = = — = = cos'' \0L-\- sin" \/2+V2 + V2, and proceeding in this way, we can calculate sin ^ tt and cos ^ tt. (2). We have sin^7r=l/2, co8^'7r = V3/2; hence from formulae (5) and (6), we have sin^7r = i(V6-V2), cos3^7r = i(V6 + V2), the values obtained for sin 1.5°, cos 15° in Art. 34; proceeding in this way we calculate the sines and cosines of all the angles TT 2". 3" THE CIRCULAR FUNCTIONS OF SUBMULTIPLE ANGLES 73 (3) We have siii^7r= 2 sin -j^ttcos^tt and sin f tt = 2 sin ^tt cos ^ir, therefore sin ^tt sin ^tt = 4 sin Itt cos ^ir sin ^^tt cos ■^•jt ; hence since sin|7r = cos35^7r, we have 4cos^'7rsin^7r = 1, or sin^7r-sin^^7r = ^, that is cos-^TT — sin33j^7r = ^, also (cos ^TT + sin -^iry = i + 1 = | ; therefore cos ^tt + sin ^-n- = ^ V5, or sinT-ig7r = i(V5-l), cos^7r = i(V5 + 1), and hence cos j^v = i V 10 + 2 a/5, sin ^tt = -^ VlO -2^5; these values agree with those giveii in Art. 34. It should be noticed that, if a is any angle of which the sine and cosine are known, then the sines and cosines of all angles of the form ma/2", where m and n are positive integers, can be found in a form which involves only the extraction of radicals; for we have shewn how to find the functions of all angles of the form 0/2", and when these are known, the formulae of the last Chapter enable . „ 1 . ma T ma us to find sm-g- and cos ^r • 66. We are now in a position to calculate the circular functions of all angles differing by 3° or 7r/60, commencing at 3°, and going up to 90°. We have sin 3° = sin (18°- 15°) = sin 18° cos 15" — cos 18° sin 15° = 1^ (\/6 + V2) Wo -l)-i (V3 - 1) V5 + V5, similarly cos 3° = ^ (^3 + 1) VsTW + tV W^ " ^2) (V5 - 1). We have also 6° = 36° - 30°, 9° = 45° - 36°, 12° = 30° - 18°, 21° = 36° - 15°, 24° = 45° - 21°, 27° = 30° - 3°, 33° = 45° -12°, 39° = 45° -6°, 42° = 45° -3°; ' hence we can calculate the sines and cosines of all the angles 3°, 6°, ... up to 45°. It is then unnecessary to proceed farther, since the sine or cosine of an angle greater than 45° is the cosine or sine of its complement, which is less than 45°. The results of the calculation are given in the following table : 74 THE CIRCULAR FUNCTIONS OF SDBMULTIPLK ANGLES y = ^n Jff{(V6 + V2)(V5-l)-2(V3-l)V5+V5} 6° = ^'r ^(\/30-6j5-V5-l) 9° = *T i(VlO + V2-2V5-V5) 12° = Jj7r |(V10+2V5-V16 + V3) 15° = T^^ i(\/6-V2) 18°=^^ i(V5-l) 21° = B%T J5{2(V3 + 1)V5-V5-(V6-V2)(V5+1)} 24°=ft7r J(Vl5 + V3-VlO-2^5) 27°=^ TT i(2V5+V5-VlO + V2; 30°= in i 33°=eT Jj{«6 + V2)(V5-l) + 2(V3-l)V5 + s/5} 36°= in JV10-2V5 39°=M,r iJ)r{(V6 + V2)(V.5 + l)-2(V3-l)V5-x/5} 42° = b'^t J(V30+6V5-V5 + 1) 45°= in iv/2 48°=A,r J(Vl0 + 2V5 + Vl5-V3) 5r=J5,r -jV {2 ( V3 + 1) V5 - ^5 + ( V6 - \/2) (\/5+ 1)} 54°=^^ i(V5 + l) 57°=Mt 3V{2(V3 + l)\/5+V5-(V6-V2)(V5-l)} 60°= in iV3 63°=^^ J(2V5+s/5+VlO-V2) 66° = ij7r J (V30- 6^/5 + ^/5 + 1) 69° = fg,r Jj{(Ve + V2)(V5 + l) + 2(V3-l)V5-V5} 72°= %n iVio+avs 75° = ^^^ i(V6 + V2) 78°=Mt J(V30 + 6V5 + \/5-l) 81°=Mt- |(Vl0+V2 + 2V5-x/5) 84° = J,^ i{Vl5 + V3 + VlO-2v'5) 87°=M7r Jj{2(V3 + l)V6 + V5 + (V6-V2)(V5-l)} EXAMPLES. CHAPTER V 75 In this table, the sines of the angles 3°, 6°, . . . up to 87° are given ; the cosines will be found by taking the sines of the complementary angles. The values of the surds in the above expressions are given to 24 decimal places in the Messenger of Math. Vol. vi., by Mr P. Gray. In Button's tables the values of these surds are given to 10 places of decimals. A complete table giving the tangents, secants, and cosecants of these angles, with the denomi- nators in a rationalized form, will be found in Gelin's Trigonometry. EXAMPLES ON CHAPTER V. Prove the relations in Examples 1 — 8, where A+B-'t-C—\&0° : tan^jl _ 1— C0S.4+C0S5-I-COS C tan^C 1 -COS C + cos .4 -(-COS 5' 2. am{A-B)s.in{A-C)-\-sm{B-C)s.m{B-A)+sm{C-A)sm{C-B) = 2cosJ(.B-C)cos^(C-il)cosi(4-5)-2sinfulsinf5sinfC. 3. cos^^A+,cos'^^B-\-cos^\C-ir'icosA-S) tan J (»— ^) tan J («— Vf), where 2g=5 + 0+i//'. 20. If ^ +^+ C+Z>= 180°, shew that sin .4 +sin 5+sin C- sin Z)=4 cos ^{A + D) cos J (5+i)) cos J {G-^D). 21. If a+j3+y=2n-, prove that sin (1 + 2 cos y) +sin y (1 + 2 cos a) +sin a (1 + 2 cos (3) =4 sin J (y-|3) sin J (a-y) sin J(;3- a). 22. If 2«=a+6 + c, prove that cos \s cos J (« — a) cos J (« - 6) cos \{s — c) +sin Ji sin J («-a) sin |(«- 6) sin J (s— e) = cos ^a cos Ji cos Jc. 23. If a+;3+y=ijr, then (1-tan Ja)(l-tan|/3) (l-tan|y) _ sin a+sinj3+siny - 1 (1+tan \a) (1 +tan J/3) (1 + tan Jy) ~ cos a+cos /3 + cos y ' 24. Prove that if a+0+y=7r, cos (|/3+y- 2a)+cos (f y +a - 2/3) +C0S (f a+;3-2y) = 4cosi(5a-2^-y)cosJ(5/3-2y-a)c08j(5y-2a-^). 25. If cos^5=cosa/cos/3, cos^fl'sscosa'/oosft and tanfl/tanfl'=tana/tana', shew that tan J a tan J a' = + tan J /3. EXAMPLES. CHAPTER V 77 26. If cos a = cos cos ^ = cos /3' cos ^', and sina=2sin J^sin|(^' ; shewthat ±tan^a=tan |/3tan^/3'. 27. U A + B + 0= 180°, and tan | ^ tan j 5 = tan | C ; shew that tan f 4 + tan}5 + tan |C=cot 1^ +cot f 5+cot I C. 28. If tan^(y+2) + tan|(0+a;) + tanJ(^+y)=O, prove that sin ^ + sin y + sin « + 3 sin (.» +y + z) = 0. 29. Prove that cos a sin 1(5 + a) sin ^ {|3 - y) + cos /3 sin 1(5 + ^) sin I- (y - a) + cos y sin -J- (5 + y) sin -^ (a - /3) = 2sini(/3-y)sin^(y-a)sin^(a-0)sin|(a+/3+y + 5). 30. Solve the equations tan^a+tan^i3 = Jl tana+tan;3 = jj ' 3^_ JJ sin(»+a)sin(,^-a) ^ sin(<^+^)sin (0-^)^^.^ Sin(^^+W) sin(^-25) shew that cos2^a+cos2^^-cos25=|^. 32. If tan(i7r+J5) = tan6(Jjr + J0), prove that sin^lssin^ (^ + '':^'°:g(^+f7.^tL ; ^ (1 +a-2 sm2 0) (H-/3-2 sm2 0) ' and find a, j3. 33. If a+;3+y=7r, shew that tan ~ 1 (tan ^/3 tan |^y) + tan ~ i (tan ^y tan Ja) + tan ~ i (tan Ja tan ^/3) -tan-' fl I Sain^nsin^ffsin^ \ sin^ a + sin^ /3 + sin^ yj ' 34. Prove that the sum of the three quantities C0s'|y-C03^ j-|3 cos^^g — cos^^y cos^ ^j3 cos^ ^y +sin2 ^/3 sin^ ^y ' cos^ Jo cos^ ^y +sin2 ^a sin^ ^y ' cos^ ^^ - cos^ ^g cob2 j)3 oos2 ^g + sin2 J3 sin2 Jg is equal to their continued product. 35. Prove that COS^O + y) COS^(y + g) C0S^(a + i3) 3cos^O + y)cos^(y + g)cos^(g + ff) cos^O-y) cos^(y-g) cos^(g-;3) COS ^ (/3 - y) COS ^ (y - g) COS J (a - 13) _ COSgCOS/ScOSy — C0S(g + j3 + y) ~C0S^(j3-y)C0S^(y-g)C0S^(g-3)' 36. Having given that cosg + cos j3 + oosy _sing + sin/3 + siny _ cos(g+j3+y) ~ sin(a+0 + y) ' prove that each fraction is equal to cos O + y ) + COS (y + g) + cos (g + /3), and also to {tang— tan J0+y)}/{tang+tan^(j8+y)}. CHAPTER YI. VARIOUS THEOREMS. 67. In this Chapter, we give various examples of trans- formations of expressions containing circular functions. Some of the theorems given are of intrinsic interest, others are given on account of the methods employed in proving them. Facility in the manipulation of expressions involving circular functions can only be obtained by much practice, but a careful study of the processes we employ in various cases will very materially assist the student in acquiring the power of dealing with this kind of symbols. » Identities and transformations. 68. Examples. (1) Prove that sin 2a sin (/3 — y) + sin 2/3 sin {y — a)+sin2y sin (a — j8) = {sin 0+y) + «m {y+a)+sin (a+0)} {sin (y-|8)4-si'i {a — y)+sin (|3-a)}. The factors on the right-hand side of the equation are the sum and the difference respectively of the two quantities sinycosj3-l-sinacosy+sin|8cosa and cosysin^ + cosasiny-f-cos^sina; hence the product of these factors is equal to (sin y cos /3 + sin a cos y -f- sin j3 cos a)^ — (cosysin/S-hcosasiny+cosjSsina)^. Now sin2ycos2^-cos2ysin2|3=sin2y — sin^^, hence the algebraical sum of the square terms is zero ; the product terms are equal to 2 sin a cos a (sin j3 cos y — cos /3 sin y) -f- 2 sin |3 cos /3 (sin y cos a — cos y sin a) -f- 2 sin y cos y (sin a cos j3 — cos a sin /3), and this is equal to sin 2a sin O - y ) + sin 2^ sin (y - a) -t- sin 2y sin (o — 13) ; thus the identity 2 sin 2a sin (|3-y)=2 sin (/3-)-y) 2 sin (y-0) is proved. VARIOUS THEOREMS 79 (2) Inthelastexample, put Jff + a, j7r+/3, Jw+y, for a, ^, y, respectively ; we then obtain the identity 2cos2asiu((3 — y) = 2cos(/3+y). 2 sin (y — ^). (3) Prove that 2 sin? a sin {^ — y)= — sin {a+ ^+y) sin {^ — y) sin (y — a) sin(a — fi). In this case, as in many others, we replace the quantities sin' a, sin'|3, sin' y, on the left-hand side of the equation, by the equivalent expressions in sines of multiple angles ; the expression on the left-hand side then becomes |2 sin a sin O — y) — J2 sin 3a sin (/3 — y) or — J2 sin 3a sin (j3— y) in virtue of Ex. (3), Art. 45. We now replace the products of sines by the difference of cosines, the expression then becomes J {cos (3a-|3 + y)-COS (3a-/3-)-y)-|-C0S (3^-t-y-a)-C0S (3|3-y-|-a) -I- cos (3y -1- a - j3) - cos (3y - a -t- 13)}, and the algebraic sum of the first and last terms in the bracket is 2 sin 2 (y — a) sin (a-t-|3-|-y) ; taking the second and third terms, and the fourth and fifth together, in the same way, the expression becomes —J sin (a+^+y) 2 sin 2 (y-a) or -sin (a-^j3-^y)sin (/8-y)sin (y-a)sin (a-|3) in virtue of Ex. (3), Art. 47. (4) Prove that 2 cos^ a sin {^ ~y)=cos (a+P+y) sin {fi — y) sin (y — a) sin {a — ^). (5) Prove that 2 sin^ a sin^ O — y) = 3 sin a sin /3 sin y sin (j3 - y) sin {y — a)sin(a — 0}; this follows from the fact that x+j/+z is a factor of x^ + j/^+:^ — Zxi/z; put .»=sinasin(|3-y), y = sin j3 sin (y - a), 2 = sin y sin (a - ^), then x+y+z=0. (6) Prove that sin (a + 0) sin (a - ff) sin (y + 8) sin {y-8)+ sin (/3 + y) sin (0 - y) sin (a + 8) sin (a-B) +sin{y + a)sin(y — a)sin(^+8)sin{P — 8)= 0. The expression (.^2 _y2) (^2 _ ^2) + (y2 _ 22) (^ _ ^2) + (jZ _ ^2) (y _ ,,,2) vanishes identically; put ^=sina, y=sin/3, 0=siny, w=smS, then remembering that sin^ a — sin^ j3 = sin (a -f- /3) sin (a - /3) the theorem follows. (7) Prove that 2 {cos /3 co« y - cos a) {cos y cos a — cos /3) {cos acos^ — cos y) -I- sin^ a sin^ j3 sin^ y — sin^ a {cos /3 cos y — cos a)^ — sin^ ^ {cos ycosa — cos j3)^ — sin^ y{oosa cos /3 — cos y)^ = (1 — cos^ a — cos' (3 — cos^ y + 2 cos a cos /3 cos y)^. 80 VARIOUS THEOREMS This follows from the known theorem that the square of the determinant a h g h h f 9 f c is equal to ho-p fg-ch fh-hg fg-ch ea-g^ gh-af fh — hg gh — af ah — h? put a=6=c=l, /=cosa, gr=cosj3, ^=0037, then 6c— /^=sin2a, ..., expanding the determinant, the theorem follows. (8) Prove that cos 2a COt\{y-a) COt J (a-|3) + C0« 2^C0< | (a-/3) COt\{^-y) ■\-COS 2y CO* J (;8 — y) CO* J (y — a) = cos 2a+co« 2/3+C04 2y + 2 cos 0+y) + 2 co« (y+a) + 2 co« (a+/3). Replace each cotangent on the left-hand side, by means of the formula l+cosfl cot|fi=- sind , then reduce the whole expression to the common denominator sin O — y) sin (y - a) sin (a — ;3) ; the numerator becomes 2 cos 2a sin (/3— y) {l + cos (y-a)} {l+cos (a— /3)}, or 2 cos 2a sin (,3 — 7) + 2 cos 2a sin (/3 — y) cos (y — a) cos (a - 13) + 2 COS 2a sin (/3 - y) {cos (y — a) + cos (a — j3)}, or {1+ 2 cos 0-y)} 2 cos 2a sin 0-y)-|2cos 2asin2(/3-y) + 2 cos 2a sin O — y) COS (y — a) COS (a — 13). Now 1 + 2 cos 0-y)^4cos J (/3-y) cos J (y-a) cos J (a -/3) from Ex. 4, Art. 47, and 2 cos 2asin 0-y) = 2 cos (/3+y) 2 sin (y-|8) =4 sin J 0-y) sin Hv -") sin J (o-;3) 2 cos 0+y). Also 2 cos 2a sin 2 - y) = 0, and 2 cos 2a sin (|3-y) cos (y-a) cos (a— j3)=i2 cos 2a {sin 2 (/3-y) -8in2(y-a)-sin2(a-^)} =12 cos 2a sin 2 (/3-y)-i2 cos 2a 2 sin 2 (|3-y), which equals sin (j3 - y) sin (y - a) sin (a - /3) 2 cos 2a, hence the numerator of the whole expression is equal to sin 0-y) sin (y-a) sin (a-j3) {22 cos (;8+y) + 2 cos 2a} ; therefore the expression is equal to 22 cos (|3+y)+2 cos 2a. (9) If a+)3+y=7r, and tan^{p+y — a)tan\{y\-a-^)tan^{a+^-y)==\, prove that 1 + cos a + co« j3 + co« y = 0. Squaring the given equation, we have sin2 (i:r-ia) sin" (iTr-^^) sin^ (jTr-^y) =cos2 (in- - Ja) cos-^ (ijr -i/3) cos2 (iTT - ly), or (1 - sin a) (1 - sin /3) (1 - sin y)=(l +sin a) (1+sin j3) (1 +sin y) ; VARIOUS THEOREMS 81 hence sin a + sin /3 + sin 7 + sin a sin /3 sin y = 0, or 4 cos ^a cos -^/S cos ^y + sin a sin ^ sin y=0 ; hence l + 2siniasin^^sin^y=0, also cos a+cos ^+cos y - 1 = 4 sin ^a sin -JjS sin Jy ; therefore cos a+cos ^ + cosy+ 1=0. (10) Prove that if tan i (P + y- a) tan^ {y + a - ^) tora ^ (a + /3-y) = l, then sin 2a+sin 2^+ain 2y = 4 cos a cos /3 cos y. We have siniO+y-a)sini(y + a-0)sini(a+i3-y) = cos^O + y-a)cos^(y + a-^)cosi(a + ^-y), or {cosO-a)-COSy}sinJ(a + 3-y) = {cOsO-a) + COSy}cos|(a + (3-y), which may be written COsO-a)cosJ(a + |3-y + ^7r) + COSysin^(a + /3-y + j7r) = 0. Now sin 2a + sin 2/3 + sin 2y - 4 cos a cos /3 cos y is equal to 2 sin (a + )3) cos (|3-a)-2cosy {cos (/3-a) + cos(a+/3)-siny}, or 2 cos (/3- a) {sin (a + /3) - sin (|jr - y)} - 2 cos y {cos (/3 + a) - cos (Jn- - y)}, which is equal to 2 sin J (a+/3 + y- -^tt) {cos (/3 - a) cos |(a+/3- y +^7r) + C0Sysin|(a + (3-y + ^7r)}, and this is equal to zero. (11) Saving given that 4 cos (y - z) cos (z - x) cos (x - y) = 1, prove that 1 + 12 eo« 2 (y - z) cos 2 (z — x) cos 2 (x - y) = 4 co« 3 (y - z) co« 3 (z — x) cos 3 (x — y). Let a=y-z, ^=z-x, y—x-y^ then a+/3 + y = 0, hence 1 - cos^ a - cos^ |8 — cos^ y + 2 cos a cos ^ cos y = 0, therefore cos2a+cos2/3 + cos2y = ^. Now cos 3a cos 3^ cos 3y = cos a cos j3 cos y (4 cos^ a - 3) (4 cos*/3 - 3) (4 oos^ y — 3) = i (4 - 27 - 48 2 cos2 j3 cos^ y + 36 2 cos^ a) = J (31 - 48 2 cos" /3 cos2 y)^ and cos2acos2/3cos2y=(2cos^a- 1) (2cos^/3 — 1) (2cos2y- 1) = (i-l + 3-4Scos2/3cos2y) = f - 4 2 cos^ /3 cos^ y, hence 4 cos 3a cos 3/3 cos 3y - 1 2 cos 2a cos 2^ cos 2y = 1 . (12) Having given y^ + z^ — 2yz cos a _ z^ + x^ - 2zx cos^ _ x^+y^ - 2xy cosy sin^a ~ sin^^ sin^y ' H. T. 6 82 VARIOUS THEOREMS prove that one of the following sets of equations holds\ 2s denoting a+/3+y ; X _ y ^ z cos(s-a) cos(s — ^) cos{B-y)' X ^ y _ z cos s COS (s — y) cos (s — /3) ' X ^ y ^ z co« (s — y) COS s COS (s — a) ' X ^ y ^ z co« (s — 0) COS (s — a) cos s " Let each of the equal fractions be denoted by k% and put .r=^cos5, y^kco&ff), s=koos\lr, we have then cos^ + cos^ 1^ — 2 cos . In order that these equations may be consistent, we must take all the ambiguous signs to be positive, or else two of them negative and one positive. In the former case we find 6 = s — a, (f)=s-^, ■\|/'=s--y; in the other cases we find the three sets of values e=s \ e=y-s\ 6=s-^\ '^=«-7p <^=» [. , thus one of the four given relations is always satisfied. The solution of equations. 69. Examples. (1) Solve the equation sin 26 sec id + cos 26= cos Qd. This equation may be written sin 26 sec 4fl+cos 25- cos 65 = 0, or sin 26 sec 45 + 2 sin 45 sin 25 = ; hence sin 25=0, or sec 45 + 2 sin 45=0, that is sin 85 = — 1. Hence the solutions are*" 6=imn, 5 = j|m7r-(-l)"||. (2) Solve the equation'- cos^ asecx + sin^ a cosec x = 1, for x. We may write the equation cos' asin ;r+sin5 a cos .r=sin x cos x, ' This example is taken from Wolstenholme's problems. VARIOUS THEOREMS 83 or sin^ a cos x — cos a sin* a sin ^ = sin x (cos :;; — cos a), hence sin^ a sin (a - x) = sin x (cos x - cos a), both sides are divisible by sin ^ (a— a*), rejecting this factor, we have 2sin2acos^(a — «) = 2sina;sin|^(a + ^)=cos^(a' — a)-cos J {3x + a), therefore cos ^{'Sx + a) = cos ^(x — a) cos 2a, or 2 cos ^{Zx+a) = cos ^ {x + 3a) + cos ^{x- 5a), which may be written cos^ (3.K + a)- cos J (^4-3a)=cos^ («- 5a) -cos -1(3^ + 0), therefore sin ^{x-a) sin (a; + a) = - sin {x - a) sin ^{x+3a); again rejecting the factor sin|^(A' — a), we have sin (^+a) = - 2 cos ^(x-a) sin ^ (ii;+3a) = - {sin (x+a) +sin 2a}, whence sin (^ + a) = —sin a cos a. The solutions are therefore x=2n7r + a, and ^ = re7r-a+(-l)''~isin-i (sinacosa). (3) Solve the equations a sin (x a sin (x+y) + b sin (x - y) =2n cos y We have a, sin (x+y) - b«m(x-y)=2m cos xl -2 {asin {x + ^) + bsm {x-y)Y ^ {a sin {x+y) — b sin (x -y)Y = 4(cos^y - cos* x) = i sin {x+y) sin {x-y). Let -: — ; ^i = t, then i is eiven by the quadratic equation sin(:i;-j') • „ , „,,. ,. , sin(x+v) tan^+tan«' Usins t for either root of this equation, we have t=—. — -. ^^ =t r — ^, ^ 1 ) sin (^ - y) tan x - tan y whence t = — r ; also dividing one of the given equations by the other, tan y t-1 we have = r ; and thence eliminating y by means of these two ncosy at + b' °^ equations and the relation sec*y — tan*y=l, we have n^ fat-by „ ft-\y. „ , mAatTb) ^^""'-KJ+V **° ^ = ^' from which we find ^L n-^ fat-by^^ln^ (at-by /i-lVr* tan .;= ±{l --, (^^ | |-, (^^ - (^j | , which gives four values of tan x, two corresponding to each root of the quadratic which determines t. Thus x is found, and then y is given by tany=— -TT tan x. t+ l: 6—2 84 VARIOUS THEOREMS Eliminations. 70. Examples. _, . . , , . cos^ 6 dn? 6 IV) Elimmate 6 from the equations ; sm = ~^~7 S2\ - ^ ' ■' ^ cos(a-36) sin(a — 3fl) ,„ , sin 6 cos' 6 + cos 5 sin' 5 sin S cos 6 We nave m = : — -, -;rr^ = -■ — -, aa\ > sin(a-2fl) sin (a -25) whence -— =sinacot 2tf — cosa. 2m cos*fl-sin*fl cos2fl Also cos 6 cos (a - Z6) - sin fl sin (a — 35) cos (n - 25) 1 cosa+sina tan25' hence |;r — |-cosa)( cosal = sin2a, V2m l\m, j or 2m2— l=mcosa, the result of the elimination. (2) Shew that the result of eliminating 6 from, the equations cos 3 (6 — a) _COsS(d + a-y) _C0s3a cos{e-^) ~ cos{e+fi-y) ~ eos^ is independent of ^. 6,y-0, and zero are independent values of a; which satisfy the equation cos 3 (a; - a) _ cos 3a . cos (^ - /3) ~ cos /3 ■ We have cos 3:r cos 3a + sin 3^ sin 3a = /; (cos x cos + sin .r sin ^) , where i=cos3a/cos/3 ; substituting for cos3.j;, sin3« their values in terms of C08.r, sin^ respectively, then dividing throughout by cos'x, we have the following cubic in tan^ ( = 0i cos 3a {4 - 3 (1 +t^)} +sin 3a {St (1 + 1^) - 4t^}=i: (cos j3 + « sin j3) (1 + 1^) or <3(Asin/3 + sin3a)H-<2(^cos/3+3cos3a) + <(/i;sin^-3sin3a) + icos^ — cos3a=0 hence tan 6, and tan (y- 5), are the roots of the quadratic <2(^sin^ + sin3a) + <(icos/3+3cos3a)+/i;sin0-3sin3o=O; ^cos/3+3 cos 3a therefore tan 5+tan(y-5) = and tan 5tan(-y — 5) = A8in/3 + sin3a ' i sin /3 - 3 sin 3a /;Bin;3+sin 3a ' , , — (/;cosS + 3cos3a) , . hence tan y = — ^^ . . „ = - cot 3a ' 4 sm 3a or y-3a=(2r + l)^n-, VAEIOUS THEOREMS 85 where r is any integer, thus the result of the elimination is independent of ^. (3) Eliminate 6 from the equations S.COS i + yj^^l^ xsine-ycose={a?nn?6 + b^cos^e)^. a b Square each of the equations, and put tan 6=t, the equations become '■('-S)-'lH-3)-». J2 (a2 _ a;2) + 2tsn/ + (b^-y^) = 0, respectively, and we have to eliminate t from them. Solving for t^ and t, we have <2 t 1 9.,,^ ^' 6!rl!\ (^^-y^)" {a'-~)^ ^{a:^-ai-^) %xy{W-y') - ^"^^K a^^~ar) ~~¥ '^~~ ah ¥■ Hence hence — |-%=a+6 OS is the result of the elimination. (4) Eliminate 6 from the equations X sin 6 ■\- J cos ^=2astm 26, -x. cos 6 -y sin 6= & cos 29. Solving for x and y, we find x—acoa6{^- cos 26), y = asm6{2 + cos 26) or a;=a!cos5(cos2 5 + 3sin2 5), y = asin fl (Scos^fl + sin^fl), therefore a;+y=a(cos d+sin^)^, a'-y=a(cos ^-sinfl)^, hence {x+yf=a^{\ + sm26), (a--y)^=cs*(l-sin25) and the result is {x +y)^ +{x-y)^=^ 2a^. Relations between roots of equations. 71. Examples. (1) Consider the equation a,cos d + hsin6 = c. Let a, P be distinct values of 6 which satisfy it, then acosa+6sin a = o, a cos (i+b sia p=c ; 86 VARIOUS THEOREMS a h therefore sin;3-sina cosa — cos/3 sinO-a)' hence tan |^ (/3 + a) = 6/a, and also -cosi(i3-a) = r sin^(i8 + a) = -cos^O + a). cod These relations may also be found as follows : put tan ^6=t, then the • given equation may be written a{\-fi') + '2,ht = o{\+t^) or «2(c+a)-26i + c-a=0. The roots of this quadratic are tan ^a, tan ^j3, hence tan i a tan hB= — ; — , 1 . . ,1 , i- cosi(j8-a) c whence we obtain the relation f .' , — ; = - . cos-J(/3 + a) a 26 Also tania+tanij3=— — , from which the other relation may be obtained. (2) Gonsider the equation a, cos'id -Vh sin'iB + c cos 6 + di dn 6 + e=0. Let t=iax\.^6, then the equation may be written as a biquadratic in t, «*(a-c + e) + i!3(_46 + 2c?) + <2(_6a!+2e) + <(46 + 2rf) + (a+c+e) = 0; if tan^^i, tan^fl2) tanj^s, tan 1^4 be the roots of this biquadratic, we have . ,. 46-2rf ^^ ,., ,. 2e-6a Stani5T = , 2 tan i 5, tan A ^2= r— i 2 ' a-c+«' ^ ^ ^ ' a-c+e' Stan Jflitan^fl2tan^53= — , tan|flitanJ52tan J53tan^fl4 = —- , and from these relations symmetrical functions of the four tangents may be calculated. If 28=61+62+63+8^^6 hsLve _ 2 tan ^ Sj - 2 tan ^ 61 tan ^ 62 tan ^ 63 ~ 1 - 2 tan ^ 61 tan ^6^ + tan ^ 61 tan J ^2 tan J ^3 tan ^ 64 45-2rf+(46+2rf) _6 ~a — c + e-{2e — 6a,)+a+c+e a' We leave it as an.exercise for the student to prove the relations a b -c —d e cos* sin« 2cos(« — ^i) 2sin(«-6i) 2 cos J (01 + ^2-^3-^4)' (3) // sinacoe{a+6) tan 2a = sin fieos{^ + 6) tan 2^ = sin 7 co« (y + 5) tan 2y = sin 8 cos (8+ 6) tan 28 VARIOUS THEOREMS 87 and no two of the angles a, /3, y, S differ by a multiple of w, skew that a + fi+y + S + 6 is a multiple of jr. Write each of the equal quantities equal to k, then a, ft y, 8 are roots of the equation sin sc cos {x + d) tan 2x = k which may be written 2 tan^ a; (cos 6 -sin 6 tan a;)=k{l- tan* x), , „ , 2 sin 5 , , „ 2 cos 5 , , „ , ,nence 2tana= — y- — , 2tanatan/3 = — y — , 2 tanatan;3 tany=0, and tan a tan 3 tan y tan 8= — 1 ; therefore tan {a + ^ + y + 8)— -r — „ ^ — r = - tan 6, IC — Ji cos v — K hence a+/3 + 7+S + 5 is a multiple of tt. (4) If a, ^,y be unequal angles each less than 2ir, prove that the equations cos (a + 6) sec 2a= cos (6+ /3) sec 2^=cos{6 + y) sec 2y cannot coexist unless cos (fi+y) + cos {y+a) + cos {a+^)=0. Writing k for each of the equal quantities, we have cos a cos 5 — sin a sin 6 — kcoa2a=0, cos j3 cos 5 — sin |3 sin 5 - ^ cos 2/3 = 0, cos y cos 6 — sin y sin 6 — k cos 2-y =0, hence eliminating cos 6, sin 0, we have 2 cos 2o sin (3 - y) = or 2 cos O +7) 2 sin (y - (3) = 0, by Example (2), Art. 68, hence 2cos((3 + y) = unless 2sin(y-^)=0, that is unless sin ;^ - y) sin -^y - a) sin ^ (a - ;8) = 0. This example may also be solved in a similar manner to Example (3). Maxima and minima. Inequalities. 72. Examples. (1) The greatest value of a, cos 6 +\) sin 6 is Va^'+b^. Put b/a = tan a, then b=\/a^+b^ ain a, a= \/a? + h^ cos a, thus a COS 6 + b sin 6 = \/a^-'rb^ cos {6 -a), now cos {6 - a) always lies between + 1, hence a cos 6 + b sin 6 lies between (2) If\i.= Va^ cos^ fl + b^ siii^ 6 + fJ&'tin^B+h^cos^e, then u lies between a + b and \/2 (a^ + b^). 88 VARIOUS THEOREMS Let x^a^ cos2 e + ¥a\n^ e=-\{a^-^h^) + k{a^-h^)ooa'ie, then u=n/x + 'i/a^ + l>'-x. m2 = a2 + 6H 2 Vi {a' + 6'')2 - {^ (a^ + 6^) - 4^ hence u is greatest when a; = ^ (a^ + 6^), or the greatest value of u is V2(a^+6^) ; also u is least whei^ ^{a^+b^)-xia greatest, that is when x is least, which will be when cos 2d = -1, in which case x=b% and then w=a + b; this therefore is the least value of u. (3) Shew that if 6 lies between and n, cot ^0 — cot 6>2. We have .1/, x/, sinfd 3-4sin2je l +2cos^g COt^d — C0tg= . ,*. . = : 3-*— = -j— a , * sinjflsinfl sin 5 sin 5 hence cot Jd — cot5 = cosec fl+coseo^fl; now cosec 6, cosec^d are each never less than unity, if 6 lies between and tt, hence cot ^6- cot 6>2. (4) If the sum of n angles, each positive and less than Jtt, is given, sheio that the sum or the product of the sines of the angles is greatest when the angles are all equal. A similar theorem holds for the cosines. Let a^, 02 ... an be the angles and s be their sum. Then we have sin a,.+sin aj=2 sin \ (a^ + ag) cos ^ {0^.-0,), now cos \ (a,. - oj) is less than unity unless 0^=08, hence sin a,.+sin a,<2 sin ^(ay + as) unless ar=ag. If any two of the angles ai, 03... a, are unequal, we can therefore increase S sin a by replacing each of those two angles by their arithmetical mean, hence Ssina is greatest when all the angles are equal; we have therefore 2 sin a:f«7i sin «/w. Again sin a,, sin og = ^ {cos (a,. — a,) - cos (0^ + a,)}, and this is less than ^ (1 —cos ((v+a,)} or sin^^- (oy + Og) unless ar=ag. Hence as before, if any two angles in the product sinai, sina2...sina„ are unequal, we can make the product greater by replacing each of those two angles by the arithmetic mean of the two ; it follows that sinoi, sin 03... sin a„ is greatest when ai = a2 = ...=a„, or the greatest value of the product is (sin s/»i)". (5) Under the same condition as in the last example, shew that the sum of the cosecants of the angles is least when tlie angles are all equal. We have cosec Or + cosec Og -sin$(,ar °«^|c(,s|(^_o^)_cos^(a,.+a,) COS J (a^ - Og) + cos ^ (a,. + a,)]' hence for a given value of 0^+0,, cosec a, + cosec a, has its least value when VARIOUS THEOREMS 89 cos ^ (a, - a,)= 1, or when ar=a,. The reasoning is now similar to that in the last example. (6) Under the same conditions as in, the last two examples, shew that the sum of the tangents or of the cotangents of the angles is least when the angles are all equal. (7) Shew that if a + /3 + 7 = jt, co« a co« /3 cos y :^ 1/8. Porismatic systems of equations. 73. A system of equations is said to be porismatic '■ when the equations are inconsistent unless the coefficients satisfy a certain relation; when this relation is satisfied the equations have an infinite number of solutions. The system acos0cosy + 6sin/3sin'y + c + a'(sin3+siny)+6'(cos/3 + cosy)+c'sin(|3 + -y) = O, acosy cosa+6sinysina + c + a' (sin ■y+sina) + 6' (cosy +cosa) + c' sin (y + a) = 0, acosacos/3 + 6sinasin|3 + c + a'(sina+sin^) + 6'(cosa + cos/3) + c'sin(a+|3) = 0, is a system of three porismatic equations. Consider the equation «cosacos5 + 6sinasin5 + c+fl!'(sin5 + sina)4-6'(cos0 + cosa) + c'sin(5 + a) = O, this is satisfied by 6=^, and by 6=y. Write this as an equation in t&n^6 = t, thus : i^ ( — a cos a + c+ a'sin a + 6' cos a - b' -c' sin a)+2t (b sin a + a'+c' cos a) + (a cos a + e + a' sin a + 6' + 6' cos a+c' sin a) = 0. From this equation we find tan 1^/3+ tan ^y, and tan ^0 tan J y, u X wo . \ 2 (6 sin a + a' + c' cos a) hence tan*(/3+y)=7r7 r, r-- ^■ 2\MT n 2(acosa + 6' + c'sma) We should find similarly , , , 6sinS+a' + c' cos/3 tani(a + y)= ^— r; J-^> -^ " acosj3+o +c sm0 we can now deduce the value of tan^(a— j3) ; we find for the numerator the value (6sin/3 + a' + c'cos/3) (a cos a + 6' + c' sin a) - (6 sin a + as' +c' cos a) (as cos ;3 + 6' + c' sin ^) or 2 sin |(a - ^) {(c'2 - ab) cos J (a - ^) + (os'c' - bb') cos ^ (a + ^) -(aa'-6'c')sin^(a+^)}, ' See Proc. London Math. Soc. Vol. iv. " On systems of Porismatic equations " by Wolstenholme. 90 VARIOUS THEOREMS and for the denominator, (6 sin a + a' + c' cos a) (6 sin /3 + a' + c' cos ;8) + (a cos a + 6' + c' sin a) or (a cos /3 + 6' + c' sin j3) (a2+c'2) cos acosj3 + (62 + c'2) sin asin/3+(a'2+6'^) + (a'& + 6'c')(sina+siu/3) + (aV + ah') (cos a + cos j3) + (a + 6) c' sin (a + 13) ; dividing this fraction by sin \ (a - j3), we have this denominator equal to (c'2 - o6) {1 + cos (a - /3)} + (aV - 66') (cos a + cos /3) — (aa' — h'cf) (sin a + sin j3), hence (a + 6){acosacos/3 + 6sinasin^+c+a'(sina+sin j3) + 6'(cosa + cos/3) + c'sin(a+^)} is equal to c'2 - a'2 - 5'^ + oa + c6 - a6. Hence unless the condition c'2_a'2-6'2 + ca + c6-a6 = is satisfied, the system of equations cannot be satisfied except by equal values of a, j3, y. When this condition is satisfied, any one equation can be deduced from the other two. The swmmation of series. 74. A large number of series involving circular functions can be summed by the method of differences. The most important example of the use of this method is the case of a series of sines or cosines of numbers in Arithmetical Progression. Let the series be cos3"a; therefore the sum of the series is ^ = |(ltan3»a;-3l^tan3n-ia;); - (—tan 3" a;- tana;). 75. The sum of a series of either of the forms Ml cos a + Ma cos (« + /8) + Us cos (a + 2y8) + . .. + M„ cos {a + (n - 1) ^], Ml sin o + 1*2 sin (a + /3) + Ms sin (a + 2^8) + . . . + m„ sin {a + (m - 1) ^}, can be found, if m, is a rational integral function of r, of any positive integral degree s. Let /S = Ml cos a + Ma cos (a + yS) + . . . + m„ cos (a 4- (n — 1) /8}, then 2 cos yS . S = Ml (cos (a - /S) + cos (a + /8)} + Ma {cos a + cos (a + 2.0)] + . . . + Mr {cos (a + r-2^) + cos (a + r/3)} + ... + M„ {cos {a + TO- 2/3) + cos (a + nyS)}, whence 2(1— cos y8) /S = (2mi — Ma) cos a + (2m2 — Mi - Mg) cos (a + /S) + . . . + (%lr — Ur-1 — Ur+i) COS (« + r — 1/S) + . . . + (2t<^i — M,j_2 - M„) COS (a + n — 20) + (2m„ — M^i) COS (a + n — 1/3) — u^ cos (a — y3) — m„ cos (a + n/8). Now 2ur — Mr_i — Wj+i is a rational integral function of r, of degree s — 1, whence excluding the first and the three last terms, we have a series of the same kind, but of which the coefficients are of lower degree than in the given series. We again multiply by 1 — cos /S, and proceed in this way s times ; the series will then be reduced to the form (1). VARIOUS THEOREMS 93 Examples. (1) Sum the series co«a+2 cos (a + ^) + 3 cos (a + 2/3) + ...+n cos {a+(n- 1)0}. We have in this case 2m,.-m,._i-«,.^.i = 0, 2mi-«2=0, whence 2(l-coS|3)/S'=(n + l)cos{a+(K-l)0}-cos(a-|3)-?icos(a+ra/3), or .S'=-^(»i+l)oos{a+(»i-l)0}/(l-cos/3) - ^ cos (a - 13)/(1 - cos ^)-^n cos (a + »3)/(l - cos j3). (2) Sum the series cos a + 22 cos (a+0)+32 cos (a + 2/3) + ... + n2 cos {a + (n-l)/3}. This series will be reduced to the last one by multiplication by 2 (1 - cos 0). 76. The series cos a + a; cos (a + ;S) + «?■ cos (a + 2/3) + ... + «""' cos {a + (n - 1) /S}, sin a + a; sin (a + yS) + a^ sin (a + 2/8) + . . . + a;""' sin {a + (w - 1 ) /3}, are recurring series of which the scale of relation is 1— 2a; cos /3 + x^, for we have cos (a + r/S) + cos (a + r — 2/3) = 2 cos /3 cos (a + r — 1/3), and sin (a + r^) + sin (a + »• — 2/S) = 2 cos /S sin {a-\- r — 1/8). The series can therefore be summed by the ordinary rule for summing recurring series. If S denote the sum of the first series we find S (1 - 2a! cos /8 + x^) = cos a — a; cos (a — /8) — a;" cos (a + n/3) + a;"+' cos {« + (w — 1) /8}. If a; < 1, we find, by making n indefinitely great, the limiting sum of the infinite series cos a + a; cos {a-\- ^) + a? cos (a + 2/8) + . . . ^ , cos a - a; cos (« - /8) -^ ... „ „ , to be — ; Ti 7^ ^ . rutting a = 0, we find 1 — ^xcosp + a? ^ X ~~ sc cos Q = 1 + a; cos /S + a;'' cos 2/8 +..... . ad inf , 1 — 2a; cos /3 + x" whence also l-a;2 1 — 2a; cos + a^ = l + 2a!cos/8 + 2a!''cos2/3 + ...adinf. (3). 77. In some cases the sum of a series may be found by means of a figure. We will take as an example the series (1) and (2) of Art. 74. Let OA^, AiA^, A^A,, ... An-iAn be equal chords of a 94 EXAMPLES. CHAPTER VI circle, and let yS be the angle between OJi produced and A^A^; draw a straight line OX so that Aj 0X= a, then the inclinations of OA^, A^A^, ...An-iAn, to OX, are a, a + yS, a+ 2/3, ... a+(w-l)/8, and that of 04„ is a + J(w— l)y3; also if D be the diameter of the circle, we have 0^1 = i) sin |j8, 0-4„ = i) sin ^n/3. Now the sum of the projections of OA^, A^A^, ... An^^A^, on OX, is Oili cos a + ^.1^2 cos (a + /8) + . . . + A„^iAn cos {a + (w — 1) /S}, or Z) sin ^/3 [cos a + cos (a + /3) + . . . + cos {a + (n - 1) ;8}], and this must equal the projection of OAn which is 0A„ cos {a + 1 (« - 1) /3), or D sin ^mj8 cos (a + ^ (n — 1) /8}, therefore cos a + cos (a + /3) + ... + cos {a + (n- 1) yS} = cos (a + |(n — 1) /8} sin ^n/8 cosec ^/3. If we project on a straight line perpendicular to OX we obtain the sum of the series of sines. Examples. (1) OA is a diameter of a circle, O, P, Q... are points on the circumference such that each angle PAO, QAP, EAQ... is a; AP, AQ, AE... meet the tangent a< O m p, q, r Find by means ofthisfigwe the sum of the series «ecma«e(!(m+l)a+sec(m+l)a«ec(m+2)a+... to n terms. (2) Prove geometrically, that if a, ^,y ... k be any number of angles, sec a sec (a+^) sin ^+ sec (a+/3) sec (a+;8+y) sin y + sefl(a+^+y)«ec(a+0+y+8)sin8+... =secasec{a+^+y\-... + K)sin{^-\-y+... + K.). EXAMPLES ON CHAPTER VI. 1. Eliminate 6 from the equations cos'5+acos6=6, a\-D^6-\-asm.6=c. 2. Eliminate 6 irom the equations (a + 6)tan(fl-<^) = (a-6)tan(fl+<^), acos2(^ + 6cos2fl = c. EXAMPLES. CHAPTER VI 95 3. Prove that {a sin ^ + 6 cos //■ + 6 cos i/f) »in (0 - ^) +(a sin i/^ + 6 cos i/r) (a sin d+ 6 cos 6) sin {-^—6) + (a sin ^ + 6 cos 6) {a sin <^+ 6 cos <^) sin {6 - (f>) + (a2 + 62) sin (0 - V') sin (i/. - 5) sin (e - 0) = ; and interpret the equation geometrically. 4. Reduce to its simplest form, and solve the equation cos2 e ^ cos^ a = 2 cos3 fl (cos 5 - cos a) - 2 sin' 6 (sin 6 - hin a). 5. Prove that the sum of three acute angles A, B,C, which satisfy the relation aos^ A + coa^ B + cos^ C=\, is less than 180°. 6. If ^ +5+0=90°, shew that the least value of tanM +tan2 5 + tan2 C is unity. 7. Find 6, from the equations sin 5 + sin (^+sina = cos 5 + cos0 + cosa1 e + = 2a j' 8. If ^+5+C=180°, shewthat8sin^4sin^JBsinJC:t*l• ^. xain ^+y8in0+«sin\/c_4sin flsin //■)+ sin ^(g + ) (cos a sin fl + sin (^) = 4 (cos a cos 5 +C0S \|^) (cos a sin d + sin yj/) = (cos (j) — cos \^) (sin <^ — sin i/r ), and prove that cos (1^ - ^) = 1, or cos 2a. tany _sin(.j;-a) , tany _sin(^— 2a) tan /3 ~ sin a tan 2/3 sin 2a ' , tany sin.^ cos.r; shew that ' sin 20 sin 2a cos 2a - cos 2/3 ' 18. Prove that the system of equations sin(2a-0-y) _ sin (20-y - a) _ sin(2y-a-/3) cos (2a+0+y) "" cos(20+y+a) " cos(2y + a+0) ' if a, ft y be unequal and each less than n, is equivalent to the single equation cos 2 0+y) + cos 2 (y+a)+cos 2 (a+j3) = 0. 19. If a;=2oos(/3-y)+cos(5+a)+cos(5-a) = 2 cos (y-a)+cos (e+/3)+C08(5-/3) = - 2 cos (a- /3) - cos {0+y)- cos (fl - y), prove that a;=sin2 5, if the diflference between any two of the angles a, /3, y neither vanishes nor equals a multiple of n. 20. li A + B+C=180'' and if 2 sin (2re+l) A sin (5- C) = 0, n being an integer, then shew that 2 siu (m- 1) A sin (n+l) {B-C) = 0. EXAMPLES. CHAPTER VI 97 21. If cot^(a+j8)(cosy-oos8)+cot^(a + y)(cos8-cosj3) + cot^(a + S) (oos/3-cosy)=0, and no two of the angles are equal, or differ by a multiple of 25r, prove that cot J (/3 + a) (cos y- cos 8)+cot J (/3+y) (cos fi- cos a) +cot J (|8 + 8) (cosa— cosy) = 0. Tf sin(a+fl) siDO+a)^ cos(a+g) cos(g+ffl_ aia{a+) cos(a+(^)'^cos(^+<^) ' shew that either a and differ by an odd multiple of Jtt, or 6 and (/> differ by an even multiple of n, 23. If eicos((^ + \;r) + 5cos(^-i/f) + c=0, acoa{6 +<|)) + 6 cos(5 - (/))+c=0, and if 6, , ^ are all unequal, shew that a^- 6^ + 26c=0. C0S{a + ^ + 8) ^ COs(y + a + a) • sin(a+g)cos2y sin(y+a)cos2 3' and j3, y are unequal, prove that each member will equal cosQ+y + g) sinO+y)cos2a' sin O + y) sin (y + g) sin (g + 13) and °°* ^"cos 0+y) cos (y+g) cos (a+^) + sin2(a+^+y) " 25. If A, B, C be positive angles whose sum is 180°, prove that cos-4+cos5 + cosC'>l and ^•3/2. 26. Solve the equation 64 sin' fl + sin 75=0. 27. If 2s = ^ + y + «, pro ve that tan («-.!p)+tan(«-2/)+tan(s-z)- tang 4 sin X sin y sin z \—cos^ X - coa^y - cos^ z—^cos xcosy cosz' tan-i(«-^)+tan-i(4-y)+tan-i(s-2)-tan-i« =tan-, ^^^ cos^ , sin 5 cosd) sin0 28. If + -■ — = ^ + -:— ^ = 1, cos a sm g cos g sin g cos 5 cos ^ , sin 5 sin _ prove that cos^a + sin^g + ^ =°' 29. If ■ 2singcos(fl+(^)=2cos(fl-(/))+cos2g, and 2 sin a cos (6 +>/') = 2 cos (i^-tf) +C082g, ■ then 2 sin g eos (<^ + \/f ) = 2 cos (^ - V') + cos^ a. H. T. '^ 98 EXAMPLES. CHAPTER VI 30. If coa{ff-2)+cos{z-x) + coa(x-y)=—SI% shew that cos' (x+ff)+coa^(2/+6)+cos^{z+e)-Zcoa (x+d) co8(y+fl) cos (2+tf)=0, for all values of 6. „, J. sinra sm(j- + l)a sin(r+2)a ol. It J — ^ = , I m n nrovP that «"« '•« _ oos(r + l)a _ cos(r+2)a prove tnat 2m^-l{l+n)- m{n-l) ' n{l+n)-2m^- 32. Prove that the equations in a = 2 y (a;H — ) sina=2H i-cos^o, \ xj z y (y-\ — ) sin a=-H l-cos*a, y/ X s sin a = - + - + 008^0, are not independent, and that they are equivalent to 111 x+y+z=- H h-= — sma. •^ X y z 33. Prove that 2 cos (|3 - y) cos (fl + /3) cos (5 + y) + 2 cos (y - a) cos (d + y) cos (fl + a) + 2 cos (a-/3) cos (fl + a) cos (fl+^) - cos 2 (5 + a) - cos 2(fl+/3)- cos 2 (fl+y) - 1 is independent of 6, and exhibit its value as the product of cosines. 34. Prove that if a, /3, y, 8 be four solutions of the equation tan(5+j7r)=3tan35, jio two of which have equal tangents, then tan a+tan /3+tan y + tan 8=0, tan 2a + tan 2j3 + tan 2y + tan 28 = 4/3. 35. If 6tan(r+^)=3tan(r+2^) = 2tan(r+2), shew that 3sin^ {x —y)-\-h slv? {jj — z)-1a\s?'{z—x)=Q. 36. Solve the equations sin~i.K — sin~iy=f ttI coa~'^ X - -yj,)+q(r-p)^ cot {^-d) + r{p-qy cot (6-) = 0. 40. Develop :; — = — : — :; l+a cos 6+0 am 6 in a series of the form Aq+AiCos {d-a.) + A^cos2 (6 -a) + .... 41. Solve the equation tan 35 - tan 26 - tan 5 = 0. 42. If cos' ^ + cos' 3^= cos 3a, sin'.j;+sin'y = sin3a, and .»+y=2/3, prove that 8 sin' 3 (a +0) = 27 sin 2/3 sin^ 4/3 cos (3a + j3). 43. If aicos=c, prove that 6c + ca+a6=0, unless a=b = e. 44. Solve the equation cos*i(:»+^)+cos-i^+cos-'(,»-^)=f jr. 45. Eliminate from the equations a^i/siQ+¥x cos <^ + a6 (a^ sin^ <^ + 6) cos* <^^ = 0, (m; sec (jj — b^ cosec = a* — 62. 46. Solve the equation cos 5fl + 5 cos 35 + 10 cos 5= J. 47. Eliminate 6 from the equations a cos 6 cos 25=2 (a cos 5— ;»), a sin 5 sin 25=2 (a sin 6—y). 48. Prove that the number of solutions in positive integers (including zero), of the equation Z!e+y=n (n integral), is *[«««-')-=^4ii±^']- 49. Solve the equation 6 cos 35-3 sin 35- 10 cos 25+5 sin 25+22 cos 5- 5 sin 5 = 10. 50. Find the greatest value of cosec^ 5 - tan* 5 cot«5+tan25-l* 7-2 100 EXAMPLES. CHAPTER VI 51. Prove that sec^a seo^a sec^a sec^a 4 - 1 - 4 - 1 -■ to r quotients is equal to 2 sin (r+l)aCOSa* 52. Eliminate 6, ^ from the equations a sin (5-a)+6 sin {0-\-d)=x sin (<^ + /3)+y sin {+y sin (l>+z+cos2 = 8sin|(a+^+y+<^)sinJ((/)-a)sin§(«^-/3)sin^((/)-y). 64. Eliminate 6, ^ from the equations tan 5 + tan = a, sec 6 + sec =b, cosec 5+cosec <^=c, and shew that, if b and c are of the same sign, be > 2a. 65. Prove that the result of eliminating 6 from the equations cos (d - 3a) _ cos (g - 3^) _ cos (g - 3y) cos' a cos' |3 cos' y is sin (/3 - y) sin (y — a) sin (a — ;3) {cos (a + ^ + y) — 4 cos a cos /3 cos y)} = 0. 66. If (l—x+a;^)~^ be expanded in powers of x, shew that the coefficient of ^is sin^(?i+l)7r/sin JjT. 67. Prove that 2 cos 4a sin ((3 + y) sin O — y) = — 8 sin (/3 - y) sin (y - a) sin (a - 0) sin 0+y) sin (y +a) sin (a+0). 68. Prove that Scos2(j3+y-a)siii(|3— y)cosa=8sin(^-y)sin(y-a)sin(a-^)cosacos^cosy. 69. If asm6+bcoa0=acoseod+bsecd, shew that each expression is equal to {J-b^){J+b^)k 70. Find the greatest value of sin (iS — y) + sin- (y - a) + sin (a — ^). 102 EXAMPLES. CHAPTER VI 71. Solve the equation cos {x- a) cos {x- h) cos {x — c) =sin a sin h sin c sin a;+ cos a cos 6 cos c cos x. 72. Solve the equation cos 2a!+cos 2 (x— o) +C0S 2 (a;— 6) +cos 2 (a; - c)=4 cos a cos 6 cos c. 73. Solve -the equation sin^ 3a + sin' 2a = sih^ a (sin 3a + sin 2a). 74. Eliminate 6 from the equations a cos 25 + 6 sin 25=c, o'cos3d + 6'sin3fl=0. 75. If ^ + £+(7=180°, shew that sin2 IB sin2 J C+ sin^ I C sin^ |4 + sin^ lAsai?\B is not less than j^ (sin^ 4 + sin^ B + sin^ C). 76. Eliminate fl from the equations ix=ba cos 5 — a cos 56, Ay = 5a sin fl - a sin b6. 77. If cos 2a sin O -y) sec (/3+ 7) = cos 2(3 sin (7 - a) sec (y + a) = cos 2y sin (a - /3) sec (a + /3), prove that cos 2a + cos 2/3 + cos 2y = 0, and sin 2 (|8+y) + sin 2 (y+a)+sin 2 (a+/3)=0. 78. Prove that m=M 2 cos (nia+/3) = cos (-Ji/a+jS) sin J (if+1) a cosec Jo, and 2 2 2 cos (»ia+»ij8+^7+ ) m=0 n=0 p=0 =cos^ (J/a+iV/3+i'7 + ...) sin i(Jf+l) a sin ^(il7^+l)/3... cosec ^a cosec ^/3.... Sum to n terms the following series in Exs. 79 — 93. 79. sin2a+sin2 2a+sin2 3a+ ^-wi^na. 80. sin^asin2a+sin2 2asin3a+ +sin2masin («+l) a. 81. cosec a cosec (a+^) +cosec (a+j3) cosec (a+2/3)+ +cosec{a+(«- 1) j3} cosec (a+ra/3). 82. sin X sin 2.» sin 3:f +8in 2^ sin 3a; sin 4a;+ +sin iix sin (« + l) a; sin (»i+2) ^. 83. sinS a+g 8in3 3a+gj sin' 3^ a+ +3;r^i sin' 3"-i a. 84 tan 6 tan 3fl+tan 25 tan 45+ +tan m5 tan (m+2) 5. EXAMPLES. CHAPTER VI 103 85. tan 6 sec 2tf +tan 26 sec 2^d+ + ta,nnd sec 2"^. 86. tan^+-tan2+4tan- + +^—^t&n^^^. 87. tan x sec^ a; + 5 tan - sec^ ^ + g2tan|sec2|+ + __ tan g^sec^g^i. 88. 1 +c cos fl COS (^ + 0^008 25 cos 2(^ + + c'^~^cos{n-l)6cos{n—l) (j). c os2g 2cos4fl 4 cos 85 2''-icoa2»5 ■ sin2 25sin2 45"*" 8^2 85"*" "*" sm''2»5 ' ._ sin 5 sin 25 sinm5 cos5+cosl25 cos25 + cos225 cos«5 + cos«25' - cot 2a cot 3a C0t(»l + l)a 1-C0822asec^a 1 — cos^Sasec^a 1 — cos^(w + l)a.sec2a' 92. 1.3sin-+3.58in — + + (2ra-l)(2w+l)sin ^^"'~^^"" . 93. 3.4sina + 4.5sin2a+ +(»i + 2)(»H-3)sin»a. 94. If 5i, 62 be two solutions of the equation sin (5 + a) + sin (5+/3)+sin (a+^)=0, where 61, 62, a, and ^ are each less than 2jr, shew that sin (5i+52)+sin 0+5i)+sin (^+ei)=0. 95. Prove that , ._,24/4 + l ,^ _,'v'4+l , icot '■ = hitan 1 — ;^-=«T> s/S \/3 4^2+1 2 4^2^+1 and ^tan-'-— ^-^tan-i , =^7r. 96. If a, ft 7, 8 are four unequal values of 0, each less than Zw, which satisfy the equation cos 2 (X-5)+cos (ju— 5)+cosi'=0, prove that a+/3+7+8-4X = 2ra7r, and that sin^ (/3+y + 8-a-2;i) + sin^(y + 8 + a-j3-2;a) + sin^(8+a+^-y-2/i)+sin^(a+^+y-8-2|t»)=0. CHAPTER VII. EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES. Series in descending powers of the sine or cosine. 78. If in the formula (40), of Art. 51, we vmte for sin'* A its value (1—coB^ Af, and arrange the series in powers of cos J., we shall obtain an expression for cos nA in powers of cos A only. Writing d for A, we have cos n^ = cos" ^ - ^^^^^ cos'" 6' (1 - cos'' 6') + . . . + (_l). '^("-l)-;-(f-^ + l) cos"--g(l-cos-g)r + .... ^ ' (2r) ! The coefficient of (— l)*" cos"""^ in this series is n(n-l)...(n-2r + l) njn-l) ...(n-2r-l) {2r)\ (2r+2)! ^ "^^ w(«-l)...(w-2r-3) ( r + l)(r+2) ■^ (2r+4)! ^ 2! '^'"'' this is equal to the coefficient of aF in the product of (1 + a;)" and (1 — l/a?)~ ''■'"", X being supposed to be greater than unity; the coefficient is therefore equal to the coefficient of af~^ in the expansion of (1 + a))"-^-^ (1 - l/a;)-'''+«. This latter coefficient is equal to (n-r-l)...(n-2r + l) f —, ^ jr + (»i-2r)(r + l) (n-2r)(n-2r-l), „, 1 + 2! \r + 2)+..j. EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES 105 and this is equal to (n-r-l)...(n-2r+l) ^^ ^^ ^ ^^_ _^ ^^ _ ^^^ ^^ ^ ^^„__j^ or to n(«-r-l). (»-2r-+l) 2„_,_,_ The coefficient of cos"^ is seen to be ^ {(1 +1)» + (1-1)"}, or 2"~^; the coefficient of — cos"~^0 is the term independent of X in the expansion of (1 + a;)"~* (1 — l/«)~°, and this is easily seen to be (1 + 1)"-" + («- 2) (1 + !)»-», or n.2»-». Hence we have cos n0 = 2"-i cos" d-^. 2"-» cos»-» + "^^~^^ 2»-'> cos""" ^. . .(1), XI ^ ! of which the general term is (_ ^yn(n-r-l)..(n-2r+l) ^_^^ ^^^„_, ^ r In a similar manner we obtain from the formula (39) of Art. 51 the series sin ne/sia d = 2»-' cos""' 6 - ^^^ 2"-" cos""' + (ii:iMLi:i)2«-=cos»-»^ - (2), of which the general term is y ! 79. If in the formulae (1) and (2) we change 6 into ^tt — 6, we obtain the formulae 2 « (- 1)2 cos nd = 2»-' sin" 6-j 2»-* sin"-=' ^ + *^^^2»-»sin'-0- (3), --1 n — 1 (-1)2 sin K^/cos 6 = 2"-' sin"-' ^ — 2'"-» sin»-» ^ ^ (^-3)^(n-4 )^„_.^.^„_.^_ ^4^_ where w is even, and (_ i)4(»-i) sin ne = 2»-i sin" d -^ 2»-= sin"-^ + 'L(!L3) 2„-5 gi^„-, ^ _ (5)_ A I 106 EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES (_ i)i(»-i) cos n^/cos = 2»-' sin«-^ - ^^ 2»-» sin»-» 6 where m is odd. Series in ascending powers of the sine or cosine. 80. In order to find expansions of cos w^, sinre^ in ascending powers of cos 6 or sin 0, we may write each of the six series we have obtained in the reverse order. It will, however, be better to obtain the required series directly. First suppose n even, we have then cos n0 ■■ = (1 - sitf 0f" - "^""-^^ (1 - sin= 0)^"-^ sin= ^ 2! ^n(n-l)(n-2)(n- 3) ^^ _ ^.^, g^in-2 ^^, g _ 4 ! expanding each power of 1 - sin'' by the Binomial Theorem,' we have 'n /n \ . _ (n n(n-l)] . ,„ . 2\2~ ) , n(n-l) fn \ cosn^ = l-|^ + -^^}sin'g + | ^, +— ^— (2"^j n(n-l)(n-2)(n-S)\ . ,, „ the coefficient of (— 1)' sin^ being ^n(^n-l)...(^n-s+l) njn-l) (jw-l) ... (^w-s+ 1) s\ "^2! (s-1)! n(n-l)(n-2)(n-3)Qn-2)...(^n-s+l) "^ 4! (s-2)! ' which may be written in the form ln(n-2)(n-4>)...(n-2s+2)(/2s-l\/2s-l \ / 2s -1 \ s"! 1.3.5...(2s-l) 11"^" A^ V-V~l *^ J }■ EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES 107 Now, taking Vandermonde's theorem' iP + 3)s =Pa .+ sps-iQi + ^^^^, ' ps-iQi +■■., where p^ denotes p(p^l) ...(p-s + 1); since this holds for all values of p and q, let p = — ^ — , q = — - — , then applying the theorem to the series in the brackets, we see that the coefficient of (- ly sin^ d is ln(n~2)...(n-2s + 2) il 1.3.5...(2s-l) ■ipi + s-l)(^n + 8-2)...(^n) nHn" - 2n (n''-4s') ... (n=- 2s - 2 n or — ^—^ -^ • (2s)! We have therefore, when n is even, cosji0=l -g-jSm^^H- ^ — ^sm*^... + (-iy -'<-'-^^)-"y-^^^% in--^+ (,). (ZS) ! this series is the series (3), written in the reverse order. 81. We have also sin w^ = cos \n(l — sin' 6y " sia ^ n(n—l)(n — 2),, . „/,>im-2 . „ /, ^^ ^ ^ (1 - sm" 0)*^^ sm=^+...-; supposing n even, we expand each term of the series in powers of sin' ; we find the coefficient of (- 1)"+' cos sin'«-i to be 1 ra(w-2)...(w-2s+2) (/ ^s-l N . / 2s-l \ / w-l \ (s-1)! 1.3.5...(2s-l) ][ 2 ),_,^'^' ^n 2 A-A 2 A (s-l)(s-2) / 2g-n M-l \ 2! I 2 A-sV 2 A + which is equal to 1 n(w-2)...(w-2s+2),, (7317! 1.3.5...(2s-l) — <^" + ^-^>-^^" + ^) n(w'-2')(?i'-4')... (w-2s-21') '^^^^ ^ (2s -1)! -■ ' See Smith's AlgOira, page 288. 108 EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES We have therefore when ?i is even smndjao&d — ^smd ^^-x-j — ' svD?d+ ... cos 82. When n is odd, we have n^=cos^|(l-sin^0)*("-'>-'^^\l-sin=0)*<"-')sin^5+...| and sin w^ = w (1 - sin'' e)*^" ~ ^^ sin n (w — l)(w-2) ,, . „a\i(n-Z) . ,^ , o I expanding in powers of sin 6, as in the last article, we find in a similar manner cos wg/cos e=\- ''^^ sin° e + ^'^^~ '^''^^'f ~ ^'^ sin^ e - + (_l/^'-10(^--B;)--;(^--2^-l|%in^,+ ...(9). smn^ = :rsin^ ^-^, ' am.'9 +^^ i^ ^sm=^— .... 1 o! o! +(- ')" °'"'"V"i)i '^"-^-^■■■<""- 83. If in the formulae (7), (8), (9), (10) we change 9 into ^TT — 0, we obtain the following formulae / i\4» /I 1 '^^ ,^ . wHra^- 2'') ^- (— 1)^ cos w= 1 — g- cos^^H ^-j^ — ^cos'p - "^^-^'-y/-^-^^) cos^^+...(ii). / ,x4n + l . /J/ ■ /) " ^ ft(w^-22) ,. (—1)^ sin nff/sm ^ = :j- cos » — — ^ cos' ^ + -5^ ^ ^ cos' ^ - . . .(12), when n is even, and (_ l)*("-^^sinn^/sin^ = 1 -^^^^cos^^ + ^-'-^°>f-^% os^^-...(13). EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES 109 (,-\r 'cos 710 = ^003 ^—^^ ^COS'^ H — ^^ dy cos^ » — . . .(14), when n is odd. These formulae are all the same as those of Arts, 78 and 79. The circular functions of sub-multiple angles. 84. If in the formulae (1) to (6), or in the equivalent formulae (7) to (14), we write 0/n for 0, we obtain equations which give cos - or sin - when cos and sin are given. We will consider n n ° the various cases. (1) Suppose cos given, then the equation obtained from (1) will give us n values of cos-. If cos is given, we should expect to find the cosines of all the angles =— , since 2kTr ± represents all the angles which have the same cosine as 0, where k is any integer. Now whatever value k has, we can put ±k = s + k'n, where s always has one of the values 0, 1, 2 ... w — 1, and k' is a positive or negative integer. We have then 2kir + /^ + 2sw , „ ,,\ 0+2sTr cos = COS + 2'irK = cos — cosl- n \ n J n thus we should expect to obtain the n values + 2ir + ^iT + 2(n-l)Tr ' cos - . cos , cos cos ^^ , n n n n and these will be the roots of the equation we obtain from (1). These roots are in general all different, since neither the sum nor the difference of two of the angles is a multiple of 27r. (2) Suppose cos is given, then the equations obtained from g (3) or (6) will give the values of sin -. Before we use (6), we must square both sides and write 1 — sin'' - for cos^ - ; thus we obtain an a equation of degree 2w, for sin- , when n is odd, and the equation 110 EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES (3) gives us an equation of degree n when n is even. We expect to obtain all the values of sin = — when cos is given : as in n ° the last case, we can shew that all these values are included in the expression sin ^— , where s has the valuesO, 1, 2...n— 1. When n is odd, all these values are different, and therefore we obtain 2n values which are the 2n roots of the equation obtained from (6). „„ . , . (n-2s)7r-d . 2s7r+0 , When n is even, we have sm ^ = sm , hence n n in this case there are only n values, these being given by the equation obtained from (3). (3) When sin 6 is given, we use the equation obtained from 6 a (2) to find cos - . this gives 2w values of cos - , for we must 6 6 square both sides and replace sin^ - by 1 — cos^ - , before using the equation. We shew as before that the expression cos ^^ — 6 has 2w values, so that we expect to find cos - given in terms of sin 6, by an equation of degree 2n. 6 (4) If sin 6 is given, sin - will be given by (4) or (5), accord- ing as n is even or odd. When n is even, the equation from (4) Q gives 2w values of sin-; these will be the 2m values of sin . When n is odd, the equation formed from (5) Q gives n values of sin - ; these will be the n different values of sm . Symmetrical functions of tke roots of equaticms. 85. The formula (1) may be regarded as an equation of the nth degree in cos 0, when cos n0 is given. Now each of the n angles - - 2ir n 4!7r « 2 (w — 1) ir . , , , 0,0 ^ ,0-\ ^ is such that .the cosine of n EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES 111 times the angle is equal to cos nd, hence since cos 0, cos id-\ J , cos [ ^ H ) cos j + -^^ ~ [ are all diflferent, they are the n roots of the equation (1) in cos 0; we can now use the ordinary theorems for calculating symmetrical functions of the roots of equations to calculate symmetrical functions of the n cosines cos (0-\ TTj , r having the values 0, 1, 2 ... n — 1. We may of course, when it is convenient, use the forms (11) and (14) which are equivalent to (1). Again the equation (2) may be used to calculate symmetrical functions of the cosines of the n — 1 angles for which sin n0/sm 6 has a given value. The equation (3) may be used in the same way to calculate symmetrical functions of the 2m sines sm^, sm^ + — , sm^ + — sm^ + , \ ml \ m j V m ) where n = 2m. In the same way the theorem (5) may be used to calculate symmetrical functions of the 2m + 1 sines where n = 2m + 1 . The equation ^ .L n(n-l). ,^ , 7i(n-l)(rt-2)(n-3) , ^. \ ta.nn0 \\ ^-^^ — ^ tan^^H — ^^ j-j — tan* y V = ntan^ ^^ ^y Ha,Ji'0+ may be regarded as an equation in tan 0, of which the roots are tan^, tan(^ + j), tan(^ + ^) tan j^ + ^-^?^:^| , and may therefore be used for calculating symmetrical functions of these expressions. 112 EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES Examples. ( 1 ) Prove that the sum, of the products of the cosecants of e,e+^-^ 6+^-^^=^, ' n n taken two at a time, is — |n^co«ec^Jnfl, n being an even integer. Using the equation (7), the required sum is the sum of the products of the sines of the angles taken m - 2 at a time divided by the product of all of them ; this is equal to the coefficient of sin^ 6, divided by the term not involving sin 6, or —Tm ;r: which is equal to --7 cosec^inA ' 2(l-coswfl) ^ 4 ^ (2) Prove that eos*^jr + CO«*f7r + CO«*fn- + CO«*|7r = 19/16 and «ec* ^ TT + «ec* fTT + «ec* f 57 + sec* fjr = 1120. If sin 96/sin 6 be expressed in terms of cos 6, and be then equated to zero, the values of cos 6 obtained by solving the equation of the eighth degree so obtained will be COsJtT, COSfTT COsfTT. We notice that cosf7r= — cos JjT, cos J7r= -cosJtt , thus +cosj7r, ±cosf7r, ±cosf7r, +cos^7r are the roots of the equation. We may either use the series (2), or proceed thus : — if sin 9fl=0 we have sin b6 cos 4fl + cos 5d sin 45 = or (sin Zd cos 25 + cos Z6 sin 25) (2 cos^ 25-1) + (cos 35 cos 25 - sin 35 sin 25) 2 sin 25 cos 25 = ; substitute the values for sin 35, cos 25 ... and reject the factor sin 5, then let .•!7=cos* 5, we obtain the following biquadratic in x {(4iB2 - 1) (2ii; - 1) + 2 (4a;2 _ 3^)1 {2 (2a; - 1 )2 - 1} + {4 (2a; - 1 ) (4^ - Sis) - 8 (4*-- 1) {\-x) x} (2a;- 1)=0 or (16a;2-12a;+l)(8a;2-8a; + l) + (e4a;3_80a;2 + 20a;)(2a;-l)=0 or, arranging according to powers of x, 256a;* - 448a;3+ 240a;2 - 40a; + 1 = 0. The sum of the roots of this equation is 448/256, and the sum of the products of the roots taken two together is 240/256, hence the sum of the squares of the ^ . 4482-2.240.256 19 , ,, ,^, „^, roots 18 . .^ = T^ ; also the sum of the squares of the reciprocals of the roots is 40^- 2 . 240, or 1120. (3) Prove that sina+iin2a + gin4a=^'>/7, where a=^iT. We find (sin a + sin 2a + sin 4a)' = sin' a+sin'2a+ sin' 4a. EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES 113 Now the roots of the equation sin T^/sin 5 = in sin 6 are ± sin a, ± sin 2a, + sin 4a ; put ;i;=sin2 6, then the equation in x is found to be 64^-n2x^ + 56x-7=0, hence sin2a+sin22a + sin24a=112/64 = 7/4 ; therefore sin a + sin 2a + sin 4a = ^ v7. (4) Evaluate sin =-= • Writing a = 27r/17, we find by the formula for the sum of the cosines of angles in arithmetical progression (cosa + cos9a+cosl3a + cosl5a) + (cos3a+cos5a + cos7a + cos lla)=-J. Also (cos a + cos 9a + cos 13a + cos 1 5a) (cos 3a + cos 5a + cos 7a + cos 11a) is found, on multiplying out and replacing each product by half the sum of two cosines, to be equal to - 1. The two quantities in brackets are therefore the roots of the quadratic z^+^z- 1 = 0, of which the roots are |^(- 1 ±\/I7). It is easily seen that cos a+oos 9a + cos 13a + cos 15a is positive, and cos 3a + cos 5a + cos 7a + cos 11a is negative, we have therefore cos a + cos 9a + cos 13a+cos 15o=J (Vl7- 1), cos3a + cos5a+cos7a + cos lla= -J(\/l7+l). We can now shew that (cosa + cos 13a)(cos9a + cos 15o)= - J, hence cosa+cos 13a, co8 9a + cos 15a are the roots of the quadratic ^2-i(Vr7-l)a;-i=0, hence cosa + cos 13a = J(-l+V17 + V34-2 ^17) ; similarly we find cos 3a + cos 5a= J ( - 1 - Vl7 + '\/34 + 2 ^'17). Now cosacos 13a=J (cos 12a + cos 14a) = J (oos3a + cos5a) ; and since we have thus found the sum and the product of cos a, cos 13a, we can find each of them. Noticing that cos a > cos 13a, we have cos a = iV {Vl7 - 1 + V34 - 2 V W + 2 Vl 7 + 3 Vl7 - \/T70 + 387T7}. We have then sin tt/I 7 = Vi (^ - cos a) = |V34-2Vl7-2V'34-2\/T7-4Vl7 + 3v'i7-v'i70+38x/17. (5) Shew''- that, if f(x, y) be a homogeneous function of x, y 0/ n — 1 dimensions, f {sin X, cos x) sin (x - ai) sin (x - 02) . . . sin (x - a„) _ '■=™ t{sinar, cos Or) r=i sin (x — Or) sin (a, - aj) sin (a^ — a2) . . . sin (a^ — a„) ' ^ This theorem was given by Hermite in a memoir "Sur rinWgration des Fonctions oiroulaires" in the Proc. Land. Math. Sac. for 1872. H. T. 8 114 EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES The expression on the left-hand side of the equation may be written ,, , ,y , — ; , . — , where t=ta,nx, a.=tana,. (t- aj)[t-a2) ... {t — a„) cos ^ cos ai cos oj . . . cos a„ Now since/(<, 1) is of degree n-1, lower than n, we have by the ordinary method of resolving into partial fractions fji, 1) '•=" /K, I) {t-ai) it-Oi)... {t-a„) r=l (t-ar) («,• - "l) (^r - "2) ••• ("r- «n) _ /(sinor, cos a,), cos .» cos ai cos a2 ... cosa„ sin (:;; — Or) sin (a,.-ai) ... sin (or — a„) ' thus the result follows. Factorization. 86. Since cos n 9 can be expressed as a rational integral function of the nth degree in cos 6, we can express cos nd as the product of n factors linear in cos 6; the values of cos 6 for which cos nd vanishes are TT StT (2n - 1) TT cos 77- , cos 7:— cos ^^ -r — '- — ; 271 ' 2m 2m ' these cosines are all different ; therefore cos ne = A f cos - cos ^ j (cos - cos -~\ (^cos^-cos^— 2^), where .4 is a numerical factor. Since the highest power of cos 6 in the expression for cos nO is 2»-' cos" 0, we see that A = 2"-' ; therefore cos n0 = 2'^' fcos 6 - cos —^ fcos - cos ~ n0 = 2'^' Uos 6 - cos ~\ Uos - , a (2n - 1) ttN — cos ^^ pr , 2n / AT '""■ V2,n—r)'n- , JNow cos 2^= -cos 2^^ , therefore this expression maybe written cos nd = 2«-' (cos^ e - cos^ ~\ Uos^ - cos^ |^ (cos»^-cos^^-^'^l|>^jcos0, when n is odd, and EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES 115 ,37r\ , (W - 1) TT^ COS nd = 2"-' cos^ - cos^ ;r- 1 1 cos'' 6 - cos^ 2nJ 2n, cos^ 6 — cos'' • In ')■ when n is even ; these expressions may also be written cos ndlcos e = 2"-! f sin^ ^ - sin"* e\ f sin^ ^ - sin= e\ when n is odd, and i6 = 2»-i [sin^ ^ - sin^ 0^ ^sin^ ^ - sin^ f3in=(!iZ.^_3in^0V V 2?!. J cos mt sin^(!ir_l)^_sin^0), 2« when n is even. In each of these equations put ^ = 0, we then obtain the theorems 2^^ smj^smji— sm^ — ^r—^ — = 1, 2?i 2m 2n when n is odd, and 2^^ ^sin^^sm^r— sm^ — ^ — = 1, In 2n 2n when n is even. I. ..(15), The positive sign is taken in extracting the square root, since the angles are all acute. If we divide the expressions for cosn0/cos0 or cosm^ by the corresponding one of the products in (15) squared, we obtain the expressions cosnO _ / sin^ cos 6 I 1- ^^^'^£. sin^g 2n sin'' ) ( 1- sin^^ sm^ .(16), when n is odd, and sin^^ cos nd = I 1 1- sin-' 2m; sin'g > • ,37r 1-. 2n sin=^ {n-l)ir ...(17), sin' 2n when n is even. 8—2 116 EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES We may write the theorems (16) and (17) thus : — cosn0/cos^= n (1 T\, \ (16), "' I '^•'^^) where n is odd, and cosn^=n/l- ;;_% \ (17), where w is even. 87. As in the last article, since sin nd/sin is an algebraical function of degree w — 1 in cos 6, we may find a corresponding expression for it in factors linear in cos 6 ; in this case TT 27r (W — 1) TT cos — , cos — ... cos — n n n are the values of cos 6 for which sin nd/sia is equal to zero. These values may be thus grouped + cos — , + cos — ; hence as before sin w^/sin 6 = 2«-i cos 6 ^cos'' 6 - cos" -) f cos" <^ - cos" — ] . . . (cos"0-cos"^-^?^|^), when n is even, and sin nejsai 6 = 2»-i ('cos" 6 - cos" -] Tcos" - cos" — ) . . . (cos^^-cos"^-^^:^], when n is odd. We can write these equations in the forms sin w0/sin = 2"-' cos [sin" - - sin" 0^ [sin" — - sin" ^"j ... (sin"(!^_,in"^), when n is even, and sin n0/8m = 2"-' (sin" ^ - sin" 0\ (sin" — - sin" 0Y.. (-■^-^--•''). when n is odd. EXPANSION OP FUNCTIONS OP MULTIPLE ANGLES 117 We shall shew in the next Chapter that sinw0/sin0 has the limit n when d is indefinitely diminished; hence Vn = 2"2~sin-sin— (18), n n the last factor being sin ^ — - — — or sin — ^-^ — , according as n is even or odd. Hence sinH^/nsin^ = cos0 H / 1 - .!iiif. ^ (19), '■=1 \^ sitf"''^ when n is even, and sin n6'/n sin 6/= U /l--Sili_\ (20), "" I sin^'J -V n / ) when n is odd. 88. The expression cos n0 — cos n^ may be regarded as an algebraical function of cos 9 of degree n, and can therefore be factorised; the values of cos 6 for which the expression vanishes '^ + i\ . r=o i n j ' (2r + 1) TT 2 (n - r) - 1 now cos ^^ — =cos — ^^ — TT, n n hence the factors are equal in pairs, except that when n is odd we have the single factor a;'' + 2a; + 1 ; hence a;'* + 1 ='^"11^ ja;^- 2a; cos ^?^^±^^ + ll (26), when n is even, and EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES 119 .-+l^(. + l)^'r^'L-2.cos (^--^^)- + ll-(27). , . ,, r=0 ( n J ' ' when n is odd. 91. In the equation (22) put a; = 1, we have then l-cos??i9=2"-i n h-cos(6l+— - r=0 [ \ ' n changing into 26 this becomes sin^ nd = 2»-= sin» 6 sin» (^ + ^) sin^ (^ + ^) • • • ^i"' (^ + ^'') ' n — Itt or sin?i0= + 2»-isin(9sinf0 + -)sinf^ + — V..sinf6l + where the ambiguous sign is as yet undetermined. It has been shewn, in Art. 51, that the form of the expansion of sin nd in terms of sin 6 and cos 6 is definite ; the sign of the product on the right-hand side is therefore always the same ; put then 6 = 7r/2w, the sign to be taken is clearly positive as each factor is positive. We have therefore sin «6'=2»-isin6'sinf6' + ^)sin(^-|-^V..sinf6' + ^-^)...(28). In (28) "change into + 7r/2w, we thus obtain cosr,^=2"-sin(^ + j)sin(^+|^)...sin(0+?!^)...(29). The theorem (18) can be deduced from (28) by putting 0=0, and taking the square root. In a similar manner, the theorem (15) may be deduced from (29). Examples. (1) Prove that ifn be an odd integer, sinn6 + cosn6 is diuisible hy sin 6 + cos d, or else hy sin B — cos 6. Let Un = sin n6+ cos n6, then M„H-M„_4 = 2cos25.M„_2 = 2(cos2 5-sin2 5)M„_2. Hence, if «« - 4 is divisible by cos 6 + sin 6 or by cos 6 — sin 6, «„ is divisible by the same quantity. Now Ui=ain6+cos6, hence %, u^, M13 ... are all divisible by sin d+ cos 5; also «_j = cosfl — sin5, hence U3, itq, Mh... are all divisible by cos 6 - sin 6. (2) Factorise tan n6 - tan nn. w 1, i a i sin n {6 -a) We have tan no - tan na = ^^ . cos no cos na 120 EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES In the formula (28) write a — 5 for 6, we then have sin«,(fl-a) = (-l)"-i2"-i n sin(d-a ) r=0 V "■ / T=n~\ / r7r\ ( f Tir\\ = (_l)»-i2n-ioos''fl n coslaH — 1 jtan^-tan (a + — jv =(_l)n-icos"dsinm(a+|) n jtan^-tan (a + — U . Again, we have from (16) and (17) r=i(»-i) / gin2 Q s r=bi I sin2 6 cos»ifl = cose n / 1 — — — 7T— — TT— 1 or n /I — - 1 sin^(2^; -^ sin^(?!:=l)^ 2>» ' ^ 2/1 according as n is odd or even. Now 1 — ;— v7: = cos* 6\\ —-, — rr; 1 . hence the sm^ ^ \_ tan^ pj expression for cos n6 may be written cos"^ n /I t;: rr— \ or cos"fl n /1-- -1 I tan^(^-^)-J -l tan^ (^'-^>- We have therefore sinmfa+^l n -^tan 5-tan ( aH ]\ XT /l tanM \ tanm5-tan7io = (-l)" ; — s-^ cosna „ /- tan'^fl the product in the denominator being taken up to r=^»i or J(?i— 1), according as 71 is even or odd. EXAMPLES ON CHAPTER VII. 1. Prove that, if n be an odd positive integer, and a= ir/n, tanjM^ = (-l)i*"~^' tan(^tan((|) + a) tan(^ + «, — la), and 7itan«(^=tan<^ + tau((^ + a) + +ta,n{(j) + n-la). 2. Prove that sin6g-cos5fl 1 -2sin 2g-4sin^2g sin 5(9 + cos 55 ~^"^ *'^h+2sin2fl-4sin2 2e" 3. Prove that n.cotrea = cota+cot (a + - j + +cot (aH — ^^-^), n being an integer. EXAMPLES. CHAPTER VII 121 4. If (^=7r/13, shew that oosd>+cos3(^ + cos9(^ = i(l+\/l3), c I— J € f /> . / ' ■ and cos5<^ + cos7(^ + cosll + ^ sin^0+ "~ " ^ 7177 = C0S — — COSTll 10. Prove that tana + tan(^-a)+tang+a) + tan(g-a)- to %n terms is equal to 2m coseo %na. 11. Prove that . 277 . 477 . n- 477 . 71-277 . JW7 sm T— sm =- sm — - — sm --r — am ^r- -- 2« 2m. 271 2?i 2m. where n is an even positive integer. 12. Prove that Y 2»-i' .,(n-l)jr sin" TT- sin" -- sm" ^ — ain2 - 2n 2?i 2m-l . , 77 ■ . 2 277 . „(n-l)7r' sm^- sin^- sm^^- '— 2m -1 2m- 1 2m -1 where n is any positive integer. 13. Prove that ainm^sinm^ sin ^ sin 6 = 2"-i|co8(<^-d)-cos('<|) + (9+^)| |cos((^-5)-cos('(^+e+^^l . the number of factors on the right-hand side being m - 1. 122 EXAMPLES. CHAPTER VII 14. Prove that to sin »5 — »i sin »ifl is divisible by sin'd, if m and n are positive integers such that »i or » is even. 15. Shew that if m is a positive integer, sec^^.d + cosec^il can be expressed in a series of powers of oosec ZA. -,„ -r, ,, , sin2asin4a sin(2?i-2)a 16. Prove that n= -. ^-^ ■■ — rk rr— i smaSinSa sm(2?i-l)a where a=ir/2n. 17. Prove that sin^ X _ sin^ a (1) sin {x - a) sin {x - b) sin {x - c) sin {x - a) sin (a - 6) sin (a — c)' . sin A- _ cos (a; - a) sin as sin (a; — a) sin (x- 6) sin (^ — c) sin (a- — as) sin (a -6) sin (a — c)" 18. Prove that the product of / , 4jr\ , , / , 4j^^n:jr\ 1+cosa, l+cos( a-^ I l+cos( oH I \ nj \ n J is 22-" {(-l)i"-cos^«aP or 2i-''(l+cos?ia), according as « is even or odd. 19. Prove that »2=(versin^) V (versin|^) ' + (versing) ' + , n terms being taken on the right-hand side. 20. Prove that (tan 7 J° + tan 37^° + tan 67^°) (tan 22J° + tan 52 J° -(-tan 82^°) = 1 7 + 8 ^/3. 21. Shew that, if m is odd, tan?7j^ = tan<^cot (tj> + -^\ta.\i(<^+— 1 22. If 28a= TT, shew that ^14 = 2'' sin a sin 2a sin 13a, and cos2a-l-cos6a-t-cos 18o=Jv'7. 23. Prove that tan — tan ;r- tan — ;r — = 1, 2m 2«. 2m ' n being any positive integer. 24. Prove that / , 2jr\ , , / 2n^n\ cosec^c-l-cosec \x-\ \ + +cosecl a;H 1 = n {cosec nx + cosec {nx + n) + -1- cosec (nx + n-l n )}. EXAMPLES. CHAPTER VII 123 25. Prove that, according as n is even or odd, 2(l + cosmfl) or (l + coswfl)/(l+cosfl) is the square of a rational integral function of 2 cos 6. Shew that 1 + cos 9(9=(1 +C0S 6) (16 cos* 5-8 cos^ 5- 12 cos^ 5 + 4 cos 6+\f. 26. Prove that 2''~^cos''5 — cosmfl is divisible by l + 2cos25, when n is of the form Qm— 1, and by (1 + 2 cos 25)^, when n is of the form 6»t+ 1, to being a positive integer. Prove that 210 cos" 5- cos 115 = 11 cos 5 (1 +2 cos 25) {(1 +2 cos 25)3 + (l + 2 cos25) + l}. 27. Prove that, if n be an odd positive integer, and tan {\n+\^) = tan" ( Jtt + ^5), 1, ,, (l +sin2 5cot^- — I r=J(?l-ll I M I then sin (^= re sin 5 n '•=' Il+sin2 5tan2 — 28. Shew that any function of the form /(sin 6, cos 5)/^ (sin 5, cos 5), where /and ^ denote rational integral functions of degree w, containing cos" 5, can be expressed in the form 4n sin ^ (5 - a)/ll sin ^ (5 - a'), where A and the quantities a, a' are independent of 5, and there are 2m factors in the numerator and 2re in the denominator. ^„ , „ ,. acos25 + 6cos5+esin5 + rf , j • iu- r If the function -; „. , ,, . . , ^-^ . ^ ' ^ expressed m this form, a COS25 + cos5 + c sm5 + o! prove that 2a and 2a' are even multiples of it. 29. Prove that . 3n- ... 277 /7^ tanYT- + 4sin — = vll. 30. Prove that 2''sin7 5 + sin75 26cos7 5-cos75 = tan 5 tan2 {e + '^ tan^ (s - ^ CHAPTER VIII. RELATIONS BETWEEN THE CIRCULAR FUNCTIONS AND THE CIRCULAR MEASURE OF AN ANGLE. 92. We shall now investigate theorems which assign certain limits between which the sine, cosine and tangent of an angle whose circular measure 6 is less than ^ir must lie. The first theorem which we shall prove is that if 6 he the circular measure of an angle less than ^ tt, then sin 6 < 6 < tan 6, unless 6 = 0. Let AOB = AOB' = 0; and let TB, TB' be the tangents at B and B', and let SA8' be the tangent at A. In Art. 11, it was shewn that the length of the arc AB does not exceed AS+SB\ and thus the arc BAB' does not exceed B8 + B' S' + SS' , .axiA therefore arc BAB' BA> BC. Consequently we have BG/OB < arc BA/OB < BT/OB. Now = a.TcBA/OB, sin 6 = BC/OB, and tan 6 = BT/OB; THE CIRCULAR MEASURE 125 therefore sin 0< ^< tan0. If 6 had been greater than \-n-, T might have been on the other side of 0, and the inequalities which we have employed would not necessarily hold. Since sin < ^ < tan 6, we have 1 < 6 Java. 6 < sec 6 ; now sup- pose 6 to be indefinitely diminished, then the limit when ^ = of sec is 1 ; hence also the limit of 0/sin 9, when 6 is indefinitely diminished, is unity. Since -2j— = (6 cosec 6)~^, and — ^ — = sec 6 . (0 cosec 0)-^, u o we have the theorems that the limits of — -^r- o,nd — ■„ — , when •'6 6 6 is indefinitely diminished, are each unity. The theorem may also be proved thus : — The triangle OAB, the sector OAB, and the triangle OBT are in ascending order of magnitude ; and iiOAB=:\OA.BC=\OA^sm6, also sector OAB=\OA''.e, and AOBT=iOB . BT=\OB^ . tan 6, therefore sin 6 ^d, we have sin^d>^6 cos ^d, hence 2 sin ^ ^ cos ^ ^ > cos^ ^ 0, or sin 6* > 6/(1- sin" i^). Now sm^8<{^ey, hence sin > 0(1 - i^"), or sin (9 > 0- i(?». Also cos 6 = 1 — 2 sin" ^0, and this is greater than l — 2(^6y; or cos 0> 1 — ^0"- Also, since sin^d >^d — K^Of, we have cosee-\eK Also cose = l-2sitf|0; therefore cos ^ < 1 - 2 (|^ - i^O^y < 1 - ^^^ + ^6\ Hence sin 6 lies between 6 and 6 — ^6^, and cos lies between 1 — ^^'^ and 1 — ^0^ + -^6'^, the angle being less than ^ir. We have also tan 5 = sin 0/cos 6, hence or ta.ud>e + ie^ + ie^-^ff', therefore tan 6x9 + ^^3. Euler's product. 96. We have sin ^ = 2 sin^0cos|^, . . . sm2 = 2sin2jCOS2i, . : 2 Sin 2-3 cos 2^ , = 2 sin 2^ cos 2^, hence sin ^ = 2" cos ^ cos 2^ • • . cos — sin g^ . 128 THE CIRCULAR MEASURE Q Now when n is indefinitely increased, the limit of 2" sin ^ is 6; hence the limit, when n is indefinitely increased, of the product e e 6 ^ . sin^ cos 2 cos 2i cos 2i..- cos 2^, is —^. In this product put 6=^7r,-we then obtain Vieta's expression for w, viz. 77 2 ■ 2 2 Examples. (1) Prove that as 6 increases from to \ir, —^ continually diminishes, and — 3 — continually increases. 6 -m- 1 11 1 ii , sind sin (5 + A) ,, . . We shall shew that -3- > — J' , ; that is ,«,,..,,,.., , /I • ,v tanfl sinA ^ ' ^ 6 fl+(l-cosA)fl ,, ,tan^ , sinA , sinA sin A . , , Now we know that — j- > 1 >—j— , and ^j— > -t-— r; ttt, since 1 - cos A A A A+(l — cosA)5 is positive, hence the inequality is established; thus —3— diminishes from 1 to 2/n-, as 6 increases from to ^tt. We shall next shew that tan(g+A) ^ta|j^ or esin(5+A)cos^>(^+A)sinflcos(5 + A); a+A p this is equivalent to . ■ , ,.„ /„.v sinA sin 6 ,. ,. dainh>hain0 coa{d + h), or — t— > . cos (6+h); now we may suppose h<6, hence by the first theorem sin A sin fl , ,, . sin 6 sin^ ,. ,. — J— > , and therefore „ > . cos (fl + A). no 00 Thus as 5 increases from to ^tt, increases from 1 to cjo . The theorems may be seen to be true by referring to the graphs of sin 6, cos 6 given in Art. 32 ; it will be seen that in the first case the ratio of the ordinate to the abscissa diminishes, and in the second case increases, as 6 increases from to ^TT. (2) Prove that the eqimtion tanx=\x has an infinite number of real roots, and find the approximate values of the large roots. In Art. 32 we have drawn the graph of the function tan;j;; draw in the same figure the graph of \x, this is a straight line through the point 0. The '2 * Tm+.+.i'nor p.nR« = l_ siTlw = . Xti small, then - coi;i/='\y+(2k + l) -^ ; putting aosy = \, siny=y, and neglecting THE CIRCULAR MEASURE 129 straight line will obviously intersect each branch of the graph of tan^, and the values of x corresponding to these points of intersection are the solutions of the equation. There is therefore a root of the equation between x={ik-\)'!^ and (2^+1) |, where k is any integer. If hX be large, then (2^ + 1) — is obviously an approxi- mate solution ; to find a nearer approximation let i» = (2/fc + l)g -fy, where y is small, then - y^, we have -l=(2*+l)^y, or y= -(2^+1)^^, therefore *=(2*+l)| - j^,^^^ is the approximate solution. To find a still nearer approximation, neglect y\ 2 putting y= - T^j — fvx— in the terms which involve y% we have i3,2-l = {x3/ + (2A+l)^}y=Xy2+y(2i+l)?^^, Xtt 4 hence y(2A + l)^= -l + (i-X) ^2^_l_ ^^2x2^2' fore *=(2*+i)|-(-2i:i^+(i-^)(-2i:j:i33x3^- 1 f) B 6 (3) Provethat ^ = cote+itan^+itan- + itan-^+... adinf. It can easily be shewn that J cot - - cot 5 =^ tan ^ , /i S B hence also i cot - - J cot ^ = J tan - , 1 fl 1 .6 1 6 hence by addition we have 1 ^1 Now when n is indefinitely increased, the limiting value of 22^ cot ^ is ^, hence the Umiting sum of the series is 3— cot d. If we put 6=^iT, we obtain the theorem -=Jtan 1^ + J tan l + ^tan ^+... . H. T. ^ 130 THE CIRCULAR MEASURE The limits of certain expressions. 97. When n is indefinitely increased, the limits of each of . the expressions cos - , is unity ; hence the limiting values n ^ ,. / sm - \ of (cos- j , I —^ j are also unity provided T is any number which \ n / in independent o/n ; but if r is a function /(n) of n, which becomes . / ^\/(») /''"«\ infinite when n does so, the expressions (cos— 1 , I — ^ I are \ n / undetermined forms of the class 1", and the values of their limits depend upon the form oif(n). To determine the limiting values of I cos - 1 , we have, denoting the expression by m, log, w = i/(n) log, \\ - sin^ - It will be assumed as a known theorem that the limit, when x becomes indefinitely small, of _ sl5^ 1 is — 1. Then, since log,(l-sin=-j Q loge u = \f(n) sitf - . ^——Q sin^- n a the limit of log^w is equal to that of ^/(w)sin^-, with its sign it changed, provided this latter limit exists. We can find the limit of loge W) and therefore of u, in the following cases : — 6 G (1) If f{n) = n; then /(w) sin'' - = w sin - . sin - , and the n n n limit of n sin - is 0, and that of sin - is zero ; therefore the n n limit of logjM is zero, or that of u is 1. THK CIKCCJLAR MEASURE 131 (2) If /"(n) = w" ; then /(«) sin" - = ( « sin - j , of which the limit is 0". Hence the limit of log^w is — i^", or that of u is e~i''. (3) f{n) = nP, where jo>2, then /(n)sin''- = n^~^ (n sin- j , and this increases indefinitely as n does so. Therefore the limit of loge M is — 00 , hence the limit of u is zero. / . ey- . /sin- \ sm- 98. To find the limiting value of I —3— 1 ; since —5— is less \ n / n .9 / ■ 6I\» sin - - / sin - \ than 1 and greater than r. or cos - , the limit of I — rr- I lies ^ tan^ ^ [ ^- n \ n / between 1" or 1, and (cos-j ; thus from case (1) in the last Article, the limit of the expression is unity. We see also that sin - \ / sin - \ the limiting values of | —3— | and of I —^ I (p > 2) lie n I \ n / between 1 and e~i^\ and between 1 and 0, respectively. Series for the sine and cosine of an angle in powers of its circular measure. 99. In the formulae (39), (40) of Chapter iv. write ^ for J, and let x = n9, we have then sin a; = n cos""' ^ sin ^ - - ' ^'^ ~ g^"" ~ ^^ cos»-= 6 sin= 6+... + (_ 1 ). ^i^^nAi^^Z^ cos— sin-+^ 0+..., cos X = cos" - !!i!|llL) cos"-" 0sm^0 + ... _^ n{n-l) {n-2s + l) ^^^„_,, Q^^^.e + .... ^ (2s) ! 9—2 132 THE CIRCULAR MEASURE We may write these series in the forms sina; = a;cos"-'^(— ^j ^^ ^^ ^cos"~'^(— ^1 + ... cosa; = cos"0 ^ ^ cos^-'g I— ^1 +... The number of terms in each of these series depends upon the value of n, and increases indefinitely as n is indefinitely increased. In order to obtain the limits of the expressions when n is in- definitely increased, it is necessary to replace each of these series by a series in which the number of terms is fixed, and does not increase indefinitely with n. The ratio of one term of the series for sin x to the immediately preceding term is (a;-2r+lg)(a;-2r-|-2g) / tang y. (2r-l-2)(2r + 3) V '+l case is seen by observing that the ratio „ ,„ — -^r\ of -7^ — -— r-, to •' ^ 2r(2r + l) (2r + l)! T^ =Yr may be made arbitrarily small, for any. fixed value of x, by taking r great enough. Similar reasoning applies to the expression for cos x. Examples. (1) Expand cos^x in powers of x. We have cos' a; = J (cos 3a; + 3 00s «); expanding cos 3a;, cos a; in powers of 3271 _i_3 X, we find for the general term in the expansion of cos^a;, ( — 1)" . . x^. It will be seen that any integral power of cos a; or sin x, or the product of two such powers, may be expanded in powers of x by putting the expression into the sum of cosines or sines of multiples of x, (2) Expand tan x in powers of x as far as the term in xJ. We have tan «;={.;- J + 3J - ^} {l-|Vg-^} \ leaving out terms of higher order than x'. Expanding the second factor, we have f «3 , a;6 a;M r, , /«2 .■B* xfi\ (a? a;*\2 lx\^~\ multiplying out and coUegting the coefficients of the terms up to x', we find \iWiX=x-'r\3?-\-^x^-'r-i^x'. /„v n- 7 xi 7- •. . «m (tore x) - tore (siji x) , (3) Fxnd the hrmt of ^ —^ ^ -, when x=0. The numerator of the expression is equal to tan a; - ^ tan' a; + 1 J^ tan* a; - 5 5^5 tan' « - sin a; - 1 sin' « - ^ sin« a; - ^ sin^ a;, using the expansion obtained in the last example. This is equal to {x+i^+^-g^+^-gX^)-ix^{l+x^+i\xi)+~{l+ix^)-^ rejecting all terms of higher order than x''; this expression reduces to - ^x''. The limit of the given expression is therefore - 1/30. THE CIRCULAR MEASURE 135 A relation between trigonometrical and algebraical identities. 100. From any trigonometrical identity in which the angles are homogeneous functions of the letters, a series of algebraical identities may be deduced, by expanding the circular functions in powers of the circular measure of the angles, and equating the terms of each order. Thus for example, in the formula sinasin6 = ^{cos(a — 6) — cos(a+ 6)}, expand each of the sines and cosines and equate the terms of the second order, we have then ab = ^{(a + by-{a-by]. In Articles 44 and 47 of Chapter IV., we have given a number of examples of analogous trigonometrical and algebraical identities ; in each case the algebraical identity is obtained, as we have above explained, from the trigonometrical one. For example, in example (11), Art. 47, which may be written S sin^ a sin (6 + c — a) — 2 sin a sin 6 sin c = sin (6 + c — a) sin (c + a—b) sin (a + b — c), if we equate the terms of the third order, when the sines are expanded, we obtain the analogous algebraical identity ^a" (b + c — a) — 2abc = (b + c — a) (c + a — b) (a + b — c). EXAMPLES ON CHAPTER VIII. 1. Prove geometrically that tan 5 > 2 tan ^6, where fl < Jjr. 2. Trace the changes in the value of tan 36 cot^ 6, as d increases from to Itt. Shew that 17 + 12^2 is a minimum and 17-12^/2 a maximum value of the expression. 3. Prove that tan 35 cot 5 cannot lie between 3 and 1/3. 4. Prove that 6>7r. 3i where 6 5 tan 35, if 6 lies between and w/lO. 6. Shew that the limiting value of -^-Tq ~ Z2 > ^^^"^ 6=Q, is \. 136 EXAMPLES. CHAPTER VIII 7. Prove that sin (cos fl) "4 be all nearly equal; shew that 6 is very nearly equal to i(ai + "2+03 + «4)- 15. Sum the series 6 6 6 6 6 6 cos- + 2cos^coSg2+22 cos - cos r2C0Sp+ to n terms. 16. Prove that the sum to infinity of the series tan - sec a; + tan =5 sec ^r+tan r=sec;ri+ is tan x. 2 2^ 2 2^ 2* 17. Shew that 6 6 6 6 6 6 — sin6cos6=isai6siv?^-\-'i^ sin -sin2- + 2' sin -sin2- + ad inf. 2 2 4 4 8 ■' 18. Prove that tan 6 = J J J ' cot - - cot -: - cot ^ — 2 4 8 2 sm-+sinjn, + + sm — ■ --- ■ 19. If 6 and so on, shew that a(„ = 6„=i i-. cos ^Y Shew that the vahie of tr may be calculated by means of this formula. 21. Find the limiting value of the infinite product (sinflcos^5)4(sini5cos^6)i(sinJ5cosp)^ 22. If tan 5 = 45, the value of 6 between and Jtt will be 2 V^n- 24773 ^48077^^ )• 23. Prove that sin5 | 1 + 2 cos 5 1 24. Prove that ?^^^^^^ = (2cos5-l)(2cos25-l) (2cos2»-i5-l). 2 cos 5 + 1 ^ '^ ' ^ 1 . 6 2" 2 e 1 25. Sum to n terms the series an 92/1 j__ 23 5 log tan 25 + =2 log tan 2^ 5 + -3 log tan 2^ 5 + . 26. Having given that the limiting value, when 5=0, of .^_ . ^„ is neither zero nor infinite, find n. 27. Find the limit, when x=0, of 1 — cos 2a' + cos ix - cos Qx + cos 8^- — cos 10.» — cos 14a; + cos \Qx 3 - 4 cos 2^ + cos 4a.' 28. Prove that the sum of the infinite series whose Hh term is (-^)'""2^i (2F^ ^^ ^«na-+i). 29. If e be very small, and 0=5-2esin5 + je2sin25, shew that 5 = (/) + 2e sin (^ + 1 e^ sin 2(^, nearly. 30. If y=z-\-k am (z+ka), expand '■ in powers of the small quantity k, as far as the term in k^. 31. From the trigonometrical identity sin (d-b) sin (a-c)+sin (6-c) sin (a — d)+sm(c-d) sin (a — 6)=0, deduce the algebraical identity (d-b){ci-c){{d-b)^ + {a-cy} + {b-c)(a-d){{b-cy+(a-df} + {c-d)ia- b) {{c - df+{a - bf] = 0. 138 EXAMPLES. CHAPTER VIII 32. Prove that d> differs from ^-, ^ ^'° '^, , by -h(b^ nearly, (b being a ^ 2(2 + cos2(p) •' ^"^ •' ^ . ° small angle. (Snellius' formula.) 33. Find the circiilar measure, to five places of decimals, of the smallest angle which satisfies the eqiiation sin {x -\-^ir) = \0 sin. x. 34. Solve the equation (sib 5)"™^* = 6, approximately, where a is positive and not large, and 6 is kiiowu to be nearly equal to a, which is itself not very small. 35. Shew that there is only one positive value of 6 such that ^ = 2 sin 6, and find its value to two places of decimals by means of a table of logarithms. 36. In the relation a sin-^.« = 6sin~'y, where a and b are integers prime to each other, prove that there are 26 values of y for each value of x, unless a and 6 are both odd numbers when there are h values. 37. Assuming that if a be the acute angle whose sine is — , sin 7a must be ^ , prove that cos a - cos = exceeds ^ ,„ by less than '0000005. CHAPTER IX. TRIGONOMETRICAL TABLES. 101. In order that the formulae of Trigonometry may be of practical use in the solution of triangles and in other numerical calculations, it is necessary that we should possess numerical tables giving the circular functions of angles, so that from these tables we can find to a sufficient degree of accuracy the functions corresponding to a given angle, and conversely the angle corre- sponding to a given function. Such tables are of two kinds, (1) tables of natural^ sines, cosines, tangents, &c., in which the numerical value of the sines, cosines, tangents, &c., of angles are given to a certain number of places of decimals, and (2) tables of logarithmic sines, cosines, tangents, &c., in which the logarithms to the base 10, of these functions, are given to a certain number of places of decimals. The latter kind of tables are those which are now used for most practical purposes; in nearly all such tables the logarithms are all increased by 10, so that the use of negative logarithms is avoided; the logarithms so increased are called tabular logarithms and written thus, L sin 30° ; so that L sin 30° = 10 + log sin 30°. Calculation of tables of natural circular functions. 102. We shall first shew how to calculate tables of the natural circular functions, which will give the values of these functions accurately to a certain specified number of places of decimals, for all angles from 0° to 90°, at certain intervals such as 1' or 10". We shall first calculate the sine and cosine of 1' and of 10". 1 Logarithms were formerly called " artificial" numbers, thus ordinary numbers were called " natural " numbers. 140 TRIGONOMETRICAL TABLES (1) To find sinl', cosl'. Let 6 = — r- — — - denote the circular measure of 1', then 180 X 60 ^ ^ 3141592653589793... ^ .ooo290888208665 10800 to 15 places of decimals, hence 16'' = \ (-0003)' = -000000000004 *to 12 places of decimals. Now from the theorem in Art. 95, sinl' lies between 6 and 6 — \d^, and these numbers only differ in the twelfth decimal place, therefore to eleven places of decimals ■00029088820 is the correct value of sin 1'. We find also 1-^61^ = -999999957692025029 to 18 decimal places, and ^^e^ = ^ (-00029 ...)* = "00000000000000029 to 17 decimal places. Now cosl' lies between 1-^^" and 1-^0^ + ^6^; and since these two numbers differ only in the 16th decimal place, we have cos 1' = -999999957692025 correct to 15 decimal places. (2) To find sin 10", cos 10". If 0= , the circular measure of 10", o4o00 ~ we find = -000048481368110, to 15 decimal places, ^0^ = -000000000000021, to 15 decimal places, hence the two numbers 8 and — 10" agree to 12 decimal places, therefore sin 10" = -000048481368, to 12 decimal places. Also ^0* is zero to 17 decimal places, thus cosl0"=l — ^5^ or cos 10" = -9999999988248, to 13 decimal places. 103. The formulae sin nA = 2 cos A sin (n — 1) A — sin (n — 2) A, cos nA = 2 cos A cos (n — 1) A — cos {n — 2) A, enable us to calculate the sines and cosines of multiples of 1', or of 10". Let ^ = 10", 2 cos 10"= 2 -A; where A; = '0000000023504, then the formulae may be written sinnA— sm(n —1) A={siii(n — 1) A — sm(n—2) A] — ksm{n — l)A, cosnA— cos (n — 1) A = {cos (n-l) A — eos(n — 2) A] — kcos (n—1) A ; TRIGONOMETRICAL TABLES 141 if in these formulae we put n = 2, we can calculate sin 20" and cos 20". We can now by letting n = 3, 4, 5, ... calculate the dififerences sin nA — sin {n — 1) A, cos nA — cos {n — 1) A, when the preceding dififerences sin {n — \) A — sin {n — 2) A, cos{n — l)A — c,os{n — 2)A, and also sin(n — 1)^, cos(n — 1)^, have been found; hence these differences can be found by a continued use of the formulae; we can then find Bva.nA, cosnA, and thus we can form a table of sines and cosines of angles at intervals of 10". We have k = -000000002354, thus in calculating hsm{n — V)A, kcos{n — l)A we need only use the first few figures of the value of sin {n — 1) A, cos {n — \)A. 104. When sin nA, cos nA are thus calculated by successive applications of the formulae, the errors arising from the use of approximate values of sin A, cos A will accumulate during the process ; it is therefore necessary to consider how many places of decimals must be used during the process, in order that with assumed values of sin A, cos A, correct to a certain number of places of decimals, we may obtain values of smnA, oosnA which wiU be correct to a prescribed number of places of decimals. Suppose m the number of places of decimals to which sin A, cos A have been calculated, and suppose that r is the number of places of decimals that is retained in the calculation of the sines and cosines of successive multiples; let «„ be the value of sin nA or cos nA, obtained by this process, and u^+Xn the corresponding correct value, we have then W„+:»„ = (2-A)(«„_l+«„_l)-K_2 + «„-2), also M„= (2 — /;') M„ _ 1 — M„_ 2, where h' is the approximate value of ^ to ?• places of decimals; let {k-k')Un-i=yn, we have then K„ = (2-*)M„_l-M„_2+y„, hence a;„ = (2-^) ^•,j_i-.J7„_2-y» or ii;„=2^„_i-^„_2-2„, where Zn=yn + *'«n-i; this may be written («„ - .»n - 1) = (■^n - 1 — •''n - 2) - ^n i whence (An-l-'*n-2) = ('^n-2 — ■'^n-3)~^n-l therefore .a?n-«n-i=^i-(^2 + «3+-"+«i.) ; the number kx^ _ j is very small compared with 2a;„ _ 1 , hence y„ + ^^„ - 1 differs insensibly from y„, hence each of the numbers e^, Z3...i!„ is less than I/IC, therefore their arithmetic mean 5„ is less than I/IO"", thus Xn-«:n-l'=«!i-{n-l)6n, •2'«-l-*'n-2 = .»l-(»-2) 5„_i, ^2~^l=*"l~^2) ^„ = »M?i- ((92 + 2^5 +...+ ?4^5„); 142 TRIGONOMETRIOAL TABLES now 62, 63... 6n are each numerically less than 1/10'', hence -(^2 + 253+...) is less than ^n (m- 1)/10'', or n n(n — 1) ^'•<10S+ 2.1(r ' «'fo'^io" ■''"^l^ + OO? ^''^• If in this formula >n = 12, ?!.= 10800, - 108 5832 •*»<10io+io?^ < -0000000108 +00 5832, where there are r-8 zeros in the last decimal, hence if r=15, *■„< -00000007, or M„ is correct to seven places of decimals; now 10800 x 10" = 30°, hence the sine or cosine of 30° will be found correct to seven places of decimals if when calculating the sines or cosines of the multiples of 10" up to 30° we retain 15 places of decimals throughout the calculation. The formula (a) may be applied in all such cases to determine the number r, so that x^ may be zero to a certain number of decimal places'. Example. Prove that in order to calculate the sines and cosines of multiples of 10" up to 45°, correct to 8 places of decimals, the values of sin 10", cos 10" being known to 12 decimal places, it is necessary to retain 17 decimal places in the calculation. 105. When a table of sines and cosines of angles at intervals of 10", or of 1', is required, it is only necessary to calculate the values for angles up to 30°, we can then obtain the values of the sines and cosines of angles from 30° to 60°, by means of the formulae sin (30° + A) + sin (30° -A) = cos A, cos (30° -A)- cos (30° + ^) = sin A, by giving A all values up to 30°. When the sines and cosines of the angles up to 45° have been obtained, those of angles between 45° and 90° are obtained from the fact that the sine of an angle is equal to the cosine of its complement, so that it is unnecessary to proceed in the calculation further than 45°. The method of calculating Tables of circular functions, which we have explained, is substantially that of Rheticus (1514 — 1576); his tables of sines, tangents, and secants were published in 1596, after his death. The earliest ' This article has been taken substantially from Serret's Trigonometry. TRIGONOMETRICAL TABLES 143 table is the Table of chords in Ptolemy's Almagest, for angles at intervals of half a degree. Historical information on the subject of Tables will be found- in Hutton's History of Mathematical Tables ; see also De Morgan's Article on Tables in the English Encyclopaedia: The verification of numerical values. 106. It is necessary to have methods of verifying the correct- ness of the values of the sines and cosines of angles calculated by the preceding method ; this may be done by the following means : (1) We. have formed in Art. 66 a table of the surd values of the sines and cosines of the angles 3°, 6°, 9°... differing by 3°; we can therefore calculate the sines and cosines of these angles to any required number of places of decimals, then the values of the functions obtained by the method of calculation above explained may be compared with the values thus obtained. If necessary, the values of the sines and cosines of angles differing by 1° 30' may be obtained by means of the dimidiary formulae, and we have thus a still closer check upon the calculations. (2) There are certain well-known formulae called formulae of verification, these are cos (36° -h ^) + cos (36° - ^) = cos ^ -I- sin (18° + ^) -f- sin (18°- ^), sin A= sin (36° -t- ^) - sin (36° - ^) + sin (72° - ^) - sin (72° -[- ^) (Euler's formulae), cos A = sin (54° + A) + sin (54° -A)- sin (18° +A)- sin (18° - A) (Legendre's formula). The verification consists in the substitution of the values obtained of the functions in these identities. Tables of tangents and secants. 107. To form a table of tangents, we find the tangents of angles up to 45° from the tables of sines and cosines by means of the formula tan A = sin Ajcos A ; the tangents of angles from 45° to 90° may then be obtained by means of Cagnoli's formula tan (45° + A) = 2Un2A + tan (45° - A). A table of cosecants can be formed by means of the formula coBecA = ta.n\A + (io\iA, and a table of secants by means of the formula sec J. = tan J. -^ tan (45° - 1 4 ). 144 TRIGONOMETRICAL TABLES Calculation by series. 108. A more modern method of calculating the sines and cosines of angles is to use the series in Art. 99; if we put in IT 1 x= — .-z- we have n 2 . (Tn sm — \n 9o°^-f^ z^Vi/"- iW^lCTi !rY_ We thus obtain the formulae sin (m/re 90°) = 1-57079 63267 94896 61923 13 min -0-64596 40975 06246 25365 58 m^jrfi +0-07969 26262 46167 04512 05 m^jn^ -0-00468 17541 35318 68810 07 m^j'n? + 0-00016 04411 84787 35982 19 m!>jn? -0-00000 35988 43235 21208 53 mH/^" + 0-00000 00569 21729 21967 93 toIS/w" -0-00000 00006 68803 51098 11 mW/wis +0-00000 00000 06066 93573 11 m"ln?^ -0-00000 00000 00043 77065 47 m^ln^ + 0-00000 00000 00000 25714 23 mP^j-n?^ -0-00000 00000 00000 00125 39 m^ln^ +0-00000 00000 00000 00000 52 rn^jn^ cos (m/n 90°) = 1-00000 00000 00000 00000 00 -1-23370 05501 36169 82735 43 nfij'n? + 0-25366 95079 01048 01363 66 toVji* -0-02086 34807 63352 96087 31 mPj-nP +0-00091 92602 74839 42658 02 m^l'n? -0-00002 52020 42373 06060 55 TOi«/mi« +0-00000 04710 87477 88181 72 m^jtiP -0-00000 00063 86603 08379 19 m"/»" + 0-00000 00000 65659 63114 98 m}'^j'n><> -000000 00000 00529 44002 01 rn^^j'n}-^ + 0-00000 00000 00003 43773 92 m^jn^^ -0-00000 00000 00000 01835 99 rn^ln^ + 0-00000 00000 00000 00008 21 m^j'n?* -0-00000 00000 00000 00000 03 m^jn^ TRIGONOMETRICAL TABLES 145 Since we need only calculate the sines and cosines of angles up to 45°, the fraction mjn is always taken less than |, so that very few terms of the series suffice for the calculation to a small number of decimal places. These series are taken from Euler's Analysis of the Infinite, where they are given to six more decimal places. Logarithmic tables. 109. When tables of natural sines and cosines have been constructed, tables of logarithmic sines and cosines may be made by means of tables of ordinary logarithms which will give the logarithm of the calculated numerical value of the sine or cosine of any angle ; adding 10 to the logarithm so found, we have the corresponding tabular logarithm. The logarithmic tangents may be found by means of the relation /y tan J. = 10 + X sin J. — i cos J., and thus a table of logarithmic tangents may be constructed. We shall in a later Chapter give a direct method by which tables of logarithmic sines, cosines, and tangents may be constructed. Description and use of trigonometrical tables. 110. Trigonometrical tables, either natural or logarithmic, are constructed as follows: — (1) They give directly the functions for angles between 0° and 90° only ; the values of the functions for angles of magnitudes beyond these limits may be at once deduced. (2) The tables give the values of the functions of angles from 0° to 45°, and from 45° to 90°, by means of a double reading of the same figures ; the names of the functions, sine, cosine, tangent, and also the degrees (< 45°), are printed at the top of the page, and the corresponding minutes and seconds are printed in the left-hand column, the angles increasing as we go down the page : again the names cosine, sine, cotangent, &c., and the degrees (> 45°), are printed at the bottom of the page, in the same columns in which sine, cosine, tangent, respectively are printed at the top; in the right-hand column are printed the minutes' and seconds for the angles which are complementary to the former ones, these latter angles of course increasing as we go H. T. 10 146 TRIGONOMETKICAL TABLES up the page. We give as a specimen a portion of a page of Callet's seven-figure logarithmic tables for angles at intervals of 10". 17 deg. t " sine dif. 655 654 654 654 6.54 654 654 654 653 654 653 653 dif. cosine dif. 68 68 68 67 68 68 68 68 67 68 68 68 dif. tangent dif. 722 722 722 722 722 721 722 721 722 721 721 721 dif. cotangent " 60 51 52 10 20 30 40 50 10 20 30 40 50 9-4860749 9-9786148 9-5074602 0-4925398 50 40 30 20 10 50 40 30 20 10 10 9 9-4861404 9-4862058 9-4862712 9-9786080 9-9786012 9-9785944 , 9-5075324 9-5076046 9-5076768 0-4924676 '0-4923954 0-4923232 9-1863866 9-4864020 9-4864674 9-9785877 9-9785809 9-9785741 9-5077490 9-5078212 9-5078933 0-4922510 0-4921788 0-4921067 9-4865328 9-4865982 9-4866635 9-9785873 9-9785605 9-9785538 9-5079655 9-5080376 9-5081098 0-4920345 0-4919624 0-4918902 9-4867289 9-4867942 9-4868595 9-9785470 9-9785402 9-9785334 9-5081819 9-5082540 9-5083261 0-4918181 0-4917460 0-4916739 ' " cosine sine cotangent tangent " / 72 deg. For example, in the third line of the column headed cosine, we find that 9'9786012 is the tabular logarithmic cosine of the angle 17° 60' 20", and reading the minutes and seconds in the right-hand column we see that the same number is the loga- rithmic sine of the complementary angle 72° 9' 40". It' should be observed that the logarithmic sines and tangents increase with the angle, whereas the logarithmic cosines and cotangents diminish with the angle. 111. In order to find the functions corresponding to an angle whose magnitude lies between two of the angles for which the functions are tabulated, we use the principle which we shall presently investigate that, except for angles which are either very small or very nearly equal to a right angle, small changes in the natural or in the logarithmic function of an angle are •proportional to the change in the angle itself. For example, if the difference between two consecutive tabu- lated values corresponding to a difference of 10" in the angle is a, TRIGONOMETRICAL TABLES 147 the difference between the values o£ the function for the smaller tabular angle and an angle greater than this angle by y" is ~ a ; the increase of the function for an increase 10" of the angle is a, and that for an increase y" (< 10") is that fraction of a which y" is of 10". In the specimen of Callet's tables which we have given, the differences between consecutive logarithms is given without the decimal points in the columns headed dif. For example, suppose we wish to find isinl7°51' 13", we find from the table L sin 17° 51' 10" = 9-4865328, L sin 17° 51' 20" = 9-4865982, t^i/. = 654; we have ^x 664 = 196-2, hence we must .add -0000196 to the first logarithm and we obtain isinl7°51'13" = 9-4865522. Again suppose we require the angle whose tabular logarithmic tangent is 9-5082032. We find from the table that the given logarithm lies between the two L tan 17° 51' 40" = 9-5081819, Xtan 17° 51' 50" = 9-5082540, dif. = 1%\; the difference between the given logarithmic tangent and the first obtained from the table is 213, hence the angle to be added to 17° 51' 40" is ^W X 10"= 2"-9 approximately, hence the required angle is 17° 51' 43" approximately. The principle of proportional parts. 112. We shall now investigate how far, and with what excep- tions, the principle of proportional increase, which we have assumed in the last Article, is true. Suppose X to denote any angle, and f(x) to denote a natural or logarithmic function of x, we shall shew in the various cases that if h be any small angle measured in circular measure, added to X, f{x + h)-f{x) = hf'{x) + h'R, where f {x) is another function of x, and iJ is a function which remains finite when h = 0. From this we see that, provided /* be sufficiently small, f{x + h) -f{x) is for a given value of x pro- portional to h, and it will appear that in general h^R will be so 10—2 148 TRIGONOMETRICAL TABLES small that it will not affect, the values of the functions to the number of decimal places to which they are tabulated; hence f(x + h)— f(x) ■^-^^ ^ — is constant to the requisite number of decimal places for a given value of x. However, two exceptional cases will arise. (1) If X be such that /' (x) is very small, then the difference f(x + h) —f(x) may vanish, to the order in the tables; the difference f(x + h)—f(x) is then said to be insensible, and in that case two or more consecutive tabulated values of f(x) may be the same. (2) If X is such that R is large compared with /' (x), the terra h^R may not be small compared with hf'(x); in this case the difference f(x+h)—f{x) is not proportional to h, and is said to be irregular. In either of these cases (1) and (2) the method of proportions fails, but we shall shew how by special devices the difficulties are obviated. The student who is acquainted with Taylor's theorem will see that the formula given above is really the special case of Taylor's theorem /(^+A)=/(^) + A/' ia:)+ih^f" {x+6h), where 6 is between and 1, thus R = \f" {x+6h), and the error made in assuming /(a; + A) —/(.»)= A/' (a;) lies between the greatest and least values which ^h?f"{z) assumes between the limits z=x and 2=.r+A. 113. First let f{x) = sin x, then sin (a; + A) = sin x cos h + cos x sin h, or sin {x + h) — sin x = cosx{h — ^h^ +...)- sin a; (^^' — -^h* + ...) = h cos X — ^h? sin x + higher powers of h ; in this case /' (x) = cos x, and the approximate value of R is — ^ sin x; thus sin(a;+ h)— sina! = h cos a; — |- A" sin a; (1) is the approximate difference equation. Similarly it may be shewn that, approximately, cos(a; + A) — cosa! = — Asina;— ^ A' cos a; (2). / J^ i. sin A Agam tan (x + n) — tan x = ^ — — rr^ ° ^ cos X cos {x H- A) ^ h cos'' x — h sin x cos x ' or, approximately, tan (x + h) — tan x = h sec" x + h' sec'' x tan x (3). TRIGONOMETRICAL TABLES 149 A 1 r • / 7 \ T • 1 sin (x + h) Also Z/Sin(iz;+ A) — //smi» = log — ^^ ° sma; = log (1 - 1^2 + /i cot x), or X sin {x + h) — Lsixix = h cot a; — ^/i,' cosec'' x (4). Similarly L cos (x+h) — L cos x= — h tan a; — ^A^ sec'' x (5), itan(a; + /0-itan« = -^— ^^ 2h^^^^. ..(%). sin X cos « sin^ 2* In each case we have found only the approximate value of R, that is to say, we have left out the terms involving cubes and higher powers of h. It appears from these six equations that if h is small enough, the dififerences are, for values of x which are neither small nor nearly equal to a right angle, proportional to h. The following exceptional cases arise. (1) The difference sin (« + A) — sin a; is insensible when x is nearly a right angle, for in that case h cos x is very small ; it is then also irregular, for J A'' sin a; may become comparable with h cos x. (2) The difference cos (a; + A) — cos x is insensible when x is small; it is then also irregular. (3) The difference tan {x-\-h)— tan x is irregular when x is nearly a right angle, for A^ sec* a; tan a; may then become comparable with hsec'x. (4) The difference L sin {x + h) — L sin x is irregular when X is small, and both insensible and irregular when x is nearly a right angle. (5) The difference L cos (x + h) — L cos x is insensible and irregular when x is small, and irregular when x is nearly a right angle. (6) The difference L tan (x + h)— L tan x is irregular when X is either small or nearly a right angle. It should be noticed that a difference which is insensible is also irregular, but that the converse does not hold. In order to investigate the degree of approximation to which the principle of proportional parts is in any case true, it is the simplest way to consider the true value of if ; in the case of sin {x + h) — sin x the true value of the second term is — \h? am {x + 6h), where 6 is between and 1; if the table is for intervals of 10", the greatest value of ^h^ is J ( eoxeo'^lSo T ""^ ^^'^^^^f' 150 TRIGONOMETRICAL TABLES this gives no error in the first eight places of decimaJs ; in the case of tan (a; + A)- tan ^ the error is (■QOO05fsec^{x + 6h)ta,n{x + eh); hence when tan.r+tan^^=40, the error will begin to appear in the seventh place of decimals. In the case of isin:i; there is no error in the seventh place of decimals if a;>5°. 114. When the differences for a function are insensible to the number of decimal places of the tables, the tables will give the function when the angle is known, but we cannot employ the tables to find any intermediate angle by means of this function ; thus we cannot determine x from the value of L cos x, for small angles, or from the value of L sin x, for angles nearly equal to a right angle. When the differences for a function are irregular without being insensible, the approximate method of proportional parts is not sufficient for the determination of the angle by means of the function, nor the function by means of the angle ; thus the approximation is inadmissible for L sin x, when x is small, for L cos X, when x is nearly a right angle, and for L tan x in either case. In these cases of irregularity without insensibility, the following means may be used to effect the purpose of finding the angle corresponding to a given value of the function, or of the function corresponding to a given angle. (1) We may use tables of Lsinx, Lta.nx for the first few degrees calculated for angles at intervals of one second, and for Lcosx, Lta,nx for the few degrees near 90°, calculated for each second ; Callet gives such a table in his trigonometrical tables ; we can then use the principle of proportional parts for all angles which are not extremely near zero or a right angle. (2) Delambre's method. This method consists of splitting L sin x or L tan x into the sum of two terms, the differences for one of which are insensible for values of x near those at which the irregularity takes place, and the differences for the other one are regular ; the difference for the first of these terms is irregular, but this is of no con- sequence, owing to its being also insensible. Thus if x be the circular measure of n" a small angle, L sin n" = (log 1- La) + log n, TRIGONOMETRICAL TABLES 151 L tan n" — flog \- La\ + log n, where a. is the circular measure of 1". Now log (w + A) - log n = log ( 1 + - j hence the differences for log n are regular, if h be small compared wit) for with n. Also the differences for log , log are insensible, OS 0(j , sin (aj + A) , sin x , sin (« + A) , x + h X , , h? ^ h h^ = n cot ie — -zr cosec^ x \- ^-^ 2 X zx^ = h [cotx I + -?r I -„ — cosec'' x ] \ xj 2 W J , , tan (x + h) , tan x and log -, — - — loc; ^ x + h ^ X. \sina;cosa! xJ 2\ sin^2i» of J' each of these differences is insensible since the coefficient of h is small when x is small. If tables of the values of log 1- La, log h La are ° X ° X constructed for the first few degrees of the quadrant, we may use these tables in conjunction with the tables of natural logarithms of numbers to find n accurately when Zsinn" or L tan n'' is given, and conversely. If L sin n" or L tan n" is given, find the approximate value of sm 00 n ; then from the table we get the value of log 1- La. or 00 log 1- La, either of which changes very slowly ; then log n is given by the value r ■ nil sin X T N L sm n — ( log V La\, or Z tan »"— flog vLa\, 152 TRIGONOMETEICAL TABLES and we find n accurately from the table of natural logarithms. If n is given, the table gives the value of log f- La, and sin n" is 3C then found by the formula. (3) Maskelyne's method. The principle of this method is the same as that of Delambre's. If a; is a small angle, we have sin X ^ Of? f^ a?\^ i • , , =1 — ^=(1— „-! = cos' X, approximately ; hence log sin x = log a; + -j log cos x ; when a; is a small angle the differences of log cos x are insensible ; hence it is sufficient to use an approximate value of cos x. If log sin X is given we find an approximate value of x, and use that for finding log cos a; ; a; is then obtained from the above equation. If X is given we can find log x accurately from the table of natural logarithms, and also an approximate value of log cos a;; the formula then gives log sin x. We can shew, in a similar manner, that log tan a; is given by the formula log tan a; = log a; — | log cos «. Example. Shew that the following formula is more nearly true than Maskelyne's : — log sin 6 =log 5 - ^ log cos 5 + If log cos \6. Adaptation of formulae to logarithmic calaulation. 115. In order i,o reduce an expression to a form in which the numerical values can be calculated from tables of loga- rithms, we must make such substitutions as will reduce the given expression to the product of simple expressions; this may be frequently done by means of one or more subsidiary angles, as the following examples will shew. (1) va' + 6« = a" sees <^, where tan = l?ja^ ; hence log Va" + 6« = 2 log a + f (X sec <^ - 10), where L tan ^ = 10 + 3 (log h — log a) ; thus Ja' + b' can be calculated by means of logarithmic tables, ^ having first been found from the tables. TRIGONOMETRICAL TABLES 153 (2) a cos a + b sin a = a cos (a — (f)) sec ^, where tan <^ = bja ; hence log (a cos a + 6 sin a) = log a + L cos (a—) — L cos 0, where ^ is found from L tan (^ = 10 + log b — log a. 116. To calculate numerically the roots of a quadratic equation supposing the roots to be real. Let ax" + bx + c = be the equation, and first suppose a and c to be both positive. We have tan^ 6 — 2 cosec 20 tan 0+1=0; now let x=y ^c/a, the equation becomes y" + by/'^ac + 1 = 0; hence if sin 26 = 2 ^/acjb, the quadratic in y will be the same as that in — tan 6, the roots of which are — tan d, — cot 6 ; thus the roots of the given quadratic are — s/c/a tan 0, — Vc/a cot 0, where sin 26=2 ^/ac/b, and hence the roots may be calculated by means of logarithmic tables. If a and c are of opposite signs, we may take the quadratic to be aay' + bai — c = 0; in this case put x = y '^cja and it reduces to 2/^ + byj^ac — 1 = 0; comparing this with the equation tan^ + 2 cot 20 tan - 1 = we see that if tan 20=2 ^/acjb, the roots of the quadratic in x are 'Jcja tan and — '^cja cot 0. 117. To calculate the roots of the cubic a? + qx-\-r = supposing them all to be real. We shall suppose q to be negative. Consider the equation sin«0-|sin0+isin30 = O; let x=y V— 4g/3, then the equation in x becomes f-iy + r{-^mf = Q; this will be the same as the cubic in sin 6, if sin 30 = 4r (- 3/42)1 = (- 27rV4?«)* J hence the values of x are V- 4^/3 sin 0, V- 4g/3 sin (0 + |7r), V- 4j/3 sin (0 + 1 tt), the condition that sin 30 :)>• 1 is the condition that the roots of the cubic are all real. 154 TRIGONOMETRICAL TABLES We shall shew in a later Chapter how to calculate the roots of a cubic when two of them are imaginary. The processes by which we have solved the quadratic and cubic equations shew that the two algebraical problems are really equivalent to the geometrical problems of bisecting and trisecting an angle respectively. It follows that a quadratic equation can be solved graphically by means of the ruler and compasses only, whereas the cubic can not in general be solved graphically by these means, since they are inadequate for solving generally the geometrical problem of trisecting an angle. CHAPTER X. RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE. 118. If ABC be any triangle, we shall denote the magnitudes of the angles BAG, ABC, AGB by A, B, G respectively, and the lengths of the sides BG, CA, AB by a, b, c respectively. We shall, in this Chapter, investigate various important formulae connecting the sides a, 6, c of a triangle with the circular functions of the angles. These formulae will afford the basis of the methods by which we shall solve a triangle in the various cases in which three parts of the triangle are given. 119. From the fundamental theorem in projections we see that the sum of the projections of BA, AC, on BG, is equal to BG, and that the sum of their projections on a perpendicular to BG is zero. Expressing these facts we have, since the positive direction of AG makes an angle — G with the positive direction of BG, BA cos B + AG cos G = a, or c cos B + b cos G = a, and BA sin B — AG sin (7 = 0, or c sin jS -^ 6 sin C = 0, 156 RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE which may be written b/ain B = c/sinC. These relations and the corresponding ones obtained by projecting on and perpendicular to each of the other sides, in turn, may be written a = b cos C + c cos B ^ b = c cos A + acos G > (1), c =a cos B +b cos A ) a/sin A = b/smB=c/ain G (2). The equations (2) express the fact that, in any triangle, the sides are proportional to the sines of the opposite angles. 120. The relations (2) may also be proved thus : — Draw the circle circumscribing the triangle ABG, and let B be the length of its radius, then the side BG is equal to twice the radius multi- plied by the sine of half the angle BG subtends at the centre of the circle, that is BG=2RsiaA or 2B sin (180°-^), hence a = 2R sin A ; similarly b = 2RsinB, and c = 2RsmG; hence a/sin J. = 6/sinjB = c/sinC=2ii. These relations (2) may also be deduced from (1); writing the first two equations (1) in the form a- 6 cos C—ccoaB=0, - a cos 'C+b-c cos .4=0, we can determine the ratios of a, b, o ; we obtain a _ b _ e cos Coos 4+ cos .B cos 5 cos 6'+ cos 4 I-cos^C" hence -. — r—. — ^= -. — ^—. — ^= . ..-, , or a/sinit=6/sin.B=c/sin C. sm.4sm(7 smBsmC am' C ' ' ' To deduce (1) from (2) we have a=-. — Tsin(5-|-C) = -: — 7 (sin 5 cos C+cos^sin C) ; sm .4 ^ sm .4 ^ ' 6 c hence a=-. — =smBcosG+-. — ^ cos 5 sin C=b coa C+ccoaB, amB amC ' which is the first of the relations (1). If we eliminate a, b, c from the three equations in (1), we obtain the relation cos2.4 + cos2 5+cos^C+2cos.4oos5cosC=l, which holds between the cosines of the angles of a triangle. RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE 157 121. If we multiply the equations in (1) by —a,b,c respec- tively, and then add, we have ¥ + c^-a''=2bccosA, which gives an expression for the cosine of an angle, in terms of the sides ; we may write this relation and the two similar ones for cos B, cos G thus a^ = 6" + c2 _ 26c cos A\ ¥ = c' + a''- 2cacosB[ (3). c^^a' + b^- 2ab cos c] 122. We may obtain these relations (3) directly by means of Euclid, Bk. II. Props. 12 and 13. If AL be perpendicular to BC, we have, when C is an acute angle, AB' = A(? + BC''-2Ba. CL, and when G is obtuse AB^ = AG' + BG^+2BC.GL; in the first case GL = AG cos G, and in the second case Ci = ^(7cos(180°-O) = -4(7cosC; therefore in either case c^ = a' + ¥- 2ab cos G. To deduce the relations (2) from (3) we have cos A = ;r^ ; 26c ' therefore . , , ib^c'-jb^+c'^-a^f i2be+b^+c^-a.^)(2bc+a^-b^-c') ^^^ ^ W? ~ 4b'c^ ... (a + b + c){b + o-a){c+a-b){a + b-c) or sinM = ^ '-^ ^p^2 ; thus — 2— is equal to the symmetrical quantity (a + b+c)(b + c-a)(,c+a-b){a + b-c) _ sin^ A sin^ B sin^ C hence "^2- - -p- - "^ ' from which (2) follows. To deduce (1) from (3), divide the first two equations of (3) by c, and then add them ; we get ^!±*!=2c+^^i^-2(6cos^+acosiS), or c= 6 cos 4 + a cos .B. 158 RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE 123. We have siii= ^4 = J (1 - cos A), oos^ ^A = ^(l + cos A) ; hence 6^ + c^ - a- or Now let 2s = a + 6 + c, then 2 (s — a) = 6 + c — a, and we have ■ ,1 A (s — b)(s — c) „ 1 . s(s — a) therefore sm *-a = f-^-f^f . -M = {^f . these formulae are more convenient than (3) as a means of determining functions of the angles when the sides are given, because they are more easily capable of being adapted to logarithmic calculation. ,-. „. siniJ sinC , 124. bince — = — = , we have c s in .B + s in g _ & + c 2 sin|(^± C) cos^(5 + 0) b±c sinA '~ a ' °^ 2 sin^(5+ (7)cos|(5+ 0)" ~a~' hence ^« _ «2ii(^ZL^) ^^^ b-c _ sm^(B-G) '^^''''^ a - cos ^{B+Cy^'''^ a -sm^(B + G)' (6 + c)sin^ ^ (&-c)cosM .K^ we obtain by division the formula tanH5-C) = |^cot^^ (5'). To prove these formulae geometrically, with centre A and radius AB describe a circle cutting AC in D and E ; draw DF parallel to BE, then CE=b+c, DC=c-b, DEB=^A, DBF=C+^A-m'=\C-\B. We have CD _ sm DBF h-c _ sm^{B-C) or CB sinCDB' a cos^^ RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE 159 E also h + c_GE _EB _ BDcoi^A _ coi\A c-b~ CD~'DF~ BDta.ni{C-B)~ta,ni{G-B)' h-c hence taD.\{B — C)=j— cot^A. The area of a triangle. 125. The area of a triangle is half that of a parallelogram on the same base and with the same altitude; if the side a is the base, the altitude is h sin C or c sin B, we have thus the expressions ^ ab sin G and ^ ac sin B for the area of the triangle ; the area of a triangle is therefore half the product of any two sides multiplied by the sine of the included angle. Using the expression for sin J, found in Art. 122, sr- V(a + b + c) {b + c - a)(c + a - b)(a + b - c), we have for the area of a triangle the expression i \/{a + b + c)(b + c - a){c + a~^b){a + b-c), or v's(s — a)(s — 6)(s- c) (6); this formula was obtained by Hero of Alexandria^ (about 125 B.C.). The formula (6) may also be written I ^26^0^ + 2cW + 20^6" -a'-¥- c\ 1 See Ball's History of Mathematics, p. 82, where the original geometrical proof of the formula is given. 160 RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE Variations in the sides and angles of a triangle. 126. We shall now investigate the relations which hold between small positive or negative increments in the values of the sides and angles of a triangle. Suppose three of the parts of a triangle to have been measured, of which one at least is a side, the other three parts will be determined by means of the formulae of this Chapter ; the relations between the increments of the parts will enable us to find the effect in producing errors in the values of the latter three parts of small inaccuracies in the measurement of the former parts. We shall suppose that the increments are so small that their squares and products may be neglected. Suppose A, B, C, a, b, c to be the values of the angles and sides of a triangle, as ascertained by the measurement of one side and two angles, two sides and one angle, or the three sides, the other three values being connected with the three measured ones by m«ans of the formulae given above. If the three parts have been measured inaccurately, there will be consequent inaccuracies in the values of the other three parts as found by the formulae; let A + hA, B + SB, C+SC, a + Sa, b + Sb, c + Sc be the accurate values of the angles and sides ; we shall obtain relations between the six errors 8A, BB, BG, Ba, Bb, Be. It will be convenient to suppose the increments of the angles to be measured in circular measure; they can however of course be at once reduced to seconds. We have c sin B — b sin 0=0, (c + Be) sin (B + BB) -(b+ Bb) sin (G+BG) = 0; since, when the squares of BB, BG are neglected, sin (B + BB) = sin 5 + BB cos B, sin (0 + BG) = sin C + BG cos G, we have, (c + Be) (sin B + BB cos B)-(b + Bb) (sin G+BGcosG) = 0; hence if we neglect the products Be, BB, Bb, BG, we have e COB B. BB + sin B. Be -b cos G.BG- sin G.Bb = 0. This, with the two corresponding equations, may be written sin G .Bb — sin B . Be =c cos B .SB — b cos G .BG \ sin A .Be — sin G .Ba = a cos G .BG —e cos A .BA\ (7). sin B .Ba — sin A .Bb = b cos A. BA — a cos B. BB) RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE 161 Also BA+BB + BC = (8), in virtue of the relations A + B + G = -n; A + SA + B + SB + C+BC = 'n: The equations (7) are not independent, as may be seen by writing them in the form X - - = cot£ . 85- cot 0. 8G, b c -- — =^cotC.8C-cotA.SA, c a J- = cot A.SA — cotB. SB, a b which shews that any one of the equations may be deduced from the other two. The system consisting of two of the equations (7) and the equation (8) is sufficient to determine any three of the six errors when the other three are given, except that one at least of the three given errors must, belong to a side. By eliminating SB, SO, between (7) and (8), we obtain an equation giving Sa in terms of Sb, So, and SA ; this may however be found directly from the formula a''=b^ + c^— 26c cos A ; we obtain aSa = (b — c cos A) Sb + (c — b cos A) Be + bc sin ASA, which, with the two corresponding formulae, becomes, in virtue of (1), aSa = a cos G .Sb +a cos B .Be + be sin A . SA] bSb = b cos A. Be + 6 cos G . 8a + casing. S^l (9). cSc = c cos B .Sa + c cos A .Sb + ab sin G . SGj Relations between the sides and angles of polygons. 127. Let ai, a^, as...an denote the lengths of the sides, taken in order, of any plane closed polygon, and let «!, a^.-.o^ denote the angles, measured positively all in the same direction, which these sides make with any fixed straight line in the plane of the polygon; then from the fundamental theorem in projections in Art. 17, we have, projecting on the fixed straight line and perpendicular to it, the two relations di cos «! + cij cos 02 + + a„ cos «„ = 0, tti sin «! + Oa sin tta + + «« sin a„ = 0. H. T. 11 162 RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE Now let the line on which the projection is made be the side a„ ; if we denote by /S, the external angle between a„ and Ui, by /Sj the external angle between Oj and ctj, &c., then ai = /3i, a2=|8i+y82, «s=/3i + A + ySs, ifec, «,, = 27r, we have then ai cos /3i + fls cos (/Si + /S^) + a^ cos (A + A + /^s) + • • • + «« = O] Oi sin /Si + tta sin (A + /S^) + a, sin (A + iS^ + A) + • • • f (10), + a„_, sin (/3i + /Sa + . . . + jS„_i) = 0^ the two fundamental relations between the sides and angles of a polygon. If there are only three sides, these relations reduce to (1) and (2) respectively, remembering that ySi = tt — A^, /Sg = tt — A^. 128. In the first equation in (10), take a„ over to the other side of the equation, then square both sides of each equation and add ; in the result the coefficient of 2aras is cos(ySi + A+ ... + y8^)cos(/3i + /S2+ ... + ;S,) + sin (/Si + ^,, + ... + A-) sin (A + A + ••• + /3«), or cos {^r+i + /Sr+2 +■■■ + ^s); this is the cosine of the angle ffrg between the positive directions of the sides a^ and a^ ; we thus obtain the formula an = a/ + eta" + . . . + a„_i^ + 2aia2 cos ^,2 + . . . + 2ay a^ cos 6rs +...(11), which is analogous to the formulae (3), to which it reduces when n = d. In the formula (11), r and s are each less than n and are unequal. The area of a polygon. 129. The area of a polygon is given by the expression ^(aia2sin^i2+ ... +arae sin Org + ...) (12), or ^"Zarttg sin d^s, the summation being taken for all different values of r and s ; if we suppose s is always the greater of the two quantities r and s, the angle 6rs is, as in the last Article, the sum of the external angles /8,.+i + /3y+2+ ... H-zSg. To prove this formula, we shall first shew that in the case of a triangle it reduces to the expression ^ajttssin^i, and shall then shew that if it holds for a polygon of w — 1 sides, it also holds for one of n sides. RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE 163 We have in the case of the triangle ^1^.2^3, in which AiAs = (Xi, 6i2 = Tr-Ai, 6^ = 17- A3, ^13 = 277-^2-^3; hence in this case ^-Sa^ag sin 0^ is equal to Koiajsin^j + Oaassinils- aiftssin^i) or ^aaassin^,, thus the formula holds when n = 3. Now suppose the formula true for a polygon of sides '"ij 0^2> Oin—i! SO that the area of the polygon is ^Xarttg sin drs + ^a'm-iSar sin ^n-i,r, where r and s are each less than n—1. Now replace the side a'n-i by two sides a„_i, a„, thus making a polygon of n sides ; we have to add ^an-iansm6n-i,n', the area of the polygon of n sides is then ^ Sftr «« sin Ors + h "''n-i 2a,. sin 0'„_i, r + i a»i-i «« sin ^„_i, „ . Now we have, by projecting the side a'„_i on a^, a'n-i sin ^',.,»-l = ftn-i sin ffr.n-^ + dn sin 0r,n, hence the above expression becomes •^ Sa^aj sin ^rs + iSay(a„_i sin 0,._„_i+a„ sin ^r,n) + iaji-ian sin ^„_],„, or ^Sttj-ttg sin^j.j, where r and s have all different values from 1 up to n, such that roa{C-A)+ A\ l+"icos.B and cx>t\(B-A)= ■. — =— . 16. Prove that, in a triangle, cos.4+cos5 + cos C>1 and :^J. 17. Prove that, in a triangle, tan2-^5tan2^C'+tan2^Ctan^^.4 + tan^J4 ta.n^^B' - a<- 6« - c<). 34. If A, B, C are angles of a triangle, and x, y, z are real quantities satisfying the equation ysinC-zsin^B _ zsin.4— .j;sinC ^— ^cosC— 2C0s£ y-zcos.4— a;cosC" ,-, ■■,■, X y z then wiU —. — -. = . „ = -: — tv. sm A sin B sin 6 35. Prove that the area of the greatest rectangle that can be inscribed in a sector of a circle of radius R is R^\&-a.\a, where 2a is the angle of the sector. 36. Shew how to construct the right-angled triangle of minimum area which has its vertices on three given parallel straight lines ; and if a, h are the distances of the middle line from the other two, shew that the hypothenuse makes with the parallel lines an angle cot~i— ---^. 37. If the angles of a triangle computed from slightly erroneous measurements of the lengths of the sides be j1, B, C, prove that if a, /3, y be the approximate errors of lengths, the consequent errors of the cotangents of the angles are proportional to cosec .d. (/3 cos C-t- y cos B — a), cosec B (y cos A+a cos - j3), cosec C (a cos 5 + j3 cos .4 — y). 38. Prove that, if in measuring the three sides of a triangle, small errors x, y be made in two of them a, b, the error in the angle C is -(-cotB+^cotAj, and find the errors in the other angles. 39. The area of a triangle is determined by measuring the lengths of the sides, and the limit of error possible either in excess or defect in measuring any length is n times the length, where re is a small quantity. Prove that in the case of a, triangle of sides 110, 81, 59, the limit of error possible in its area is about 3'1433m times the area. 40. Prove that the cosines Cj, e^, C3, C4 of the four angles of a quadri- lateral satisfy the relation CHAPTEE XI. THE SOLUTION OF TRIANGLES. 130. We shall now proceed to apply the formulae obtained in the preceding Chapter to the solution of triangles, that is, ■when the magnitudes of three of the six parts are given, to find the magnitude of the remaining three parts ; one at least of the three given parts must be a side. We shall generally select such formulae as can be used for numerical computation by means of logarithms, as these formulae only are of use in practice. The solution of triangles is made to depend upon a knowledge of the numerical values of circular functions of the angles, hence since such circular functions are the ratios of the sides of right- angled triangles, it is seen that the solution of all triangles is really performed by dividing up the triangles into right-angled ones. The solution of right-angled triangles. 131. Suppose the angle C of a triangle to be 90°, then this is one of the given parts, and we can solve the triangle in the various cases in which there are two other parts given, one at least being a side. (1) Suppose the two sides a, b to be given; then the angle A can be determined from the formula tan A = a/b, and B is then found as the complement of A ; also c = a cosec A, which deter- mines c, when A has been found; the logarithmic formulae for solving the triangle are then L tan .4 = 10 -I- log a - log b, B = 90°-A, log c = log a — LsinA + 10. ]68 THE SOLUTION OF TRIANGLES (2) Suppose the hypothenuse c and one side a to be given ; then the angle A is determined by means of the formula sin A = a/c, B is found as the complement of A, and h is found from the formula 6 = ccos^, or from h^ = c^ — a'^. The logarithmic formulae are i sin ^ = 10 + log a - log c, 5 = 90° - A, and log 6 = log c + i cos J. — 10 or log 6 = i log {c + a) + \ log (c - a). (3) Suppose the hypothenuse c and one angle A are given, then B is found at once as the complement oi A; a is found from a = csin^, and h as in the last case. The formulae are log a = log c + i sin J. — 10, 5 = 90°-^, log h = log c + Z cos J. — 10 or log 6 = ^ log {c + a)-ir^ log (c - a). (4) Suppose one side a and one angle A to be given, then B is 90° — ^, c is acosec J., and b is found as in the last two cases; the formulae are log c = log a — Z sin J. + 10, 5=90°-^, log h = log c-\- L cos A — 10 or log 6 = ^ log {c + a) + ]^ log (c - a). 132. In certain cases, the formulae of the last Article are inconvenient, for example in case (2) if the angle A is nearly 90°, it cannot be conveniently determined from the equation sin A = ajc, since the differences for consecutive sines are in this case in- sensible, we therefore use another formula ; from the theorein (4) of Chap. X. we obtain 6tan^5 = c — d. &cot ^5 = c + a, hence ia,-D?\B= , thus we have tan(45° — i.4)= ( -) , and this formula, being free from the objection, may be used to determine A. Again, in cases (3) and (4), the formula h = ccos A is in- convenient if A is very small; we may then use the formula & = c — c sin ^ tan \A. THE SOLUTION OF TRIANGLES 169 133. Various approximate formulae may be found for the solution of right-angled triangles. Let us denote by a, /3 the circular measures of the angles A, B respectively. (1) An approximate form of the formula a = o cos B is which is obtained by taking the first three terms of the expansion of cos B in powers of the circular measure of B ; this formula may then be used for approximate calculation of a, when c and B are given, provided /3 is not too large. (2) Since &n\A=alc, we have a — \a^-'r^\-^a' = alc, approximately; to obtain a in terms of ajc, we have as a first approximation a=ajc, and as a second approximation « = ~ + fi(") i ^^^ third approximation is a 1 fa 1 faY\ ^ 1 fa\^ a 1 (ay 3 faV- which may be used to calculate a. (3) From the equation t&.n\B = { — — J we can obtain the approximate '— »^=(^)'{'-Ks)-i(k)}- (4) Using Snellius' formula 6 = ^77^-. ^r^ > ^°^ *^s circular measure of ^ ' ° ^ 2(2-)-cos2^) an angle (see Ex. 32, p. 138), in which the approximate error is ^^ put 2(A = a we then obtain the formula ^= , and the error is approximately yJ^^^ ; thus B is given in degrees by the approximate equation 5=;r^x57°-2957. 2c + a The solution of oblique-angled triangles. 134. To solve a triangle when the three sides are given; any one of the formulae , , {{s-h){s-c)\^ ^ I s{s-a) J with the corresponding formulae for the other angles, may be used; these formulae are adapted for logarithmic calculation. 170 THE SOLUTION OF TRIANGLES Example. The sides of a triangle are proportional to 4, 7, 9 ; Und the angles, having given % 2 =-301030, Z taw 12° 36' = 9-349329, diff. for 1' = -000593, X torn 24° 5' =9-650281, diff. for 1'= -000339. We find s=10, s-a = 6, s — 6 = 3, « — c=l, and hence tan J .4 = \/ 1/20, tan^5=V2/10, thus Xtan J .1 = 10-1(1 + -301030) = 9-349485 and Ztani5 = 10+|(-301030-l)=9-e50515. To find A, we have 9-349485 - 9-349329 = -000156, and i||.60" = 15"-8 approximately, hence i4 = 12°'36' 15"-8, or ^=25° 12'31"-6. To find B, we have 9-650515 - 9-650281 = -000234 and §^f . 60" = 41"-4 approximately, hence Jfi = 24° 5'41"-4, or 5 = 48° ll'22"-8; also C= 180° -A-B= 106° 36' 5"-6 ; thus we have found the approximate values of the angles. 135. To solve a triangle when two sides and the included angle are given. Suppose h, c, and A are the given parts, then B and C may be determined from the formula ta.ni(B-G) = l^cot^A, ~\~ c together with 3 + 0=180" — A; the logarithmic formula is Lta.ni(B-C) = log(b-c)-hg{b + c) + LcotiA. Having found B and G, the side a may be found from any one of the three formulae log a = log c + i sin .4 — i sin C, loga + X-cos ^{B-G) = log {b + c) + L sin^ J., log a + i sin ^{B-G) = log (b-c) + L cos ^A. We may also determine a thus : — Since a'' = b'' + c''— 2bc cos A we have a^ = (6 + cf - 46c COS" ^A, hence a = (6 + c)cos^, where by the logarithmic formulae Zsin^= log2 + Jlogft + ^logc + Zcos^J. — log(6 + c), and then determine a by the formula log a = log (6 + c) + L cos ^ — 10. Example. If a=123, c = 321j B = 29° 16', find A, C, b, having given log 99 = 1-9956352, Lsin29°l&= 9-6891978, log 123 = 2-0899051, Z«ml5°42'= 9-4323285, diff. for l" = 74-87, log 2220 = 3-3463530, Z cot 14° 38' = 10-5831901, % 2221 =3-3465486, Z tow 59° 39' = 10-2324552, diff. for l"=48-27. We have Xtani(C-4) = Zcot 14°.38' + log99-log222 = 10-5831901 + 1-9956352 - 2-3463530 = 10-2324723. 171 Now 10-2324723- 10-2324552 = -0000171, and 75:27=3-5 approximately, hence ^(C-.4) = 59° 39' 3"-5, also -^((7+^) = 75° 22', therefore 4 = 15° 42' 56" -5, C=135°l'3"-5. Again log 6 = 9-6891978 + 2-0899051 -Z sin 15° 42' 56"-5, and 56-5x74-87 = 4230-155, hence Z sin 15° 42' 56" -5 = 9-4327515, therefore log 6 = 2-3463514, so that 6 = 222-iJf5 = 221-992. 136. To solve a triangle when two sides and the angle opposite one of them are given. This is usually known as the ambiguous case. Suppose a, c, and A are the given parts, then sin C is deter- mined from the equation sin = - sin J. ; when sin G is thus found, there are in general, if c sin J. :^ a, two values ^f G less than 180°, the one acute and the other obtuse, whose sine has the value determined ; we must consider three different cases : — (1) if c sin A >a, we have sin C > 1, which is impossible, and indicates that there is no triangle with the given parts ; (2) ifcsin^=a, thensin(7 = l, and the only valueof(7is 90°. If -4 < 90°, there is one triangle with the given parts, and that one a right-angled triangle. If A > 90°, the value (7=90° is inad- missible, and there exists no triangle with the given parts. 172 THE SOLUTION OF TRIANGLES (3) if c sin ^ < a, then sin (7 < 1, and there are two values ot G, one acute, the other obtuse ; (a) if c < a, we must have G < A, hence C must be acute, thus there is only one triangle with the given parts; (/8) if c> a, the angle G is not restricted to being acute, and both values are admissible, provided A < 90°; but ii A> 90° neither value is admissible since C> A. There are two triangles or none with the given parts according as J. < 90° ox A> 90°; (7) if c = a, then G = A or 180° — ^ ; for the latter value of G two sides of the triangle are coincident, the first then gives the only value of G for which there is a triangle of finite area, but this is only admissible when A < 90°. We may state the above results thus : c sin ^ > a, no solution csxaA = a, A< 90°, one solution c sin A = a, A> 90°, no solution c sin J. < a, c 90°, no solution j c > a, A< 90°, two solutions \c>a, A >90°, no solution When C is nearly 90°, it cannot be conveniently determined by means of its sine ; in that case we may use one of the formulae tanC=±^^ , tan(45°+|C)=±./^ omnA V(a + csin4)(a-csinjl)' V a — cskaA' 137. It is instructive to investigate geometrically the different cases considered in the last Article. From B draw BD perpendicular to the side h, then BB = c sin .4 ; witt centre B and radius a, describe a circle ; then if a is less than csin-il, this circle will not cut the side AG and no triangle with the given parts can be drawn, but if a>csin^, the circle will cut AG m two points, Gi and G^. In the case a90°, A will be beyond C,, and no triangle with the given parts exists. If a>c, then Gi and G^ are on THE SOLUTION OF TRIANGLES 5 173 opposite sides of A, and only the triangle ABG^ has the given parts. The triangle ABG^, in this latter case, has the angle at A not equal to A, but to 180° — J., and therefore does not satisfy the given conditions. If a = c sin J., the circle touches AG sA, D, and the right-angled triangle ADB is the one triangle with the given parts, provided A < 90°. We remark that since, in Fig. (1), AD = c cos A, and OiD = G^D = 'Jo' - & sin^ A, the two values of h are c cos J. + Va' — c^ sin^ A and c cos -4 - s/a^ — d^sin^A, these values being both positive when there are two solutions; we may also obtain these values of h as the roots of the quadratic equation in 6, a2 = 62 + c''-26ccosJ. 174 THE SOLUTION OF TRIANGLES 138. To solve a triangle when one side and two angles are given. Suppose a the given side, and 4, the given angles, then B is determined from the equation B = 180° -A-G, and the sides h, c will be determined by means of the formulae log h = log a + LshiB — L sin A, log c = log a + LsinC — LamA. Example. 7/ a =10, A=51°30'40", B=76°, Jmd b, having given log 12396=4-0932816, Lsin 76° = 9-9869041, log 12397 = 4-0933166, L sin 51° 30' = 9-8935444, Z«in 51° 31' = 9-8936448. We have log b = 9-9869041 +1-L sin 51° 30' 40" and i sin 51° 30' 40" = 9-8935444 + ^x -0001004 = 9-8936113, hence log 6 = 1-0932928, therefore 6=12-396+Jigx -001, or 6 = 12-3963 approximately. 139. The expression ccosA±'Ja' — c^sm^A for b may be adapted to logarithmic calculation; let sind) = -sin.4, then 5= r~^ — -, thus 4 . Denoting by a, ^, y the circular measures of the angles A,B,C, respectively, and by a, j3', y the complements of a, j3, y, we obtain the following approxi- mate formulae for the solution of triangles. (1) Suppose A, C, a are given, C not being large; then from the formula c= — : — ;- , we get the approximate formula sm 4 c = ascoseo^ {y- J/+_|^yi}_ Also if A and C are both not large, we have hence c is given approximately by which may be used for calculating c. (2) Suppose, as in the last case, that A, C, a are given ; also suppose C is nearly 90°, then c=— ; — ^, therefore c=-. — -ji^—^y'^+^y'*) may be used to determine c approximately. THE SOLUTION OF TRIANGLES l75 If both A and C are nearly 90°, we have cos a' l_Ja'i+... ' therefore e=a{l-^{y'^-a'^)} gives c approximately. 140. We shall give a few examples of the solution of triangles, when instead of sides and angles there are other data. (1) Suppose the three perpendiculars from the angles ou the opposite sides given; denote them by Pi, P2, Ps, we have then api=bp2 = cpg = 2 area of triangle. Now since cos ^A = we have cos iA = .Ap^P^+P^P^ +P^P^\^ -P2Ps+PsPi +PiP,) which determines A ; also ^2 = " sin A, hence c is determined when A is known. (2) Suppose the perimeter and the angles of the triangle given. We have « = iJ(sin 4+sin J5+sin C), hence R is determined, and the sides are then 2IiBmA, ZRsmB, 2/JsmC, or a = ^ sin 4 + sin 5+ sin C" «sini.4 with similar values for 6 and c ; this value of a reduces to =~^ j-r; 1 cosJiJcos^C which is adapted to logarithmic calculation. (3) Suppose the base, height, and difference of the -angles at the base given. Let a be the base, p the height and B—C=2a the given difference : then since B + C=\80°-A, we have ^=90° + a-J.4, C=90°-a-|^, also a=p {cot B + cot C) =p {tan {^A—a)+ta,xt.{\A-\-a)}, therefore - = -. ;r- 1 hence cos A is given by the quadratic p cos^+cos2a o J -s. «2 (cos A +COS 2a)2=4p2 (1 _ cos^ A) or cos^ A {cfi + 4p^) + 2a^ cos 2a . cos A = 4:p'' — a^ cos^ 2a, the solution of which is g^ cos 2a 2p (4p^ + g^ sin^ 2a)^ COS^--^2 + 4p2± a2+4j92 ' these are two values of cos A corresponding to two solutions of the problem. Solve the triangle with the following data: (4) C, c, a + b. (5) B, a, b + c. (6) The area and the angles. (7) C, c+a, c + b. (8) The angles and the height. 176 THE SOl.UTION OF TRIANGLES The solution of polygons. 141. The relations between the sides and angles of polygons, and the methods of solving a polygon when a certain number of sides and angles are given, have been considered by Carnot^ L'Huilier', Lexell', and others. The two fundamental equations in this so-called Polygonometry have been given in Art. 127. In order that a polygon of n sides may be determinate, 2ra — 3 of its 2n parts must be given, and of these at least w — 2 must be sides. To prove this, suppose the polygon divided, by means of a diagonal, into a triangle and a polygon of « — 1 sides ; if the sides and angles of the latter polygon were determined, we should only require to know two parts of the triangle in order to determine the figure completely, since one side of the triangle is already determined as a side of the polygon, hence to determine a polygon of n sides we require to know two more parts than for a polygon of w — 1 sides ; since therefore for a triangle three parts must be given, one of which is a side, for a polygon of n sides we must have 3 + 2 (w — 3), that is 2w — 3 parts given. If of these 2n — 3 parts, only w - 3 were sides, we should have n angles given ; but if n—1 angles are given, the nth is also given, so that only 2n — 4 independent parts would be given, thus at least n—2 of, the given parts must be sides. In some cases, a polygon can be conveniently solved by dividing it by means of diagonals into triangles, taking the diagonals for parts to be determined ; this method is however not always con- venient, as may be seen, for example, by considering the case of a quadrilateral when two opposite sides and three angles are given. 142. To solve a polygon of n sides, when n—1 sides and n — 2 angles are given. (1) Suppose the angles to be found are adjacent to the side to be found. We shall, as in Art. 127, use the external angles ;8j, ^2...^n between the sides, instead of the internal angles; 1 Camot, Geometrie der Stellung. ^ L'Huilier, Polygonomitrie. Geneva, 1789. ' Lexell, Nov. Comm. Petrop., Vols. xix. xx. THE SOLUTION OF TRIANGLES 177 suppose a„ the side to be found, then from the second equation (10) of Art. 127, we have sin /3i {cii + Oa cos /Sj + as cos (A + A) + • • ■ + «»-i cos (0^+... + ;S„_i)} = - cos ^1 [a^ sin ySj + as sin (A + /Ss) + . . . + a-n-i sin (/Sa + . . • H- ^n-i)\, hence tan /3 = - tta sin ^Q^ + a, sin (A + A) + • ■ ■ + a«-i sin (ft + . . . + /3„_i) ^ ai+a2Cosft+a3COs(/82 + y8s) + ... + a„_iCos(y82 + ...+/3^i)' this determines /3i in terras of the given angles ft, ft ... /3,i_i and the given sides a^, as ... a^-j; it should be noticed that this equation is found by projecting the sides on a perpendicular to the unknown side; the remaining angle ^n is then determined from the relation ft + ft + . . . + /3„ = 27r. Having found ft and /S„, we can determine a„ from the equation obtained by projecting the sides on a„, ^K = — (oi cos ft + as cos (ft + ft) + . . .} , or by means of the equation (11) of Art. 128, which gives an' in terms of the squares and products of the other sides and of the cosines of the angles between the sides. (2) Suppose the angles to be found are adjacent to one another but not to the side which is to be found. We shall take a„ as the side to be found, and ft., ft+i the angles to be found, then ft + ft+, = 27r-(ft + ft+...+ft_i + ft+s,+ ... + /3„), thus ft + ft+i is known ; also from the second equation (10) ttr sin (ft + ft + . . . + /3r) = - Oi sin ft - a^ sin (ft + ft) - . . . -aMsin(ft + ft + ... + ft_i) - Or+i sin (ft + . . . +ft.+i)-... - a„_i sin (ft + . . . + ^„), hence /3i + ft + . . . + ft can be determined, and therefore ft. The side an is then determined as in the last case. (3) In the case in which the two unknown angles are not adjacent to one another, let H, K be the angular points at which the angles are unknown; join HK, then the polygon is divided into two polygons, in one of which all the sides except one are known, and all the angles except the two which are adjacent to the unknown side. We can solve this polygon as in (1), determining HK and the angles H and K. TT T. 1^ 178 THE SOLUTION OF TRIANGLES In the other polygon we now have all the sides except one given, and all the angles except two adjacent ones ; this polygon can therefore be solved as in (2); we have then all the sides of the given polygon determined, and the angles at H and K are determined by adding the two parts into which they were divided by HK, and which have been separately found. 143. To solve a polygon of n sides, when n — 2 sides and n — 1 angles are given. We determine the remaining angle at once frorn the condition /3i + ^2+...+/3„ = 27r. To determine an unknown side a^, use the equation a, sin /3i + cia sin (/8i + /Sa) + ... + «„_, sin (/8i + /32+ ... +;8„_i) = 0, obtained by projecting perpendicularly to the other unknown side a„. We can then determine an in a similar manner, or use the other fundamental equation. 144. To solve a polygon of n sides, when the n sides and n — 3 angles are given. Let P, Q, R be the angular points at which the angles are not given; join PQ, QR, RP, then the polygon is divided into four parts, one of which is a triangle. In each of the parts except PQR, all the sides except one are given, and all the angles except those adjacent to those sides; we can therefore determine PQ, QR, RP, and the angles at P, Q, R. We can then find the angles of the triangle PQR, of which the sides have been determined. We obtain now by addition the angles at P, Q, R, of the given polygon. Heights and distances. 145. We shall now give some examples of the application of the solution of triangles to the determination of heights and distances. For fuller information on this subject, as for the de- scription of instruments for measuring angles, we must refer to treatises on surveying. The angle which the distance from any point of observation to an object makes with the horizon is called the elevation or the depression of that object, according as the object is above or below the horizontal plane through the point of observation. THE SOLUTION OF TRIANGLES 179 146. To find the height of an. inaccessible point above a hori- zontal plane. Let P be the inaccessible point and C its projection on the horizontal plane, let PG = h, and suppose any line AB = a, measured on the horizontal plane, if possible so that ABC is a straight line ; let the elevations of P at 4 and B be measured, denote them by a and ;8; then a = J.6' — 5C = /i(cota — cot/8), therefore , _ a sin a sin /3 ~ sin (/3 - a) ' which determines h. If it is impracticable to measure the base line directly towards G, let it be measured in any other direction ; let the elevations a of P be measured at A, and also the angles PAB=ry, and PBA = S, then PA = AB ■^^°^., ' sm(7+o) therefore h = a —. — -. gr , thus h is determined. sm (7 + 6) and h=APsma, 147. To find the distance between two inaccessible points. Let P and Q be the two objects, and let any base line AB = a be measured, the points A, B being so chosen that P and Q are 12—2 180 THE SOLUTION OF TRIANGLES both visible from each of them. At A measure the three angles PAQ = oi, QAB=0, PAB = 'y; it should be observed that the angles PAQ, QAB are in general not in the same plane. At B measure the angles PBA = S, and QBA = e. From the two triangles ABP, ABQ, we have sin S AP = a and AQ = formulae sin 6 ' sin (/8 + e) ' sin(7 + S)' Thus AP, AQ are determined by the log AP = log a + Lsin B — L sin (y + S), log AQ = log a + Lain e — Lsm(^+ e). In the triangle PAQ, we now know AP, AQ, and the angle PAQ — a, we find then the angles APQ, AQP by means of the formulae Lta,n^{APQ-AQP) = Lcot^a + \og(AQ-AP)-\og(AQ + AP), APQ + AQP =180° -a. We then find PQ by means of the formula log PQ = log AP + Z sin a - Z sin ^QP. 148. Pothenot's Problem. To determine a point in the plane of a triangle at which the sides of the triangle subtend given angles. Let a, ^ be the angles subtended by the sides AC, GB of a triangle ABC at the point P, and let x, y denote the angles PAC, PBG respectively; the position of P is found when the angles x and y are determined, for the distances PA and PB can be found by solving the triangles PAG, PBG. We have Also THE SOLUTION OF TRIANGLES x + y = 2'ir-a-^-G. 181 b sin X a sin y sin a sin ^ Assume ^ to be an auxiliary angle such that a sin a tan = J—. — ;5 , sm /3 therefore ^ — = tan ~ ^^°) = tan (45° - ) tan ^ (a + /3 + C) ; thus « — y can be found, and since x + y is known, we can find X and y. 149. Examples. (1) It is observed that the elevation of the top of a mountain at each of the three angular points A, B, C, of a plane horizontal triangle ABC, is a; shew that the height is J a tan a cosec A. Shew also, that if there he a small error n" in the cos C sin I elevation at C, the trite height is very nearly - —. — i- ( 1 +- ' ^ -^ ■^ 2 nnK \ s !2a/' sin A sin B ' sin ; Let be the projection of the top of the mountain on the plane ABC, we have then, if A is the height of the mountain, A = 04tana=05tana=0C1(ana, thus is the centre of the circle round ABC, hence 04=^acosec4, or A=^atanacosec.4. When the measurement of the elevation at C is a + 7i", let 0' be the projection of the top of the mountain, then since the elevations at A and B are equal, 00' is perpendicular \x> AB; let /j + a; now be the height of the mountain. We find geometrically, 0'4 = 04 + 00' cos 0, 0'C=OC-00'oo&{A-B), 182 THE SOLUTION OF TRIANGLES when 00' is so small that its square may be neglected, hence h-\-x=0'A tan a=0 'Ctan (a + n!') = {0 A + 00' cos C) tain a={0G- 00' C0& {A -B))t&n{a+n"), hence a; = 00' . cqs C . tan a= -00' coa {A- B) tan a + OC seo^ a . sin n", since tan(a+ra") = tana + sec'''a. sinm", approximately; eliminating 00', we iC cos {A — B) tan a=cos Otan a (OOsec^ a . sin n'' —x), hence 2a; sin 4 sin B=OC sec^ a cos sin ji", iu -./^ iu i 1. • i-i I . • la tan a/, cos O sin?i"\ therefore the true height A + « is -= —. — r- ( 1 + -; — ; — ; — = . . — ;— 1 . 2 sin 4 \ sin ^ sin £ sin 2a/ (2) The sides of a triangle are observed to he a = 5, b = 4, c = 6, hut it is knoion that there is a small error in the measurement of c ; examine which angle can he determined mth the greatest accuracy. Let e+x be the true value of the side c; let A+8A, B+8B, G+SChe the angles of the triangle, the parts SA, SB, SC depending on ^; we suppose x so small that its square may be neglected. We have approximately, hence sin 4 . 8^ = — ^x. Also cos(5+85)=?i±i|^gll^ = |(l+^), hence sin^.S5=-A^, and cos {G+dO)= ^ — ^=-(1 -— j, hence smC 80=^ «. , , sin ji sin B sin G Also __=^ = __, so that 24. 8^ =40. 85= -15. 80. Thus 8B is numerically smaller than 6.4 and 80, hence the angle B can be determined with the greatest accuracy. EXAMPLES ON CHAPTER XI. 1. The sides of a triangle are 8, 7, 5 ; find the least angle, having given log 112=2-0492180, i cos 19° 6' = 9-9754083, diflf. for 60" = -0000437. 2. If in a triangle a=65, 6=16, 0=60°, find the other angles, having given log 3 = -4771213, L tan 46° 20' = 10-0202203, log 7 = -8450980, L tau 46° 21' = 10-0204731. EXAMPLES. CHAPTER XI 183 3. The sides of a triangle are 3, 5, 7 feet ; find the angles, having given log 13-5 = 1-1 303338, log 14=1-1461280, L cos 10° 53' = 9-9921175, L cos 10° 54' =9-9920932. 4. If .5=45°, C=10°, a = 200ft., find 6, having given log 2 = -3010300, log 172-64 = 2-2371414, Z sin 55° = 9-9133645, log 172-65=2-2371666. 5. If in a triangle 6 = 2-25 ft., c=l-75 ft, 4=54°, find B and C, having given log 2 = -301030, i cot 27° = 10-292834, Ztan 13° 47' = 9-389724, Ztan 13°48'=9-390270. 6. If the ratio of the lengths of two sides of a triangle is 9 : 7 and the included angle is 47° 25', find the other angles, having given log 2 = -3010300, L tan 66° 17' 30" = 10-3573942, Ztan 15° 53' = 9-4541479, diff. for l' = 4797. 7. An angle of a triangle is 60°, the area is 10 ^3 and the perimeter is 20 ; find the remaining angles and the sides, having given log 2 = -3010300, L sin 49° 6' = 9-8784376, log 7 = -8450980, L sin 49° 7' = 9-8785470. 8. In a triangle ABC, it is given that a=10 ft., 6=9 ft., C=tan~i(^); find c. If errors not greater than 1 in. each are made in measuring a and 6, and an error not greater than 1° in measuring C, shew that the error in the calculated value of c will be less than 2-7 in. 9. In the ambiguous case, a, h, B being given, where a > 6, if c, c' be the values of the third side, shew that c^— 2cc'cos2S+c'2=46''cos2Z. 10. In the ambiguous case in which a, h, A are given, if one angle of one triangle be twice the corresponding angle of the other triangle, shew that a\/3 = 26sin J, or 463sinM=a2(a + 36). 11. The base of a triangle is equal to its altitude, and the two other sides are of known length ; determine the remaining parts of the triangle by formulae adapted to logarithmic calculation. Shew that the ratio of the given sides must lie between ^(\/5-l) and J(\/5 + l). 12. A triangular piece of ground is 90 yards in its longest side, and 100 yards in the sum of the other two sides, and one of its angles is 46°. Determine the other angles, having given X tan 23° = 9-6278519, itan 13° 15' = 9-3719333, itan 13° 16' = 9-3724992. 13. An angle of a triangle is 36°, the opposite side is 4, and the altitude Vs-l; solve the triangle. 184 EXAMPLES. CHAPTER XI 14. Shew that it is impossible to construct a triangle out of the perpendiculars from the angles of a triangle on the sides if any side is < I (3 - V5) X perimeter ; and it is ceriiainly possible to construct such a triangle if each side is > ^ perimeter. 15. If a triangle be solved from the parts C=75°, 6 = 2, c = J% shew that an error of 10" in the value of C would cause an error of about 3"'44 in the calculated value of B. 16. Having given the mean side of a triangle whose sides are in a.p., and the angle opposite it, investigate formulae for solving the triangle, and find the greatest possible value of the given angle. Solve the triangle when the mean side is 542 feet, and the opposite angle is 59° 59' 59". 17. Solve a triangle, having given the length of the bisector of a side, and the angles into which this divides the vertical angle. 18. Solve a triangle, having given one side, the angle opposite it, and the perpendicular from that angle on the side. 19. A triangle is solved from the given parts a,h, A. If the values of a, h are affected by small errors x, y respectively, find the consequent error in the value of the perpendicular from A on the opposite side, and prove that this error is zero if xs\x^ B = 2 cos ^fl . Vsin a sin /3 . (sin a-l-sin /3)~'. 37. A man on a hill observes that three towers on a horizontal plane subtend equal angles at his eye, and that the angles of depression of their bases are a, a, a" ; prove that, c, d, c" being the heights of the towers, sin (a' - a") sin (o" — a) sin (a - a') _ c sin a c' sin a c" sin a" 38. A gun is fired from a fort, and the intervals between seeing the flash and hearing the report at two stations B, C are t, t respectively ; Z) is a point in the same straight line with BC, at a known distance a from A ; prove that if BD=h, and CD=c, the velocity of sound is \^ — hl'^- t^ — ^1 ' ^^^'"^''^ the case when a'^=be. 39. From a point on a hill-side of constant inclination, the angle of elevation of the top of an obelisk ou its summit is observed to be o, and a feet nearer to the top of the hill to be ^ ; shew that, if h be the height of the obelisk, the incUnation of the hill to the horizon will be (a sin a sin (3] (h ' sin(j3-a)J ' 40. On the top of a spherical dome stands a cross ; at a certain point the elevation of the cross is observed to be a, and that of the dome to be ; at a EXAMPLES. CHAPTER XI 187 distance a nearer the dome the cross is seen just above the dome, when its elevation is observed to be y ; prove that the height of the centre of the dome above the ground is ."f^y . sin a cos y - cos a sin /3 sin (y — a) cos y - COS ^ 41. At noon on a certain day the sun's altitude is a. A man observes a circular opening in a cloud which is vertically above a place at a distance a due south of him ; he finds that the opening subtends an angle 25 at his eye, and that the bright spot on the ground subtends an angle 2(^. Shew that if x is the height of the cloud ^ (cot2 a tan^ (^ - tan^ ff) - lax cot a tan^ <^ + a2 (tan^ ^ - tan^ fl) = 0. 42. From a point on the sloping face of a hill two straight paths are drawn, one in a vertical plane due South, the other in a vertical plane at right angles to the former, due East ; these paths make with one another an angle a, and their lengths measured to the horizontal road at the foot of the hill are respectively a and 6. Shew that the hill is incUned to the horizontal at an , . , (a^ + 6^ - 2a6 cos a\4 angle sm-M . . . \ ab sm a tan a / 43. The breadth of a straight river is calculated by measuring a base of length a along one side of the river and observing the angles made with this base by lines joining its extremities to a mark on the opposite bank. If the instrument by which the angles are measured gives each a value which is (1 + w) times the true value, n being very small, shew that the error in the computed breadth is nearly equal to na . - — t-j-t ; u, ^ being the circular measures of the above angles. 44. An observer from the deck of a ship, 20 feet above the sea, can just see the top of a distant lighthouse, and on ascending to the mast-head, where he is 80 feet above deck, he sees the door which he knows to be one-fourth of the height of the lighthouse above the level of the sea ; find his distance from the lighthouse, and its height, assuming the earth to be a sphere of 4000 miles radius. 45. Three vertical posts are placed at intervals of one mile along a straight canal, each rising to the same height above the surface of the water. The visual line joining the tops of the two extreme posts cuts the middle post at a point eight inches below the top; find to the nearest mile the radius of the earth. 46. Borings are made at three points A, B, C in a, horizontal plane, and the depths at which gault is found are a, b, a respectively; also AB = h, BO=k, ABC=a. If the upper surface of the gault be a plane, shew that its inclination <^ to the horizon is given by tan2,= {(^^2(--^i^cosa.(^}cosec2.. 188 EXAMPLES. CHAPTER XI 47. The angular elevation of a column as viewed from a station due north of it being a, and as viewed from a station due east of the former station and at a distance c from it being |3, prove that the height of the tower is e sin n sin fi {sin(a-/3)sin(a+0)|4' 48. A lighthouse stands 9 miles due N. of a port from which a yacht sails in a direction E.N.E., until the lighthouse is N.W. of her, when she tacks and sails towards the lighthouse until the port is S.W. of her, when she tacks again and sails into port. Shew that the length of the cruise is 16 miles nearly. 49. A circular pond of radius a is surrovmded by a gravel walk of uniform width 6, and the whole is enclosed by a fence of height d. A person of height h stands just inside the fence. Shew that the portion of the fence whose highest points can be seen by reflection from the water is -th, where 1 2 , ( h + d 'Jb^+2ab] - = — COS" n ■^ l2 V^flf ( h + d ^/^+2ab\ d provided h ., . 50. The width of a croquet-hoop, the thickness of its wires, and the diameter of a ball are given ; the ball being in a given position, shew how to find the conditions that it may just be possible for it to go through the hoop (1) straight, (2) by hitting one wire, (3) by hitting both wires ; assuming that the angle of incidence is equal to the angle of reflection. 51. Three mountain peaks. A, B, C, appear to an observer to be in a straight line, when he stands at each of two places P and Q, in the same horizontal line ; the angle subtended by AB and BC at each place is a, and the angles AQP, CFQ are (/> and -^ respectively. Prove that the heights of the mountains are as cot 2a + cot yjf : ^ (cot a + cot \/f ) (cot a + cot <^) tan a : cot 2a + cot (f>, and that if QB cut ^Cin D, 4C=Ci3 sin 2a(coti//-+cot 2a). 52. A man standing at a distance c from a straight line of railway sees a train standing upon the line, having its nearer end at a distance a from the point in the railway nearest him. He observes the angle a, which the train subtends, and thence calculates its length. If in observing the angle a he makes a small error 6, prove that the error in the calculated length of the train has to its true length a ratio -. ; : — r . sm a (c cos a — a sm a) 53. The height A of a mountain, whose summit is .4, is to be determined from the observed values of a horizontal base line BC{a), the angles ABC, ACB, and the angle {z) which AB makes with the vertical. Shew that ,_a cos z sin '7 sin(Z( + C) ■ EXAMPLES. CHAPTER XI 189 If h be kiiown approximately, shew that the best direction of BC in order that an error in measuring C may have least effect on the accuracy of the above value of h, is given by 5=2tan~i( r I. ^ ■' \acos0 + Ay 54. Three vertical flag-staffs stand on a horizontal plane. At each of the points A, B and C in the horizontal plane, the tops of two of them are seen in the same straight line, and these straight lines make angles a, /3, y with the horizon. The plane containing the tops makes an angle 6 with the horizon. Prove that their lengths are jBC/( Vcot^ ^ - cot^ 6 + Vcot^ y - cot^ 6), and two similar expressions. Explain how the signs of the roots must be taken. 55. A tower AB stands on a horizontal plane and supports a spire BC. An observer at a place ^ on a mountain, whose side may be treated as an inclined plane, observes that AB, BC each subtend an angle u at his eye; he then moves to a place F, measuring the distance £!F{=2a), and observes that AB, BC again subtend angles a at his eye ; he then measures the angles AFE{=^) and CFE{=y). Shew that if x and y are the heights of AB, BC respectively, ^C cos B cos 7 cos^ a "la = VC0S7=a-^l „ , ,„ , — . ., 1 , . ;(■ . ^ y \ cos2i(/3 + y)cos''i((i-y)J Also if O is the middle point of EF, and H is the point on the line of greatest slope through O, at which AB, BC subtend an angle 8, and &H is measured (=6), prove that the inchnation 6 of the mountain to the horizon is given by \iJ^^-\-W)\ ^"'^+-2^'^°^^= ^.^+y^-2^ycos2S - CHAPTER XII. PROPERTIES OP TRIANGLES AND QUADRILATERALS. 150. In this Chapter we shall for the most part assume without proof the theorems in Euclidean Geometry which are necessary for our purpose, referring to works on pure Geometry for the investigation of those theorems. The circumscribed circle of a triangle. 151. We have already, in Art. 120, obtained the formula It = \a cosec A, for the radius of the circle circumscribing a triangle, or as it is now frequently called, the circum-drcle. This formula may also be obtained as follows: PROPERTIES OF TRIANGLES AND QUADRILATERALS 191 Let be the circum-centre ; draw OD perpendicular to the side BC of the triangle ABG, then B is the middle point of BC, and the angle BOD = A. Since BD = OB sin BOD we have ^a = RainA, or R = ^acosecA (1), If S denote the area of the triangle ABG, we have /S = ^ 6c sin 4, thus we have the expression B = abcl4!S...{2). Also 0D= OB cos A = Rcos A. The irtscribed and escribed circles of a triangle. 152. We know that four circles can be drawn touching the three sides of a triangle ; the inscribed circle, or in-circle, touches each side internally, let I be its centre ; the escribed circles each touch one side of the triangle and the other two sides produced, let Ii, li, Is be the centres of these circles ; we know that I A, IB, IC bisect the angles A, B, C, respectively, and that I A bisects the angle A, and IiB, I^C bisect the angles B, 0, externally; it follows therefore that AI^, BI^, CI3 are the perpendiculars from /j, Zj, /j, on the opposite sides of the triangle I^I^Is, and that / is the orthocentre of the triangle IiIJ^. The circum-circle of the triangle ABC is the nine-point circle of the triangle /1/2/3, and therefore passes through the middle points of the sides I^I,, I3I1, Iil^, and also through the middle points of III, II2, III- 192 PROPERTIES OF TRIANGLES AND QUADRILATERALS 153. Let H, K, L be the points of contact of the in-circle of the triangle ABC, with the sides BG, GA, AB, respectively. Then A IBG + A IGA +AIAB = S. Now AlBG = ^IH.BG=^ra, AlGA^^rb, AlAB = \rc, where r denotes the radius of the in-circle, hence \r{a + h + c) = S, whence we have the formula r = S/s. ..{3), for the radius of the in-circle. Also a = BH+HG=r{cot^B + cot^G), hence 7- = a sin ^5 sin ^Csec^4 (4), another expression for r, which might of course be deduced from (3). Combining the formulae (1) and (4) we have the symmetrical expression r = 4Rsm^Asm^Bsm^G (5). Again, since AK + BG = ^ (BG +GA+ AB), we have AK = AL = s — a, and similarly BH = BL = s-b, GH=GK=s-c, hence since r = AK tan \A = BH tan \B= GK tan ^ G, we obtain the expressions r = (s-a)tan^^ =(s-6)tan^£ = (s- c)tan^C' (6), which may also be deduced from (3) or (4). PROPEKTIES OF TRIANGLES AND QUADRILATERALS 193 154. Expressions corresponding to those of the last Article may be found for the radii r-j, r^, r^ of the escribed circles. Let H^, Ki, Li be the points of contact of the circle whose centre is I^, with the sides of the triangle ABG. Then AliAB + AI^AG -AI^BG = S, therefore ^r^{b + c-a) = S, thus we have the formulae S S 8 r.= r.= - •, — c s — a' ' s — b' for the radii of the escribed circles. Also a = BE-, + Hfi = n (tan 1 5 + tan ^ C), therefore r-j = a cos ^ -S cos ^ C sec \A ■(7), ■(8), (9), whence we obtain the formula ri = ^Rsai\Acos\BGOs^G with corresponding expressions for r^ and r^. Again, since BH, = BL„ and GH, = GK„ and AK.^-AL,, we find BU^ = s-c, GH^ = s-b, AK^ = ALl = s, thus we obtain the formulae r, = staJi^A = (s-c) cot ^B = is-b) cot ^G (10). H. T. 13 194 PROPERTIES OF TRIANGLES AND QUADRILATERALS Examples. « (1) Prove that ri+r2+r3 — r=4R, r2r3 + r3ri + rir2 = S2/r2, rri+r2-> + r3-i=r-». (2) Prove the following formidae for the sides and angles of a triangle, in terms of the radii of the escribed circles: (a) a=-^M^i^, (,3) siniA=-—= 3 , Vrgra+rsri + rira y(ri+r2)(ri+r3) ^^^ (ri + r2)(ri+r3) (3) Prove that r=1 (fVH^t^illlf^. ^ ' 4 r2r3 + r3ri + rir2 (4) Prove that 16E2rrir2r3=a2b2c2. , , r, , A 2R+r-ri (5) Prove that cosA = ^= — . (6) If the escribed circle which touches a is equal to the circum-d/rcle, prove that co« A=cosB+cosC. (7) Prove that ri(r2+r3)co«eoA=r2(r3+ri)cosecB = r3(ri + r2)co«ecC. (8) If a, ai, a^, a^ are the distances of the centres of the inscribed and escribed circles from A, and p is the perpendicular from A on BC, prove that (a) aaia^a^ =4R2p2, (b) 02 + 012 + 022+032 =16E2, (c) a-2 + ai-2 + a2-2 + a3-2 = 4p-2. (9) Shew that the area of the triangle formed by joining the centres of the escribed circles is -5—, or SWcos^Acos\Ecos\Q. (10) Shew that the radius of the circle round any of the fov/r triangles fwmed by joining the centres of the inscribed and escribed circles is double of R. (11) Prove that the a/reas I1I2I3, I2I3I, I3I1I1 I1I2I o.re inversely as r, r„ r2, rj. (12) Prove that (a) Mb^ + W + W^S^, ^ ' ^ ' r2r3 ran ^rj r (b) r3 . Ill . II2 . II3 = IA2 . IB2 . IC2. (13) If di, d2, d3 be the distances of I from the angular points of a 7 1 .1 ^ did2d3 r triangle, shew that — r- — = - . (14) If a,', V, c' are the sides of the triangle formed by joining the points of a2 — a'2 b2 — b'2 c2 — c'2 contact Hi, H2, H3 of the escribed circles, shew that = — t- — = . PROPERTIES OF TRIANGLES AND QUADRILATERALS 195 (15) Prove that the sides of the triangle formed hy joining the centres of the circles BOO, COA, AOB are as sin 2A ; sin 2B : sin 20. (16)' Prove that the cireum-circles of the two triangles in the ambiguous case, when a, b, B are given, are equal in magnitude; shew also that the distance between their centres is (^ cosec^H-aF)^. (17) In the ambiguous case of the solution of a triangle, prove that the distance of the points of contact of the inscribed circles with the greater of the two given sides is equal to half the difference of the values of the third side. (18) If p\, Pa, Pi be the radii of the circles described about IBC, ICA, lAB, prove that iW-'B.{px^ + pi+p^^)-pipipi = 0. (19) Prove that the radii of the escribed circles of a triangle are the roots of the cubic x^ - x^ (4R + r) + xs^ - rs^ = 0. The medians. 155. The lines AD, BE, CF, joining the angular points of a triangle to the middle points of the opposite sides, are called the medians. The length of AB is given by the well-known geo- metrical theorem AB^ + AG' = 2(AD'' + BD% thus the squares of their lengths are given by m,'' = ^¥ + ^c'-ia^ m,' = ic'' + ^a'-ib^ m3» = K-f-i6^-ic= (11). Let Ml denote the angle ADO, then cot if, = DL/AL = ^{BL- CL)/AL, where AL is perpendicular to BG, therefore Mi is given by cot ifi = Hoot 5 - cot (7) (12). 13—2 196 PROPERTIES OF TRIANGLES AND QUADRILATERALS The point G, where the medians intersect one another, is called the centroid of the triangle. It is well known that divides each of the medians in the ratio 2:1. Examples. (1) Prove that cotAQF+cotBQI>+cotCGE=cotA+cot'B+cotG. (2) If a, ^, y are the centres of the circles BGC, OGA, AGB, and A, A' are the areas of the triangles ABC, a^y, prove that 48AA'=(a2+b2+o2)2. (3) // Ri, R2) E,3 be the radii of the circles BGC, CGA, AGB, prove that a^(b2-c2) bg(c^-a^) c^sfi-V) _ Ri^ + R^a + R32 -"• (4) If the angles BAD, CBE, ACF are a, ft y, and the angles CAD, ABE, BCF are a', /3', y, prove that cot a + cot ^ + cot y=cot a +eot^+ cot y'. The bisectors of the angles. 156. Let a and Oj be the points in which the internal and external bisectors of the angle A meet the opposite side BG. Let /, g, h be the lengths of the internal bisectors Aa, 5/3, Cy, and /', g', h' the lengths of the external bisectors Aa^^, B^i, Gji. To find the positions of a and a^, we have Ba/Ga=BA/GA =Ba^jGai, whence R„ «c n„- "'^ Tir, *" n» "^ b + c' b + c c-b' c-b To find the lengths /, /', we have AABa+AAGa=S = AAaiB-AAa,G, hence / (6 + c) sin ^4 =/' (c - 6) cos ^.4 = 28, PROPERTIES OF TRIANGLES AND QXTADRILATERALS 197 therefore / and /' are given by J. 2bc , . ., 26c . , . .(13). Examples. (1) If a, 13, y are the angles that Aa, Bft Cy make vdth the sides a, b, o, shew that a«i}i2a + bai'?i2j3+c«j?i2'y=0. (2) If fj, gj, h] are the lengths of the bisectors of the angles, produced to meet the circum-eircle, shew that f->co«^A+g-ico4p+h-icosiC = a-i + b-i + c-i, and fieo«^A+giCO«^B + bi cos ^C=a + b + c. (3) Prove that a^ cuts Cy in the ratio 2o : a+b. The pedal triangle. 157. The triangle LMN formed by joining the feet of the perpendiculars AL, BM, GN, from A, B, G, on the opposite sides, is called the pedal triangle oi A, B, G. Let P be the orthocentre of the triangle ABG, then since PMA, PNA are right angles, a circle whose diameter is PA circumscribes PMAN, hence MN is equal to PA multiplied by the sine of the angle in the segment MN, or MN — PA sin-d; now if is the centre of the circum-eircle, and OD is perpendicular to BG, it is well known that AP =20D, and we have shewn in Art. 151 that this is equal to 2BcosA: hence MN=2Rsm. A gos A ■= a cos A. Also 198 PROPERTIES OF TRIANGLES AND QUADRILATERALS the angles PLM, PLN are each the complement of A, or MLN='7r—2A; the sides and angles of the pedal triangle are therefore respectively a cos 4, 6 cos 5, ccos(7] ,-.> TT-IA, 7r-25, 7r-2C) *■ '' It should be remarked that ABG is the pedal triangle of I1I2I3. The pedal triangle of LMN is called the second pedal triangle of ABG, and so on. We have assumed that the triangle is acute-angled; if the angle A is obtuse, it can be easily shewn that the angles of the pedal triangle are 2A — TT, 25, 2(7, and that the sides are - a cos A, b cos B, c cos C. Examples. (1) Prove that the radius of the circle inscribed in the triangle LMN is 2R cos A cos B cos C. (2) If a, ft y are the diameters of the circles MPN, NPL, LPAI, shew that be ca ab (3) Prove that if r', rj', r2', rs' are the radii of the inscribed and escribed circles of the pedal triangle, then ^ 3 ^ = — k|-^ ■ (4) If AL, BM, CN meet the dreum-circle in L', M', N', shew that AL' BM' CN'_ AL "•■ BM "•" CN ~ The distances between special points. 158. Let P be the orthocentre, the centre of the circum- circle, / of the in-circle, /j of one of the escribed circles, the centroid, and U the centre of the nine-point circle of the triangle ABG. According to Euler's well-known theorem, the three points 0, Q, P lie on a straight line, and PG = 20G ; the point U is also on OP, at its middle point. Each of the angles lAO, lAP is equal to|(JSo..C); also AO = E,AP = 2R cos A, AI = r cosec ^ J. = 4J? sin ^fi sin J (7, AI^ — 4i? cos J 5 cos ^ G. We can now find expressions for the distances of the points 0, I, P, Ii, U irom one another. PROPERTIES OF TRIANGLES AND QUADRILATERALS 199 (1) To find 01 = L We have ^ = AO^ + A1^-2AO.AIcobOAI, hence S^ = iJ^l + 16 sin'' ^5 sin'' i^C- 8 sin ^5 sin ^Ocos i£^^) or 8' = ii^(l-8sini^sini£sin^O), we thus obtain Euler's formula (2) To find Oil = Si . We have .(15). Si» = J?''(l + 16cos^i5cos=JO-8cos^5cosiOcosi5-(7) or S,=' = i2^(l+8sini^cosii?cosJC), which gives h^ = m + tUr^ (16). (3) To find OP. From the triangle OAF we have OF"^ = 04" + AT'' -20A.AP cos OAP or OP " = E 2 (1 + 4 cos" ^ - 4 cos A cos 5 - G), which gives OP" = iJ" (1 - 8 cos 4 cos £ cos 0) (17). 200 PROPERTIES OF TRIANGLES AND QUADRILATERALS (4) To find IP. We have IP" = 4,B^ cos'' A + 16B^ sin^ ^B sm^G -16B" cos A sin^Bsia^G cos i(B-G), hence /P" = 4^" {cos« A+(l- cos B) (1 - cos G) - cos AsinB sin G — cos A{1 — cos B) (1 — cos G)}, or /ps = 4iJ2 {(1 - cos ^ ) (1 - cos B) (1 - cos 0) — cos^ cosjBcosO} (18), or /P^ = 2r^ -472^008^ cos 5 cos C. (5) To find lU. We have /f^'' = i/P=+iJO^-iOP^ hence IU"=-7^+^R''-Br-lB' = (^B-ry; hence IU=^B — r; in a similar manner it can be shewn that IiU = ^R + ri; now ^i2 is the radius of the nine-point circle, hence the expressions we have obtained for lU, IJJ shew that the inscribed and escribed circles touch the nine-point circle. We have then a trigonometrical proof of Feuerbach's theorem, of which a considerable number of geometrical proofs have been given. Examples. (1) If ti, t2, ts are the lengths of the tangents from the centres of the escribed circles to the circum-drole, prove that 111 ^ a + b+c ti> ■*■ ta^ "^ tjS abc ' (2) Prove that the area of the triangle lOP is -2W sini(B-C) siniiC- A) sini{A-B). (3) Prove that QV='ifR^ {2 sin^ p sin^ ^C - JjS sin^ A} and GP+4Rr=i(bc+ca + ab)-J(a2+b2+c2). (4) Prove th^ Qp.^SaVa^-bp (a^-c^)_ (5) If a, ^, y be the distances of the centre of the nine-point circle from the amgidar points, and g its distance from the orthoeentre, shew that a2+;32+y2 + g2=3R2. (6) Prove that the nine-point circle does not cut the circum-cirde unless the triangle is obtyse, and in that case they cut at an angle co«~ ^ (1 + 2 CO* A cos B cos 0). PROPERTIES OF TRIANGLES AND QUADRILATERALS 201 (7) Skew that, if the distance between the orthocentre and the centre of the circum-circle is ^a, the triangle is right-angled, or else toraBto«C = 9. (8) If Qis the centre of the nine-point circle, shew that (Ql2-QIs)(QIi-QI)=b2-c2. (9) IfOTPisan equilateral triangle, shew that cos A + cos B + cos C = § . (10) If the centre of the in-drcle he equidistant from the centre of the circum-circle and the orthocentre, prove that one angle of the triangle is 60°. Expressions for the area of a triangle. 159. A very large number of expressions for the area of a triangle, in terms of various lines and angles connected with the triangle, have been given. Large collections of such formulae will be found in Mathesis, Vol. iii. and in the Annals of Mathematics, Vol. i. No. 6. We give here a few of these expressions, leaving the verification of them as an exercise for the student. (1) \/r?vv^, (2) ^J^Rplp^p3, (3) ^\/o-(o--mi)(o--m2)(o--«i3) where 2'' - c% hence le^Sf" = 4 (ad + bcf - (a' + d^ - 6^ - c^' - Wahcd cos" a, or 16S" = {(a + dy - (6 - c)"} {(6 + cf - (a - df] - 16a6cd cos" a ; hence 8^ = {s- a) (s -b)(s — c) (s-d)- ahcd cos^ a (19), where 2s = a + 6 + c + d. In the case of a quadrilateral inscribable in a circle we have 2a = 7r, thus S^ = {s-a){s-b){s-c){s-d) (20). The expression (19) shews that the quadrilateral of which the sides are given has its area greatest when a^^ir, that is, when the quadrilateral can be inscribed in a circle. The theorem (20) was discovered by Brahmegupta, a Hindoo Mathema- tician of the sixth century. 165. Expressions for the area of a quadrilateral can be found, which involve the lengths of the diagonals and the angle between them. The area of the quadrilateral is the sum of the areas of the four triangles into which the diagonals divide it ; the area of each of these triangles is half the product of the two segments of the diagonals which are sides of it, multiplied by sin ; hence by addition we have S=\xys\n<\) (21). Also WA.OB cos ^ = 0^2 + OB^ - a\ 20G . OD cos = OG' + OD^ - c\ 20 A . OD cos ^ = d^-OA''- 0D\ 20B.0G cos = b^-OB^- 0G\ hence 2*^ cos ^ = 6^ 4- d'' - a" - c'' (22), therefore S = \{b^ + d^- a? -&) ta.-D.il> (23), PROPERTIES OF TRIANGLES AND QUADRILATERALS 205 and eliminating S=i{4a^2/'-(^' + '^'-a'-cO'}* (24), which expresses the area in terms of the diagonals and the sides. If a circle can be inscribed in the quadrilateral, we have a+c=b + d, hence the formulae (23), (24) become S=^{ac-bd)ta.ii ^ A.'^D'T' We have p'T'_qt) ="qp' = ^' ^lence PL={R-a)coaA + -sPL, with similar expressions for PM, PN; now 'ihLMN=PM. PNainA+PN. PLainB+PL. PMamC ={R-d)^SamAcoaBcosC+^^sP'M' . P'N'ainA +^(R-d)2P'L'aiaA; also ^SP'M' . P'N' sin A is the area of the triangle L'M'N', which is zero, and ^P'L'smA=^^a.P'L'=\^hP'BC=\hABC, and 2 sin 4 cos 5 cos (7= sin 4 sin 5 sin 0; hence 2 A LMN= {R - d)^ sin A sin £ sin C+ 2rf {R - d) sin A sin 5 sin C = {R^ - dF) sin A sin 5 sin C. (3) //^ A, B, C 6e any three fixed points, and P osmy point on a circle whom centre is 0, shew that AP^. ABOO+BP^. ACOA+CP^. A AOB is constant for all positions of P on the circle. Denote the angles BOC, GOA, AOB by a, ft y, then a+0+y=27r, and let the angle POA be 5. We have AP'^=OP^+OA^-WA.OPooa6,a,TA similar expressions for BP^, CP^, hence the expression above is equal to OP^ . i^ABC+sOA^ . hBOC-'iOPs.OA . hBOG .coae ; PROPERTIES OF TRIANGLES AND QUADRILATERALS 211 the first two terms in this expression are independent of the position of P on the circle, and the coef&cient of %0P in the last term is \0A. OB. OC{oosflsina+cos(5-|-y)sin/3 + co8(/3-5)sin'y} or ^OA. OB. (9Ccos5(sina+sin(3coS')/ + cos/3siny) which is zero ; thus the theorem is proved. Particular cases of this theorem are the following : (a) P^2sin24+P£2sin2£+PC2sin2C is constant if P lies on the circum-circle ; (6) PA^ wa. A +PB^ sin B-k-PC^ SCO. C is constant if P lies on the in-circle. (c) PA^ sin A cos {B-G)-\- PB^ sin Bcos(C-A) + PC^ sin C cos (A - B) is constant if P lies on the nine-point circle. (4) Shew that the length of the side of the least equilateral triangle that can he drawn with its angvlar 'points on the sides of a given triangle ABC is 2aV2 VaHb2+c2-|-4'\/3A where A is the area of ABC. Let DEF be such an equilateral triangle, and let the circle round DEF cut BC and AG in H and O respectively ; the angles FOA, FEB are each 60° thus FO, FH are in fixed directions ; also the angle HFG is 120° - C. We have, if AF be denoted by x, FG =xsm Ajsin 60°, FH={c- x) sin Bjsin 60°, hence ^6*2= cosec2 60° {x^ sin^ A + {c- xf sin^ B-'2,x{c-x) sin .4 sin 5 cos (120° - C^}. Now the radius of the circle is HOj^ sin (120° — C), hence the circle is least when HO is least. The least value of a quadratic expression 'Kai^+'ifM + v, 14—2 212 PROPEETIES OF TKIANGLES AND QUADRILATERALS in which X is positive, is i/-^, for Xx^ + Sux+v may be written in the form A X (^ + ^ j + ./ - ^ . We find therefore for the least value of HG am 60°, fj • a„ (csin'^ + c3in^sin^cosl20°-C)' ) i (" ^'° ain^A + sin^ £ + 2 sin 4 sin B cos (120° - C)J ' which is equal to c sin J. sin 5 sin (1 20° — G) {sin2 A + sin2 B+2amA sin 5 cos (1 20° - C)}4 ' \/2 b" sin A sin B sin (120° - G) sinC'Va2+62 + c2+4\/3A Now the side of the equilateral triangle is ^(Tsin607sin(120°- C), thus the least value of the side is 2A^2 Va2+6Hc2+4V3A' (5) Describe three circles mutually in contact, each of which touches two sides of a given triangle. Let pi, p2, p3 be the radii of the circles, then MN=2'\/p2P3, hence a=BM+G^+MN=piCot^B+psCot^G+2 ^/pips, with similar equations for b and c. Let ^=piCot^4, y^=p2Cot^B, z^=p3Cot^G, Vtan^5tauiJ(7= -cosa, Vtan^CtanJ^ = -cosjS, \/tan^yltan^5=— cosy; we find sin2a = l-tan^5tan JC=a/«, and similarly ain^ p=b/s, ain^y =cjs, hence we have the equations y^+z^- %yz cos a _^-\-a?' — %zx cos |3 _ a:^ +y* — 2a^ cos y _ sin^a sin^^ "" sin^y ~ ' PROPERTIES OF TRIANGLES AND QUADRILATERALS 213 these have been considered in Art. 68, Ex. (12) ; adopting the first solution there found, we have x=\/s ooa((r-a), y = \/s cos(o--/3), «=V« cos ((r-y), where 2o-=a + 0+7, hence pi=stan J4cos^((r — a), p2 = «tan^5oos^(o--/3), ps=sta,ii^Ocoa^(a--y) are the required radii of the circles. The other solutions give the radii of three sets of circles which are such that two in each set touch two sides of the triangle produced ; of one such set, the radii are stan^^cos^*, stan^5cos^(s-y), stan^Ccos^(«— /3). There are altogether eight sets of circles which satisfy the conditions of the problem. This solution is founded on that of Lechmiitz given in the Nouvelles Annales, Vol. v. A geometrical solution of this problem, which is known as " Malfatti's Problem," will be found in Casey's Sequel to Euclid. A history of the problem will be found in the Bulletin de I'Academie Royale de Belgique for 1874, by M. Simons. EXAMPLES ON CHAPTER XII. 1. If 6 be the angle between the diagonals of a parallelogram whose sides a, h are inclined at an angle a to each other, shew that tan 5 = — = — 5^—. ' a^ — h^ 2. If a, /3, y be the distances, from the angular points of a triangle, to the points of contact of the inscribed circle with the sides, shew that 3. The area of a regular inscribed polygon is to that of the circumscribed polygon, of the same number of sides, as 3 : 4 ; find the number of sides. 4. From each angle of a parallelogram a line is drawn making the same angle, towards the same parts, with an adjacent side, taken always in the same order ; shew that these lines will form another parallelogram similar to the original one, \i a^~V'=%ahcoaB, where a, h are the sides, and B is an angle of the parallelogram. 5. The straight lines which bisect the angles 4, (7 of a triangle meet the circumference of the circum-circle in the points a, y ; shew that the straight line ay is divided by CB, BA into three parts which are in the ratio Biv?\A : 2 sin ^-4 sin 1^5 sin ^C: sin^^C. 214 EXAMPLES. CHAPTER XII 6. If / be the centre of tte in-circle of a triangle, la, lb, Ic perpendiculars on the sides, pi, p2, ps the radii of circles inscribed in the quadrilaterals Able, Be la, Calb, prove that Pi . Pi . P3 ^ a+h + c r-pi r-p2 r-p3 2r 7. Prove that the line joining the centres of the circum-eircle and the . , . . , , ■., T.^ , . ,/ sinjB~sinC \ m-cu:cle of a triangle makes with BC an angle cot~i =- 7= — 7 . ° ^ \cos B + coaC-lJ 8. If, in a triangle, the feet of the perpendiculars from two angles, on the opposite sides, be equally distant from the middle points of those sides, shew that the other angle is 60°, or 120°, or else the triangle is isosceles. 9. If ABC be a triangle having a right-angle at C, and AS, BB drawn perpendicularly to AB meet BG, AC produced in B, D respectively, prove that tanC^Z>=tan3£.4(7, and hEGD=hACB. 10. If a point be taken within an equilateral triangle, such that its distances from the angular points are proportional to the sides a, b, c of another triangle, shew that the angles between these distances wiU be in + A, in + B, in + O. 11. The points of contact of each of the fovu: circles touching the three sides of a triangle are joined; prove that, if the area of the triangle thus formed from the inscribed circle be subtracted from the sum of the areas of those formed from the escribed circles, the remainder will be double of the area of the original triangle. 12. If ABCD is a parallelogram and P is any point within it, prove that A APG . cot APG- A BPD . cot BPD is independent of the position of P. 13. Three circles touching each other externally are all touched by a fourth circle including them all.' If a, b, c be the radii of the three internal circles, and a, ft y the distances of their centres from that of the external circle respectively, prove that \bc ea ah) a^ b^ c^ 14. P, Q, R are points in the sides BC, CA, AB of a triangle, such that :?.^ = ^ = 41 ; shew that AP^+BQ^+CR^ is least, when P, Q, R bisect the PC (^A BK sides. 15. On the sides a, 6, c of a triangle are described segments of circles external to the triangle, containing angles a, ft y respectively, where a+j3+y = 5r, and a triangle is formed by joining the centres of these circles; shew that the angles of this triangle are a, ft y. EXAMPLES. CHAPTER XII 215 16. Through the middle points of the sides of a triangle, straight lines are drawn perpendicular to the bisectors of the opposite angles, and form another triangle ; prove that its area is a quarter of the rectangle contained by the perimeter of the former triangle and the radius of the circle described about it. 17. P is a point in the plane of a triangle ABG, and L, M, N are the feet of the perpendiculars from P on the sides ; prove that if MN+NL+LM be constant and equal to I, the least value of PA'^ + PB^ + PC^ is- ?V(sinM + sin2 5+sin2C'). 18. Lines B'C, C'A', A'B' are drawn parallel to the sides BO, CA, AB of a triangle, at distances ri, r^, r^ respectively; find the area of the triangle A'B'C. If eight triangles be so formed, the mean of their perimeters is equal to the perimeter of the triangle ABC, but the mean of their areas exceeds its area by (aVi2 + 6V22 + cV32)/4A. 19. On the sides of a scalene triangle ABG, as bases, similar isosceles triangles are described, either all externally or all internally, and their vertices are joined so as to form a new triangle A'B'C ; prove that if A'B'C be equi- lateral, the angles at the base of the isosceles triangles are each 30° ; and that if the triangle A'B'C be similar to ABC, the angles are each , _, 4A *™ a^ + V + o'' where A is the area of ABC. 20. A straight line cuts three concentric circles va A, B, C, and passes at a distance p from their centre ; shew that the area of the triangle formed by . . „ ^. BC.CA.AB the tangents ax, A, B, U is . 21. If N is the centre of the nine-point circle of a triangle ABC, and D, E, F are the middle points of the sides, prove that BG oos. NDC+CA cos NEA+AB cos NFB=Q. 22. On the side BA of a triangle is measured BD equal io AC ; JSCand AD are bisected in E and F ; E and F are joined ; shew that the radius of the circle round BEF is ^BCcoseo\A. 23. If A', B', C be any points on the sides of the triangle ABC, prove that AB'.BC .CA'-\-B'C. C'A . A'B=^4.R.AA'B'C'. 24. If X, y, 2 denote the distances of the centre of the in-circle of a triangle from the angular points, shew that a4^+64y4+c*z*-f.(a-l-6-fe)2ii;2y222=2(62cy22-(-c2a232ii;2-l-a262a;y). 25. D, E, F are the points where the bisectors of the angles of the triangle ABC meet the opposite sides ; if oo, y, z are the perpendiculars 216 EXAMPLES. CHAPTER XII drawn from A, B, C, respectively, to the opposite sides of DEF, pi, P2, Ps those drawn from A, B, C, respectively, to the opposite sides of ABC, prove that „ 2 », 2 « 2 ^+^+%=ll + 8sini4sini5sinift 26. Shew that the distances of the orthocentre of a triangle from the angular points are the roots of the equation x3-2{R+r)a;^ + {r^-4:W+s^)x;-2R{s^-{r+2Rf}=0. 27. If each side of a triangle bears to the perimeter a ratio less than 2 : 5, a triangle can be formed, having its sides equal to the radii of the escribed circles. 28. ABC is a triangle inscribed in a circle, and from 2), the middle point of BC, a line is drawn at right angles to BG, meeting the circumference in £! and F; AE, AF are joined. If triangles be described in the same way by bisecting AB, AC, shew that the areas of the three triangles thus formed are as sm{B-C):sm{C-A):sin{A-B). 29. Three circles, whose radii are a, b, 0, touch each other externally ; prove that the radii of the two circles which can be drawn to touch the three are abc (bc+ca+ab) + 2'i/abc(a + b + c) 30. ABC is a triangle; on its sides equilateral triangles A'BC, EC A, CAB are described without the triangle; prove that (1) AA', BB', CC meet in a point 0, (2) OA'=--OB+OC, (3) hA'B'C'=ihABC+'^{BC^+CA^-¥AB^). a 31. A', B' are the middle points of the sides a, 6 of a triangle ; D, E are the feet of the perpendiculars from A, B oia the opposite sides ; A'D, B'E are bisected \nP,Q; prove that PQ=^\/a^+l^-2aboosZC. 32. The perpendiculars from the angular points of an acute-angled triangle meet in P, and PA, PB, PC are taken for sides of a new triangle. Find the condition that this is possible, and if it is, and a, /3, y are the angles of the new triangle, prove that cos a cos^ cosy 1 > D ^ 1 -I 7 H a H 7V=? sec A sec^sec G. cos A cos B cos C 33. Two points A, B are taken within a circle of radius r, whose centre is C. Prove that the diameters of the circles which can be drawn through A and B to touch the given circle are the roots of the equation a;2 (rV - ffl2 52 aiii2 c) _ 2^c2 (?-2 - a6 cos C) + c2 (r* - 2r2 a6 cos C+ a2 62) = 0, where the symbols refer to the parts of the triangle ABC. EXAMPLES. CHAPTER XII 217 34. If a triangle be cut out in paper, and doubled over so that the crease passes through the centre of the circumscribed circle and one of the angles J, shew that the area of the doubled portion is ib^am^OcoaOcosec(2C-B)aeo{C-B), where C>B. 35. From the feet of the perpendiculars from the angular points A, B, G of a triangle, on the opposite sides, perpendiculars are drawn to the adjacent sides; shew that the feet of these six perpendiculars lie on a circle whose R (cos2 A cos2 B cos2 C+sin^ A sin^ B sin^ C)*. 36. Prove that if P be a point from which tangents to the three escribed circles of the triangle ABC are equal, the distance of P from the side BC will be ^{b+c) Bec^Asin^B sin^C. 37. If X, y, z be the sides of the squares inscribed in the triangle ABC, on the sides BC, CA, AB, shew that -H \-- = - -\---\ (--. X y z a r 38. AA', BB', CC are the perpendiculars from A, B, G on the opposite sides of the triangle ABC ; Oj, 0^, O3 are the orthocentres of the triangles AB'C, BG'A', CA'B'. Prove (1) that the triangles O1O2O3, A'B'G' are equal, and (2) that '2,riB^=BaRhRc, where R^, Rt, Re are the radii of the circles OiA'Oz, OsB'Oi, OiC'O^, and rj is the radius of the circle inscribed in A'B'C, and ^1 of the circle about A'B'C. 39. If X, y, z are the distances of the centres of the escribed circles of a triangle, from the centre of the in-circle, and d is the diameter of the ciroum- circle, shew that xyz + c? (a;2 +3/^ + ^) = 4(^'. 40. The lines joining the centre of the in-circle of a triangle, to the angular points, meet that circle in A^, B^, C^; prove that the area of the triangle A^B^G^ is ^r'^{cos^A+coa^B+coa^C). 41. If each side of a triangle be increased by the same small quantity x, shew that the area is increased by Rx (cos J + cos 5+ cos C), nearly. 42. AA', BB', CC are diameters of a circle, D, E, F are the feet of the perpendiculars from A', B', G' on BC, GA, AB respectively; prove that AD, BE, €F meet in a point, and that the areas ABC, DEF are in the ratio 1 : 2 cos A cos B cos G. 43. If ID, IE, IF are drawn from the in-centre / of a triangle, perpen- dicular to the sides, find the radii of the circles inscribed in lEAF, IFBD, IDGE; if they are denoted by pi, p^, pg respectively, shew that (»--2pi) (r-2/j2) (r-2p3)=r3-4pip2P3- 218 EXAMPLES. CHAPTER XII 44. Shew that the radii of the circle which touches externally each of three given circles, of radii a, b, c which touch each other externally, is given by ^/Rbc{b + c+R) + \/Eoa{c+a+£) + ^Iiab{a+b+E) = \/abc(a + b + c). 45. Perpendiculars AA^, BB^, CG^ to the plane of a triangle ABC are erected at its angular points, and their respective lengths are a,b, c ; shew that if A and Aj be the areas of ABC and A-^B^G^, then i^^^-A^ = i{a^{x-y){x-z) + b^{y-z)(il-3(;) + c^{z-x){z-y)} =l{ai^{x-y){x-z) + bi^i7/-z){y-x) + Ci^{z-x){z-y)}. 46. Three circles are described, each touching two sides of a triangle, and also the inscribed circle. Shew that the area of the triangle having their centres for angular points bears to the area of the given triangle the ratio 4 sin \A sin \B sin \ C (sin \A-^ sin ^^+sin ^C) : cos^4cos^5cos^C(co8^4+cos^5+cos^C). 47. If the lines bisecting the angles of a triangle meet the opposite sides in D, E, F, prove that the area of the triangle DEP is 2r2 cos ^A cos ^5 cos ^C/cos i (5 - C) cos \{C-A) cos ^{A- B), and that {a+bf{a + cfEF^ + {b + cf(b + afFD^ + {c-\-af{c-irbfDE^ = \QA^R{\\R+^r), where A is the area of ABG. 48. is the centre of the circum-circle of a triangle, K is the ortho- centre, and OK meets the circle in P and P', and the pedal lines of P and P' in § and §' ; prove that OQ . OQ; ='i.R^ coa A cos B cos G. 49. N is the centre of the nine-point circle of a triangle ; B, E are the middle points of CB and CA ; prove that the area of the quadrilateral NDGE is J/)^(sin24 -|- sin 25 -1-2 sin 2 C), where p is the radius of the nine-poinb circle. 50. A triangle is formed by joining the centres of the escribed circles, a third from this, and so on ; shew that the sides of the nth triangle are A TT-A 3rr+A (2^-^-l)w + (-l)''-^A a cosec -^ cosec ^ cosec — ^ — cosec ^^ _^ , and similar expressions. 51. If iV is the centre of the nine-point circle of ABG, and AN meets BC in D, shew that BN-.BA ::cos(5-C):4sin5sinC, and that the area of BNC is \R^sia.A cos {B—G). 52. Shew that the radius of the circle which touches the three circles BGE, EAF, FBB, where B, E, F are the feet of the perpendiculars from A, B, G on the opposite sides, is 2iJ sin .4 sin 5 sin Ccos^^ cos 5 cos (7 (sin ^-)- sin 5 -t- sin C) s\v? A sin^ B sin^ C- 2 sin''' A cos^ A + i cos A cos B cos CS sin BsmG' EXAMPLES. CHAPTER XII 219 53. If from any point 0, perpendiculars OD, OE, OF are drawn to the sides BO, GA, AB of a triangle, prove that coiADO+cotBEA + coiOPB=0. 54. If h, c, B are given, and there. are two triangles with these given parts, shew that their inscribed circles touch, if c2 (cos2 5 + 2 cos fi- 3) + 25c (1 -cos B) + h^=Q. 55. If , E, F are drawn straight lines B'G', C'A', A' El, equally inclined to BC, GA, AB respectively, so as to form a triangle A'B'G' similar to ABC. Prove that the radius of the circumscribed circle of A'B'G' is {EF cos a + FD cos /3 + DE cos ■y)/4 si n .4 sin ^ sin C, where a, ft y are the inclinations of AA', BB', CG' to BC, GA, AB re- spectively. 66. If P be a point on the circum-circle whose pedal line passes through the centroid, and if the line joining P to the orthocentre cuts the pedal line at right angles, prove that PA^+PB^+PG^ = 4:R^ {1-2 COS A COS B cos C). 67. D is a, point in the side BG of a triangle ; if the circles inscribed in the triangles ABD, ACD touch AD in the same point, prove that D is the point of contact of the in-circle of ABC with BG ; but if the radii of the circles be equal, then CD : BD : : coseo D + cosec C : coseo D + cosec B. 68. From a point within a circle of radius r, three radii vectores of lengths Ti, r^, r^ are drawn to the circle, and the angle contained by any pair is Stt/S ; shew that 3»-2(r2r3+r3ri + ?-ir2)2=()-22+»-2'-3-t-»-3^)W+»-3n+»-i'')(»-i''+»-i»-2+»-2*), and that the distance of the point from which the radii are drawn, from the centre of the circle, is d, where (r2-cP)(j-2r3-|-r3ri+r,r2)=rir2r3(ri-(-j-2+r3). 69. Circles are inscribed in the triangles DiEyF-i, D^E^Fi, DsE^Fg, where D^, Ei, Fi are the points of contact of the circle escribed to the side BG; shew that if pi, p2, ps be the radii of these circles -:-:- = l-tanJ^:l-tan^5:l-tanJC. Pi Pi Pi EXAMPLES. CHAPTER XII 221 70. In a triangle ABC, A', B', C are the centres of the circles described each touching two sides and the inscribed circle ; shew that the area of the triangle A'B'C is tan ^(jr-A) tan ^{m — B) tan \{n-C) {cosec J(7r — 4)cosec J {n- B) cosec J (ir-C)+4}f2. 71. The three tangents to the in-circle of a triangle which are parallel to the sides are drawn ; shew that the radii of the circles inscribed in the three triangles so cut off from the corners are given by the equation 42^_„2^2_j^2(„2 + 52^.c2_26e_2ca-2a6)^-r6=0. 72. The perpendiculars from the angular points of a triangle on the straight line joining the orthocentre and the centre of the in-circle are p, g,r; prove that psin^ _ gsin.B ^ rsinC sec 5 — sec C sec C— sec 4 seo A — aeoB' a convention being made as to the signs of ^, q, r. 73. A point is taken within an equilateral triangle, and its distances from the angular points are u, j3, y. The internal bisectors of the angles between (ft y), (y, a), (a, fi) meet the corresponding sides of the triangle in P, Q, B respectively ; shew that the area of PQR is to that of the equilateral triangle in the ratio 2a^y:(/3+y)(y + a)(a+0). 74. If I, m, n are the distances of any point in the plane of a triangle ABC, from its angular points, and d the distance from the circum-centre, prove that l^ sin 24 +»i2 sin 'i.B + 'n? sin %C=i{R^-ir=Q. 76. Each of three circles, within the area of a triangle, touches the other two, touching also two sides of the triangle ; if a be the distance between the points of contact of one of the sides, and ft y be like distances on the other two sides, prove that the area of the triangle of which the centres of the circles are angular points is ^{^^y^+y^a^ + a^^^Y . 77. If a, b, c, d be the perpendiculars from the angles of a quadrilateral upon the diagonals c^i, d^, shew that the sine of the angle between the , , { {a+c){h+d)\ h diagonals is equal to V" J^ j ■ 222 EXAMPLES. CHAPTER XII 78. If A BOD be a quadrilateral, prove, in any manner, that the line joining the intersection of the bisectors of the angles A and C with the intersection of the angles B and D makes with AD an angle equal to _j ( • sin.4-sini) + sin(^+.B) 1 (,l+cos^+cos/)+cos(J+5)J ■ 79. ABODE is a plane pentagon; having given that the areas of the triangles EAB, ABC, BOD, ODE, DEA are equal to a, h, c, d, e respectively, shew that the area A of the polygon may be found from the equation A^-{a + b + c + d+e)A+{ab + bc+cd+de + ea)=0. 80. Shew that if a quadrilateral whose sides, taken in order, are a, b, c, d be such that a circle can be inscribed in it, the circle is the greatest when the quadrilateral can be inscribed- in a circle, and that then the square on the radius of the inscribed circle is -. , ,. , ,, . (a+c)(6+d) 81. A polygon of 'in sides, n of which are equal to a, and n to b, is inscribed in a circle ; shew that the radius of the circle is w ^ + 2a6 COS — + 6^ I cosec — . n J n 82. A quadrilateral whose sides are a, b, c, d can be inscribed in a circle ; its external angles are bisected ; prove that the diagonals of the quadrilateral formed by these bisecting lines are at right angles, and that the area of this , ., , , . 1 s^{ab + cd){ad+bc) quadrilateral is ^ — , ^ 2 (a + c) (fi+d) Kj{s-a) (s-b) («-c) (s-d) where 2s=a+b+c+d. 83. A quadrilateral ABOD is inscribed in a circle, and EF is its third diagonal, which is opposite to the vertex A ; prove that if the perpendiculars from A on BO, OD meet the circles described on AD, AB respectively as diameters, in F, then FQ sin D = EFiaiii' A- siv?D). 84. The power of two circles with regard to one another, is defined to be the excess of the square of the distance between their centres, over the sum of the squares of the radii. Prove that for a triangle ABO, the power of the inscribed circle, and that escribed circle which is opposite A, is ^{«^ + (6 — c)^}, and hence verify that if the escribed circle touches an escribed circle, the triangle must be isosceles. 85. The sides, taken in order, of a pentagon circumscribed to a circle are a, b, c, d, e; prove that its area is a root of the equation x*-x^s{i2a'(b+e-c-d)-iSa'+i2acd} + {s-a — e)(s-b-d){s-c — e){s-d-a) (s — c-b)s^=0, where 2« is the sum of the sides. EXAMPLES. CHAPTER XII 223 86. If a, b, c, d be the distances of any point on the circumference of a circle of radius r, from four consecutive angular points of an inscribed regular polygon, find the relation between a, b, c, and d, and prove that 2_ {ab — cd){bc — a(r){ca — bd) ~{a+b-c-d){b + c-a — d){c+a-b-d){a-\-b + c+d)' 87. The perimeter and area of a convex pentagon ABODE, inscribed in a circle, are 2s and S, and the sum of the angles at ^and B, at A and G, are denoted by a, ^, ; shew that «2(sin2a+ + 8in2e) + 2/S'(sina+ +sine)2=0. 88. ABCD is a convex quadrilateral of which the sides touch one circle, while the vertices lie on another; tangents are drawn to the circumscribed circle at A, B, C, D so as to form another convex quadrilateral ; prove that the area of the latter is {s(r — 2abcd) (abed)^ a- {a- - bed) (o- — add) (o- - dab) (or - abc) ' where r is the radius of the circle ABCD, 2s=a+6 + c+ci, and '^riyi, oo^-iriy^, then 6y;\-6^ is not necessarily the principal value of the argument of the product. We can now obtain a geometrical construction for the product of two numbers ; let A, P, Q represent the three numbers + 1, ""i + *yij "'s + iyi ', join AP, on OQ describe a triangle QOR similar to AOP, and so that the angle QOR is equal to +0^, then ROA = 01 + 01, and also OR:OQ::OP :0A; hence the length of OR is equal to the product of the lengths of OP and OQ; it follows that the point R represents the product (so,, + iy,) (oo^ + iy^. If we now introduce a third factor x^ + iy^ = r^ (cos ^3 + 1 sin ^3), we have («i + iyi) («2 + iy^ («s + iys) = n»'a»'s {cos (^1 + ^a) + i sin (^1 4- ^2)} {cos 0s +i sin ^3} = r,r-2r-3 {cos (^1 + 0^ + 0,) + i sin {0^ + 0^+ 0^)], and we obtain, in a similar manner, the product of four or more COMPLEX NUMBERS 231 complex numbers. In the case of n such numbers, we obtain the formula («i + iyi) («2 + iy^) ...(«„ + iyrt) =Vi-rn{oos(e, + e, + ... + e„)+isia(e, + 0, + ... + en)]...(i). Or the modulus of the product of any set of complex numbers is the product of their moduli, and the argument of their product is the sum of their arguments. The product may be obtained geo- metrically by a repeated application of the construction we have given for the product of two numbers. Division of one complex number by another. 179. The quotient (^i + iy^) (x^ + iy^ is equal to — 3{«A + yly2-^■(«ly!!-«22/l)} or - {cos (61, - ^2) + i sin ((91-612)}; thus the modulus of the quotient is the quotient of the moduli, and the argument of the quotient is the difference of the argu- ments of the two numbers. To construct the quotient geometrically, join the point Q 232 COMPLEX NUMBERS (Xi + iy^ to the point A (+ 1), and draw a triangle ORP similar to the triangle OAQ, the angle ROP being measured equal to — ^2; then the angle ROA is 6^-6^, and OR = OP/OQ, therefore the point R represents the quotient. The powers of complex numbers. 180. If in equation (1), we put all the factors on the left- hand side of the equation equal to a; + iy, we obtain the formula (x + iyY = ^" (cos nQ + i sin n6) ; thus the modulus of the nth power of a complex number is the nth power of the modulus, and the argument is n times that of the given number. The number n here denotes any positive integer. To construct such a power geometrically, let P^ {x + iy) be joined to J. (+ 1) ; on OPi draw the triangle OP^P^ similar to OAP^, on OP^ draw OP^P^ similar to the same triangle, and so on; then the lengths of OPi, OP^, ... OPn are r,r'\ ... r", respectively, and the angles PfiA, P^OA, . . . PnOA are 0, 29,... nd, respectively, therefore the points Pj, Pi,...Pn represent the numbers (a; + iy), (x + iyY, ... {x + iyY- In the particular case r = \, we have (cos ^ + i sin QY = cos nd +i sin nd, and if Qi represents cos + i sin 6, then the points Qi , Q2, . . . Q», which represent the different powers of cos d + isin 6, are all on the circle of radius unity, and so that the arc between any two consecutive points of the series subtends an angle 8 at the centre 0. 181. In accordance with the theory of indices, supposing n to 1 be a positive integer, the expression (x + iyY denotes a number of which the nth power is a; + iy. Now since the nth power of the modulus of a number is the modulus of its nth power, and since the modulus of any number is real and positive, the 1 modulus of (jK + iy)^ is {/r, where ^r is the real positive nth root 1 of r. Suppose that {/r (cos t^ + i sin ^) is a value of {x + iyY, then we have r (cos <^ + i sin <^)" = r (cos 6 + i sin 6), COMPLEX NUMBERS 233 or COS n0 + i sin n<^ = cos ^ + i sin ^ ; therefore cos n^ = cos 6, and sin n^ = sin 0, orn^ =0 + 2s7r, where s is any positive or negative integer including zero ; hence a value of 1 (x + iyY ^, { e + 2s7r . .• + 2s'ir IS ^r < cos h I sm n n since the nth power of this expression is equal to x + iy. The 1 above reasoning shews that every value of (x + iyY must be of this form. If we give s the values 0, 1, 2, ... n — 1, the expression + 2s'7r . . + 2sir cos h I sin ■ n n has a different value for each of these values of s, for in order that it may have equal values for two values Sj, s^ of s, we must have d+2si-rr 0+2ss,Tr , . ^ + 2si7r . 0+28^^ cos = cos , and sm = sm , n n n n whence — = 2Ar7r = ^— , or Si — s^ — nk, n n where k is some positive or negative integer ; this cannot be the case if Si and s^ are both less than n, and unequal, therefore the values are all different. If we give s other values not lying between and n—1, we shall obtain no more values of (cos + i sin ^)", for if Sj be such a value of s, it is alwfiys possible to find a number Sj lying between and n — 1, such that Si — s^^ is a multiple of n, and therefore the value of the expression for s = Si is the same as for s = s^. We see then that all the values of (x + iy)" are given by the series of n numbers / . . 0\ ^, ! ^ + 27r . . (? + 27r\ P/r (cos- + isin- , ^r cos |-^sm , ^r -^cos ^ '— + % sm ^^ '—\ , ^ [ n n ) where {/r is real and positive. 182. If be the principal value of the argument of a; + iy, that is, that value of the argument which lies between — ir and ir, we 234 COMPLEX NUMBERS may regard y/r ( cos - + i sin - ] as the principal valve of {x + iy)". We may consider e . . e e + lir . . e + ^-n- e+^-n- . . ^ + 47r cos ~ + 1 sin - , cos 1- ^ sin , cos h t sm n n n n n n as the principal values of the wth roots of cos^+isin^, cos(^+27r)+isin(0+27r), cos(^+47r)+tsin(5+47r) * i respectively. The dififerent values of {x + iyY are then the principal values of the corresponding expression in r and d when n different values of the argument 6 are taken, the principal value 1^ of {x + iyY being considered as that expression in which 6 has its principal value. The two values of a*, where a is a positive real quantity, are ^/a(cosO + tsinO) and ,^c8(oos7r+i8in7r), that is v/a and — "~^ Vib + iy. Examples. (1) FiJid all the values of (- 1)^ and o/ ( — 1)*. (2) Find the values of (1 + V^)*- COMPLEX NUMBERS 235 184. We shall now shew how to represent geometrically the nth roots of a complex number; the method will give an intuitive proof of the existence of n different values of the nth root. Without any loss of generality we may take the modulus to be unity, so that we have to represent the values of 1 (cos 6 + isin Oy. Let a point P describe the circle of radius unity starting from A, at which 0=0, then in any position of P for which the angle POA described by OP is 6, the point P represents the expression cos 6 + i sin 6. Let another point p start from A at the same time as P, and let its angular velocity be always equal to 1/n of that of P, so that the angle pOA is always equal n Q to Qjn, then p represents cos - + i sin - . When P reaches any position Pi for the first time, let p be at pi, then the angle PJ)A is n times the angle piOA, therefore Pi represents the nth. power of the number represented by pu or conversely pi repre- sents an wth root of cos d^ + i sin 6^. Now let P move round the circle until it again reaches Pj, so that it has described the angle 6i + 2ir, then p will be at p^, where p^OA is equal to (^i + 27r)/n ; if P proceeds to make another complete revolution, when it again 236 COMPLEX NUMBERS reaches the position Pi, p will be at p^, where p^OA — (di + 4!ir)ln, and so on. The points Pi, p^, ■•■ Pn are the angular points of a regular polygon of n sides inscribed in the circle. When P makes more than n complete revolutions round 0, the point p will again reach thepositions ^1,^2. •••■ Each of the points j)i, ^2, ...pn repre- sents a value of (cos 6^ + i sin ^1)", since the wth power of the expressions represented by any one of these points is the expression represented by the point P. The point pi represents the value for the smallest argument 6^. We have thus obtained the J. n values of (cos 0i + i sin ^i)", and we see that these values are the different values of cos — 1- i sin — , when s = 0, 1, 2 , . . . n n w-1. 185. To obtain graphically the ?ith roots of any number sc + iy, we must be able (1) to divide an angle into n equal parts, and (2) to inscribe a regular polygon of n sides in a circle, and (3) in order to construct the modulus, we must be able to construct a straight line whose length is the nth root of the length of a given line. In order to obtain all the nth roots of unity, it is only necessary to solve the second of these geometrical problems, since in this case the angle to be divided into n parts is zero. The problem of inscribing a regular polygon of n sides in a given circle is therefore equivalent to that of obtaining the numerical values of the roots of the equation «" — 1 = 0. This geometrical problem can be solved by a method involving the construction only of straight lines and circles in the following cases: (1) When n is a power of 2 ; for example w = 4, 8, 16, 32. (2) When n is a prime number of the form 2™ + 1 ; for example, when n = 3,'S, 17, 257. This was proved by Gauss in his Disquisitiones arithmeticae. (3) When n is the product of different prime numbers of the form 2'" + 1, and of any power of 2 ; for example, when n=15, 85, 255. The proof of Gauss' theorem would lead us too far into the theory of numbers ; we have however considered the special case re = 17 in Art. 85, Ex. (4), where sin7r/l7 is found in a form involving radicals. COMPLEX NUMBERS 237 De Moivre's theorem. 186. For all real values of m, cos md + i sin md is a value of (cos 6 + i sin O)"". This theorem, known as De Moivre's theorem, has been proved in Arts. 180 and 181, in the two cases m = n, and m=l/n, where n is a positive integer. To complete the proof, we have to consider the cases when m = p/q, a positive fraction, when m is a positive irrational number, and lastly when m is any negative real number. p 1 It is clear that (cos 6 + ism 6)1 = (cos p6 + i smpOy, and one value vd . . p6 of this is cos-^-— + i sin ^—. Therefore the theorem holds when m q i is a positive rational number. p It should be remarked that all the values of (cos 6 + i sin ^)* are given by the expression p(e + 2s'ir) . . p(6+2sTr) cos^-^^ ^^ + I sin-^-^^ -, q q where s = 0, 1, 2,...q — l, when p/q is a rational fraction in its lowest terms. When m is not a rational number, it can always be defined in an indefinite number of ways as the limit of a convergent sequence of rational numbers wii , nza , . . . wig , — Such a convergent sequence is characterized by the property that, if e be an arbitrarily chosen rational number, as small as we please, s can always be so deter- mined that nig differs arithmetically from each of the subsequent numbers mj+i, mg+j, ... by less than e. If r is any positive real number, the principal value of r™ is defined as the limit of the convergent sequence r™', r'"', . . . r™*, . . . , when each of the numbers is real and positive, r™» having its principal value. It is known' that this sequence is convergent, and that it has a limit which is independent of the particular sequence of rational numbers em- ployed to define the irrational number m. If z denotes the complex number r (cos 6 + i sin 6), a value of z"\ when m is an irrational number, is defined as the limit of the sequence of numbers r"*- (cos 6 -\-i sin 0)'^\ r™'^ (cos 6 + i sin 6)^^, . . . ' For a proof of this, see the author's Theory of functions of a real variable, p. 44. In Chapter i of that work, a full discussion of the theory of irrational numbers is given. 238 COMPLEX NUMBERS r™' (cos ^ + 1 sin &)'"'', ..., where r™' has its principal value, and corresponding values for all values of s are assigned to (cos d + i sin 0)™'. In accordance with this definition, one value of z'^ is the limit of the sequence r™' (cos mid-\- i sin mi d), r'^ (cos m^d + i sin m^6), . . . j-™« (cos mgd + i sin nigd), .... Since r™' converges to r*", and cosms^+isinmj^ converges to cos md + i sin m0, on account of the fact that cos m.6, sin mO are continuous functions of m, we see that one value of ^ is r"* (cos m,6 + i sin m^) ; and one value of (cos 6 + i sin Oy"' is cos md + i sin md. Thus De Moivre's theorem is established for a positive irrational index. The general values of (cos d + i sin 6)"^ are cos m{6 + 2s7r) + i sin m{6 + 2sir), where s denotes any positive or negative integer. Since m (sj — s^ can never be an integer when m is irrational, we see that (cos d + i sin 0)™ has an indefinitely great set of values. It can be shewn that the definition of 0™, in accordance with which its values are those of r"* [cos m(d + 2s7r) + i sin m,(6+ 2s7r)} is such that the laws of indices applicable to real indices still hold for irrational indices. In case m has a negative rational or irrational value — k, we have (cos ^ + isin 0)'" = l/(cos ^ + isin^)*; and one value of this is always l/(cosM +isin/<;^), or cosA;^— i sinA;^, which is equal to cos m^ + i sin m^. Thus De Moivre's theorem holds for any negative index. 187. The theorem (cos 6i + i sin ^i) (cos O^ + i sin 6^) ... (cos 0„ + i sin 0n) = cos(^i+^2+ ... + 0n) + ism{di + 0^+ ... + 6^), used in the proof of De Moivre's theorem, affords a proof of the theorems (28), (29), (30) of Art. 49. We may write the left-hand side of this identity in the form cos 61COS62... cos0„(l + ttan^i)(l-f-itan^2)... (1 +itan^„); hence equating the real and imaginary parts on both sides of the identity, we have COS(0i + ^2+ ... + ^„) = cos ^1 cos ^2 ... cos 0n{l — t, + ti-...), sin(0i + 02+ ••• + ^n) = cos ^1 cos ^2... cos 0„(^-<3+ «5-...), COMPLEX NUMBERS 239 where tg denotes the sum of the products of the n tangents taken s at a time. The theorems (39), (40), (43), of Art. 51, are obtained at once from the theorem cos nd + i sin nO = (cos 6 + i sin 0)", by expanding the right-hand side of the equation by the Binomial theorem, and equating the real and imaginary parts on both sides of the q nation. When n is a positive integer, we have {cosd + iamd)'^ = cosn6+isinn6, and therefore also (coad—ism.6y^ = coan6 — iainn6; thence we obtain the formulae cos n6—^ (cos B +i ain 6)" +i (cos 6 — iamd)^, i sin nd=^ (cos 6 + i sin fl)" - J (cos d-i sin 6)". The first of these equations is really an expression of the fact mentioned in Art. 51, that \+xGoa6+x^ooa^6 + ...+ii^coand + ... is a recurring series of which 1 — 2x cos 6+x* ia the scale of relation. Denoting cos n6 by m„, we have M„ — 2cos 5. tt„_i+M„_2=0; to solve this equation assume, as usual in such cases, M„=.4^, then we obtain for k the quadratic A^ — 2icos fl+l = 0, of which the roots are k=ooa6±iava.6, hence M„=^ (cos fl+isin ey+B (cos d-isin By is the complete solution of the equation for u^. Putting n = \, and Ji=2, we find A = B=\, and thus obtain the expression given above for ooanQ. The expression for sin nd may be found in a similar manner. Factorization. 188. We are now in a position to resolve «"—(» + {&) into n factors linear with respect to x. The expression vanishes if x is 1 equal to any one of the values of {a + i6)" ; if g'l , ^'j, . . . g'„ denote the n values of this expression, we shall have a?" - (a + ih) ={x- q^) (x-q^) ...(x- qn), for since «" — (a + ib) vanishes when x — qs = 0, x — qg must be a factor without remainder; thus we obtain n different factors and there can obviously be no more. Put a = r cos d,h = r sin 6, then the expression for «" — (a + ih) in factors becomes s=n—\ n 8 = e + 'ZsTT . . d + 2s7r\) x — p[ cos — 1- 1 sm I y , n where p = ^r={a' + 6^)i»- From this result several of the factorizations already obtained in Chap. VII may be deduced. 240 COMPLEX NUMBEES (1) Let a= 1, 6 = 0, we then obtain »=»-i/ 2s7r . . 2s7r\ «" — 1 = n « — cos — - I sm , s=o \ n n J J . 2s7r 2 (n — s) TT _ and since 1 — ^^ '— = 27r, n n this gives us, if n is odd, *=i(»-i)^_ _2s7r ...„._ 2s7r\/„ _._ 2s7r . , _,_ 2s7r\ «=i « — 1=(«— 1) 11 a;— cos ism — «— cos (-ism 1 «=i\ w n /\ n n / «=i(»-l)/ 2sTr \ ={x-\) n (a;^-2«cos=^+l) s=i \ n J and fl;»-l=(a;-l)(a;+l) H ( «'' - 2a; cos ^=^ + 1 1 , if n. is even. (2) Let a = — 1, 6 = 0, then we obtain the formulae «'-. + l=(a; + l) n (a;''-2a;co8 ^ ^ +1), («, odd), «»+l= n (a:''-2a;cos '^ ^^ +1), (n even). (3) «'»'-2a;»cos^ + l = {x^ — cos ^ — i sin 6) (a;" — cos 8 + i sin 0) «=«-!/ 0+2s7r . . e+2s'jr\ ( ^+2s7r . . ^+2s7r\ = 11 he— cos tsin I ir— cos l-tsin s=o\ w n I \ n n J = n [x^ — 2x cos :: hi). »=»-i/ g + 2g 7r or writing a;/y for oc, and multiplying both sides by y'", we have s=^~*-( V + 2s7r \ ^2n_2a;'ycos0 + 2/'"= TI U'-2a;ycos + 2/' • (4) From the last result we have s=«-i / ^ _|_ 2s7r a;" + a;-" — 2 cos = 11 a; + a;-' — 2 cos Put X = cos <^ + i sin ^, then a;~' = cos (^ — i sin (^, and a;" = cos n + 1 sin n<^, a;"" = CQsn(f) — i sin n^, therefore, changing ^ into n0, cosri' ^-cosm^=2'^i n jcos<^-cosf^ + =^j[, COMPLEX NUMBERS 241 Properties of the circle. 189. Certain well-known properties of the circle may be ob- tained by means of the factorization formulae of the last Article. Let .4i-42^3...^„bea regular polygon ofn sides inscribed in a circle of radius a, and let P be any point in the plane of the circle, its distance from 0, the centre of the circle, being denoted by c. Let the angle POA^ be denoted by 0, then the angles POA^, POA3, ... are 6 + 2irjii, 6 + 4!Tr/n, . . . respectively. Then PA,\ PAi. PA^\..PAn^ =*1n''|a» - lac cos (d + '^\ + c4 , hence we have the theorem PA^'' . PA^" . PAs^ . . . PAn^ = cv^ - lal'c'' cos nQ + c"", which is known as Be Moivre's property of the circle. In the case when P is on the circumference, the theorem becomes p^^ p^^ .PA^... PA^ = 2a» sin i nO. In the case when P is on the radius OA^, we have ^=0, and the theorem becomes PA^.PA^...PAn=a''~c\ Again if P lies on the bisector of the angle AnOA^, we have 6 = tt/w, and the theorem becomes PA,.PA^...PAr, = a^ + c\ The last two cases are known as Cotes' properties of the circle. 190. Examples. (1) Express x^'VCl+x") in partial fractions, m being an integer less than n. If a be a root of the equation a;" + 1=0, the partial fraction corresponding „m-l \ X „ire-n to the factor x-a\a ——-, ■ i or 5 taking the two fractions cor- ma" -' x — a n x — a 2r + l , . . 2»-+l ^ ,, responding to the conjugate values of a, cos jr±tsm— ^ n-, together, we obtain the fraction 27*+ 1 27"+ 1 2.j;cos (n—m) 7r-2cos (n-m+l)7r 1 n n ^ ii;2_2a;cos?^7r + l n cos(2r+l) 7r-^cos(2r+l)- 77 2r + l xfi — 2x cos TT + 1 H. T. 16 242 COMPLEX NUMBERS if ji is odd, we have the additional fraction ^ , ,. ; hence when n is odd ^-i_ (_l)»-m 2r=t(^) "°«(2'- + l)^-^-^cos(2r+l)^^ 1+:b"~ w(a;+l) ■*■« ^?o , „ 2j-+1 ,^ ^ ' a;2 - 2a; cos n- + 1 TO and when n is even /„ ,v «i — 1 ,„ ,> w ™ I n^-A« ,cos(2r + l) 7r-a;cos(2r + l)— n- l+x^^n ^=0 r"!i 27-+ 1 ~ • a;^-2^cos tt + I TO (2) Express x™- i/(x" — 1) m partial fractions, m ftem^" ?es« iAara n. (3) Prove y~ r=o {cos6 — cos{ + 27t/q)}^' The expression on the left-hand side in (a) is an algebraical function of cos 6, and can therefore be resolved into partial fractions, as in Ex. (1) ; the equation (6) is obtained by differentiating both sides of (as) with respect to +cos-^=0, and sind+sin + iff)=O, and sin30+sin3(|)+s^n3^lr — 3sin(6 + (j)+■^) = 0. This is an example of the general method of deducing trigonometrical theorems from algebraical ones, by substituting complex values for the letters. If a + b+c=0, we have a^ + b^+c? -3abc=0 ; let a=cos fl+isind, b=coa (jy+i sin tf>, c=cos-\|(--|-isin-\Jc, then we have given that if (cos 5+C08 (/)-l-cos i/')+* (sin fl+sin +^)+ism (d+ + iir)}=0 ; equating to zero the real and imaginary parts separately in each equation, the theorem follows. (l+a;)''-(l -a;)" 2a; EXAMPLES. CHAPTER XIII 243 EXAMPLES ON CHAPTER XIII. in j.1. J. /l+sinrf)+icoa + i(sin6- sin (^)}" + {cos 6 - cos +pi3s-^p2X^+ prove that Pn — Pi+Pi— =22"cos Jreir, Pi-Pi+Pb- = 2i"sin JwTT. 7. li x^, ^2, ... Xn be the corresponding roots selected from the conjugate pairs of roots of the equation x'"^ - 2«" cos »fl+ 1 =0, and if /(a)= S ^,.cos(a + — 1, prove that /W/W /W = (i«)*-' [/{^ (a, + a2+...+a,)|J. 8. If a, ^, y, S, € be any five angles such that the sum of their cosines and also the sum of their sines is zero, shew that 2 cos 4a = -^ (2 cos 2a)2 - J (2 sin 2a)2, 2 sin 4a =2 sin 2a . 2 cos 2a. 9. If e two concentric and similarly situated regular polygons of 2m sides, then PA^.PA^. ...PA^_^ _ PB^.PB^. ...PB^_^ PA^.PA^....PAi^ PBi.PBi....PB^ ' where P is anywhere on the concentric circle whose radius is a mean propor- tional between the radii of the circles circumscribing the polygons. 20. A point is taken within a circle of radius a, at a distance b from the centre, and points Pi, P-i, ... P„ are taken on the circumference so that P1P2, P2P3, ■■■ PnPi subtend equal angles at 0; prove that OPi-l-OP2+- + OP„=(a2-62)(OP,-i + OP2-i + ... + OP„-i). 21. Prove that if » is a positive integer . , „ . 5 fl + TT , w(ra-l)„, . „5 2(5-l-ir) cos 116 = 1 + 2m sin 5 cos — g — 1 — ^-^ — -' 2^ sm^ ^ cos — ^-^ — - , m(»-l)(ra-2)„, . .e 3(fl-|-7r) , ■V— gy -'23sm3-cos-i-2 — i+.... 22. Shew that the number m of distinct regular polygons of n sides which can be inscribed in a given circle of radius r is equal to half the number of integers less than n and prime to it. _ Shew also that the product of their sides is equal to 1^ >\/nj\/n - 2m, or r", according as n is, or is not, the power of a prime number. CHAPTER XIV. THE THEORY OF INFINITE SERIES. 191. We shall, in this Chapter, give some propositions con- cerning the convergence of infinite series in which the terms are real or complex numbers, or variables. Anything like a complete account of the theory of such series would be beyond the limits of this work ; we shall therefore confine ourselves to what is absolutely necessary for the purpose of discussing the nature and properties of trigonometrical series. The convergence of real series. 192. Let Oi, Oa, Os, ... an, ... be a sequence of real numbers formed according to any prescribed law, and let Sn = (h + ct2 + as+ ... +an. If Sn has a definite finite limit S, when n is indefinitely increased, the infinite series Oi + Oj + 03 + . . . is said to be convergent, and S is said to be its limiting sum, or simply its sum. We shall, in this Chapter, use the notation LSn to denote the limit of Sn when n is iudefinitely increased, whenever that limit exists. The condition that LSn = S is that, corresponding to each arbitrarily chosen positive number e, as small as we please, a value Me of n can be determined such that the arithmetical value oi S — Sn is less than e, for every value of n which is S Wj. When the series 01+02 + 03 + ... + «« + ... converges to S, the series a„+i + o„+2 + --- is convergent, and its limiting sum ia S—8n, ^hich may be denoted by i?„. The number i2„ is called the remainder of the convergent series 01 + 02+... +a„ + ..., after n terms, and the remainders Bi, R^, ... R„, ... form a sequence of numbers such that LRn = 0. It should be observed that it is THE THEORY OF INFINITE SERIES 247 only on the assumption of the convergence of the series that the remainders i?„ have any meaning. The number a„+, + a^+i +...+ a„+,„ may be denoted by Rn,m ; and the numbers iJ;„,i, B„^^, R^„ ... are called the partial re- mainders of the series after n terms. It will be observed that these partial remainders i2„,^ exist as definite numbers for all values of n and m, whether the given series is convergent or not. The limiting sum of a convergent series ai + cij +•••««+■• • is frequently denoted by 2a. 1 193. A series osi + aj + as + . . . + a„ + . . . may be such that the numbers Sn have no definite limit as n is increased indefinitely. The following cases may arise : (1) It may happen that, corresponding to each arbitrarily chosen positive number k, as great as we please, a value n^ of n can be determined such that all the numbers S„ , Sn+i, ... S„+m, ••■ are of the same sign, and are all numerically greater than k. In this case 8n increases indefinitely with n, either in the positive or in the negative direction ; the series is then said to be divergent. The fact of the divergence is then sometimes denoted by LSn = oo , or LSn = — oo , as the case may be. (2) If, as in the last case, Sn increases arithmetically in- definitely with n, but however great wj, may be chosen there are both positive and negative numbers among Sn^, Sn+i, ... S^ +m, ..., the series may be said to oscillate between indefinite limits of indeterminancy. It is however, in this case, usually spoken of as divergent, and its behaviour may be denoted by LSn = + oo . (3) It may happen that, although Sn has no definite limit as n is indefinitely increased, it is possible to select a sequence of increasing values of n, say n^, n.^, ... rip, ... so that Sn converges to a definite limit provided n is restricted to have only the values in this sequence. In this case the series is said to be an oscillating series ; but oscillating series are sometimes spoken of as divergent. An oscillating series in which Sn is for every value of n numerically less than some fixed positive number is said to oscillate between finite limits of indeterminancy. It is easily seen that if the terms of a series have all the same ■248 THE THEORY OF INFINITE SERIES sign, the series is divergent in accordance with case (1), unless it is convergent. The series l + 2 + 3 + ... + ?i + ..., 1/1 + 1/2 + .. . + l/w+... «ire both divergent, since in each case S„ increases indefinitely with n, and is of fixed sign. The series 1-2 + 3 — 4 + 5—... oscillates between indefinite limits of in- determinancy. For S„= —^n, when n is even, and »S„=J(?i + l), when n is odd; thus a ; then take q negative terms, q being so chosen that Sp — S'q-i > a, and Sp — S'q< a; next take p' positive terms such that (Sf^+y-i — 8'q a, then q' negative terms such that Sp+p' — S'q+g' < a, and Sp+^y — S'q+q'-i > a, and so on. Proceeding in this way, we obtain a series such that its sum differs fi:om a by less than its last term, hence when we make the number of terms indefinitely great its sum will converge to a. It can also be shewn that the terms may be so re-arranged that the new series diverges, or that it oscillates. The convergence of complex series. 197. Suppose ^1, ^^2, ^^3, ... Zn, ... to be a sequence of complex numbers ; thus z„ denotes Xn + iyn, where Xn and yn are real numbers. Let 8n = z-, + Z2+ ■••+Zn, s« = «i + a!2 + . ..-!-«„, «'n = 2/i +2/2 + ■•■ + y« ; thus Sn = Sn + is'n- If Sn has a definite limit S, itself a complex or real number, when n is indefinitely increased, the infinite series Z1 + Z2+ ■•■ +Zn + ••■ is said to be convergent, and >Si is called its limiting sum, or simply its sum. The condition that 8 = LSn is that | /S — ;Si„ | converges to zero as n is indefinitely increased ; thus if S — 8n = pn (cos 6n + i sin 0n), 252 THE THKOBY OF INFINITE SERIES we must have Lpn = 0. If S = s + is', when s and s are real, we have s — Sn = pn cos On, s' — s'n = Pn Sin 0n ', it then follows that, if Lpn = 0, we also have L{s — s„) = 0, L (s — s'„) = 0, or s„, s'„ converge to s and s' respectively. It thus appears that in order that the series 2i + Z2 + Zs + ... may be convergent, it is necessary that the two series Xi + Xs + Xs + ..., yi + y2 + ys + ••■ should both be convergent. Conversely if these latter series are convergent, the series of complex numbers is also convergent, for I (S + is') - (Sn + is'n) | ^ | S - S„ ( + | s' - s'„ | J if now Lsn = s, Ls'n = s', we can choose a value n^ of n so large that |s — s„|<^6, \s' — s'n\<\e, provided n^n^. It follows that I (s + is') — {sn + is'n) | = 6, if w S w^ ; and since e is arbitrary we therefore have L (s„ + is'n) = s + is', and thus the series of complex numbers is convergent. In case the limiting value of either of the sums %x, ^y is not finite, or in case either of these series oscillates, the series Xz is not convergent. Suppose Zn = fn (cos dn + i sin 0„), then we shall shew that the series liZ is convergent provided the series %r, in which each term r„ is the modulus of the corresponding term Zn, is convergent. The given series 2r„ (cos dn + i sin 5„) is convergent provided each of the series 'LvnCosdn, XrnSmOn is convergent; now each of the numbers VnCosdn, rn&vaOn lies between the numbers ±r„; also the number Sn+m — Sn is for either of the series Sr cos 6, Xr sin 6 numerically less than the corresponding partial remainder for the series 2r. If then the latter series is convergent, so is each of the former ones ; hence the series Xzn is convergent. The converse is not necessarily true ; thus the series 2r„ (cos On + i sin 9n) may be convergent, whilst Sr^ is divergent. If the series Xvn formed by the sum of the moduli is convergent, then the series 2r„ (cos 6n + i sin dn) is said to be absolutely con- vergent. For example, the series of which the general term is n~^(coand+ismnB) is absolutely convergent, since the series Sm"^ converges, whereas the con- vergent series of which the general term is n~^(coan6+iaianff), (2i7>6>0), is not absolutely convergent, since the series Sn''- is divergent. THE THEORY OF INFINITE SERIES 253 Continuous functions. 198. Suppose /(«) to be a function of the complex variable z = x-\-iy, which has a single finite value for every value of z which lies within any given limits ; this function will then have a single value for every point in the diagram, which lies within a certain area ; this area may be any finite portion of the plane of repre- sentation of z, or the whole of that plane. Such a function is said to be continuous at the point z = Zi, if a positive number rj can always be found such that the modulus of f (z) — f (Zi) is less than an assigned positive number e, taken as small as we please, for all values of z which are such that the modulus of z — Zi is less than rj. For each value of e a value of ij must exist. A function which satisfies this condition at every point within any given area, is said to be continuous in that area. The boundary of the area niay, or may not, be included. Uniform convergence. 199. Let fn{z) be a function of or a; + iy, which is continuous in any area ; then if the series f{z)+f{z) + ...+f^{z) + ... is convergent, we may denote its limiting sum by F{z). Suppose Mz)^f{z) + ...+f^{z), where n is any fixed number, is equal to Sn{z), then the limiting sum oi fnj,i{z) + fn+i{z) + ■•• is called the remainder after n terms, and may be denoted by Rn{z) ; we have therefore F{z) = Sn{z) + Rn(z). Now suppose that, corresponding to any given positive number e, however small, a value of n, independent of z, can be found, such that for all values of z represented by points lying in any given area, the modulus of Rm(z) is less than e. where m is equal to or greater than n, the series is said to converge uniformly for all values of z represented by points in that area. The integer n will depend in value upon e. If as z approaches indefinitely near any fixed value z^ in the area, in order that the moduli of all the remainders R'm{z) may be less than e, it is necessary to suppose n to increase indefinitely, 254 THE THEORY OF INFINITE SERIES then in the neighbourhood of the point Zy, the series does not con- verge uniformly and is said to converge infinitely slowly. A point z^ for which e can be so chosen that this happens is said to be a point in the neighbourhood of which the convergence is non-uniform, or sometimes simply a point of non-uniform con- vergence in case the series converges at that point itself For any space including such a point it is impossible to assign any fixed value of n, such that for all values of z within that space, the moduli of Rm are less than the sufficiently small positive number e; and thus the series does not converge uniformly throughout that space. When z is equal to z^, the series may be either convergent or divergent. We may state the matter as follows : Suppose that as z approaches some fixed value z-i a positive number e can be assigned such that the number of terms n of the series /i(^:) +/i.(^^) -I- ... which must be taken, in or^er that mod. Rm{z) < e, where m is equal to or greater than n, depends on the modulus of z — Zi in such a way that n continually increases as mod. {z — Zi) diminishes, and becomes indefinitely great when mod. {z — z^ becomes indefinitely small, the series is said to converge non-uniformly in the neighbourhood of z^. In the neighbourhood of such a point, the rate of convergence of the series varies infinitely rapidly, and when mod. {z — z^ is diminished indefinitely, the series converges indefinitely slowly. It should be observed that a convergent numerical series cannot converge infinitely slowly ; thus when z is equal to z^, the convergence of the series /i(^i) 4-/2(^1) + ..., if it is convergent, is not indefinitely slow; it is only when ^r is a variable such that mod. {z — Zi) is indefinitely diminished, that the series converges infinitely slowly. It is consequently more exact to speak of the non-uniform convergence of a series in the neighbourhood of a point, than at the point itself. The number of terms n that must be taken in order that the modulus of the remainder i2„(^^) may be less than the sufficiently small number e, increases as z approaches the value z^, becomes indefinitely great when mod. (z — z,) becomes continually smaller, and then, if the series is convergent at the point Zi, suddenly changes to a finite value; this number n is therefore itself discontinuous in the neighbourhood of such a point. THE THEORY OF INFINITE SERIES 255 If in any area A we have, at every point of the area, |/i(«) I S (h, |/2(«) I S tta, ... \fn{z) I S Cf„, ..., ■where Oi, a^, ... an, ... are fixed positive numbers such that the series aj + as +•••+«»+•• ■ is convergent, then the series is uniformly convergent in the area A. This theorem affords a test of uniform convergence which is of great value in application to particular cases ; it is known as Weierstrass's test. To prove it, we observe that, if e be an arbitrarily chosen positive number, We may be so chosen that a„+i + a„+2 + • • • + a»+m is, for every value of iJi, less than e, where n = ne. The modulus of |/»+l(«) +/«+2(^) + ••• +/«+m(^) I is, for every value of z, not greater than a^+i + an+i + ... + a«+m, and is therefore less than e. Since this holds for every value of m, we see that the complex series is convergent, and that for every value of z, | i2„(^:) | < e, provided nSrie. Therefore the series converges uniformly in A. By some writers, a series is defined to be uniformly convergent in a given area, when a number n can be found suoli that for all values of z, the modulus of the remainder R^ is less than e. The definition given in the text is more stringent than the one here mentioned ; it is possible to construct series which converge xmiformly according to the latter but not according to the former definition. 200. If the functions /i(0), /^(z),... are continuous for all values of z represented by points lying in a given area A, then the function F{z) which represents the sum of a convergent series 'S,f(z), is a continuous function for all values of z represented by points lying in the area A, provided the series 2f(z) converges uniformly/ in the whole area A. For we have F{z) = Sn + Rn,n being such that for all values of z to be considered, the modulus of -K„ is less than e; let ^ receive an increment hz, and let BF(z), BSn, S-B„ be the corre- sponding increments of F(z), Sn , and jB„ . Then, since by supposition the moduli of i2„ and J?„ + SJ?„ are both less than e, the modulus of SRn is less than 26. Also since Sn is a continuous function of z, if the modulus of Bz be small enough, the modulus of SSn is less than e ; hence, provided mod. Sz is less than a certain value, the modulus of S8n + SRn or of SF(z) is less than Se, since the 256 THE THEORY OF INFINITE SERIES modulus of SSn + Si?„ is not greater than the sum of the moduli of BS„ and Si?„. Now Se can be made as small as we please, there- fore mod. BF(z) can be made as small as we please by making mod. Sz small enough; that is to say the function F(z) is continuous. It will be observed that for this proof, the less stringent definition of uniform convergence, given in the note to Art. 199, is sufi&cient. 201. For a value z^ of z, for which the series converges non- uniformly in the neighbourhood, the sum of the series is not necessarily continuous; in this case the reasoning of the last Article fails. The limiting value of the function /„(^^), when z = z^, 00 is fni^i), but it does not follow that 2 {/»(^) —fn{zi)} converges to n zero as z converges to z^. We may denote the sum 2 {/(^) —/(■»!)} 1 by F(n,z — Zi), a function of n, and of z — z^; now the limiting value of F{n,z— Zi) when z is first made equal to Zi, and then n is afterwards made infinite, is zero; but if n is first made infinite, and afterwards z — Zj is made zero, the limiting value of F(n, z — z^ is not necessarily zero. As an example of this phenomenon, Stokes considers the real series 1 + 5^ x{x-V%')'rfi-\-x{^-x)n-^\-x + ••• + „/- . IN f/_ — Tx_ , II /__ , 1N + ---; 2(1+^) n(»i+l){(K-l)x+l}(?u; + l) when A'=0, this series becomes 1 1 ;+... + - 1.2 " ra(re+l) Now the general term is 1 2« Ji(9i + 1) {{n-\)x+\\{nx-V\y \n (?i-l)^+lj [« + l Tix-VX) ' therefore the sum of the series is 3, whatever value different from zero x may have; the sum of the series = — 5 + 5—5 + ... is however unity, and thus the sum of the series is discontinuous in the neighbourhood of the value of a:=0. r find 1 2 The remainder after n terms is =■ H =- ; putting this equal to e, we 71 + 1 nx->r\ ' ^ D T^ M = {^+2-f(^+l) + \/{e(«+l)-(^+2)}2-4f^(6-3)}/2€a;, which increases indefinitely as x becomes indefinitely small ; thus the series converges infinitely slowly when x is infinitely small ; this is the reason of the discontinuity in the sum of the series. THE THEORY OF INFINITE SERIES 257 The discovery of the distinction between uniform and non-uniform con- vergence of series has usually been attributed to Seidel, who published his " Note iiber eine Eigensohaft der Reihen welche discontinuirliche Functionen darstellen " in the Transactions of the Bavarian Academy for 1848 ; the theory had, however, been previously published by Stokes, in a paper "On the Critical Values of the sums of Periodic Series*," read on Dec. 6, 1847, before the Cambridge Philosophical Society. Although the theory is in some respects stated more fully by Seidel than by Stokes, the latter must be considered to have the priority in the discovery of the true cause of dis- continuity in the functions represented by infinite series 2. The distinction between uniform and non-uniform convergence has played a very important part in the modern developments of the subject. The matter is summed up by Seidel in the following theorem : — Having given a convergent series, of which the single terms are continuous functions of a variable z, and which represents a discontinuous function of z : one must be able, in the immediate neighbourhood of a point where the function is discontinuous, to assign values of z for which the series converges with any arbitrary degree of slowness. The geometrical series. 202. Consider the geometrical series 1+ 2 + z^-'r ... + ^"~\ where z = x + iy = r (cos 9 + i sin 6). We have for the sum of this series the value l-«" l — r'^(cosn6 + iainnd) 1—z 1—r (cos 6 + i sin 6) ' put l — rcosd = pcos(j}, r sin = p sin + i sin (fj) jcos {nd + ) + i sin {n6 + 1, it becomes infinite. Thus the infinite series l+z + z'' + ...+z'^'+... converges if the modulus of z is less than unity, and its sum is then 1 ^ , . . , . 1 — r cos d + i.r sin - (cos

)= — z — ^ ^- — - — . p^ ^ , ^' 1 — 2r cos p -t- r^ If the modulus of z is greater than unity, the series is divergent ; 1 See Stokes' Collected Works, Vol. i. 2 On the history of this discovery see Eeiff's Gesehichte der unendlichen Seihen. H. T. 17 258 THE THEORY OF INFINITE SERIES and if mod. z is unity it is also not convergent, since the sums of the two series 2 cos nQ, S sin nQ, which have been found in Art. 74, do not approach a definite value when n is indefinitely great. We have, by equating the real and imaginary parts of the series and the sum, \—r cos 6 1 -2r cos ^H-r^ = 1 + r* cos ^ + r^ cos 2^ + . . . + r" cos n^ + , = r sin + r^ sin 2^ + . . . + r" sin m^ + . . . ; 1 — 2r cos 6 -Vt^ these series hold for all values of r \jia.^ between + 1, excluding r = 1 and r = — 1, for which the series are not convergent. To see that this is the case, we need only write — ^ for ^r in the original series. The geometrical series is uniformly convergent for all values of z of which the modulus is Si—?;, where 17 is any fixed positive number, arbitrarily small. For the remainder after the first n ^" (1 — 7j)" terms is :; , and the modulus of this less than ^^ — ; the \—Z 7} series will then be such that for all values of z of which the modulus is S 1 — 77, \Rn{z)\< e, if (li:^<,,orifn> ^°g^+^°g^ V log (1- 17) Hence, since it is possible to choose n so that for all values of z of which the moduli are S 1 — 17, the remainders after n terms are less than e, and since this clearly holds for all greater values of n, the series converges uniformly for all such values. It has thus been shewn that the geometrical series is uniformly convergent in the area bounded by any circle concentric with and interior to the circle of radius unity with the centre at the origin. Series of ascending integral powers. 203. We shall now consider the general power-series tto + aiZ_+ cii^" +-.. +ariz^ + ..., where ag, a-i, a^, ... are complex numbers independent of the com- plex variable z. Let r denote the modulus of z, and a„, a,, Oj, ... the moduli of ao, Oi, aj, . . . . The series of moduli is Ko + «!'" + «2'"' + . . . + a„r-" + . . . ; THE THEORY OF INFINITE SERIES 259 when this series is. convergent the series in powers ofz is absolutely convergent. If the series of moduli converges for any value of r it is convergent for all smaller values of r ; and if it is divergent for any value of r it is also divergent for all greater values of r. As regards this series a„ + Oir + 0,7-^ + ..., three cases may arise. (1) The series may converge for some values of r dififerent from zero, and diverge for other values ; there then exists a positive number p such that the series converges when r< p, and diverges when r > p. Wh«n r* = p the series may either converge or diverge, as the case may be. (2) The series may converge for all values of r ; it is con- venient to express this by p=oo . (3) The series may diverge for all values of r except r = 0; this may be expressed by p = 0. In order to determine the number p in any given case, we consider the values of 0^"- It may happen that, as n is indefinitely in- creased, Km" converges to a definite limit A ; in that case, if e be an 1 arbitrarily chosen positive number, as small as we please, a„" lies between -4-1-6 and A — e for all values of n with the exception of a finite number of such values. More generally, it may happen that a positive number A exists, such that, for all values of n 1 except a finite set, a„» 1/A. For all values of n except a finite set a„?-"< (J. -(-6)"r", where e may be arbitrarily chosen ; if r has a value < 1/A, we can choose 6 so that (A + e)r 1/A, we can choose e so that {A — e)r > 1, and thus cinr"' > (^ - e)"r-" > 1, for an infinite number of values of n ; the series is consequently divergent. 2_ If a„" converges to the limit zero, as n is indefinitely increased, the series converges for every value of ?■. For, in that ease, 17—2 260 THE THEORY OF INFINITE SERIES Hnr" < 6"y, where e may be so chosen that er < 1 ; and this holds for every value of n except a finite set of such values. Each term of the series, with the exception of a finite number, is then less than the corresponding term of a convergent geometric series; consequently the series is convergent. In this case p = oo . If a,i" has indefinitely great values, that is, if no number exists which is greater than all the numbers «„", the series diverges for all values of r except r = 0. In this case p = 0. For, if r have any given value except zero, there are an infinite number of terms of the series each of which is greater than unity, and thus the series is divergent. 204. In the last Article it has been shewn that a number p exists, which may however be zero, or may have the improper value 00 , which is such that the series 0,^ + a^r + a^r^ + ... is con- vergent for each value of r which is < p, and is divergent for each value of r that is > p. About the point ^; = as centre, describe a circle of radius p ; this circle is called the circle of convergence of the series and its radius is called the radius of convergence of the series. The radius of convergence may be finite, or zero, or infinite. It will be shewn that the series ao + c^i^ + pia seen from the fact that the necessary condition of convergence L \ anZ^ | = is not satisfied. For |a„^:"| = (r//3)"a„/j"; and for an infinite number of values of n, OnP^ > (1 — P^Y ! hence if e be so chosen that '^p-¥^' we see that I QnZ'^ I > 1, for an infinite number of values of n. THE THEORY OF INFINITE SERIES 261 205. It will next be shewn that the series «» + ^I'S' + ci^z^ + • • • converges uniformly in any circle of which the radius is less than the radius of convergence, and of which ^ = is the centre. Suppose p — A; to be the radius of this circle, and let p^ be a fixed number between p and p-h\ let p-k = p-^-h. The modulus of the limiting sum of a„£:" + a„+i2:»+i+ ... does not exceed the limiting sum of the series a„r'"+«„+ii-»+i+..., or a„pi" {rlp^^ + a„+,/),"+^ (r/pi)"+i+ .... Now the numbers a^p^, 0^+1 Pi""*"'. ••• are all less than some fixed number K, since the series is convergent when r = pi ; thus the sum of the series is less than ^{(r//3i)"+(r//0i)"+' + ...}, or than -K'(^//3i)"(l - rlpxy ; and this is less than ^(1 - h/p.fp./h. If e be an arbitrarily chosen positive number, a value rii of n can be determined such that K{1 — h/piy^pjh < e, for n S Hj. Hence the modulus of the remainder Bn(z) of the series ao + aiZ + a2Z^+ ... is less than e, for niui, and for all values of z such that mod. z & p — k; therefore the convergence of the series is uniform in the circle of radius p — k. This is true however small the number k (> 0) may be taken to be, but it would be incorrect to assert that the convergence is necessarily uniform in the circle of convergence. . Denoting by F(z) the sum of the series ao + aiZ+ a^z'+ ... for values of z of which the moduli are less than the radius of con- vergence, it follows from Art. 200 that F{z) is a continuous function of z, for all points lying in the interior of the circle of convergence. If the radius of convergence is infinite, F(z) is continuous for all finite points in the plane. The series l+z + z^ + z^ + ..., z z^ 2' have the radius of convergence unity ; their sum-functions are continuous functions of z in the interior of the circle of radius unity. z z^ ^ Theseries ^'^T\'^'^\'^'"'^n\^"' has the radius of convergence infinite ; the sum-function is continuous for all finite values of z. Theseries H-1 Iz-fSlzH... -1-re !«"+... has the radius of convergence zero. 262 THE THEORY OF INFINITE SERIES 206. Tbe convergence of the series on the circle of conver- gence itself has not yet been considered ; we may without loss of generality, take the radius of convergence to be unity. It can be shewn that the series a„ + a^z + a^z" + ... , when the coeflScients are real, converges for points on the circle of conver- gence, with the exception of the point z = l, if the coefficients are all positive, and with the exception of the point z= - 1, when the coefficients are alternately positive and negative, provided in both cases the coefficients a„, ai, a2, ... are in descending order of absolute magnitude, and provided the limit of a„, when n is indefinitely increased, is zero. Let Sn = Qo + <^i'^ + a^^^ + . . . -I- an-iz"'~^ and suppose the coefficients all positive, then /S„(l - a) = a„ - Un-iZ'^ — z {(a„- a^ -1- (oj - aj) « + (a^- a^z'^-'r ... ; -l-(a„_2-a„-i)«"~^}; now the series (aj - aO -1- (tti - aj) -I- (aa - as) + ... is convergent (see Art. 194, note), therefore the two series (fflo — ai) + (oi — aj) cos Q-\-{a^ — a,) cos 2^ -I- . . ., (tto —ch) + (o^i — "s) sin 6 + (ai — as) sin 20 + . . . are alsb convergent, since the cosines and sines all lie between it 1, thus the series {uo- Ui) + (ori- cQ z + (a^- a^) z^ + ... is convergent when mod. z=l; since |a„_i^"| has the limit zero when n is infinite, we see that i/S„ (1 — z) is finite when mod. z=l; hence unless, z.= l, LSn is finite. If the coefficients in the series are of alternate signs, change z into — z, then this case iS' reduced to the last. Whether the series is convergent when ^ = 1, or in the case of coefficients of alternate signs, when z=—l, has not been determined, and depends upon the particular, series. The series may be only semi-convergent on the circle of convergence. If the coefficients of the series are complex, we can divide the series into two, in one of which the coefficients are real and in the other imaginary; the two series can then be considered separately. THE THEORY OF INFINITE SERIES 263 z The series ^''"T"'"2"'"3'''-" is convergent when mod. z=\, except when 2 = 1. Thus the two series 2 — cos n6, S - sin nd are both convergent, except that the first is divergent wheii 6 is zero or an even multiple of tt. 207. Suppose F{x) is the continuous function of x which is represented as the sum of the series a^ + a^x + a^x^ + . . . , with real coefficients, which converges for real values of x, less than unity. Let us assume that the series diverges when x>l, but that the series ao + ^i + «2 + • • • > corresponding to a; = 1, is convergent. It will then be shewn that the sum of the series a,, + Oj + aa + . . . is the limit oi F(x) when x increases from values less than unity to unity as its limit. Thus the continuous function F(x) defined for a; = 1 by ^(1) = L F (x) continues to represent the sum of the series when x=l. This theorem was given by Abel'. Let s„ = tto + 0^1 + «2 +••• + «» 3 So = fflo- In virtue of a theorem which will be proved in Art. 209, since the two series a, + «!« + a^x^ + . . . + a„a;" + . . ., l + x + x^+...+x'^+ ... are both absolutely convergent when xx =Xi, since 1—x and 1 — a;" may be taken as small ' See Crelle's Journal, Vol. i. ^64 THE THEORY OF INFINITE SERIES as we please by properly choosing x. Since 26 is an arbitrarily small number, it follows that s is the limit oiF{x) for «= 1. If tto, a,, tta, ... are complex numbers, we may divide the series into two parts, one real and the other imaginary, and the theorem applies to each part separately ; hence it holds for the whole series. Next let F{z) be the continuous function which represents, when mod. z<\, the sum of the series «„ + Or^z + a^z^ + . . . , where z is the complex number r (cos Q ■\-i sin 6). The series may be divided into the two parts tto + 0^1 ^ cos d + a^r' cos 1Q -\- ... i (oi r* sin ^ + a^r'^ sin 20 + . . .), and the theorem holds for each of these two series. Therefore if the series a^ + a^z + a^z^ + . . . is convergent when z = cos d-\-i sin 6, its sum is the limit for r = 1, of F{z), the value of being kept constant. The function represented by the series is then con- tinuous at the point on the circle of convergence with the values on the radius of the circle of convergence through the point. In order that the necessity for the investigation in this Article may be seen, we remark that a similar theorem would not hold for the series obtained by altering the order of the terms in the series ai,+aia;+a2X^+.... For example, consider the two real series 3!-^a;'^+lifi-lx*+... and 3;+^ifi-^a!^+^afi+^^x'-ix*+...; as long as ^ < 1, the series are absolutely convergent, and they have the same sum ; when however x=l, the sums of the series are not equal, as has been shewn in Art. 195. The sum of the first series is continuous up to the value «= 1, of X, but that of the second is not so. 208. There cannot be two distinct series of powers of z, tto + Oi^ + a^z^ + .... bo + biZ + b2Z^+..., which both converge to the same value F(z) for all points in a circle of radius k (> 0). For since they converge to the same value for z = 0, we must have ao = &oj and thus the series OiZ + a^z' + ..., biZ+ b^z^ + ... converge to the same value when mod. z ^k. This is impossible unless the two series are both convergent and have the same limiting sums for < mod. z ^k. The radii of convergence of these two series are each ^ k, and their sum-functions are both continuous within their THE THEORY OF INFINITE SERIES 265 circles of convergence. Since their sum-functions are identical for each value of z except z = ^, in the circle of radius k, it follows from the continuity of those functions that they are identical when «= ; therefore aj = fej. By proceeding in this manner, it can be shewn that all the corresponding coefficients in the two series are equal, and thus that the series are identical. Convergence of the product of two series. 209. Let 8, S' denote the limiting sums of two absolutely convergent series ai + ai + as+ ... +an+ ..., bi + bi + bs+ ... + bn+ ...; then it can be shewn that the series ffli6, + (a^bi + a^bi) + . . . + (aj 6„ + a^bn-i + ...+ a„6,) + . . . obtained by multiplying together the given series is convergent, and that its limiting sum is SS'. Denote by s„ the sum of n terms of the product series, and let a, /3 be the moduli of a and b respectively. Since the series S, S' are absolutely convergent, the series of moduli are convergent; denote their sums by S, S', and let a-n. = KiA + (ai/3.2 + aa/Si) + ...+(aA + «A-i +■■■+ a„/3i)- We have >S„/S„' — «« = ciafen + a^bn-i + . . . + anbn ; hence mod. (S^A' - ««) S a A + «3^n-i + ■■■+ Onffn Now (7n < 2n2„' < o-are, because a.^ contains more terms than the product 2ji2n', whereas "liSj ••• Wl,«> "'2,1) 0(2,2, ^2,3, ••• ^,a> ^,1> ^,2j Let us assume that the numbers in each row when added together have a definite limiting sum; and let s^, s,, ...Sr, ... denote the values of this limiting sum for the first, second, ... j-th rows. Let it be further assumed that the series S1+S2+ ...+Sr + ■■■ is convergent, and has S for its limiting sum. It will be shewn that the series «i,g + «2,« + ••• +«»•,«+ •■. obtained by adding the numbers in any one column is convergent, and that if its limiting sum be denoted by \ , the series Sj + Sj + 2, + . . . is convergent, and has S for its limiting sum. That ai,s+ Hj^sH- ... + o,._g+ ... is convergent follows from the fact that each term is less than the corresponding term of the convergent series Si + Sj + . . . + s^ + An integer p may be so chosen that the r numbers ^=p n=p n=p Sj — i *n,i, Sj— 2, dn,!, ...Sy— i an,r n=l n=l n=l are all less than e/r*. Therefore 21 + 22+... +s, is less than e + Si + S2+ ... +Sp, or than e-\-S; and since this holds for every value of r, the series s, + 22 + . . . is convergent and its limiting sum is S S, since e is arbitrarily small. Also the integer q may be so chosen that the r numbers n=q n^-q n-=q Si — ^ ^1,113 ^2 ^ ^2,n, ■•• ^r ■i ^r, n »=1 ' «=1 »=1 are all less than e/r. Therefore the limiting sum of the series 21 + 22+ ... is greater than Sj + S2 + . . . + Sy - 6 ; and as this holds for each value of r, the limiting sum is S /S — e. Since e is arbitrarily small, the limiting sum of the series 2i + 22+ ... is ^8; and it has been shewn to be S 8 ; consequently it is equal to ;S. When the positive numbers a^^s are such that each of the series a,,i + a,.,2 + ... converges to a number Sr, and so that the series Si + Sa + .. . is convergent, the numbers Or.a are said to be THE THteOKY OF INFINITE SERIES 267 terms of a convergent double series of positive numbers and S is said to be its sum. In accordance with the theorem proved, the limiting sum of the series is the same whether the summation be taken first with respect to s and then with respect to r, or in the converse order. Thus CO QO 00 00 c=l«=l s=ir=l If the numbers o^^j are no longer restricted to be of one sign, then if the numbers | ar,« I are the terms of a convergent double series, the numbers «,.,g are said to be the terms of an absolutely convergent double series. If the double series, of which the terms are «,,«, is absolutely convergent, then 00 00 00 00 2 2 0^,8= 2 2 a^,i. • r=l s—l s—1 r=l For let ar,s = ^r,s — yr.s, where jr.s = when a^.s is positive, and /8r,s = when 0^,8 is negative. The series may be regarded as the difference of two series of which the terms are the positive numbers ^r,s and y^^g. Since the series of which ^r.s + Jr.s is the general term is convergent, the two series of which ^^.s, Jr,/ are the general terms are both convergent, and their sums may be taken in either order ; it follows that the sum of the series of which ar,s is the general term may be taken in either order without affecting the result of summation. X OO 00 00 The theorem S 2 ar,s= 2 2 a^,s r=l s=l s=l r=l is also valid when the numbers «,._, are complex, in case the series of moduli I Or,s| is absolutely convergent. For if o^r,s = 'Yr,s + iK,s> the series of which yr,s, K,s are the general terms are both absolutely convergent, whence the result follows. The general theorem may also be stated as follows : If Oi + ffls + as + . . . be a convergent series of real or complex numbers, and if each term a, be expressed as the limiting sum of an absolutely convergent series then the given series may be replaced by the series 00 00 2 03,,!+ 2 ap_:i+ ..., i)=i p=i 268 THE THEORY OF INFINITE SERIES without altering its limiting sum, provided the series is convergent, where iSf^ denotes the limiting sum of |ar,i| + |ar,2| + |ar,3| + .... An important case of this theorem, of which we shall afterwards make use, is the following : If a„ + aiZ + a2Z^+ ... be a convergent series of which the limiting sum is F{y,z), and ii a^, (h, a^, ... are the limiting sums of the absolutely convergent series &o, + 60, 1 y + &o, 22/' + 60, 32/' + • • • 6,,o + 6,,i2/ + 61,22/' + 6i,s2/' + --- Ko + KiV + K^y'' + ^2,32/' + • • ■ then, if the series J-o + ^1 1 2^ | + ^2 1 ■« |' + ■ • • is convergent, where Ar denotes the sum of the series |.6r,o| +|&r,iyl + 1 6i-,2y°i + •••. the series (60,0 + &i,o« + Ko^ +...) + (60,1 + &j,i« + K-^z^ + ■••) 2/ + (io,2 + 6l,2^ + &2,22'+...)2/' + ---. which is obtained by substituting for a^, Oi, a^, ... in the given series, and arranging the terms as a series in powers of y, is convergent, and its limiting sum is F(y, z) the same as that of the original series. The Binomial theorem. 211. A very important case of series in ascending integral powers of a variable is the series m(m— 1) „ m (m — 1) (m — 2) , l+mz + —^^ — U'+—^ ^ ^z^+.... In the particular case in which m is a positive integer, the series is finite, and its sum is (1 +z)'^, the ordinary proof of this being applicable to a complex value of z. We shall suppose ^ to be a complex number, but shall confine ourselves to the case in which m is real. In this case On/on+i is 71 4- 1 equal to , the limiting value of which is unity ; the radius of convergence of the series is therefore unity. The series con- verges absolutely at any point z interior to the circle of radius THE THEORY OF INFINITE SERIES 269 unity, and uniformly in any circle of radius less than unity. Denoting the limiting sum of the series by f{m), and applying the theorem of Art. 209, we find for points within the circle of convergence y^^^) ^y(^^) ^f^^ + ^^)^ and thence f{m^f{m^ . . .f(mq) =f(mi + m^+ ... + nig). First suppose m to be a positive fraction p/q in its lowest terms, then putting 'm^ = m2 = ... = mq = p/q, we have [fip/q)]''=f(pl therefore f(p/q) is a g'th root of f(p), that is of (1 + z)p. Let 1 H- r- cos ^ = r-i cos (j>, r sin ^ = rj sin (j), then (1 + zy = r/ (cos p^ + i sinp(j)), and the values of the gth roots of this are f j «(i + 2sTr . . p6 + 2s7r r*!* icos ^-^ h I sm-^-^^ ( q q where s has the values 0, 1, 2, ... g- — 1. We have r-i = + Vl + 2r cos + r^, V sm and we may suppose ^ to be that value of tan~' = 3 which is acute (positive or negative) ; such a value exists, for cos ^ is positive for all points within the circle of convergence. We see then that/(p/g) is a value of ^n^ jcos*^^ \-is\n~ L and s must always have the same value, since we know that f{p/q) is a continuous function for all points within the circle of convergence. To find the value of s, put ^ = 0, then f(pjq) is real, and must therefore be equal to a real value of „, „f 2s7r . . 2s7r) ^r-fi -^cos h I sin > , and therefore s = 0, or s = ^q in case q is even; if r is sufficiently small, / ( - ) is certainly positive ; hence s cannot be equal to ^g' and must therefore be zero. We have thus proved that the sum of the series, when m is a, positive rational number p/q, is the principal valu« of (1 +^)p'«, (1 + 2r cos + r^)^'"* I cos *-^ + 1 sm ^-^j, 270 THE THEORY OF INFINITE SERIES where the expression (1 + 2r cos d + r^)Pi'3 has its real positive value, t* siu and 6 is the numerically smallest value of tan~^ :; ;;, where ^ ^ 1 + 7- cos 5 s = r (cos 6 + 1 sin 6). Next let m be a positive irrational number ; we consider it to be defined as the limit of a sequence mj, mj, ... rrir, ... of positive rational numbers. It will then be shewn that/(m) is the limit of the sequence /(wii), /(m.^), . . . /(w^), . . . , or f(m) = Lf{mr). We have, for any point z in the interior of the circle of convergence, m,(m,-l)...(m,-« + 2) + {n-\)\ " +^«(^;, where | i?„ (^) | is less than the limiting sum of the convergent series i\r(i^+l)...(iy + n-l) N(B+r)...(N + n) V\ 1^' "^ (n +. 1) ! ' ' +•••' where i\ris a positive number greater than all the numbers mi, OTa, ... mr, For all sufficiently great values of w, we have |iJ„(^:)|<6, for all the numbers m^, where e is an arbitrarily chosen positive number. It is clear that the limit of the sum of the finite series 1 , ^ , . ^r{^r-r) ^ m,(m,-l)...(m,-«+2) i + m^^i- 2! ^-i--"-i- (w-1)! ' as my converges to m, is m (m — 1) „ m (m — 1) . . . (m — « + 2) and this is therefore the limit of f{m^ — i2„ (z). In accordance with the definition of an irrational power given in Art. 186, the limit of the principal value of (1 + zj^ is (1 + z)^. Since |i?„(^:)|<6, for all the numbers mi, m^, ... m^, ...., i|iJ„(^)], which must have a definite value, is & e. It follows that m(m— 1) „ m(m — 1) ... (m — n. + 2) ,2! (n — 1)! difiers from the principal value of (1 + z)^ by a number of which the modulus is not greater than e, for all sufficiently large values THE THEORY OF INFINITE SERIES 271 of n; therefore the convergence of the Binomial series to the principal value of (1 + z)'^ has been established for the case of a positive irrational number. Lastly, let m be a negative number — mj. We have then /(m)/(mO =/(0)=l. Hence /(m) = l//(mi), or /(m) is the reciprocal of the principal value of (1 + z)^, or is the principal value of (l+z)"". We may state the complete result as follows : The sum of the series l + mz+ "^(^-l) z-+...+ "^("^-l)-("^-" + l) z»»+... ^ ! n ! for all values of z of which the modulus is less than unity is the principal value of (1 + z)™, which is (1 + 2r cos + T^)i™ (cos m^ + i sin mcf>), when m is any real number, r being the modulus and 6 the argument of z, and A being that value of tan~^ ~ ^r which 1 +rcos lies between +^7r. This result was obtained by Oauchy, and will be found ia his Analyse Algibrique. 212. It now remains for us to consider the case when mod. z = l. Denoting the terms of the series 1 , ^-, , rnjin-l) m{m-l){m-2) , i+m+ 2l + ^ 3] +••■ by a,,, Oi, as, . . . , we have £i„+i/a„ = (m- n.)/(n + 1) ; wheii n>m this ratio is negative, therefore the terms of the series are alter- nately positive and negative, after a fixed term ; the series is, by Art. 194, convergent if the terms diminish in absolute magni- tude and become ultimately indefinitely small. This will be the case ifn — m — 1 ; thus the series is a semi- convergent one, if m > — 1, whereas if m :^- — 1, it is divergent, since the absolute magnitudes of the terms increase indefinitely. From the proposition in Art. 206, it follows that the series \ + mz -\ ^^^-j z'^+ ... converges when mod. z=l, provided m > — 1, and =|= — 1. . 272 THE THEORY OF INFINITE SERIES When z = —\, all the terms of the series are, after a certain term, of the same sign ; applying the known test i7i(l + a„/ajj_i)>l, the series will be convergent if Ln{l-{n-m- l)/n} > 1, or if m > 0. According to the theorem in Art. 207, whenever the series 1 + mz-\ —-^ ^z^ + ... converges on the circle of convergence, its sum is the value of (1 + 2r cos 6 + r^)^" (cos m + i sin nKJi) at the point. We may state the complete result as follows : The series T , , m(m-l) „ , , m(m-l)...(m-n + 1) „ 1 + mz + —~ — ^ z^ + ... + —^ ^ — ^^ -'' z" + . . . i ! n ! converges when mod. z=\,if xn. is positive, for all values of z; also if m is between and — 1, for all values of z except z = — 1, in which case the argument of z is tt. The series diverges when m = — 1, and when m < — 1. For all values of z for which the series converges, its sum is (2 + 2 cos 6)^™ (cos ^m^ + i sin ^m6), where 6 has a value between + tt. The Binomial Theorem has been considered generally, for complex values of m, by Abel, in a memoir published in Crdl^s Journal, Vol. i. The circular functions of multiple angles. 213. An important application of the Binomial Theorem in its generalized form, is the expansion of (cos +i sin d)'", of which, by De Moivre's Theorem, the principal value is cos md + i sin m0, if d lies between + tt. Writing (cos d + i sin 0)™ in the form cos"* 0(l + i tan ^)™, we have cos md + i sin md = cos*"^ f, m(m — 1), „- ) {l L___^tatf^+...| .f ^ . m(m-l)(m-2) ^ .. V + iimta,n0 '^ ^ ^ ta,D!> + ...[ provided the series is convergent ; this condition will be satisfied if 6 lies between the limits + Jtt, whatever be the value of m, and also when 6= ± ^tt, provided m > — 1. THE THEORY OF INFINITE SERIES 273 (1) Suppose m positive, then we have cos m6 = cos'»0 \ 1 ^^-^-j — '- tan^ Q TO(m— l)(m — 2)(m — 3), ^. ] ,,. + -^ ^4, ^^ -^an*^-...f (1), sm in me = cos"'^ |m tan g - "" ^"^ g^,^"" " ^^ an^g + . . .1 (2), for all values of m, provided ^ lies between + ^tt, and they hold for 6= ±\'tr. These results are an extension of those obtained in Art. 51, for the case of m a positive integer, in which case there is no convergence condition. (2) Suppose m negative, then changing m into — m we have cos me cos'"^ = 1 - ^^y -^^ tan=^ ^ m(m + l)(m + 2)(m + 3 ) ^^^^^ _ _ 4! ■(3), sin me CDS'" ^ = m tan ^ - "» ("^ + ^H"^ + 2) tan»^ + (4), which hold for all positive values of m, provided 6 lies between + \ir. These results hold for ^= ± \'ir, only if m lies between 1 and 0. 214. The formulae (1) and (2) of the last Article have, in the case when m is a positive integer, been applied in Chapter vii to obtain expressions for cos m^, sin m^, in series of ascending powers of sin 0. We proceed now to find similar expressions, when m is not a positive integer. We have proved that, when m is an even positive integer, cosmij} =1-2"! ^^"^^ + — 4! ^^m*^ mUm^'-2^)(m^-4?) . , , , ,., ^^ qY ^sm«^+ (5), and that, when m is an odd positive integer, sm m is positive, which will be the case if , sin m^ each contains only a finite number of terms, by Art. 210, we may arrange the result of the expansions in a series of powers of sin" m",. . ,,, . , m(m"— 1"),. . ... , l + m(i sm ^) + n-j (i sm ijif <\> -I g-^ '- {i sm <^/+ ... , where the series on the right-hand side is obtained by adding the series (5) to the series (6) multiplied by i. When m is a positive integer, we have /(m) = cosm^ + isinm<^, if <^ lies between + \ir. Now when rrii and m^ are positive integers, we have /(jTii) xfim^) = (cos Tiii^ + i sin TOi(/)) (cos nh(f> + i sin ms) = cos (mi + ma) ^ + i sin (wi, + m,) ^ The product of the two series /(mi), /(ma) will be of the same form, whatever Tn^, m^ may be ; thus, employing the theorem of Art. 209, we conclude that the equation /(mi) xf{m^) =/(m, + m^) holds for all values of mi and m^, since the series are absolutely convergent. We have consequently f(ini)f{m2) . . . f{mq) =/(mi + m^ + . . . + m,) ; let mi = m2... =mq = plq, where p and q are positive integers, we sret then THE THEORY OF INFINITE SERIES 275 1 hence f(p/q) is a value of {f(p)}^, and is therefore of the form pd) + 2s7r _ . . p, as s is indefinitely increased, is cos mcj) + i sin m^. It then follows that 1 + m {i sin ^) + g-j (i sin ^)^ + . . . mHm=-20...(m''-2r-2|n .. . .,„ + ^ ^-(2;!yi — (* sin .^)='- differs from cos m^ + i sin m^ by a number of which the modulus 18—2 276 THE THEORY OF INFINITE SERIES does not exceed e. Since e is arbitrao-y, it has thus been shewn that for each value of + i sin ini^) = cos vi^ + i sin m^. We have shewn thus that the two series , , "*=.,, m? (w? - 2=*) . , J .^, cos 7W0 = 1 — ^ sm"

= m sm 9 ^~^\ ®™ r + — '^ gj ^sin=0- (6), hold for all values of ^ lying between + ^tt, whatever real number m may be. The series (5), (6) converge absolutely when <}> = ± \-n: For, denoting by a, the absolute value of the general term of the first series, we have g, _ (2>- + l)(2r + 2) ^/ , 1 , _1_\ /, _ ^\-\ a,.+i (2r)»-m« V 2r- 2W V 4W ' therefore L r [ — - — 1 ) = ^ , \ar+i / 2 and thus in accordance with a known test, the series is con- vergent. The series (6) may in a similar manner be shewn to converge. In accordance with Abel's theorem iu Art. 207, the series (5) and (6) converge to the values cos J mvr, + sin ^m-ir, when A similar proof will shew that the two series m^ — 1" cos m^/cos = 1 g-j — - sin"* <\> + ^ ^^j ^-sm'4>- (7), ,, , . , m(m=-2'i) . , , sm m^/cos 9 = m sm 9 „ sm' 9 + —^ 5! ^sm»0- (8), hold for all real values of m, provided lies between + ^tt. The series (7), (8) are not valid when = + ^tt. THE THEORY OF INFINITE SERIES 277 The series (7) terminates only when m is an odd integer, and (8) only when m is an even integer. 215. If we take the series for cosm<^+isinm(|),from(5)and(6), and put e = i sin ^, we have, since (cos ip + isin ^)™ = (Jl + ^^ + «)'", the expansion + m (m^ - -P)... . (m^ - -2s- ■3^). (2s -2^).. -1)! m^ (mJ' ■ -2s -2h) (2s)! In a similar manner we have from (7) and (8) {Jl + z'' + zy<'lJ\+z^=l+mz+ z^+ ^ ' z^ + 2! ' 3! m(m2-2'i)...(m»-2s-2P) „ ^ (2s- 1)! , (OT'-r)(m'-3')...(m'-2s-l|') „^ , + j2sy\ ' + It can be shewn that these expansions hold for all real values of m, provided the modulus of z is less than unity. By some writers, these expansions are investigated directly, and then the series (5), (6), (7), (8) are deduced. It is however not easy to investigate these series by elementary methods, except when the modulus of zjj\ + z^ is less than unity ; we should, with that restriction, obtain the series for cos m^, sin m^, only when lies between + ^tt, which is the same restriction that applies to the series (1) and (2). However, by employing the principle of con- tinuity, it is seen that the above expansions are valid in the region | ^ | < 1 of convergence of the series. 216. If in the series (5) and (6), we change ^ into ^tt - ^, we obtain the following series which hold for values of ^ between and IT, cosm (^2 - "^j = 1 - 2l "^ + ~4! ^cos^"^ -... (9), sinm f^-0l=mcos0 ^^-^-j -cos^4> + (10). We can now find series which express cos m(f>, sin m^, when (^ 278 THE THEORY OF INFINITE SERIES has any value^. If = r-7r + <^o.' where <^i, lies between +|-7r, and r is an integer, we have cos m^ = cos rririr cos mc^o — sin mnr sin m^o j also sin ^ = (— l)*" sin ^^, thus we have, if ^ lies between (r + ^) ir, f-, m^ ■ . , \ cos wz^ = cos mr-ir 1 1 — „-: sm^ ip + ... 1 — sin (m — 1) »-7r -jm sin <^ — ■ — ^^-^-^ ' sin' ^ + . . . S . . .(11). Similarly sm m

+ sin(m-l)(2r' + l) ^jwicos^ ^-^^ --^003'^+ ...\ (14), where <^ lies between rir and (r + 1) tt. 217. Series of some interest may be derived from (5) and (6), (7) and (8), by giving m particular values^ Let <^ = ^tt, we have then, writing x for m, in (5) and (6), , a? xHx^-2^) „,, cosi7ra;=l-2^ + ^^ (15), . , x(a?-V) a;(a!^-P)(a;'-3n sm\irx=x- ^ g, ^ +-^ g^ji ^-...(16). Again letting m = 2x, ij)= ^ir, in (5) and (8), we have , x" a^ix'-m xUx-'-m(a?-2') ,,^, cosK«'=l-2!+ 4! ^ 6! " + - (^^^' , ,„f a;(a!'-P) a;(a;2- in (a!''-2'') 1 ,,„, smi7ra; = iV3U- ^ 3, ' + — ff '--...[ (18). 1 The formulae (11), (12), (13), (14) were given by D. F. Gregory in the Cambridge Mathematical Journal, Vol. iv. 2 The series in this Article were obtained by Shellbaoh, see Crelle's Journal, Vol. XLvni. ; they have also been discussed by Glaisher in the Messenger of Mathe- matics, Vols. n. and vn. Series equivalent to (15) and (16) are given by M. David in the Bulletin de la Soc. Math, de France, Vol. xi. THE THEORY OF INFINITE SERIES 279 Various series may be found for powers of w, by expanding cos^Tra?, sin ^ IT X, ... in powers of x, and equating the coefficients of the powers of x to those picked out from the above series ; for example from (16) we have, by equating the coefficients of ifi, ^-'^ U^ ll_3^+i^+l 1-3-5 / , 1 , 1\, 48 3 ■ 2 "^ 5 ■ 2 . 4 V S^y^ 7 ■ 2 . 4 . 6 V 32 "^ sV Eocpansion of the circular measure of an angle in powers of its sine. 218. If in the expansions (5) and (6), for cos m^, sin m^, in powers of sin^, we arrange the series as series of ascending- powers of m, as we are, by Art. 210, entitled to do, since the series , m" . . , m^ (m^ +2") . ^ , 1 + 2-,sin^ + —^^ ^ sin>+ ..., , m (m^ + P) . , , m sin + — ^^-;g-j sin' 9 + • • • are convergent, we may equate the coeflficients of the various powers of m, to the corresponding coefficients in the expansions of cosm^, sin»i0, in powers of ^; we thus obtaiu from (6) '•'">■' p. 13^ . , 1 sin' rf) 1.3 sin" 6 1..3.5-(2r-l)sin"+^.^ + 2.4.6...2r 2r + l ^ ^ ^' and from (5) . „ , 2 sin^ d) 2 . 4< sin'' 6 <^^=«^°''^ + 3 2 +375 3 +■•• 2.4...(2r-2) sin^-<^ + 3.5...(2r-l)^^+ • ^^^^' these hold for values of (j) between + ^tt, or when <^ = ± ^tt. We may also write them sin-* = ^+2 3+2;4 5 + (19). (sin-^)^ = a.H|.f+|^f + (20), where sin"^ x, in either equation, is the positive or negative acute angle whose sine is equal to x. The series (19) was discovered by Newton; the method of proof is that of Cauchy. 280 THE THEORY OF INFINITE SERIES 219. By changing x into x + h ia the series (20), and equating the coefficients of h on both sides of the equation, which process is equivalent to a differentiation with respect to x, and may be justified by employing the theorems of Arts. 210 and 208, we obtain the series sin~i X 2 „ 2 . 4 . ,„, . VT3^ = ^ + 3^ + 3T5^ + (21), or putting sin for x, 2 2 4 (^/sin^cos^ = l +5sitf^ + K-^sin*^+ (22), or writing 2(j> = 6, ^/sin0=l+|(l-cos<^) + ^(l-cos6»)»+... which may be written 1 12 d cosec 6=1 + ^ vers + ^-^ vers" 6+ (23). Again, in (22), put tan <^ = y, and we obtain the series Expression of powers of sines and cosines in sines and cosines of multiple angles. 220. We shall now shew how expressions of the form cos"* sin" ^ may be conveniently expressed in cosines or sines of multiples of 6. We shall in the first instance confine our- selves to the case of positive integral values of m and n. Let z = cos 6 + i sin 0, then z~'^ = cos — i sin 0, hence 2 cos = 2 + z~^, 2ism0 = z — z~^, and (2 cos 0y (2i sin 0)" = (z + z-^f (z - z'^-y ;___ if we expand the expression in z, in powers of z and z~^, we can arrange the result in a series of terms of one of the two forms k{£^ + z~^), k {f — z'^'), where i is a multiplier depending on m, n, and r ; now jH' = cos r0 + i sin r0, and z'^ = cos r0 — i sin r0, by De Moivre's Theorem, hence k{:^-\- z-^) = 2k cos r0, 2k (a' - z-^) = 2ik sin r0, thus we have the required expression for cos"* 5 sin" ^ in a series of cosines or sines of multiples of 0. THE THEORY OF INFINITE SERIES 281 Example. Express sin^Q cos^d in series of mvltiples of 6. We have (2isine)^ {'2cosdf={z- z-'^f {z + z-^)« = {z^ - e-^f(z+z-^) which is equal to {z^0-5^ + lOz^-l0z-^ + 5z-«-z-'^0){z+z-^), or a" + 29-5zf-5z8 + 10«3 + 102-10z-i-10s-3 + 5z-6 + 52-'-«-8-2-ii, which is equal to 2t (sin 115 + sin 9fl- 5 sin 75 -5 sin 55 + 10 sin 35 + 10 sin 5), therefore sin55 cos«5 is equal to ^ (sin 115+sin 95-5 sin 75-5 sin 55 + 10 sin 35+ 10 sin 5). This process may also be arranged thus, writing c for cos 5, s for sin 5, (2c)6 = l+6 + 15 + 20 + 15+ 6+ 1, {2is) (20)6=1+5+ 9+ 5- 5- 9- 5- 1, (2M)2(2e)6=l+4+ 4- 4-10- 4+ 4+ 4 + 1, (2is)3(2c)6=l+3+ 0- 8- 6+ 6+ 8- 0-3-1, (2w)4(2e)«=l + 2- 3- 8+ 2 + 12+ 2- 8-3 + 2 + 1, (2is)6(2c)6 = l + l- 5- 5 + 10+10-10-10+5 + 5-1-1; here the powers of z are omitted on the right-hand side, and a figure in any line is obtained by subtracting from the figure just above it the one that precedes the latter. This very convenient mode of carrying out the numerical calculation is given by De Morgan in his Double Algebra and Trigonometry. 221. We can obtain formulae for (2 cos 0)'^ and (2 sin 6)^, when m is a positive integer, in cosines or sines of multiples of 0, by the method we have employed in the last Article. We have ^ ! hence fvi (7)1 -^ 1 ) 2"^' cos*"^ = cos m9 + m cos {m-2)d+ ^ — - cos (m - 4) ^ + . . . , where the last term is 1 m m ! . s 71 — TTTT — r; or — ^=^ ^ — cos a, 2(im)!(im)! (^m- 1)! (fni+ 1)! according as m is even or odd. From we obtain similarly (2isin^r=(0-^-O"'=^"-m^-'+ ~^ — ' z'"-' -...+(- l)'^z-^, ^ 1 2»-i (_ 1)2 sin""^ = cos md - m cos (m-2)6 m(m — 1) . .. „ / inT in I when m is even, 282 THE THEORY OF INFINITE SERIES or 2"^i(-l) '^ sin'" = sin m^-m sin (m — 2)^ m(m — 1)., .. n ,, , -—x- ml . - + -i^sin(m-4)^-... + (-l) ^-==^^^=^s.n^ when m is odd. These formulae have already been obtained in Chapter Vll. 222. We shall next consider the expansions of cos™^, sin*"^ in cosines and sines of multiples of d, when m is any real number greater than — 1. We have from Art. 212, 2™ ( + cos ^^)™ cos m (^^ - k-rr) , m(m-l) „. , m(m— ])(m-2) _, , = 1 + m cos ^ + —~ — cos 2^ + --^^ gY^ cos 3^ +... , 2" (+ cos ^ lies between (2A; — 1) tt and (2fc + 1) tt. Let ^ = 20, then corresponding to the two cases of k even (= 2s), and k odd (= 2s + 1), we have 2m cQgm cog (a _ ^^ ^ 2mS7r) / n/i\ m(m — 1) , ... = cos a + m cos (a — 20) H ^-— cos (a -40)+..., ^ ! where lies between 2s7r — ^tt and 2s7r + ^tt ; and 2'" (- cos 0)"* cos (a - m0 + m 2s + 1 tt) = cos a + m cos (a — 20) -\ ^-^ — - cos (a — 40) + . . . , where lies between 2s7r + ^ir and 2s'n- + f tt. In these results, put a = md, then we have 2'"cos'"0cos2ms7r = cos m0 + TO cos (to - 2) + ™ ^'^ ~ •' cos (m - 4) + ... (25), THE THEORY OF INFINITE SERIES 283 where 6 lies between 2s'jr - ^ir and 2sir + ^v ; also 2™ (- cos ey^ COS (2s + 1) WITT = COS m9 + m cos (m - 2) + '' — ^ cos (m - 4) ^ + . . . (26), where lies between 2s7r + ^tt and 2s7r + f tt. Again, put a = md + ^ir, then we have 2™ cos*" ^ sin 2ms7r ^y^ { 7/i — \\ = sm mg + m sin (m - 2) g + ^ — ^ sin (m- 4)0+ ... (27), where d lies between 2s7r — ^tt and 2s7r + -I-tt ; also 2'" (- cos ey^ sin (2s + 1) m7r = sin m6 + m sin (m -2)0 + — ^— — ' sin (m - 4) + . . . (28), where 6 lies between 2s7r + ^tt and 2s7r + f tt. Next change into — ^ir, and then put a = m0, we then have 2m giQm ^ cog ^ (2s + i) TT .y« (vtr 1 ^ = cos me-m cos (m - 2) + '' — ^ cos (m - 4) - . . . (29), where 6 lies between 2s7r and (2s + 1) tt; also 2'" (- sin 0)™ cos m (2s + f ) tt = cosmg-mcos(m-2)g+ '■ — ^ cos (m- 4)0- ... (30), where lies between (2s + 1) tt and (2s + 2) tt. Lastly, put a = mO + ^tt, and change into — -^tt, we have then 2™ sin"* sin m (2s + J) tt = sinm0-msin(m-2)0 + "^^^~^^ sin(m-4)0-... (31), where lies between 2s7r and (2s + l)7r; also (- 2 sin 0)™ sin m (2s + f ) tt = sinm0-TOsin(m-2)0H ^-— — ^ sin (m- 4)0- ... (32), where lies between (2s + 1) it and (2s + 2) tt. These series are convergent for all values of 0, if m is positive. If m lies between and — 1, the extreme values of 0, 2s7r + ^ir or 2s7r, (2s + 1) TT must be excluded, as the series cease to be conver- gent for those values of 0. The eight formulae of this Article wore given by Abel, in his memoir on the Binomial Theorem, and appear to have been overlooked by subsequent writers. CHAPTEE XV. THE EXPONENTIAL FUNCTION. LOGARITHMS. The exponential series. 223. Let us consider the infinite series the limiting sum of which we shall denote by E{z\ where is a complex number x + iy. If r is the modulus of z, the series 1 + ^ + 2!+- is convergent for all values of r, since the ratio of the (n + l)th term to the nth is r/n, which diminishes continually as n increases ; consequently the original series is absolutely convergent for all values of z. This series is called the exponential series, and is uniformly convergent in any circle with centre at ^ = 0. 224. If we multiply together the two exponential series corre- sponding to ^1 and 02, the term of the mth degree in ^i and z^ is mTr (m- 1) ! f! "^ (m - 2)! 2 r ■■■ "^ ml which is equal to — (z^ + z^™', by the Binomial Theorem for a positive integral index. We have therefore for the product of the two series, the series which converges to E{z-i,-\-z^. Now by the theorem in Art. 209, since the exponential series are both absolutely convergent, the product of their sums is equal to the sum of the product series as above formed, therefore E{z^^E{z^ = E{z^ + zi) (1). THE EXPONENTIAL FUNCTION. LOGABITHMS 285 From this fundamental equation we deduce at once E{z,) X E{z,) ... X E{z^) = E{z^ + z^ + ... + ^^„) and thence [E{z)Y' = E{nz) (2), where n is any positive integer'. 225. If in the equation (2), we put = 1, we have E{n)={E {!)]-, where ^(1) denotes the limiting sum of the series l + l + 2-! + 3!+-- It will later on be shewn that the number ^(1) is an irrational number 2-718281828459...; it is usually denoted by e. We have therefore when n is a positive integer, E{n) = e". Again in (2), let z=plq, where p and q are prime to one another, and let n = q, we have then {E(pjq)]9= E(p), hence E{pjq) must be a g'th root of E{p) or eP ; since E{plq) is real and positive, it follows that E(p/q) is the real positive value of S!/eP, which we call the principal value of e*'*. The exponential series is a particular case of the power series con- sidered in Arts. 203 — 208. Its radius of convergence is infinite, and consequently the series converges uniformly in any fixed circle with its centre at the point z = 0. Moreover, in accordance with the theorem proved in Art. 200, the function E(z) is continuous at any point z. If x be any given irrational positive real number, it can be defined as the limiit of a sequence ^i , as,, ... x^,--- of positive rational numbers. In accordance with the definition in Art. 186, the principal value of e* is the limit of e*"" when the integer m is indefinitely increased; it is known that this limit exists and has a value independent of the particular sequence of rational numbers employed to define the given irrational number x. Since E{z) is a continuous function, it follows that E(x) is the limit of E(xm) when m is indefinitely increased. Hence since e*™ = E(xm), for every value of m, it follows that ^= E (x), when e* has its principal value. Next if X be any negative real number, since E(x)E(-x) = E(0) = l, we have E (x) = ^ /e"* = e*, where e*, e"* have their principal values. ' This investigation is due to Cauchy, see his Analyse AlgSbrique. 286 THE EXPONENTIAL FUNCTION. LOGARITHMS We have thus proved that fw any real number x, the sum of x' . the limiting sum of the series 1 + x + „:+.. . is the principal value of e*, where e is defined by E(l) = e. This is the exponential theorem for a real exponent. 226. We shall now shew that whatever complex number z is, the -number JS(z), the limiting sum of the exponential series in powers of e, is equal to the limiting value of (1 + z/m)™, where m has positive integral values, when m is indefibaitely increased. We have (1+zjmy z m(m—\)z^ m(m— l)...(m — s+1) ^^ ■m 2\ m'' s\ m* = l + .s+ 1 mj 2\ \ ml \ mj \ m J s\ Now if a, b, c, . . . be any positive real numbers, less than unity, we have (l-a){l-b)>l-(a + b) (1 - a) (1 - 6) (1 - c) > (1 - a - b) (1 - c) > 1 - (a + 6 + c) Hence (l-a)(l-6)(l-c)..., <1, and >l-(a + b + c+ ...), say =l-6(a + b + c+ ...), where 6 is some number between and 1. Hence we have \, mJ \ mJ \ mj \m m m/ where 0s is some number between and 1. We have now (l+zjm)"' = l+z + - + ...+- + ...+—+R, 2! s\ ml where R denotes z' { Z Z^ Z" ^m~i 1 ~2^r + ^='-T+^''-2!+-+^'+'^+- + ^"-^0iirr2y!}- The sum of the series in the bracket has a modulus less than the \z\ I ^ P limiting sum of the convergent series 1 + ^-J + ^-^ + . . . ; and when THE EXPONENTIAL FUNCTION. LOGARITHMS 287 TO is indefinitely increased, z^l2m converges to zero. Therefore the limiting value of (1 + z/m)™ when m is indefinitely increased, is the function E(z). The number e is the limiting value of (1 + l/m)"*. 227. The theorem proved in the last Article gives us the means of finding the value of E (z), where z = x + iy, a complex number. We have ^(^ + ^•2/) = z(l+■^J^ put 1 + xjm = p cos (/), yim = p sin (^, then 1 1 -j ^ 1 = p'^ (cos ^+i sin ^)'" = p'^ (cos m + i sin m<])), by De Moivre's theorem. Also -/ 1x a? + «^ 1 + — +— ^ Ttl and (^ is the principal value of tan"' — ^ — . The limiting value of jo"" is that of m) X ^ (■ or of E{x) ^^ m) X^ix^mj] 1+ ^ m{hj'm-^x\isl'inf] now suppose that r is a fixed positive number less than \/m + a;/Vm, then the limit of 'if ^ m (Vto + xl^jmj is between unity and that of or between 1 and e*^ ''' ; now r may be made as large as we please, subject only to the condition r < \lm + xjis/m, hence the limit of I ^ (a; + myj is unity, and therefore that of p™ is E{x), which is the principal value of e^. The limiting value of m tan"' -^-— is that of — -— , which is 2/ ; hence we have L [l + —^) = «* (cos 2/ + t sin y), where e* has its principal value; thus E{x + iy) = e* (cos y + i sin 3/). 288 THE EXPONENTIAL FUNCTION. LOGARITHMS Expansions of the circular functions. 228. If in the last result we put a; = 0, we have E (iy) = cosy + i sin y, - ■ v" .v' hence cosy + ismy = l+iy—~^ — i^^ + ..., or, equating the real and imaginary parts on both sides of the equation, we have cosy = i-^, + |;-... + (-i)«(g-, + (3), sin2/ = 2/-|-; + f,...-f(-iypf^ + (4), the series for cos y and sin y expanded in powers of the circular measure y ; these series have already been obtained in Art. 99. We may also write these results in the form cosy = ^{E{iy) + E(-iy)} ^ 1 I (5). sin y=^.{E(iy)-E{-iy)]^ The exponential values of the circular functions. 229. If ^^ is a real number, the expression &', as defined in Algebra, is multiple-valued except when z is a, positive integer. If is a fraction pjq in its lowest terms, e^'* has q values, the gth roots of e^ ; of these values, that one which is real and positive is called the principal value of e*, and is equal" to E (z). When z is an irrational real number, the principal value of e^ is defined, as in Art. 186, as the limit of the sequence formed by the principal values of e^', e^s ... e^', ..., where z^, z^, ... Zr, ■■. is a, sequence of rational numbers of which z is the limit. We shall in general understand e^ to have its principal value E {z). When z is not a real number, no definition of e^ Aos as yet been given, and it is so far a meaningless symbol. It is convenient however to give by definition a meaning to the symbol e^ or e*+'^. At present we give only a partial defi- nition of the meaning we shall attach to e* ; we define only what may be called its principal value, and shall shortly proceed to a more general definition. THE EXPONENTIAL FUNCTION. LOGARITHMS 289; The principal value of the function q" we define to be the function E (z), or\ what amounts to the same thing, the limit of(l+ z/m)'", when m is indefinitely increased through positive integral values. It should be observed that this definition of the principal value of ^-^iy is such that the function satisfies the ordinary indicial law this follows from the theorem (1) of Art. 224. We shall in general, when we use the symbol ^, understand it to have its principal value E {z) as just defined. 230. With this understanding as to the meaning of the symbol e»i+w we have, by Art. 227, gic+ij/ _ gas ((jQg y + i sin y), and putting x = Q, e^ = cos y + i sin y. The theorem (5) may now be written cosy = \{e^y + e-^y) \ 1 . .1 (6). smy = ^. {e'y - e-^v) j These are called the exponential values of the cosine and sine. The student should bear in mind that these theorems (6) are nothing more than a symbolical mode of writing the equations (3) and (4) which have also been written as in (5). The only advantage of the symbol e'" over the symbol E{iy) is that the former one reminds us more readily of the law of combination given in Art. 224. The theorem (1) is of the same form as that for the multiplication of real exponentials ; we therefore find it convenient to introduce exponentials with imaginary indices, for which the law of combination shall be that expressed by (I). 230 '^'. The function e' being defined as above, for any complex value of z, as the limiting sum of the exponential series 1 +^ + ^72! + 2r»/8 ! + ..., Z^ 0* we see that e" = \+ z + -^ + ...+—. + Rg, where | iJg | is not greater than the sum of the infinite series J — L_^^ + J — L^ + . . . . It (s + 1) ! (s + 2) ! ^ The latter form of the definition is that introduced by Schlomilch, see Zeitschrift fur Math. Vol. vi. H. T. 19 290 THE EXPONENTIAL FUNCTION. LOGARITHMS follows that \Rg\ is less than /^ZTTi I''" "^ ' '^l "*" V" "*" Si""*" ■■■!' \z or than^^-' ^. je'^'. In case | «]< 1, we see that !-R|<; or I jK I < \ + \z\ + \z\^+^ Is+l 1 (s + 1) ! 1-1^1 Z^ .2* We have thus shewn that e^ = 1 + a: + jr-. + ■ ■ ■ + — , (1 + w«), z ! S ! where | Wg | < - — L- e''^' ; and thus | Wg | converges to zero as | ^ | does so. In particular, by taking s = 1, we have the theorem e^ = l +2(1 +tti), where |iti | <^|2|e''''; and thus \ui\ converges to zero as 1 2^ I does so. We may express this result in the form |2:!=0 ^ gZ + A gs From the last result we have L j =ef, and thus the function ^ is ft=o A such that it is equal to its own diflferential coefficient. The function ^ may be introduced into Analysis by defining it as that function u which satisfies the conditions t-=m for every value of z, and M=l when 2=0. If it be assumed that there exists a aeries ao+ai2+a22^+"- which is convergent for every value of z, and such that the derivative series 01 + 202^+30532^ + ... has the same property, both series converge uniformly in a circle of any finite radius. Denoting by u the sum of the first series, that of the second series is, in accordance with a known theorem, -=- . If then -J- =u, we can equate the coefficients of corresponding terms; thus dz ai = a^, 202 = 01, ... *Mtn = <''n-i j and thence we find 0,1 = 00/%!. It follows 1+Z + — I + ...H | + -"f j and it is easily seen that this series actually satisfies the assumed conditions of uniform convergence. It follows that the sum of this series satisfies the condition -^=u. li u=\ when dz z=0, we must have ao=l. In this manner we are led to the series a2 2" I+2+-J + ... + -+..., with the investigation of which we have commenced in the text of this Chapter. THE EXPONENTIAL FUNCTION. LOGAKITHMS 291 Periodicity of the exponential and circular functions. 231. We have shewn that E {z) = fF (cos y+i sin y) ; now cos y, sin y are unaltered if ikir be added to y, k being any positive or negative integer, consequently E{z) = E{z + likir) ; or E {z) is a periodic function, of period 2i7r. Since e' = e^+*'^, the exponential c* is periodic, with the imaginary period 2i7r; also e'* = e*'*+^"'', or e^ as before defined, is a periodic function of z, with a real period 2ir. We have thus seen that each of the two functions e', e^' is singly periodic, the first having an imaginary period 2iir, and the latter a real period 27r. The student who is acquainted with the elements of Elliptic Functions will know that it is possible to construct functions which have both a real and an imaginary period; such functions are called doubly periodic. 232. The circular functions cos y, sin y were first introduced by means of a geometrical definition, and we have regarded them, in the earlier part of this work, as functions of an angular magnitude measured in circular measure. We can however drop the idea of the angular magnitude, and regard them as functions of a variable ; a value of the variable of course measures the magnitude in circular measure of an angle by means of which they were defined. The main importance of these functions in Analysis is derived from their property of single periodicity ; it has been shewn by Fourier and others that all functions having a real period can, under certain limitations, be represented by means of a series of these circular functions. It would however be beyond the scope of the present work to enter into this most important branch of Analysis. Analytical definition of the circular functions. 233. It is possible to give purely analytical definitions of the circular functions, and to deduce fi-om these definitions their fundamental analytical properties, so that the calculus of circular functions can be placed upon a basis independent of all geometrical ■considerations ; these definitions will include the circular functions of a complex number. 19—2 292 THE EXPONENTIAL JUNCTION. LOGARITHMS We can define the cosine and sine of z by means of the equations cos 2: = i [E (iz) + E(- iz)} ■in sin ^ = ^. {E (iz) - E {- iz)] z^ where E {£) denotes the limiting sym of the series \ +z+-^^-\- .... In other words, we define cos^^ as the limiting sum of the series 1 — a"] + r", • ■ • > ^^i^ sin z as the limiting sum of the series z — -^ + -^.... We may regard this then as the generalized definition of the cosine and sine functions, and it includes the case of a complex argument, which was not included in the earlier geometrical definitions. For real values of z, the functions cos z, sin z are in accordance with the earlier geometrical definitions, because the series which they represent agree with those obtained, in Art. 99, from the geometrical definitions. z^ z' By employing the theorem e'=l+z + -^^+ ... H — T + Rg, where I i?g I < -i — L_^^ e'^l, proved in Art. 230'^', we see by changing z into iz and - iz, letting s=2m + l and adding the expressions so obtained, that Z^ Z* Z^""' where | Rm \ < 7^ — — oTi * ' " '■ ^^ particular, we have cos 2: = 1 + i^o', \z\' 1 where \Ro\<^-^e^'\ and cos2:= 1 —-^z^ + R^, where I 7?' I <- 111 e^i\ In case U I < 1, we have also I ^0' I < ^ /J 1 in , and ' ' 2(1-|2'|)' 4>\(\-\z\y Similarly we see that sin. = ^-3j + 5j-...+(-ir^2^^-^-3^, + S„', THE EXPONENTJAL FUNCTION. LOGARITHMS 293, where 1 8^ I < -^ — ! — — — e ' « I ; and in particular sin = ^^ + i?o', where (zm + d) ! I-Ko'l < tf e'", and sin = ^ - ^ ^ + K, where i S/ 1 < ^4f e'"- If i 1 < 1, we have also | /S„' j < ^ '_' .^ , | /Sf/ 1 < g., ' _|_- ,, ■ 234. From the definitions given in Art. 233, we can now deduce the fundamental properties of the two functions. We have cos z->ri sin z = E (iz), and cos z — i sin z = E{— iz), hence cos" z + sin= z= E {iz) E (- iz) = E (0) = 1. Also cos (01 + ^3) = ^ {£' (i^i + i^a) + E{- izi - iz^)} = i{E (iz,) E (iz,) + E(- iz,) E (- iz,)] = i [E (iz,) + E(- iz,)} {E(iz,) + E(- iz,)} + i {E(iz,) -E(-iz,)}{E(iz,)-E(-iz,)} or cos (z, + z,) = cos z, cos z, — sin z, sin z,. Similarly sin (z, + z,) = sin z, cos z, + cos z, sin z,. Thus the addition theorems follow from our definition. 235. Let us now consider the equation E(z) = 1. In the first place this equation has no real root except z = 0; for it is clear from the definition of E(z) by means of the exponential series that the equation has no positive real root; and it can have no negative real root —x since the positive number x would then also be a root, as is seen from the relation E(—x)E (x) = 1. Also the equation E(z) = l can have no complex root a + i/3, where | a | > 0. For, if a + i/S were a root, so also would be a — i/S, and therefore E (2a) = E (a + i^)E (a- i^) = l, which is im- possible, since 2a cannot be a root. It thus appears that, in case the equation E(z) = l has roots other than z = 0, they must be purely imaginary. In order to shew that the equation has such a root it will be sufficient to shew that the equation E (i^) — E (— i0) = 0, or sin/3 = 0, has a real root other than zero ; for, if /3 be such a root, we have E(2i0)={E(i0)}' = l, and thus 2ifi would be a root oi E(z) = l. 294 THE EXPONENTIAL FUNCTION. LOGARITHMS It will be shewn that, if /(/3) denote the continuous function — 5— represented as the limiting sum of the series ^2 ^ ^ 3"r5! 7!"^"" then /(/8) is positive for all values of ^ such that S /8 S 3, and that it is negative when /3 = 4. From this it may be concluded that f{^) is zero for one value of /3 between 3 and 4, or for an odd number of such values ; and in any case that the numerically smallest positive root of /(/8) = is between 3 and 4, in ease the equation has more roots than one. If /8 is positive and < \J20, each term in the series for /(y8), with the exception of the first, is numerically greater than the fla Qi as next following term. We have therefore /(/S) >^ — k~,+^. — ^., for values of ^ between and some number greater than 3. Denoting l-|^+|^-^j by ^(/3), we find that ^(3) =17/560, which is positive, and ^ (0) = 1 ; also the derived function ^'(y8)s — 2;8 (^| — -^ +'-yj j is negative when ^ is between „ , n • i 2/8\ 3/8-' 1 2^^ 1 2 . 3= _ „ and3,sinceg^--gy+yy>g-,--g-|->gj--gj->0. Hence ), and since E(i) = cos <^ + i sin ^, i is a value of Log (cos <}> + ishi ^), THE EXPONENTIAL FUNCTION. LOGARITHMS 297 and log p + 2ikTr is the general value of Log p, we have therefore Log u = log p + i( is restricted to be between the values — ir and nr, we shall call log p + i(ji the principal value of Log u and shall denote it by log u ; we have then the general value Log u given by Log vi = logn + 2ik7r, where log u is its principal value, and k is any positive or negative integer. We may write this result Log {x + iy) = \ log {a? + y^) + i Ttan-^ ^ + 2k-ir\ (8). The principal value of the logarithm of a real negative number — X has not been sufficiently defined, since the argument of such a quantity may be either tt or — tt ; we shall however suppose, for convenience, that for its principal value the argument is tt, so that its principal value is log x + iir, and the general value of its logarithm is log x + (2k +1) iir. The general value of the logarithm of a real positive number X is given by Log x = log x + Log 1 = log x + 2ikir, the principal value being log*. The principal value of Log i is i^iri, hence Log i = i2k + \) iir ; the principal value of Log (— i) is — ^ iri, hence Log (— i) = (2k— \) iir. It is also possible to consider the logarithm of m as a single- valued function of the modulus p and the argument <^, the latter being supposed to go through all values from - oo to + oo , not being restricted as above to lying between rr and — IT ; the logarithm of u is then the single- valued function of p and , and every time increases by Stt, the logarithm increases by SiV, and the numerical value of the number u becomes the same as before. The student who is acquainted with the theory of Riemann's surfaces will appre- ciate the full force of this mode of considering a multiple-valued function as converted into a single-valued one. The general exponential function. 240. If a be any number, real or complex, the symbol a? may be defined to mean E (z Log a), where Log a has any of its infinite number of values; when Log a has its principal value logos, we shall call E (z log a) the principal value of a'. 298 THE EXPONENTIAL FUNCTION. LOGARITHMS Since ^(.Loga)=l + ^ + <^^+..., we have the general exponential theorem a -1+ J + 2] +••■' and the principal value of a' is given by 1! 2! In the case in which a and z are both real, we have the ordinary form of the exponential theorem ^ , a; log a a^HogaV 1! "*" 2' which gives the principal value of a*. 241. In the particular case a = e, we have Log e = log e + 2ikir = 1 + 2^A^7^, and the general meaning of the symbol e' is ^(^Loge) or E{z + 2ikirz); the principal value of e' is E{z), and this is in accordance with the definition of the principal value of e^ given in Art. 229. The general value of e' is therefore ^(0)(cos llcirz + i sin 2hirz). We shall still continue to use the symbol ef for its principal value. 242. The general value of a", as above defined, is equivalent to E{z(\.ogr + i6 + 2ikTr)], where a = r(cos^ + isia^) = a + i/S, d being between — ir and tt ; writing z = a} + iy,we thus have for the general value of (a + i^y+*v the expression E {x log r — dy— ^kiry + i (y log r + a!d+ 27rkx)] which is equal to ^logr-ey-uny jg^g (^ j^g ^ + ^J^ + 277 A;*) + 1 Sill (y log r + x0 + 2irkx)}. The principal value of (a + 1/3)*+*" is therefore gxiogr-ej^jcos (y log r + a;0) + i sin (y log r- + xd)}, where }• = Va" + 0\ ^ = tan-'/3/a. The value of tan~'jS/a, to be taken, is not necessarily its principal value as defined in Art. 38. If r= 1, we have for the principal value of (cos d+iain d^*:*', the function £! {i6 {x+ii/)} which maybe written coa(x!+ii/)d + aia(x+ii/)d; this is the extension of De Moivre's theorem to the case of a complex index. THE EXPONENTIAL FUNCTION, LOGARITHMS 299 243. In order that the equation a^> x a^^ = a^'+^s may hold, we must suppose that the values of a^', a^», a«i+*2 are those corresponding to the same value of Log a ; in that case we have a«. X a"^ = E{z^ (log a + 2ik-n-)] x E {z^ (log a + 'iikir)] = E{{z-^ + z^ (log a + 2*7r)} = a''^+\ but this will not hold if we take different values of k in the two functions a*-, a'\ In particular, the equation a^' x a"'' = a''^+^' is true of the principal values of the functions. 244. The expression (a^')*'» is not necessarily a value of a^'*», but every value of a"^"' is a value of {a^'Y', for a^i^a = ^(01^2 Log a) = E{ziZ^ (log a + 2ifc7r)} and (a^i)'''' = ^ {•22 Log a^'} = E {z.^ (z^ Log a + 2ik'-7r)} . = -&{^ri«2 (log a + 2ikTr) + 2i . k'lrz^, hence the values of a^'^^ are only those of (a*')^" in the case k' = 0. If in every case we take the principal values, then the equation a2i2a = (a2.)aa holds. If we use the symbols a", e' as equivalent to their principal values E{z log a), E(z), as is usually done in practice, then we may, as we have just shewn, perform operations in expressions in which these symbols occur, according to the ordinary rules for indices, as in common Algebra. Example. If A, B, C, D, ... be the angular points of a regular polygon of n sides, inscribed in a circle of radius a and centre O, prove that the sum of the angles oTi S'ivy Yin that AP, BP, CP, ... make with OP is tan~^—- ^ -, where OP=r, and ' ' a" cos nfl — r" ' the angle AOP=fl. We have »•« -«•'«"'= n {r-ae*^ » / }, 8=0 hence taking logarithms, log (»•" - o" cos nd - ia^ sin nff) = "!/°^ f ~'"'°' (^^?)"^''" (^"^^)} ' and equating the coefficient of i on both sides of the equation, • //, 2s7r\ /. . ™ , asm e-\ tan-*-r z ■:= 2 tan-i- a cos (»-'?)-'■ 300 THE EXPONENTIAL FUNCTION. LOGARITHMS corresponding values of the inverse functions being taken ; the expression on the right-hand side is the sum of the angles OP makes with AP, BP, ... , a" sin n6 hence this sum is tan " a"cosMfl — r"* Logarithms to any base. 245. If the principal value of a* is equal to u, then z is called a logarithm of u to the base a, and may be written Logo u. Now the principal value of a' is E{z loge a), where logj a is the principal logarithm of a to the base e, and if E{z loge a) = u, we have z loge « = Loge u = loge « + '^ihiT, therefore Loga u = Loge w/loge a = (loge u + 2iA7r)/loge a. The principal value of Loga u we regard as loge u/loge a, and can denote it by loga u ; hence the general value is Loga u = loga u + 2ikir/loge a, a multiple-valued function in which the different values differ by multiples of 2i7r/loge a. In the particular case a=e, the above definition accords with that in Art. 238, giving loge ** + ^ikir for the general value of LogeW. Generalized logarithms. 246. We may give the following definition of a logarithm, which is more general than that given in the last Article. If any value of a' is equal to u, then z is a. logarithm of u to the base a, and may be written [Loga u] to distinguish it from Log^ u as used in the last Article. The most general value of a" is ^(^: Loge a), and if this is equal to u, we have z Loge a = Loge u, or z (loge a + 2ik'ir) = loge u + 2ikir, where k and k' are integers. Hence the general value of [Log^ u\ is Loge w/Loge a or (loge u + 2tA;7r)/(loge a + ^iMif), which is multiple- valued to an infinite extent, in two ways. The logarithms Log,, u are therefore included as the particular set of values of [Loga m] obtained by putting k' = 0. We may call [Log,, m] the generalized logarithm of u to the base a. 247. If a = e, we have [Loge «] = (loge u + 2ikTr)/(l + 2ik'7r) which is the expression for the generalized logarithm of u to the THE EXPOMENtlAL FUNCTION. LOGAttlTHMS 301 base e. In the more restricted logarithm Log« u, we have defined 2 to be a value of Lbge u when the principal value of e* is equal to u, but in the generalized logarithm [Log^w], we consider z to be a value of [Log^ u] when any value of e" is equal to u. The generalized value of [Log« 1] is 2ik7rj(l + 2ik'Tr), and of [Loge (- 1)] is {2k + 1) iirl{l + 2*V). The expression (logeM + 2i/5;n-)/(H-2i^7r) maybe considered from another log«+2a;ir point of view. The principal value of {-£'(l+2t^7r)} i+at*""- is, by the theorem (2), ^(logM + 2i^7r) which is equal to u, hence (logM+2i/i;7r)/(l + 2iyi:'7r) may be regarded as the logarithm according to the definition in Art. 238, of w to the base ^(1 +2i^7r) which is the principal value not of e but of ei+2**'"-^ go that we have in fact [LogttJ equal to the values of ^og -^ ^,^^^J^^^u, for diflferent values of k'. Thus we may regard the generalized logarithms to the base e, as ordinary logarithms to the base not e but gi+^ift'n-^ which though numerically equal to e, has different arguments according to the value of if. 248. The question was at one time frequently discussed, whether a negative real number can have a real logarithm ; thus for example whether ^ can be regarded as the logarithm of — Ve, the fact being borne in mind that e^ has the values +ije. The answer to this question depends on the definition we take of a logarithm ; if we take the ordinary definition in Art. 238, that z is a logarithm of u when the principal value of e" is equal to u, then a negative real number can never have a real logarithm ; but if we define a logarithm as in Art. 246, that « is a logarithm of u, when any value of ^ is equal to u, then a negative real number may have a real logarithm. If ?• be a positive real number, we have . • ., Jogr+{ik+l)iw __ {log>-+2F(2,{;+l)7r'}+t{(2^+I)B--2yi;'irlogr} \>^% n- i + 2A'jW ~ 1 + 4A'V2 and this is real if log ?• = (2i + 1)/2^. If therefore ;• be such that log r is of the form (2/; + l)/2A', where h and Fare integers, a value of [Log(-r)] is real; if logr is not of this form, we can always find a number r^ differing as little as we please from r, such that [Log ( - ri)] has a real value ; for a fraction ; it also converges on the circle of convergence, except at the point z = — 1, when > m > — 1. Now it has been shewn in Art. 210, that we are entitled to arrange this series in powers of m, without altering its sum, provided the series II 1 |wil(|m I + 1), I, l + |mj|^| + l '^'^,' '\z\'+... | m|(|TO| + l)...(|m|+s-l) ,, is convergent ; and this is the case if | 2^ | < 1. Since E {m loge (1 + z)} stands for the sum of the series l + mlog,(l + z)+ ^ ^'^^ -^+..., we are, by Art. 208, entitled to equate the coefficients of powers of m in the two series ; hence loge{l+z) = z-^z' + iz^-... + (-l)'-^^z' + (9). This series, which gives the principal value of Log^ (1 + z), is called the logarithmic series ; it has been proved to hold when mod. z-+l (2r + l)(2>-l) (2>-+l)(2r-l)...l ) 2?-+l \ 2 "^ 2.4 ~ ■^*' '' 2.4.6...2r J' the expression in the brackets is the sum of the first r+1 coefi&oients in the expansion of (l-y)4(2r+l) jjj powers oiy, and this is equal to the coeflScient of y'' in (l-y)-i(l-^)4(2'-+i) or (l-3^)4(2'-i), which is equal to (2r-l)(2r-3)...l '' ' 2.4.6...2»- Hence the coeflScient of a*"''' in the expansion of sin~'a; is 1 1.3.5...(2r-l) _ 2J-+1" 2.4.6...2J- ' therefore . , 1^-3 1.3^ 1.3.5...(2»--l)^2'- + i ^"^2" 3 "''2.4 5"*" "*■ 2. 4. 6. ..2?- 2>-+l"''" this proof only shews that this series holds for values of x between + 1/\/2, but by employing the fact that the sum of the series is continuous within its circle of convergence, it can be shewn to hold if x is between + 1. The quadrature of the circle. 251"'. The fkmous problem known as that of "squaring the circle," that is, of constructing a square whose area is equal to that of a given circle, is equivalent to that of constructing a straight line of length equal to that of the circumference of a given circle. The construction to be employed in solving the problem is a Euclidean one, involving only the construction of circles and straight lines, in accordance with a Euclidean system of postulates. The problem may be stated as that of the construction of a straight line whose length is represented by the number tt, a given finite straight line being taken to have the length represented by unity. The fact, proved by Lambert, that the number tt is irra- tional, that is not representable in the form pjq, where p and q are integers, is not of itself sufficient to establish the impossibility of H. T. 20 306 THE EXPONENTIAL FUNCTION. LOGARITHMS constructing the line of length tt, because a certain class of straight lines of irrational length is capable of Euclidean construction. A step of fundamental importance in this connection was taken when Liouville^ established the existence of transcendental numbers, as distinct from algebraical numbers. An algebraical number is one which is a root of an algebraical equation of any degree n, with coefficients which are rational numbers; without loss of generality these coefficients may be restricted to be all integers, positive or negative. A transcendental number is one which cannot be a root of any algebraical equation with rational (or integral) coefficients. Liouville himself gave examples of transcendental numbers, but the first case in which a number, well known in Analysis, was shewn to be transcendental, was that of the number e, the transcendency of which was established by Hermite. Following Hermite, Lindemann" gave a proof that tt is a transcendental number. He proved the more general theorem that, if e* = 2/, the two numbers x and y cannot both be algebraical, except in the case a; =0, y=l. Simplified proofs that e and ir are transcendental numbers were afterwards given' by Hilbert, Hurwitz, and Gordan. A modified form of Gordan's proof will be here given. The proof that tt is a transcendental number is equivalent to the establishment of the impossibility of squaring the circle by means of any geometrical construction iu which straight lines and circles are alone employed ; or more generally when any algebraical curves may be employed. For any such construction amounts to the exhibition of tt as a root of some algebraical equation obtained by combination of the cartesian equations of straight lines and circles or other algebraical curves. The fascination which the problem of " squaring the circle '' has exercised for centuries upon many minds is such that Lindemann's proof of the impossibility of the problem under the assumed conditions is a result of great im- portance in relation to a problem of historic interest. 251'^'. To shew that the number e is transcendental, let us assume, if possible, that e satisfies the condition A, + A^e + A^e''+... + Ane'^ = 0, ^ Liouville's Journal, Vol. xvi. 1851. ' Mathematische Annalen, Vol. xx. 1882. ' Ibid. Vol. XLiii. 1893. THE EXPONENTIAL FUNCTION. LOGARITHMS 307 where A^, A^, ... An are positive or negative integers, and A^ is a positive integer. In order to shew that this assumption leads to a contradiction, it will be shewn that a number K can be deter- mined such that KA, = I, + f„KA,e = I, +f, , KA,e' = /,+/„... KA^e'^ = /„ +/„, where Ia,Ii,Ii, ... In denote positive or negative integers, and fitfi, ••■ fn denote numbers numerically less than unity, and such that /i 4-/2 +.•■+/» is numerically less than unity, and where ■^0 + A + • • • + -^Ti is not zero. On multiplying the original equation by K, we have the sum of an integer and a number numerically less than unity equal to zero, which is impossible. To determine the number K, let us consider the expression ,v.j)— 1 <^ C-^) = (^Txyi Ki - ^) (2 - *) (3 - «^). . .(»i - ^'M ^ where p is a prime number greater than n and greater than A^, We may denote ^ («), when expanded out in powers of x, by Cp_i ajP-i + CpXP + . . . + c„j,4^_i a!"^+^-i. Denoting by ^'{x), ,f>"{x), ...^<"(a;), ... {x), we see that ^(0), ^+^0). ...{m), '{in), ... <^P~'(m) all vanish, and {m) + ^'(m)+ ... + ^«p+p-^ (m)} r=p-l the first term on the right-hand side is a positive or negative integer divisible by p, and the second term is numerically less r=np+p-l than I -4^ I e™ 2 | c, | m'', or than r=p-l ' ^" ' ^" {p-l)l {(1 + »*) (2 + »*) • • • (w + ™)) *. which cannot exceed By choosing p great enough, the number {n (n+l)(n + 2)...(n + n)}P/(p - 1) ! may be made as small as we please. Let K be the value of Kp when p is so large that (jZ:Yy{(l + n)(2 + n)...{n + n)]P{\A,\e+\A,\^+...+\A„\d^] is less than unity. We have then K {Ao+ A^e + A^tf' + ... + Ane'^) equal to the sum of an integer which is not divisible by p, an integer which is divisible by p, and a number numerically less than 1 ; and this is impossible. Since e cannot be a root of an equation Ao + AiX+... + AnX'^=0, with integral coefficients, it is a transcendental number. 251 ®. If TT were a root of an algebraical equation with integral coeflScients, iir would also be a root of such an equation. Let us assume that tV is a root of the equation (« - Oi) (a; - Wa) . . . (a; - Oj) = 0, with integral coefiScients; thus iV is one of the numbers ai,a2,...as. Since e'"= — 1, we have (1 + 6°') (1 + e<^)...(l + e''«) = ; on multiplying out the factors, this is of the form A + e^' + e^' + ... + e^^ = 0, where -4 is a positive integer. THE EXPONENTIAL FUNCTION. LOGARITHMS 309 It will be observed that all the symmetrical functions of Okj, Ga^, ... Ca, are integers, therefore all those of (7y9i, G^i,...C0n are integers. We take "I" (*) = (Trryi ^""'^"'^ {(^ - 'SO («=- 13.) ■•■{«>- ^n)Y, "where p is a prime number greater than all the numbers, A,n, G, C"|A;S,...;8„|. Denoting '(^m), ...V-'i^m) are all zero, and <^^(/3^), 0^+H/Sm). ... fj^p+np-i (^^^y are integers divisible by p. Let r=p~l thus KpA is not a multiple of ^J. Also r=np+p-l ( O r+i Kpe^rn= 2 c,. |8™'- + r^^'-^+...+r! + ^ r=p-l (. r+i r=p-l where the numbers ^/ all lie between and 1. r=np+p—l The number 2 c,. 0/ ^^ is numerically less than r=p—l r—np+p—l S |c,| 1/8^1'- j.=p-i or than _^f^ OP+i>-i {(^3 + I A|)(^ + I /8,|)...(^ + |/3„|)p, where yS is the greatest of the numbers | A | , | /Sa | , . . . | /3„ | . We now choose p so great that {|es.| + |eS.!+... + ie^»|}(-^rXy-,C'"^'"^-M(/3 + |/8i|)(/S + |Ai)...}^ is less than unity. Taking the value of iiTpfor such a prime p as the value oiK, we see that K{A + e^'+e^» + . . . + e^'*)is expressible as the sum of a multiple of p, an integer not divisible by p, and a number numerically less 310 THE EXPONENTIAL FUNCTION. LOGARITHMS than unity; it is therefore impossible that it can vanish. It has thus been shewn that it cannot be a root of an algebraical equation with integral coefficients, and it is therefore a transcendental number. The approximate qiuadrature of the circle. 252. The problem of the quadrature of the circle, which is equivalent to the determiaation of tt, can be solved to any required degree of approximation, by taking a sufficient number of terms in any one of a large number of series which have been given for IT. The simplest series which we can obtain is got by putting = Jtt, in Gregory's series ; we have then i'r=l-i + i-f+... which however converges much too slowly to be of any practical use for the calculation of tt. 253. If we use theidentity \'t= tan~' J+tan~' J, and substitute for tan~^ \, tan~' ^ their values from Gregory's series, we have |=4-ia)'+i(i)'-- +*-i(i)^+*a)°--- This is called Euler's series. Another series may be obtained from the same identity by substituting for tan~^ \ and tan~^ ^ their values from the series *^'^^=rT^f + 3 1+^+375 lr+^'j + -} which we have obtained in Art. 219. We have then 1 4f, 22,2.4/ 4 10( ^3'10 ' 3.5V 3 f 2 1 2.4 "^10r"^3"10"''3T5 254. Other series obtained in a similar manner have been used by various calculators. Clausen^ obtained his series from the identity ^tt = 2 tan~^ ^ + tan~^ f , using Gregory's series ; Machin's series is obtained from ■^TT = 4 tan~' \ — tan~' -^ ; 1 See a paper " On the calculation of tr " by Edgar Frisby in the Messenger of Math. Vol. II. THE EXPONENTIAL FUNCTION. LOGARITHMS 311 Dase used the identity ^TT = tan"' 1^ + tan~' ^ + tan~' ^ . A more convenient form of Machin's series was used by Rutherford, who used the identity ^tt = 4 tan~' ^ — tan~' ^^ + tan~' -^. Hutton' gave the series „ . f, 2 1 2.41 ■^■''^'4-m "•"s.sUooj "''•■•J' this is obtained from the expansion of a!tan~'a; in powers of r^ -, J. ~T" iC by putting x = ^ and a; = f , and using Clausen's identity. Euler has given the series 28 f_ 2 / 2 \ 2 . 4 / 2 V 1 '"=10^ + 3I100J + 3T5I100J+-} 30336 [, . 2/ 144 \ 2.4 ^ 144 V ) 100000 1 "^ 3 VIOOOOOJ "^3.5 UOOOOOJ + -"j ' which can be deduced from the identity TT = 20 tan-i 1+8 tan"' ^. The value of tt has been calculated by W. Shanks to 707 decimal places^. 1 12 32 52 The continued fraction - — ^r— 5— 5— =Jir was given in 1658 A.D. by Lord Brouncker, the first president of the Royal Society. It is obtained by transforming Gregory's series \ — ^+\ — \ + ... according to the usual rule. I 12 2 3 3 4 Stem' has given the continued fraction ^jt = 1 + :r— - y^ yZ' TX " An interesting account of the history of the subject of the quadrature of the circle will be found in the article " Squaring of the Circle" in the Enoyclo- p'_e-i«-)') or -^-) Z/. ■ ) *^{cos2(d-a)+ism2(5-a)}; sm(o-/3)sm(a-y)^ ^ ' transforming each fraction in this manner and equating the coefficient of i to zero, we obtain the identity to be proved. The summation of series. 256. When the sum of a finite or an infinite series a^ + OiX + a^ + ... is known, we may deduce the sums Si and 8^ of the series tto cos a + ayX cos (a + 0) + a^^ cos (a + 26) + . . . , tto sin a + Oja; sin (a + ^) + a^ sin (« + 26) + For suppose /(a?) = ao + *i« + o^^ + ..., then e^f{x^) = 8^ + 18^, THE EXPONENTIAL FUNCTION. LOGARITHMS 313 and also e"'"/ (we~^) = 8^- iS^ , therefore Si= ^ [e'-f^xe^) + e-^ f {xe'^)} , and S^ = 2^. {e^ f (xe'^) - e-^''f(xe-'^)] , the values of Si, S^ thus obtained, can now be reduced to a real form. Examples. (1) Sum the series cosa + xcos(a+^)4-x2co«(a + 2/3) + ... + x''-ico«{a+(n-l)^}. \ — X'"' We have ^j = l+^ + i(;2 + ...+.r"-i. Change x into x^^ and multiply by ^ ; we have then l-aie'^ and similarly we have 1-xe-^^ therefore the sum of the given series is 2 1 ■ \-xe'^ ■ 1-xe-^^ J 1 e'»(l -^e'"^) (1 -a;e-'^)+e-'''(l -a:"e-"^) (1 -a?e'^) 2 (l-^e'0(l-a;e-'^) which is equal to cosa-^cos(a-|3)— ;r"co8(a + w/3)+a^"'"'cos(a+w-l^) 1— 2^cosj3+^ (2) Sum the infinite series sina + Ksin(a+^) + ;, ^^ + ... + V^ + —- We have e"^=l+^+5-:+... + -;+... , put ^«'^ for X, and multiply by e'", we have then 2! ^ ^nl gj;e»?+ja_gia^^gi(a+/3)^_^gi(a+2/3)^__ .|.^gi(a+«p) ^ __ and similarly e»«-'^-^=e-'«+a;e-»('"+^) + |^e-*("+*^' + ... + ^ e-'("'+»^) + ... ; hence the sum of the given series is 2ir ~ / 314 THE EXPONENTIAL FUNCTION. LOGARITHMS or igsoosp /g»(a!sinp+o) _g-i(a;sin/3+a)i which is equal to ga!cosPain(a+a;sin/3). 257. We shall now give some examples of the application of the exponential expressions for the circular functions to the expansion of expressions in series. (1) To expand (1 — 2a! cos 6 + a?)~^ in a series of powers of x, where x is less than unity ; we have (1 - 2x cos e + a;")-! = (1 - «e'»)-' (1 - a;e-*»)-S which expressed in partial fractions is equal to 1 f e^ er^ V 2t sin \1 - a!e^ ~ 1 - xe-*V ' expanding each fraction in powers of x, we have 1 (e'* + xeF*^ + a?^ + . . . + a;"-^ e"* + . . .) 2i sin 6 1 2i sin 6 which is equal to cosec (sin ^ + a; sin 2^ + a? sin 3^ + . . . + a;"-' sin »i^ + . . . ). It may be shewn, in a similar manner, that 1 — X? Tj ; = 1 + 2a! cos ^ + 23?" cos 2^ + ... +2a;»cosn0+ .... 1 — 2a; cos + a;^ (2) To expand loge (1 + 2a; cos 6 + a;") in powers of x, where x is less than unity ; we have loge (1 + 2a; cos 6 + 0?)= loge (1 + xe^) + log^ (1 + xe-^^ ; hence expanding each logarithm on the right-hand side, we obtain the formula (9) of Art. 250. (3) To expand e** sin (bx + c) in powers of x, we may write the expression 2i^ ' ' '' If we expand e«»+»*)* e(«-«»)a; in powers of x, we find the coefficient of a;" to be ^. ^, {e^ (a + ibr - eric („ _ ij)»} . THE EXPONENTIAL FUNCTION. LOGARITHMS 315 let b/a = tan a, then the expression becomes ^. — ; (a^ + &^)i" [eiic+na) _ gH-c+na)] 2* n ! ^ ' '■ > or ~{a' + b')i''sm{c+na.); this is the coefficient of «" in the required expansion. (4) Having given amx = n sin (x + a), to expand x in powers of n, when n3 d=<^ + 2Xsin^ + — sin2<^+-g-sin30 + ..., x=|.(|)v.(!yH..(|)v.... 5. If tan d = x-f tan a, prove that d=a+a;cos2a — ia;2cos2asin2a— Ja^cos'aCOs3a+J^cos*asin4a4-.... 6. If (l+ni)tanfl=(l-jn.)tan0, when 8 and (jj are positive acute angles, shew that 5=0-msin2(^+Jni^sin4(/)- J?re'sin6(^ + .... 7. If tan a = cos 2(b tan X, she w that X - o=tan2 a sin 2a+| tan* m sin 4a+ J tan' m sin 6a+ ... . 8. If sin x=n cos (x+a), expand x in ascending powers of n. 9. Shew that the coefficient of a^ in the expansion of (1 —2^ cos 5+^)"" ^^ 2 {Op cos p6+aiap_i cos (p - 2) d+a^ap^i cos (^-4) 5+ ...}, where aSm is the coefficient of af^ in the expansion of (1 — ii;)~". 10. Provethat jr2 = 18 2 «=o(2ra + 2)!" 11. Prove that in any triangle loec=loga — cos C-=-sCos2(7-;r-5COs3C- .... supposing 6 to be less than a. 12. If the roots of the equation ax^+bx+c=0 be imaginary, shew that the coefficient of .r" in the development oi {cufi+bx+e)~^ in powers of ^ is a^"sin(m + l)^ c*"+isin5 ' where 6 is given by 6 sec 5 + 2 \/ac = 0. 13. If p'=hr-, — r» i., , ,\ (, ■ „. , expand log, p m a senes of cosmes of even multiples of 6. 14. Expand logj cos {d+^ir)va.a. series of sines and cosines of multiples of 6. EXAMPLES. CHAPTER XV 317 15. Prove that (■_ 1)1+1 (2 I TT^l? 713 , (-1)"-^ f2 4 21 81.343"^ 16. Prove that 7 9 15 17 23^25 ■■■ 8 17. Find all the values of (V - 1)"^"^ 18. Prove that (a+a ^-lUn ^;'"«e '' d its value. 19. Iiaooa6 + bsmd = e, where c>\/a^+b'', shew that 18. Prove that (a+a ^-1 tan^/oSef"^^''*)-''' V-i is a real number, and find its value. 5 = (4» + l)_+^log,— ^== tan >^. 20. From the expression for afl+l in factors, deduce that , sin n9 tan-'r— ^ 1+ cos me o sin 26 — 2 cos — sin sin 26 — 2 cos — sin 6 =tan-i +tan~' k v.... 1 + cos 25 -2 cos — cos fl 1+COS25 — 2cos — sind n n 21. From the identity y = , r-. jt deduce •^ x—a x — b {3;-a){x—b) cos {6 + a) sin {e-^)-co&{6+ /3) sin (6 - a) = sin (a - ^) cos 26, sin (5 + a) sin (5 - ;3) - sin (0 + ^) sin ((9 - a) = sin (a - ^) sin 26. 22. Prove that tan-la tan-i0 tan-iy tt , ^3, 2+^3 ,1,1 1,1 where a, /3, y are the three cube roots of unity. 23. Express the logarithms of c-\-di to the base a-\-bi, in the form A+Bi. 24. If tan'»(j7r+iVf)=tan"(j7r+i<|>), shew that ,sin>lf , _,sin^ »itan-i — H-=»itan ' — r^. % t 25. In any triangle, shew that a" cos nB+b" cos nA=cl^-nabc'^-'^ cos (A-B) _^w^»--3)^262c«-icos2(4-5)-.,., n being a positive integer. 26. If logelogeloge(a+l/3)=^ + l2', then e«* ""^ « cos (e» sin q) = i loge (a^ + iS^), and ««''«i"«sin(e«'sin?)=tan-i^. 318 EXAMPLES. CHAPTER XV 27. Shew that the coefficient of al^ in the expansion of e^cosa; in ascending powers of ar is — r cos — . to! 4 28. Prove that 7j— ^ ^=sec' 2\ +... + (-!)» 2 sec3 2\tanX(l+?i cos 2X) cos »fl + ..., where 2X is the least positive value of sin~i e. 29. Prove that the series 1 1 ,.f 1.3.5... (2m + l) 3.5.7...(2m+3)'^" '^■'- can be expressed in the form '"'^ — •-^, where A^, B^, 0^ are whole numbers, and ^^^^ .3.5...(2^_,), c^JMl^, 5„=(2m-l)5„_i-2(TO-l) !. 30. Prove that - sin" 6 cos n^ = sin" ^ cos wfl + w sin" ~ ' cos (m - 1 ) 5 sin (fl — (/>) + ^^^^^^sin»-2 (^ cos (m-2) (9 sin2(fl - <^) + ... +sin" (d- <^), n being a positive integer. 31. Prove the identity ^ sini(«-^)sinT(a-y)sinMa-8) =^"^°^^°+'^+y+^^- 32. Prove that 1 + 1-1-1 + ...=^. 33. Reduce tan ~ i (cos fl + 1 sin 6) to the form a + hi, and hence shew that 11 TT cosfl + -cos3fl + = cos5fl-...= + — , 3 5 4 the upper or lower sign being taken, according as cos 6 is positive or negative. 34. Prove that one value of Log, (1 + cos 2d+isin 2fl) is log, (2 cos 6)-\-id, when 6 lies between -\ir and \n. Deduce Gregor/s series. Prove that one value of sin -i (cos fl+isin 6) is cos-' Vsin d+iloge (Vsin 6 + \/l + sin 0), when 6 lies between and J jr. 00 35 Find the sum of the series 2 .4„ e(2" + 1)^ sin (2« + 1 ) y in which + 1 2n-l 2n+3 36. In any triangle, shew that if a , where 6 and (j) vanish together, and c < 1. 52. Prove that cos Jjr+^cos§jr+Jcos§n- + ... ad inf. = 0. 53. Shew that the series 1 „ 1.3 , 1.3.5 ^ . C08a;+;r— ^cos3a;+;r — ;— ^cos5a; + „ , „ _ cos7^+... 2.3 2.4.5 2.4.6.7 assumes the following values, (1) sin~i(cos^;j; — sin-^.3;), when 7r>a;>0, (2) — sin~i(cos Ja;+sin^a;), when 2w>x>w. 54. If c = cos2d-Jcos35cos35+|cos6eeos55-..., shew that tan 2c = 2 cot^ $. 55. Shew that e°<=°^^sin(asin^)+e°™^^^sin(asin20)+...«''«»^'»-i'^sin{asin(«-l)/3}=O, if ^=27r/n. 56. Prove that sin 6 . sin fl- ^sin 26 sin^ 5+^ sin 36 sin^ 6- ... = cot-^i (1 + cot 6 cot^6). EXAMPLES. CHAPTER XV 321 57. Prove that log(coseca!)=2(cos2.»-^sin2 2^+^cos2 3iB-isin2 4!B+ ). 58. Prove that ^ ' " \ ^3 2 V2/^ ^2w + l' 2.4.6...2m W i" 59. Shew that the sum of the series l-icose+J-l|cos2e-14-^os35+ is -^M=, 2 2.4 2.4.6 V2cos^^ where 6 lies between ±7r. Sum to infinity the series in Examples 60—71. 60. cos^-icos3d+^cos55- -, , cos 2B cos 46 bl. 1 ^r-. h- 2! 4! „„ . , coseod „. , coseo^^ „. , 62. cosfi+-^pj — cos25H 5-T — cos3d + 63. cos 6 cos 26 + cos 25 cos 35 + ^ cos 36 cos 46 + ^ cos 45 cos 55 + . 64. sin 5 -s-f sin 35 +5-: sin 55- COS ^ COS 2^ cos 3^ ^^- 17273+27374 + 374:5 + , cos(a + 2/3) , cos(a+4/3) , C03(a + 6^) , 66. cosa-l gj 1 g^ 1 Y'l '' 67. cos 5 cos ^ — ^ cos 25 cos 20 + J cos 35 cos 30 - „ , . „ tan^ a sin 3x tan' a sin 4» , 68. tanasm2«H g-j h ^-j ^ 2 cos 8 gSoosS 69. 1 + e*^^ * cos (sin 5) + -^-j— cos (2 sin 5) + —gy- cos (3 sin 5) + . 70. Sin5.sin5-isin2 5.sin25+j8in3 5.sin35- 71. msin2a--^»i2sin2 2a + J«i'sin2 3a- , where m^l+x^ X cosh«= \/l+«2 'JX-x'- ^a^-\ ^|\^x'^ X 1 X 1 -Jx^~\ X Jl-^ ^/1 + x^ X 1 X X 1 X /»nt,h ?/ — Vi+A-2 \/^2-l Vi-^^ 1 1 X ^\-x^ X X X sl\+x^ "Jl+X^ 1 X 1 X X. X cosecli y. — ^/x^-\ Vl-^2 The addition formulae. 260. We have cosh (m + ■y) = cos i (u ± v) = cos iu cos it; + sin iu sin ii; ; hence cosh (u ±v) = cosh u cosh ^ + sinh u sinh t; (4). Similarly we have sinh {u ±v) = sinh u cosh « + cosh u sinh t; (5). These are the addition formulae for the hyperbolic cosine and sine; they may, of course, be verified by substituting the expo- nential values of the functions. From (4) and (5) we deduce , , , tanh u + tanh v ^"°'^(^±^> = l±tanh"z.tanh. (^)' coth u coth V + 1 coth(M + «)= ,, ,, coth V ± coth u 261. Since sinh (u + v) + sinh (u — v) = 2 sinh u cosh v, sinh (m +■*) — sinh (u~v) = 2 cosh u sinh v, cosh (u + v) + cosh (u — v) = 2 cosh m cosh «), cosh (u + v) — cosh (m — i;) = 2 sinh u sinh i;, .(7). 21—2 •(8), 324 THK HYPERBOLIC FUNCTIONS we have, by changing u, v into J (m + v), ^(u — v) respectively, sinh u + sinh v = 2 sinh ^(u + v) cosh ^ (u — v) sinh u — sinh v = 2 cosh ^(u + v) sinh ^(u — v) cosh u + cosh t; = 2 cosh ^{u + v) cosh ^ (m — w) cosh M — cosh v = 2 sinh ^ (m + «) sinh ^{u — v) which are the formulae for the addition or subtraction of two hjrperbolic sines or cosines. Formulae for multiples and submultiples. 262. From the formulae (4), (5), (6), and (8), the relations between the hyperbolic functions of multiples or submultiples may be deduced, as in the case of the analogous formulae for circular functions. We find sinh 2w = 2 sinh u cosh u, cosh 2u = cosh" u + sinh" m = 2 cosh" m — 1 = 1 + 2 sinh" u, , „ 2 tanh u tanh 2m = _— -— — r^— .- 1 + tanh" u sinh 3m = 3 sinh w + 4 sinh' u, cosh 3m = 4 cosh' m - 3 cosh u, 3 tanh u + tanh' u tanh 3m = 1+3 tanh" u , , /l + cosh M . , 1 /coshM — 1 cosh^M = ^ 2 ' smh^M = ^ 2 ' /cosh M — 1 sinh m tanh |M= ^ ^^^g^^^^p^ = j-:j:^^^^ . (Series /or hyperbolic functions. 263. We have e« = cosh u + sinh m, e"" = cosh u - sinh u ; thus the series for cosh u, sinh «, in powers of u, are _ m" m* cosh M = l + 2-i + T-i+-"> . , u' u" smhM = M + ^ + ^+ .... As in Art. 233, we see that coshM = l+iJ, sinhM = M+/S, where |ii|^ //// / ^ / / r VaS Vh --£" .^ M / a. a > ^ X / W\ ^ ^ - a ^^^\> \ i\^ «? ^^ ,«' From iV draw iVQ perpendicular to OA and equal to i\^P, then ON^ — NQ^ = a^; therefore the locus of Q is a rectangular hyper- bola of semi-axis a. Now denote the area of the sector OAQ by ^ahb, then as we have proved in the last Article, we have OiV= a cosh M, iVQ = asinhM. Thus we see that, just as the ordinate and abscissa of a point P on the circle are denoted by asin^, acos0, respectively, where ^a^d is the area of the circular sector OAP, so the ordinate and abscissa of the point Q on the rectangular hyperbola are denoted by a sinh u, a cosh u, re- spectively, where ^a^u is the area of the sector OAQ. Thus the hyperbolic sine and cosine have a property in reference to the 1 The figure in this Article is taken from a tract by Greenhill entitled "A Chapter on the Integral Calculus." 330 THE HYPERBOLIC FUNCTIONS rectangular hyperbola, exactly analogous to that of the sine and cosine with reference to the circle. For this reason the former functions are called hyperbolic functions, just as the latter are called circular functions. 269. We have, from the figure of the last Article, when we consider the point Q on the rectangular hyperbola, corresponding to the point P on the circle, a tan = NQ = a sinh u, and a sec = ON = a cosh u ; therefore the arguments 0, u, for corresponding points, satisfy the relations tan = sinh u, sec = cosh u. Since sinhit tanh |m = 1 + cosh u ' , . i 1 tan sm . ■, a we have tanhiit = := 7,= ^ >, = tani^, ^ 1+sec^ 1+COS0 ^ or and u satisfy the relation tanh ^u = tan ^0. Since AO§JI/'< sector OAQ< AOAQ, we have tanh M < M < sinh u. It follows that the limits of , , when u is indefinitely u u diminished, are each unity, since cosh 0=1. 270. We have e" = cosh u + sinh u = sec + tan ; therefore u = log« (sec + tan 0) = loge tan (^v + ^0). Various names have been given to the argument 0; it is called by Cayley the Gudermannian function of u, and denoted by gd u, so that 0=gdu,u = gd~^ = log tan (^tt + \0)', this name was given in honour of Gudermann, who however called the function^ the longitude of u. By Lambert, was called the transcendent angle, and by HoiieP the hyperbolic amplitude of u (written amh u). A table of the values of logtan(^7r+ \0) for values of from 0° to 90° at intervals of 30', and to 12 places of decimals, is to be found in Legendre's TMorie des Fonctions Elliptiques, Vol. ii. Table iv. The table which we give at the end of the Chapter, for intervals of one degree, was extracted^ from Legendre's table by Prof. Cayley. 1 See Crelle's Journal for 1833. 2 See " Th^orie des Fonctions complexes.'' ' See the Quarterly Journal, Vol. xx. p. 220. THE HYPERBOLIC FUNCTIONS 381 The table enables us to find the numerical values of the hyperbolic functions of u, by means of the relations sinh u = tan d, cosh u = sec 0, using a table of natural tangents or secants of angles. Those who desire further information on the subject of the hyperbolic functions and their applications, may refer to Laisant's " Essai sur les Fonc- tions Hyperboliques " in the M4moires de la Soci^e des Sciences de Bordeauie, Vol. X., also the treatises "Die hyperbolischen Functionen" by E. Heis, and "Die Lehre von den gewohnlichen und verallgemeinerten Hyperbol-funk- tionen" by Gunther. Expressions for the circular functions of complex arguments. 271. The circular functions with a complex argument may, by the use of the notation of the hyperbolic functions, be conveniently expressed in the form a + i/8, where a and jS are real quantities. Thus sin (x + iy) = sin x cos iy + cos x sin iy ; hence sin (« + iy) = sin a; cosh y + i cos ;» sinh y (9). Similarly we find cos {x + iy) = cos x cosh y — i sin x sinh y (10). ., i / , • \ sin (a; + w) cos (a; — w) Also tan (x + 1«) = ) ^4 7 H '■' cos (x + ly) cos {x — ly) hence _sin2a; + sin2iy cos 2a! + cos 2iy , . , sin 2a; + i sinh 2v /, ,x *^^<^ + ^y>= cos2^+cosh2y (">• The inverse circular functions of complex arguments. 272. We shall first consider the function sm~^(x + iy). Let sin"' (x+iy) = a + i^, then x + iy = sin (a + 1/3) = sin a cosh /3 + i cos a sinh ^, OT x= sin a cosh ^, y= cos a sinh yS ; we have therefore, for the determination of /S, the equation a;^/cosh' j8 + y7siiih^/3= 1, or x" (cosh^ /8 - 1) + 2/" cosh= ^ = cosh^ /3 (cosh= /3-1). 332 THE HYPERBOLIC FUNCTIONS If we solve this quadratic for cosh" y8, we find cosh= /3 = i (/»= + 2/2 + 1) + l'J{a^ + y^ + iy-4,a^ ; therefore cosh jS = ± ^'^a^ + y^ + 2x+l ± ^>/a^+ y''-2a!+l, and since cosh ^ is positive, we must have, if x is positive, cosh ^ = ^'^(x + ry+f ± ^^(x-iy + yK The corresponding value of sin a is aj/cosh/S or ^'/(x + 1)" + y' + ^ V(a; - 1/ + y' ; now cosh /8 > 1 > sin a, hence we have cosh ^ = ^^(x + If + y^ + ^'J(x -ly + y" = u, sin a =iV(a; + ly + y^ - ^^/{x - 1)» + 2/' = u These are the values of cosh j8, sin a, whether x is positive or negative. The quadratic cosh /3 = m gives /3 = + log {u + '^u' — 1 j ; we have therefore sin""^ (x + iy) = k-ir + (— 1)* sin~' v ±i log {u + Vu^ — 1 } , where A; is an integer, and sin~' v is the principal value of a, which satisfies the condition sin a = v. To determine the ambiguous sign, put x = 0, then sin~' iy = kir ±i log (Vl + y- + y); hence iy=± cos kTT sin [i log (Vl +y' + yj] hence the ambiguous sign must be that of (— 1)* or sin-^ (x + iy) = k-7r + (- 1)* sin-^ v + {- 1)* i log {u + Vw"-!}. . .(12), where u = ^ ^(x+iy + y" + ^ V(^"^ny+^, and I) = i'\/{a!+iy + y' - ^'^{x-iy + f. If we consider sin"~i v+i log \u + Vm" — 1} as the principal value of sin~' (x + iy), and denote it by sin"' (x + iy), the general value is ^7r + (— l)*sin~'(a! + iy), which is the same expression as for real arguments. A special case is that of x>l, y = 0; in this case u = x,v=l, and the principal value of sin~' a; is ^tt + i log {x + Var" — 1], We know a priori that sin~'a! can have no real value when x>l. THE HYPERBOLIC FUNCTIOKS 333 273. Next let cos~^ {x + iy) = a + i/3, we have then, as in the last case, « = cos a cosh /8, 1/ = — sin a sinh /3, and we find, as before, cosh /8 = I'JJxTTf + f + \'^(x^^Vf+f = u, cos a = ■^\/(a; + 1)^ + y^- iV(a; -Vf-iry^ = v ; hence cos~^ (« + iy) = ^kir ± cosr^ v +i log {u + Vm^ - 1}. To determine the sign of the last term, we put a; = 0, then iy = cos [+ ^TT ± i log (y + 'Jy" + 1)] = + sin {+ i log (y + Vy^ + 1)} = (+)(±iy); hence we see that the second ambiguous sign must be the opposite of the first, or COS"' (oc + iy) = 2k'7r ± {cos~' v-i log (u + ^/v? — 1)} . . .(13). If cos~'t) — ilog(M + \/w^ — 1) denotes the principal value of COS"' (x + iy), then the general value is 2kir ± cos~' {x + iy). 274. Let tan~' {x + iy)'= a + i/3, then sin 2a + i sinh 2/3 ^ + '^y= cos 2a + cosh 2/3' sin 2« sinh 2/3 bence a' = „^„o^ _..„„„>, otf ' 2/ = ' cos 2a + cosh 2/3' "^ ~ cos 2a + cosh 2^ ' we have , sitf 2a + sinh' 2/3 _ cosh' 20 - cos'' 2a _ cosh 2^8 - cos 2a *'■•" ^ ~ (cos 2a + cosh 2/3)' ~ (cos 2a + cosh 2^8)^ ~ cosh 2,8 + cos 2a' 2 cos 2a , , „ „ 2 cosh 2/3 or 1 -a? — ifi = — , no . S-' aid 1 + a;' + 2/'= — , t>o , sri ■^ cosh 2/8 + cos 2a -^ cosh 2,8 + cos 2a 2a; 2"!/ therefore tan 2a = ^_^,_^, , and tanh 2/3 = ^_,_^,^^, • e'^-e-'^ 2y , ,„ «' + (2/ + l)' S-ce ^^,-^^ = ^,^^, we have e^^ = ^,^(^, fl 1 1.„ ( ^^ + (y + i)n ^=il°gj^.+(y_l).l' hence the values of tan~' (a; + iy) are given by 2a! fa;' + (v 4- 1)') tan-'(«j+i2/) =A;,r+itan-' ^_^,_y, + i^'log { ^. + (y _ iy }-(l^)- TAe inverse hyperbolic functions. 275. If sinh a = z, then a is called the inverse hyperbolic sine of z, and is denoted by sinh"' z. A similar definition applies to cosh~'0 and tanh~'^^. 334 THE HYPERBOLIC FUNCTIONS If 2 = sinh a = — i sin ia, we have iz = sin ia, or a = - sin"' (iz). Similarly if z = cosh a = cos ia, we have a = -r cos~' z ; we find also if = tanh o, a = - tan~' (iz). We have therefore the inverse 7/ hyperbolic functions expressed as inverse circular functions by the equations sinh~' z = — i sin~' (iz), cosh"' z = — i cos""' (z), tanh~' z= — i tan~' {iz). 276. By means of the expressions we have found for the inverse circular functions of a complex argument, we may find the values of the inverse hyperbolic functions. We shall however find the expressions for them independently. (1) li z = sinh a, we have e" — e~" = 2z ; solving this as a quadratic for e", we find e' = z ± Vl + z\ hence a=2ik7r + loge(z+ ^/l + z'') or 2ik-7r + lege (^ - Vl + z^), both values of a are included in the expression ik-TT + (- 1)* log (z + \/l+z^). Thus the general value of sinh~'y is ik7r+(— l)''loge(z + Vl + 1^), and its principal value is log^ {z + VT+z") ; this principal value is the one which is usually denoted by sinh~' z. (2) li z= cosh a, we have e° + e~" = 2z ; hence we find e'' = z± Va^-1, thus a = 2ikir ± loge {z + '•J s? - 1), hence 2ikir ±\ogi{z + i^ z^ — \) is the general value of cosh""'^; the principal value, which is the one generally understood to be denoted by cosh~' z, is loge {z + >Jz^ — 1). gSa -1 "1 _l_ » (3) If £r = tanha, we have ^^ = g, or ^ = t-Z^' ^©i^ce a = ikiT + \ loge ( , _ j ; this is the general value of tanh~' z, the principal value being ^loge f _ j. (4) We find for the principal values of coth~'g, sech"'^', cosech""'0, the expressions ,- lz->r\\ , H-Vl-gr2 l+VrT7^ 41og«1^^3Tj' ^°^« ~z ' ^°^« i respectively. THE HYPERBOLIC FUNCTIONS 335 The solution of cubic equations. 277. We have shewn, in Art. 117, that when the roots of the cu bic af> + qx + r= are all real, and q is negative, they- are V'-^g'sin^, V^^g sin (0 + f tt), '/-fq sin (0 + ^tt), where sin 30 == f — -^ j . We shall now shew how to solve the cubic in the case when two of the roots are imaginary. In this case, the condition 21r'^ + 43^ > is satisfied. (1) Suppose q positive; consider the cubic 4 sinh' If + 3 sinh u = sinh 3m, let a; = a sinh u, then x satisfies the equation a? + |a^ . X — \a^ sinh 3m = ; this will coincide with the cubic a?+qx + r=0, if q = \a?, /27 r^\^ T= —\a^ sinh 3m, or sinh 3m = — 4 ( ^ — j . Now the roots of the cubic 4 sinh^it + 3 sinh u = sinh 3m are sinhw, sinh(M + |7ri) and sinh (m + f 7ri), hence the roots of the cubic a? + qx + r=0 are Vlgsinhu, V|5 sinh(M+|7ri), \/fg'sinh(M + |7ri), or Vfg'sinhM, V^g (— sinh m + i V3 cosh m), where sinh 3m = — § f 27 — j . We find the number 3m from a table of hyperbolic sines, when the numerical values of q and r are given, and then sinh m, cosh u from the same tables ; thus the numerical values of the roots will be found. (2) When q is negative ; consider the equation 4 cosh' M — 3 cosh m = cosh 3m, we find, as in the last case, that ii q= — \a^,r = — \a? cosh 3m, the cubic which a cosh m satisfies is a? + qx + r = 0; thus the roots required are V— IgcoshM, V — Ig' cosh(M + §7ri), V— |g'cosh(M +f7ri), or "^ —^1 cosh u, V— ^g (- cosh 11 ± \/3 sinh m), where cosh 3m = — ^ f — 27 — 1 . Hence, as in the last case, we can 336 THE HYPERBOLIC FUNCTIONS employ tables of hyperbolic functions to find the numerical values of the roots of the cubic, when the values of q and r are given. 278. Table of values of u for given values of 6. 6 M = log,tan(j7r+^fl) 6 M=logetan(^7r+^5) 0° •0 •0 46° •8028515 •9062755 1° •0174533 •0174542 47° •8203047 •9316316 2° ■0349066 •0349137 48° •8377580 •9574669 3° •0523599 •0523838 49° •8552113 •9838079 4° •0698132 •0698699 50° •8726646 1-0106832 5° •0872665 •0873774 51° •8901179 1-0381235 6° •1047198 •1049117 52° •9075712 1^0661617 7° •1221730 •1224781 53° •9250245 1 •0948335 8° •1396263 •1400822 54° •9424778 1^1241772 9° •1570796 •1577296 55° •9509311 1^1542,346 10° •1745329 •1754258 56° •9773844 1^1850507 11° •1919862 •1931766 57° •9948377 1^2166748 12° •2094395 •2109867 58° 1-0122910 1^2491606 13° •2268928 •2288650 59° 1-0297443 1 •2825668 14° •2443461 •2468145 60° 1-0471976 1^3169579 15° •2617994 •2648422 61° 1-0646508 1^3524048 16° •2792527 •2829545 62° 1-0821041 1 •3889860 17° •2967060 •3011577 63° 1-0995574 1^4267882 18° •3141593 •3194583 64° 1-1170107 1^4659083 19° •3316126 •3378629 65° 1-1344640 1^5064542 20° •3490659 •3563785 66° 1^1519173 1^5485472 21° •3665191 •3750121 67° 1^1693706 1^5923237 22° •3839724 •3937710 68° ri868239 1^6379387 23° •4014257 •4126626 69° 1^2042772 1^6855685 24° •4188790 •4316947 70° 1^2217305 1^7354152 25° •4363323 •4508753 71° 1-2391838 r7877120 26° •4537856 •4702127 72° 1-2566371 1^8427300 27° •4712389 •4897154 73° 1-2740904 1^90O7867 28° •4886922 •5093923 74° 1^2915436 1 •9622572 29° •5061455 •5292527 75° 1-3089969 2-0275894 30° •5235988 •5493061 76° 1-3264502 2-0973240 31° •5410521 •5695627 77° 1-3439035 2-1721218 32° •5585054 •5900329 78° 1-3613568 2^2528027 33° •5759587 •6107275 79° 1-3788101 2^3404007 34° •5934119 •6316581 80° 1 •3962634 2^4362460 35° •6108652 •6528366 81° 1^4137167 2^5420904 36° •6283185 •6742755 82° 1^4311700 2^6603061 37° •6457718 •6959880 83° 1 •4486233 2^7942190 38° •6632251 •7179880 84° r4660766 2-9487002 39° •6806784 •7402901 85° 1 •4835299 31313013 40° •6981317 •7629095 86° 1 •5009832 3^3646735 41° •7155850 •7858630 87° 1^5184364 3-6425334 42° •7330383 •8091672 88° 1-5358897 4-0481254 43° •7504916 •8328406 89° 1-5533430 4^7413488 44° •7679449 •8569026 90° 1^5707963 00 45° •7853982 •8813736 EXAMPLES. CHAPTER XVI 337 EXAMPLES ON CHAPTER XVI. 1. Prove that 8 sinh nx sinh^ ,r = 2 sinh (ra + 2) a; - 4 sinh me +2 sinh {n-2)3!. 2. If cos(a+i3) = cos(^ + isin) cos (a + i^) = 1 , prove that tanW (^ cosh^ |3 = sin^ o, and tanh" /3 cosh^ = sin^ ft 4. If tan.y=tanatanhj3, tan2=cotatanh;3, shew that tan (y + 2) = sinh 2/3 cosec 2a. 5. Reduce e™ '"+*^' to the form A+iB. 6. If logeSin(5+i0) = a+i/3, shew that 2 cos 26 -=2 cosh 20 - ie^, and cos(e-0)=e^oos(fl+^). 7. If tan {x+M/) = sm{u + iv), shew that coth « sinh 2y = cot m sin x. 8. Express {cos {6 + icj)) + i sin (^ - 80)}"+'^ in the form 4 + iB. 9. Prove that ,/tan25+tanh2d)\ , , , /tanfl-tanhdN ^ ,, ,^ , , ^"""\ tan2g-tanh20 J+^"° ' (t^^m^j=*"° (°°*^'=°*^*)- 10. If w=cosa — J cos3a+^cos 5a— , v= sin a— J sin3a+J sin 5a— , prove that cot 2u tanh 2v = tan a. 11. Prove that the sum of the infinite series cos i6 cos 85 cos 126 "^~Tr"''~8T~ 12! ■'' is J {cos (cos 6) cosh (sin 6) + cos (sin 6) cosh (cos 6)}. 12. Prove that ''="(-l)"sin(2m + l)refl ^^t^r , „n 1. , • .,>. „!o (2^)! ^in>»/ =2^^^{cos(cospg)cosh(smy5)}+cosa, where a is the unit of circular measure. 13. From Enter's theorem sma; , , = cos *x cos ix cos \x. . X 2 4 8 deduce that (1) -i-=^ + l-^.+l^, + .^-^+ loge.« x-\ 2 1+a;f 4 1+«4 8 1+a;' 1 111111 (2) -5=cosech2^+-5sech2-.»+-5secy7»+55sech2-.!i;+ ^ ' 0^ 2^ 2 4^ 4 8'' 8 H. T. 22 CHAPTER XVII. INFINITE PEODUGTS. The convergence of infinite products. 279. Let Zi,Zi,...Zn,... be a sequence of real or complex numbers formed according to any prescribed law, and consider the product Pn=ZjZ^... Zn of the first n of these numbers. If Pn converges to a definite limit P, different from zero, as n is indefinitely increased, P is said to be the limit, or limiting value, of the infinite product ^i^^^g ■•• % •■•> and that infinite product is said to be convergent. It is convenient to exclude the case of those products for which Pn converges to zero firom the class of convergent infinite products. If P„ = I P„ I (cos dn + i sin ^»)> where | P„ | denotes the modulus of Pn, it is necessary and sufiicient for the convergence of the infinite product that both | P„ | and dn should converge to definite values as « is indefinitely increased. In case | P„ | increases indefinitely, as n iS' i'fidefinitely increased, the infinite product is said to be divergent. In other cases in which the product is not convergent it is said to oscillate, but oscillating products are frequently spoken of as divergent. The necessary and sufficient condition that the infinite product z^e^ ...Zn-.. should converge to a definite value (other than zero) is that, corresponding to each arbitrarily chosen positive number e, an integer n can be so chosen that | Zn+iZn+2...Zn+r — 1 1 < e, for all values 1,2, 3, ... of n... To shew that this condition is necessary, let us assume that P„ converges to P, a number different fi:om zero. All except a finite set of the numbers j Pj | , | Pj | , . .,. | P„ | . . . are greater than \P\—7], where tj is an arbitrarily chosen positive number such that j P | — »? > ;, also none of these vanishes, there- fore tliere exists a positive number k which is less than all the INFINITE PRODUCTS 339 numbers j Pj | , | Pg | , ,. . . | P„ [ ,. . . Since P„ converges to a definite limit, n maybe so chosen, corresponding to e, that [P«+,- — P» | < ke, forr-=l,2,3,.... Hence we have | Zn+iZn+z • • • Zn+r - 1 1 < A;e/ 1 ^i^a . . . 0„ j < e, and therefore the condition stated is necessary. To shew that the condition, is sufficient, let us assume it to hold. For an assigned value of e, n can be so fixed that Zn+iZn+1. • ■ ■ Zn+r = 1 + Rn.r, where | /3„,^ | < €, for r = 1, 2, 3, . . . . We have then P,i+,. = P„ (1 + pn,r), and therefore | P„+^ | < | P„ | (1 + e), for all positive integral values of r ; it follows that all the numbers i-Pi|. I Pal, ••• |Pm i ••• are less than a fixed positive number \. From I Zn+iZn+a • • ■ Zn+r — 1 1 < ^ , we have I Pn+r — Pm I < \6, for r = 1, 2, 3, ... , and since Xe may be chosen as small as we please by choosing e small enough, we see that P„ must converge to a definite limit. A convenient method of considering the convergence of the infinite product z-^z^ . . . 2„ . . . , is to consider the series logeZi + lQgeZ^+...+logeZn+ ■■. If this series is convergent the infinite product converges to a value other than zero, and conversely. If the infinite product converges to zero, the series diverges to — oo , and for this reason, as before, we exclude this case. To prove that the convergence of the infinite series and of the infinite product are equivalent, we observe that the necessary and sufficient condition for the convergence of the series is that n can be so determined, for each e, that \\oge(zn+iZn+2 ■■■ Zn+r)\ or \loge(l+pn,r)\ for r = 1,2,3, ...; therefore the ipfinite product is convergent. Con- versely let us assume that n can be so chosen that \ pn,r\< e, for r = 1, 2, 3, .... It has been shewn in Art. 249 <" that if | ^ | < 1, |log,(l+^)|<|^|(l + ij4q7|)' therefore . |.loge (1 + pn,r) | < e ( 1 + f T^) ' 22—2 340 INFINITE PRODUCTS or log, {Zn+iZn+2 •■■ Zi^r) < V, provided e {l + i j-^^j < V, and if rj is prescribed, e can be so determined as to satisfy this condition. Therefore the condition of convergence of the series is satisfied. 280. Suppose MdMj, ... m„, ... tobe a sequence of real positive numbers each of which is less than 1 ; it will be shewn that the infinite products □o (l+u,)(l+u^)...(l+Un)... or 11(1 +m) 1 00 and (1 — iti) (1 — M2) . . . (1 — Un) ... or 11 (1 — m) 1 both converge, or not, according as the series ttj 4 Mj + • . . + m» + ■ • • is convergent or divergent. Since (l+iti)(l +M2)...(1+M„) >1 +1*1 + 1*2+ ... +M„, it is clear that the product II (1 + m) diverges if the series 1I1 + U2+ ... does so. Also hence if 2m diverges the product (1 — «i) (1 — ttj) . . . (1 — m») con- verges to zero, and is therefore considered as non-convergent. Next, if Sm converges, let e be an arbitrarily chosen positive number less than 1, then n can be so chosen that **«+! + 1^n+i + . . . + Un+r < ^ j for r = 1, 2, 3 We have, as in Art. 226, (1 - M„+i) (1 - Ur^) ... (1 - Un+r) > 1 - (%+i + M„+2 + . . . + Un^) > 1 - 6, and therefore | (1 — u^+i) (1 — Mb+b) ... (1 — Un+r) — 1 | < e, and thus the condition obtained in Art. 279 for the convergence of the infinite product 11 (1 — it) is satisfied. Also (1 + M„+,) (1 + M„+2) . . . (1 + Un+r) (1 - M«+i) (1 - Wn+j) ... (1 - Un+r) 1 " «' and thus 1(1 + «„+])(! +M„+j)...(l + M^,)-l|< --^. If ^ be INFINITE PRODUCTS 341 arbitrarily assigned, we can determine e so that 6/(1 — e.) < ?;, and thus n can be so determined that I (1 + M„+i)(l +M„+2) ... (1 +M„+r)- 1 1 < 1?, for r = 1, 2, 3, ... . Hence the product 11 (1 + it) is convergent. It is clear that the condition that Mj, itai ••• w„, ... should all be less than 1 can be replaced by the wider condition that all except a finite set of these numbers are less than 1. For we can remove a finite set of factors in 11 (1 + u) or in 11 (1 — u) without affecting its convergence. 281. Next let us consider the infinite product (1 + Mi)(l + M2) ... (1 +M„)..., where Mi,t<2, ... m», ... are complex numbers. We shall shew that if the series of moduli of Mj , M2, ...«„,..., i.e. the series, |Mi| + I Ma|+ ... + |m„|+ ..., is convergent, then the infinite product is also convergent. In this case the infinite product is said to be absolutely convergent. We see that |(1 + M„)(1 +M„+i) ...(l + M„+r)-l j = (l+|M»|)(l + |M»+i|)---(l + |M»+r|)-l, since the modulus of the sum of any set of numbers cannot exceed the sum of their moduli. Now if the series 2 | m | is convergent, the infinite product II (1 + 1 it | ) is convergent, in accordance with what has been shewn in Art. 280 ; it follows that, corresponding to any assigned e, n can be so determined that (1 + I W„ |)(1 + i W„+, I) ... (1 + i Un+r I) - 1 < e> for r = 1, 2, 3, ... . It follows that 1(1 +M„)(l + M„+,)...(l + M„+^)-l |„ = \b„\, and the upper or lower sign in + i<^„ is taken according as 6„ is positive or negative. If 7] be an arbitrarily chosen positive number less than unity', we have n >{l—7j) tan (^„, for all sufficiently large values of n ; and therefore 20m cannot converge. It follows that 11 (1 + ibn) cannot converge, although 11 (1 + bn')^ will converge in case the series Xbn" is convergent. It is clearly sufficient for the validity of the theorem that all the numbers 6„, with the exception of a finite set, should be of the same sign. If be a complex number a; + iy, and the numbers a^, O2, ...a„,... be all positive and such that 2a»i is divergent, the product n (1 H- Kn^) is certainly divergent if the real part of z is positive. For the product of the moduli of the terms 1 + a„^ is greater than n (1 + anx), and this is divergent when as is positive. The product (l + ^l (^ + m)-"(1"' — s) ■••) when .r.isareal number, does not converge in case p g 1, but converges if ^> 1. For 2 — is divergent when p^l, and is convergent il p>l. The product (i+t) (1+5 )... (i + - j... is certainly divergent if the real part of z is positive, and it does not converge if the real part of z is zero. When the real part of z is negative the product converges to zero, and is therefore considered as non-convergent. For loge(lH — ) = ^"aO^+VnX where | ijn | is less than a fixed number for all sufficiently large values of n ; the real part of 2 logj ( 1 H — j consequently diverges to — qo when the real part of « is negative, whence the result follows. This depends on the facts that 2 - is n divergent and 2 —^ convergent. INFINITE PRODUCTS 343 Expressions for the sine and cosine as infinite products. 282. We shall now find expressions for sin x, cos x as infinite products involving the circular measure x ; we first suppose x to be real and positive. We have - . X . X + TT sma; = 2smT-sin^-Tr — 2 2 „„ . X . x + ir . x + 2-jr . x + Stt = 2'sinTSin — - — sin — ^ — sin — -. — , 4 4 4 4 and continuing this process, we obtain n„ , . X . x + TT . x + 2ir . x + (n — l)Tr sm X = 2"~' sm - sm sin ... sin ^^ — , n n n — n where n is any positive integral power of 2 ; hence _„ , . a; a; / . „ TT . „ a;\ sm » = 2"""^ sm - cos - sm'' sin^ - n n\ n n) / . „ 27r - „ «\ / • !, '^ - 27r ■ „oi\ sin'' — — sm^ - ... sm^ — ^ sm^ - : \ n nj \ 2n n) since L sin x cosec - = w, we have IT . 27r n = 2"-^ sm'' - sm" — ... sitf „ n n In hence, by division we find sm" — sma; I , n ('sin^ ^ -27r 2n ^n — 2'ir . X X \ • „ TT nsm — COS- \ sitf — , , _ , . „„, n n \ n/ \ n / \ 2n This is the particular case of the theorem (19), of Art. 87, when M is a power of 2. We might, of course, assume the general theorem. Let |(n. — 2) = r, then if m be any number less than r, we have / sin" sina!=wsin-cos- 1 n n\ . „7r where iJ = I 1 — 344 INFINITE PRODUCTS Now, n being taken greater than 2a;/7r, m may be so chosen that a; < («i + 1) TT, then R is positive and less than unity ; also, as in Art. 226, B is greater than , . ,x { m + lTT rTT 1 — sin^-^cosec'' - + ... + cosec^ — n ( n n Now we have shewn in Art. 96, Ex. (1), that if ^ < ^tt, , sin 6 sin i^r then — ^— > , , „«7r 'n? . . ■ ,x x^ hence cosec" — - < t— „ ; also sin' — < — ; , n 4/3" n v? hence i? > 1 — -r 4 1 (m + 1)^ (m4 2y r\ - - f ^ 1 1 ] ^ 4 tm(m + l)'^(m+l)(m + 2)"^'""'"(r-l)rj ' a? fl _\\ -, _^ 4 V m r*/ 4wi ' a? 6a? Since iJ is between 1 and 1 — j— , we may put i2 = 1 — - where 6 is between and 1 ; we have then / sin? -\ I sis? — . X xl « W T '>^ sma; = nsm — cos — I 1 — 11 1 s— V sm" - / \ sm= 71/ \ n '^'^ \r, ea?\ SVD? n l(-s). where m is any number less than n, such that a; < (m + 1) tt. Now let n become indefinitely great, m remaining fixed, we have then, since each sine in the product may be replaced by the corresponding circular measure, and since cos - has the limit unity, »in.=.(-S)('-.-e^)-('-;^)(-a. where O^ is the limiting value of 6, when n is indefinitely in- creased, and is thus such that S ^i S 1. INFINITE PRODUCTS 345 Now by increasing m sufficiently, we may make the factor 6 x^ 1 — -J— as nearly equal to unity as we please, hence we have the expression sin a; -(l-S(^-2t-)0-3-t.) W. ■for sinaj as an infinite products The restriction that x should be positive may clearly be removed. 283. From the formula (17), in Art. 86, if n is even, sin" — \ / sin" - \ / sm" - II n M ^ n \ I , n cosa! = l 1 — ■ we may shew that '" V W V 3V"/ ■••V i^^Ij^J V'^j' cosa;! where m is any finite number such that 2x < (2m + 1) rrr, and is between and 1 ; hence we obtain for cos x as an infinite product, the formula /, 4a;''\ /, 4a;2 \ /_ 4 + cos" -.x tl (1 — — ) , - n r=i \ rVV' and the two expressions for cos x shew that ±cos.<±n (i-^-2^,-3^). and + cos a; > + cos" - 11 ( 1 — «*■?.*"/, 435" ;os" X .)■■ n r=i \ 2r- l|"7rV now we know that cos" - = 1 — 6„, where e„ is a number which con- n verges to zero as n is indefinitely increased ; we have therefore ™...(i-5)...(i--^)(i-9,). INFINITE PRODUCTS 347 where 9n, On are numbers which converge to zero when n is in- definitely increased ; we thus obtain the expressions (1) and (2). If we had used the formulae x=nsui- n I 1 I, ^ -1 V sin.!!!:/ ^ n r=i I . 2r-W ), COS ^ = COS - which hold for an odd value of », and the formulae sin^ = cos''- . tan- n I 1-- ^ -1 \ ^^^rn n. so (tan2- \ 1 ^- I tan^?!:ziz:/ cos;j;=cos' n T=i \ ^ ,2; 2?l obtained from them, similar reasoning would have led to the same results. 285. We shall next consider the case of a complex variable z = x-\-iy; we find, as in Art. 282, I sin' - \ / sin' sin^ = wsm-cos-( 1 11 1 =^ I... I 1 ^ l-B, n n\ where R = \ 1 ■ „in \ sm=i — \ n where n is an even integer, and r = ^(« — 2) ; we have to determine limits for the value of R. Let p denote the modulus of sin-, then as in Art. 281, since the modulus of the sum of any numbers is less than the sum of their moduli, we see that the modulus of jR — 1 is less than I . m + Itt I -2 'TJH I 343 INFINITE PRODUCTS Now we know that e^*"' > 1 + Ap', if A is any positive number, hence the modulus of i? — 1 is less than 2 I n^w-^^'y. ' I p' ( coseo'' h . . . + cosec' e \ " ,.^\ «/_1 and this is less than , '' " (m m + 1 m + 1 m+2 rj -i or than therefore the modulus of i2 — 1 is less than g W '•/-I, or thane or than e "* — 1 ; m _ thus the modulus of i? — 1 lies between zero and e "* — 1. Now y. p^ = sin" - cosh" ^ + cos" - sinh" ^ = sin" - + sinh" ^ , "^ »i w n n n n hence the limiting value of /3"n" is a?-\-if; therefore the limit of the modulus of i2 — 1, when n is increased indefinitely, lies between zero and e *™ — 1 ; now e *™ may be made as near unity as we please, by taking m large enough, thus | i2 — 1 1 may be made as small as we please, by taking m large enough. When n is in- definitely increased, each of the sines in the expression for sin^^ becomes ultimately equal to its argument, therefore =in. = .(.-5)(l-^)(l-3e-.).... The formula COS0: may be proved in a similar manner. 286. We remark about the formulae (1) and (2), that they satisfy the condition of absolute convergency given in Art. 281, since the two series — S - and — r 2 r^ ^r- are convergent. tt" 1 n" tt" 1 (2r — 1)" ° INFINITE PRODUCTS 349 Each quadratic factor in either product may be resolved into two factors linear in x, thus which may be written in the forms sva.x=xYl 1 H (3), cosa!= n ( 1 + I (4.). -00 ,V 2r-\TrJ In these latter forms, the products are semi-convergent, since the products 1 \ rirj' 1 V rirj' i V 2r-l-n-l i \ 2r-\ir} 00 1 GO T are divergent, the series S - , 2 „ r l^eing divergent. A semi- convergent product has the property analogous to that of semi- convergent series, that a derangement of the order of the factors affects the value of the product ; we are entitled to consider the formulae (3) and (4), as correct, only when it is understood that an equal number of positive and of negative values of r are to be taken; thus (3) and (4) must be regarded as an abbreviation of the forms '"' I X \ " / 2a! \ sin a; = «-£„=„ 11 1 H , cosi»=i„=a> 11 (1 + ■ -« \ «""■/ -n\ 2r-lv/ 287. It has been shewn by Weierstrass^, that the divergent product '(>+!) ('-If) (i- si)- may be made convergent, by multiplying each factor by an exponential factor; thus the product is absolutely convergent. ' See the Abhandlungen of the Beilin Academy, for 1876. 350 INFINITE PRODUCTS We have, as has been shewn in Art. 230 '"', e~»-= 1 - — + ^4— " (1 + "»)' where | m„ | converges to zero as n is indefinitely increased ; there- fore, if 6 be an arbitrarily chosen positive number, | m„ | < e, for all values of n which exceed some fixed value dependent on e. We have now The series of which the general term is in" TT" is absolutely convergent, since the series X-^, 2— are convergent, and I M„ I < 6, 1 1 + M„ I < 1 + 6, for all sufficiently large values of n. Therefore, in accordance with the theorem proved in Art. 281, the infinite product of which the general term is ^-2^»(^-''"> + 2^(l+"»>' orflH ]e '*^,is absolutely convergent. Hf(z) denote the limit of the absolutely convergent product n f 1 + — ^ e~^, and /(- z) that of H (l - -— ") e«^, we have IS The above result may be employed to evaluate the limiting value of the expression *w=(i-3(i-i)...(i-i)(i+j)(i.^) •(■ rmrj when m and n are made indefinitely great, but so that their ratio has a definite finite limit. If s„ denotes the series 1~^ + 2"' + 3~^ + . . . + n-^, we see that sm z = z L(p (z) . e" ; INFINITE PRODUCTS 351 now it is well known that the limit, when n is infinite, of s„.— log^ n is a finite number 0'5772156..., called Euler's constant, hence the limiting value of Sn — Sm, when m and n are infinite, is that of losfe— ■ We have therefore, L<\){z)=k' ——, Z sm 2f where h = Lmjn, and the value of L^ (ji) is only when m and z n become infinite in a ratio of equality. 288. The formulae (2) or (4), for cos x, may be deduced fi:om (1) or (3), by means of the formula cosi»= sin Ixjl sin x. ■: We have sin 1x n " /, -lx\ / „ ^ /, x\ ^r-. — = 2a;ni + — /2a;ni + — , the factors in the numerator, for which r is even, cancel with those in the denominator, hence if we consider the product in the numerator to be the limit of 11 ( 1 H ) , and that in the -2b \ rvj denominator to be the limit of II ( 1 H ) , when n is infinite, -n \ fir) * / 2a; \ . we see that cos a; = 11 ( 1 + ^ 1 which agrees with (2) or (4). The condition of convergence of the products shews that taking In instead of n, in one of the products, does not affect the limiting value of that product when n is indefinitely increased. 289. We may deduce the product formula for sin a; fi:om that of cos X, or vice versa, by means of the formulae sin a;=cos {\-k — x), cos a; = sin (^TT — x). From the formula (4) we have sin a; = n 1 1 + w-2x\ " /2r-7r-2a;' 2r - Itt " /Irir — 2x\ -ooV2r-l7r/ " 2r- -^/i « = n ^ ij-.a;!! 1 _„ 2r- 1 _oo \ rir. where the factor x corresponds to r = 0; taking the limit of sin X °° 2/* for. x=0, we see that we must have 11 ^ r = 1. X _oo2r--l °° / X \ hence sin a; = a; IT 1 1 ) . -00 V rirj 352 INFINITE PRODUCTS 290. The product formulae for sin x and cos x may be easily made to exhibit the property of periodicity which those functions possess. Let fi.) = xU[l^£), then /(. + .) = (.-H.)(l + ^)(l-H^)... /j X \ w+1 \ n-lirJ n mr — x now when w is indefinitely increased, we have Lf {x + ir) = ~ Lf {x), which is the equation sin (a; + tt) = — sin a; ; the formula (4) may be made, in a similar manner, to exhibit the property cos (x + 7r) = — cos X. The function sin a; vanishes when a!=0, ±ir, +27r..., and these values correspond to the factors ^k, 1± — , 1±;7— ... in the formula (3); also it has been proved in Art. 235, that sin x does not vanish for any imaginary value of X, thus if it be assumed that sin x can be expressed in the form of an infinite product A ~ , ^^, the values of a, 6, c... must be 0, IT, — TT, 2jr, — 2n-.... The value of A is then determined by putting x=0, and using the theorem L = 1, we obtain the formula (1) or (3). This is of course worthless as a proof of the formula, since we have no right to assume without proof that sin x is capable of expression in the required form. 291. It is important to notice the forms which the formulae (1) and (2) take in the case of an imaginary argument iy; we obtain in that case, the expressions for sinhy, coshy as infinite products siBh,.j,(l+9(l + J^)(l + ^) (5), INFINITE PRODUCTS 353 The formulae (1), (2), (5), (6) were first obtained by Euler, by means of the identity s2m_l=m(02-l) n 1-20 cos — +22 putting 2=H — it becomes 2*+?:. 1+ li: 1 + — 2m sin-r— if m be now made to increase indefinitely, this becomes 71=00 / r'^ \ which is the formula (5). This evaluation of the limit requires an exact investigation, as in Art. 285. The formula (1) was deduced by changing x into ix. The formulae (2), (6) were obtained in a similar manner, from the expression for z^" + 1 in factors. Examples. 292. (1) Investigate Wallis' expression, for ir. In the expression for sin a; in factors, put a;=^ir, we have then the approximate formula ■=I(-i)(-i) (-i). where n is large ; this may be written 2.4.6...2ra Vi-(2« + l) = j-3^^2^_j^, which is Wallis' formula. (2) Factorise cosh y — cos a, cosx — cos a. We have cosh y - cos a = 2 sin ^ (a + ly) sin ^ (a - ii/) =«--«?('-m(-'-sf}^ putting y=0, l-coaa=ia^n\l-^^^A , hence coshy — cosa 1— cosa therefore Ci V'^ay^V'^2nn+a)v 2n,r-a)\^ 2n,r + a) {} '^ 2nw - a) ' :oshy-cosa=2sinHa.(l+g)n{l+^^i^}{^ + (2^^}- H. T. 23 354 INFINITE PRODUCTS Writing ix for y, we have oosii; — cosa=2sin*Aa. I 1 — -„ ) n -^1 -y^ ; — x-A il — 75 r,} ■ (3) Prove that = J n- — toJi-1 ( tank -j^.cot-^j. 00 f (x-\-zi/*)^) We have sin(a; + i2/) = (;!;+iy) II U— ^^ — 22" f • ^^^^^S logarithms, this becomes 00 f nj2 ^^ «/2 9 ^''W I log(sina-coshy + icos:!;sinhy) = log(«+ty) + 2log jl 22" *'• ~z2f ' equating the imaginary parts on both sides of the equation, we have tan"i(tanh«cota;) = tan~i - — 2 tan~i-s— = — ^= ^ ; let x=y = ll^2, we have then » 1 / 1 1 \ 2 tan-i -5— „=!:«• -tan- 1 tanh -75 . cot -^ . Representation of the exponential function by an infinite product. 292"'. A representation of the exponential function e', in the case in which | 2^ | < 1, has been given by Mathews ^ Let us assume that z is the limiting sum of a convergent series S kn loge (1 + «"). We find then that ki = 1, and for w > 1, where S is any proper integral factor of n, and S' = n/B, each such value of 8 giving one term. From this it follows that nk„ = t{- If Sks = 2 (- !)»'« Bh ; and the values of all the numbers /<;„ are to be determined from the set of equations of which this is the type. It can be shewn by induction that (1) If n = 2'», then A„ = 1/2. (2) If n is the product pip^.-.p^ of fi different odd primes, then kn = {-lYln. 1 Proceedings of the Cambridge Philosophical Society, Vol. xiv. p. 228. INFINITE PRODUCTS 355 (3) Iin=2"'p,p^...p^, then K = (-iy-2«'-^ln. (4) If n has the square of an odd number as factor, then That, with the values of A;„ so determined, the series 2A;„log,(l + 0") converges when | «: j < 1 is easily seen. The exponential function e^ is consequently represented, for all values of z such that | ^ | < 1 , by the infinite product n (1 + 0»)*n = (1 + ^) (1 + zjl' (1 + z^)-'l' (1 + zj" ...; 1 or, since 1 = (1 - zyl^ (1 + z^ (1 + z^f'^ . . ., we have by division -(i^:rnG-^r" i^^i< 1, where p is the product of /t unequal odd primes, and all values of p of this form are to be taken. Series for tlie tangent, cotangent, secant, and cosecant 93. Sin multiple of TT, 293. Since sin0 = ^II (1 — 'r-— ), we have, when z is not a 1 \ n'lT^j log, sin ^ = log, 2r + 1 loge (l - ~^ . Let A be a positive real number ; changing z into z + h, and .subtracting the two expressions, we have sin (z + h) loge smz h mr Now, employing the theorem given in Art. 249'^*, we have h\ h ,h;- = loge f 1 + -) + 2 {log, f 1 + -^) + log, (l + -^ ^ \ zj n=i (. " \ z-nirj ° V z + 'i log, 1+ = 4; r-^{l+Vn), ° \ z — mr) z — n-TT \z — nirf ^ 1 /, h \ h 1 ^^ /-, log, 1 + = ^ A-T ^ (1 + Wn), ^'\ z + nirJ z+rnr ^{z + niry^ ' 23—2 356 INFINITE PRODUCTS where | Vo | , | ''n | , | w„ | all converge to zero when h is indefinitely diminished. Moreover, z having any fixed value which is not zero or a positive or negative integral multiple of tt, for all sufficiently small values of h the numbers | t'o | > i ''i 1 1 1 ''a | • • ■ ^^^ I Wil i|w2 1 ... are all less than an arbitrarily chosen positive number e, since the moduli of 1 | , | a — jitt | , | ^^ + wtt | are greater than some fixed number independent of n. We have now 1 , sin (z + K) T loge ^ - h ° smz l + Wn fl 1^/1 , \"1 _L V r 2« l + Vn ,, {z + n-irf where the series on the right side converges when z is not a multiple of tt. Let us assume that z is such that (r — l)7r< \z\< r-ir, where r is a positive integer ; then if z^lr'ir'' = 1; < 1, we have | z ^'jri'ir^ S 7), for all values of n which are ^ r. We have now 1 nV^-^" provided n'S.r; it follows, since the series of which n~^ is the general term is convergent, that the series of which n^Tr" — z^ is the general term is absolutely convergent. Since the two series of which the general terms are 2^ 2g 1^ l + Vn 1, 1+Wn z'^-n^Tt^' z^-n^ir' ^ {z-nirf ^'"(z + nirf are both convergent, it follows that the series of which the general term is {z - nirf * (^ + nirf is also convergent. If h be sufficiently small, the modulus of this general term is less than now 1 - WTT I > WTT - 1 1 S nTT- (r- + 1) TT, hence 1 ^ 1 INFINITE PRODUCTS 357 where n>r + 1, and it then follows that the series of which the general term is -. — — r^ is convergent. Similarly the series of which the general term is -. is convergent. We now see that the modulus of the sum of the series of which the general term is ^h-. ^ + ih, ^. does not ° {z- niTf ^ {z + mrf exceed a number ^h (1 + e) A (z), where A (z) is a positive number dependent only on z ; this modulus diminishes indefinitely as h is indefinitely diminished. It now follows that „ . sin (z + h) , . , , ,,,,-,,., oince -. = cos/i + sin/icot0=l + /icot5;(l + f), sm z \ = /) where | ^ | converges to zero with h, we have il0ge^^^^ = Jl0g.{l+Ac0t.(l + ?)! =cotz(i+o{i+n where | f ' | converges to zero with h ; hence _ 1 , sin (z + h) L T logs — ^ — cot z. 7t=o ft sm ^ It has now been shewn that when z is any real or complex number which is not an integral multiple of tt, cot^^ is the sum of the convergent series 11111 /^N z z + tr z — ir z + zir z—zir or i + 20 2 , \ , (8). In the form (7) the series is semi-convergent, and in the form (8) it is absolutely convergent, except for z = 0, + tt, + 2-ir, ..., for which values the series is divergent. In order that the student may appreciate the necessity for the investigation in the text, we remark that iif{z) be the sum of an infinite convergent series Ml (0) + M2 (2) + . . . + Mn (^) + • ■ •) we are not entitled to assume that R=0 h 1 ft=o A 358 INFINITE PKODUCTS Suppose R^ {£) is the remainder of the series after m terms, then f{z)=Ui{z)+Ui{z) + ... + u,^{z)+R^{z), f{z+h) = Ui{z+h) + U2{z+h) + ...+u,n{z+h) + Rm{z+h); \\ ftn p.ft ^ /(z+^)-/(z) ^g ^ t<,(z + A)-«^(z) ^ ^ R^(z+h)-E„,{z) . A 1 ft=0 «■ 7l=0 now since the given series is convergent, Rm(z), Rm{z+h) become indefinitely small when m is indefinitely increased ; it does not however necessarily follow that L -^^ 1 ^^-^ does the same, and it is only when it does that h=o "' we are entitled to employ the derived series to represent the derived function of /(a). If for example Rmi^) were of the form — sin»i«, we should find which does not converge to zero when m is indefinitely increased, but oscillates between the values + A. 294. From the expression we obtain, by a method similar to that of the last Article, the infinite series 1111 - tan 2 = , , + -^ — J— H — r- + r— + . .. 1 1 '^z + ^{2m-l)'7r'^z-^{2m-l)-7r'^ ^^^' 00 I or ta,n z = Szl, -T^ -^ HOV the series (9) is semi-convergent, but (10) is absolutely convergent for all values of z except + Itt, ± f tt .... 295. We may find a series for cosec z by means of either of the formulae cosec z = cot^z- cot z, cosec z = ^cot^z + ^ tan J z ; using the first of these formulae, we find on substituting the series for the cotangents cosec z = f- ^ 2 2 2 ] 1 ^ 1 ^ 1 ^ 1 ^ 1 1 z z + TT z - TT z + 2ir z-2ir z + 3ir z INFINITE PRODUCTS 359 hence coseca^ ^1__1 1_ 1 1 _J. 1_ z z + TT z-TT z'+'lir'^ z-2-jr z + ^w z-'^ir^'"^ ^' or cosec^ = -+.S;Ai — ^"T^ (12). In the formula (11), change z into z + ^ir; we have then secz-^Z (2r-l)=,r^-4^=> ^^^>' this series, when r is large, has its general term approaching the value „ _ , therefore the series is only semi-convergent. The cotangent and tangent series may also be obtained as follows : Using the expressions for sin (2 + A) and sinz as infinite products, we find by division sin(g+A) ^ / h\ f n^-z^-h^-U2 \ / 2^n^-z^-h^-Uz \ sin« V 2/ V T2-a2 )\ 2^n^-z^ )'"' if we assume that the product on the right-hand side can be expanded in powers of h, by multiplication, and put the left-hand side in the form cos A -I- sin A cot z, then expand in powers of h, and equate the coefficients of h on both sides of the equation, we find 1 20 2« COt0 = -+ ^ 2+ 2 02 2 + (8). Z Z'~n^ 02_22^2 \ / The justification for our assumption that the infinite product may be arranged in a series of ascending powers of h, the coefficients of which are the infinite series obtained by ordinary multiplication, would require an investigation of the conditions that such a process gives a correct result ; to do this would however require certain general theorems for which we have no space. The tangent series may be obtained in a similar manner from the infinite product cos (zH- A) _ /7r2-4z2-4^--8A0\ / 3 V^ - 40^ - 4^^ - 8fe \ COS0 ~V ^2-402 J\ 3V^-4s2 /•••• If the cotangent of z is expressed in the form / 402 \ / / _ _02_\ \ 2m-i|Vv ^ \ '"'■^■^y and this expression be transformed into partial fractions, the denominators of which are the factors in 011 ( 1 ^—A , we should obtain the series (8) ; a similar remark applies to tan0, seo0, coseca The series have been obtained ^ by Glaisher, directly, by carrying out this transformation. 1 See Quarterly Journal, Vol. xvii. 360 INFINITE PRODUCTS Expansion of the tangent, cotangent, secant and cosecant in powers of the argument. 296. We have shewn in Art. 293 that 1^ TO 2^ where | Rm \ is a number which may be made as small as we please by taking m large enough. Now if the modulus of z is less than ttt, we have hence if we suppose that the modulus of z is less than tt, we may expand each of the fractions l/(r^7r'' — z') in this manner, and we have, arranging the result in powers of z, as we are entitled to do since each of the series is absolutely convergent, 1 2z/l 1 1\ 2ir»/l 1 IN "°*^ = ^-^b + 2i+- + ,^J-^b + 2i+-+^J-- let /San denote the sum of the convergent series 1271 + 2"» vv^ + ■■■, then (SsK = Y^ + Hi^ + . . . + — ^ + 62„, where em is a number which may be made as small as we please, by making m large enough ; we have then 1 2z y 2^3 2z^- COt0 = j/Sa 7S4-... z:r- S^-.- „ 2z 2^ 22'»^i We see that ea > 64 > ee ..., hence the modulus of 2z 2^ -aea + -e4+... is less than e^ multiplied by the sum of -^-r^ -\ — !— r-L + . . . which is a convergent series, since mod. z<-ir, therefore the modulus of INFINITE PRODUCTS 361 S — ^ e^n may be made as small as we please, by making m large enough. We have therefore the infinite series for cot z, ^o^' = ^--A-:^s,---s,- (15), which holds for all values of z such that mod. ^ < tt, and in particular for all real values of z between + tt. From the theorem m ^ tan.= 8S ^^^_^^,^,_^^, + i?.-, we may obtain, in a similar manner, the series for tan z in ascend- ing powers of z. This series may however be deduced from (15), by means of the identity tan z = cot 2 — 2 cot 2z ; we find 2(2^-1)^ „ 2(2^-1)2= „ 2(2'-l)z' ,,„, Unz = -^ — ^- S, + ^ 8, + -^ ^-!— Se + ...(16), IT TT TT which holds if the modulus of z is less than ^tt, and in particular for real values of z between + ^tt. Substituting for cot ^2^, cot^^ their values from (15), in the formula cosec z = cot ^z — cot z, we have 1 ,„ ^.z„ 23-1 ^„ 2^-1 3»„ ,,^, coseoz = - + i2-l)-S,+-^^.-S,+ -^.^S,+ ...{n), which holds if mod. zktt. 297. To obtain a formula for sec z, in powers of z, we use the formula _ / 1 3 sec^ = 3^2 _ 4^2 5V2-422 (-l)'"-'(2m-l) N „. '^{2m-lfir'-iz')^^"' ' supposing the modulus of z to be less than ^v; we have on expanding each fraction .-?!A_l + l_ (-i)m-i. 2^ n 1 1 '"ttIi 3 5 ■■•^ 2m-lJ ^7r» [P 3»^5' "• sec^ = f 1 1 / 1 \m—l ) 02»+2 ^ (2m - l)'j 7r^+' J2n+i gan+i + "*'(2m-l)»+^K---"*"^'"- .562 INFINITE PRODUCTS Now let 2m+i denote the sum to infinity of the infinite series 111 and let the remainder after the first m terms be e^+i, then we have 92 94 92J1+2 SeC2r = -2i + ^2r%+ ... + ^;^^Z^1^^ + 2" 2* TT 7r' let e' be the greatest of the numbers e,, £3, ..., then the modulus 2" 2* . of — 61 H — -z^e^ + ... is less than e times the sum of 92 94 96 which last series is convergent when the modulus of z is less than ^TT. We have thus shewn that the remainder of the series we have obtained for sec 2: is a number of which the modulus diminishes indefinitely as m increases, hence we have for sec^ the infinite series 2" 2* 2' sec£: = — 2i + -^^%+— ^£5+ (18), which holds if mod. z < ^tt. 298. It is a well-known theorem in Algebra, that the function zjie' — 1), where e? has its principal value, can be expanded in a series of the form 2^2! 4! ^--^^ ' (2n)!^ +■■•' where B^, B^, ... 5„, ... are certain numbers called BernouilU's numbers, and that this expansion holds for all values of z for which the series is convergent. If we multiply by e^ — 1 we have ,^ + 2! + -+(2^+-}f- 2^+21^^-41^+- D + ^ ^^ (2n)!^ +■•• I z I being taken so small that both the series on the right-hand INFINITE PRODUCTS 363 side are absolutely convergent, we may multiply them together, and arrange the product in a series of powers -of z ; the resulting series will be absolutely convergent, hence equating the coefficients of the powers of z above the first, on the right-hand side, to zero, we have a series of equations ^_ll+i=0 -^ + 1^-11 + 1 = 2! 22!^3! ' 4!^3!2! 4!2^5! ' the general type of which is ^n 1 Bn-. , (-1)" B, (-l)"-a (-1)" ^ (2n)! 3!(2n-2)!"^""""^(2?i-l)!2! (2m)! 2"^(2n+l)! By means of these equations, the numbers J5i, B^, Bs, ... may be calculated ; we find B,= \,B, = i^,B, = ^, B, = i^,B, = ^, B, = #,V. Br = h &c. 299. The coefficients in the expansions of cot z, tan z, cosQcz, in powers of z, may be expressed in terms of Bernouilli's numbers. w u ^ .e^'+e-i^ ,/. 2 We have cot z = i -^ r, = ^ 1 + - hence, if mod. z is small enough. Also cosec z = cot |^ — cot z ; hence we have the series 1 2 (2-1) A , 2(2»-l)A .^ cosec^ = -+ ' ^/ ^+ 4! ^+- ^2J^:--_1)5,^__^__^20). Again, since tan ^^ = cot ^ — 2 cot 2z, we have the series 2^(2^-1)5, 2^2^1)5, tan^= 2l ^+ 41" -^+ ••• It has been shewn that the series (19) and (20) are convergent if mod. z<7r, and that (21) is convergent if mod. z < Jtt. The series in (19), (20), (21) must be identical with those in 364 INFINITE PRODUCTS (15), (16), (17), respectively; hence equating the coefficients in (19) to those in (15), we have 2 o_2^ R 2 2^ 2 __2!l5 . hence using the values of Bi, B^, ... in Art. 298, we have ^^-Q' ^'-QO' ""945* '^'~9450'"' ^^n — (2n) ! ■Bn, thus Sai may be calculated by means of the formulae which give The series (19) and (21) give a ready means of calculating the tangent or cotangent of an angle, the first few terms of the series are 1 X x^ 2afi cota;=- 3 45 945 , x^ 2^ , nx! , tan^=^+^ + ^+3jg+.... The calculation of tan — 90°, cot — 90° may be carried out as follows : n n tan(m/ra90°) = 2mnl(n^-m^) x -6366197723675 +«i/»ix -2975567820597 +m?ln» X -0186886502773 +m'/nfi X -0018424752034 +ni'ln' X -0001975800714 +«i»K X -0000216977245 +»i"/m" X -0000024011370 +mi3/mWx -0000002664132 + m>6/mw X -0000000295864 +m^''y X -0000000032867 + m"/?ii'' X -0000000003651 + m^ln^^ X -0000000000405 + m'^ln^^x -0000000000045 +m^ln^ X -0000000000005 In these expressions, the terms cot (m/n 90°) = n/m X -636619772367581 - imnjiAn^-m?) x -3183098861837 -m/m X -2052888894145 -m?l-n? X -0065510747882 - m^jn^ X -0003450292554 -rri'ln' x -0000202791060 - m8/>i» X -0000012366527 -m^yri^^ X ■0000000764959 _to13/^i3 X -0000000047597 -ot15/m16 X -0000000002969 - m"'/)i" X -0000000000185 -mi9/«i9 X -0000000000011 8z which occur in the formulae (10) and (8), are first calculated separately, the series being then more rapidly convergent. These series are taken from Euler's Analysis of the Injmite ; they are however given by him to twenty places of decimals. INFINITE PRODUCTS 365 Series for the logarithmic sine and cosine. 300. We have shewn in Art. 285 that si„,..(l_5)(l-^)...(l-^)(l-..), --(-?)(i-^)...(i-j^)a-«, where 6m, ^m are numbers whose moduli may be made as small as we please by taking m large enough ; taking logarithms, we have log sin ^ = log^; + log (l - ^ + log^l - 2^J + ... log COS 2 . log (^I - -^j + log (^1 - p^j + . . . expanding the logarithms, we have, assuming that | a^ | < tt in the first case and < -= in the second case, so that the logarithms may be expanded in absolutely convergent series of powers of z, logcos. = -J^(-j^ + 3,„+...+|=^)^^+log(l-^.')- Now 1 1 1 1 1 1- .. _ / 1 Jl Jl ^ J_ f JL Jl Jl V _ 111 2«'-l„ hence jm+3i^+5ii+ ^m "^' 366 INFINITE PRODUCTS we have therefore ain 1 OSM -271 log COS ^ = - 2 ^^^^^ Z^S^r, + 2 ^^^^.„ + log (1 - dj), where e^n, Vm are the remainders after m terms in the two series 1 1 ' 02n ~r • ' ■ ? T 2?l ' -Jgn "■"■'* Z"" • , ,, /^hS' \m The modulus of 2 — -e^ is less than e'2J — '--, and that of ^anig^n 2^ I z I'" 2 ;;r- Van is less than 7j' 2 -Jr- , where e, i) are the greatest values of e^n, Vm respectively; hence , sin^r „ ^^' „ log = - 2 — -^ S^, 22» _ I log cos « = 2 jr- Z^S.^. Since S.m= , Bn,'vie have the following infinite series for , sin 5 , log , log cos z, z ^°S ^ " '^12!'' 24! - ^ «(2n)! ^^^^' where mod. zktt, \ogcosz = -2(2^-\)^f -2^2^-1)^^-... -2^-\2^-l)^y^- (28), n {2n) ! ^ where mod. zk^tt. The first few terms of the series (22), (23) are sin 2 z^ z* z' log- 6 180 2835 z'' z* zfi logcos.= 2 12 45 •••' hence also g2 *j^ 62^* logtan^ = log^+3 + 3^ + 2g35+. INFINITE PRODUCTS 367 The series (22), (23) may be employed to calculate tables of logarithmic sines and cosines; it is best to calculate separately the first logarithms, log (1 2 ) ) log ( 1 2 ) ) *s we thus obtain the series in a more convergent form than in (22), (23). We have log sm — =log . + log ^+log (^1 - _j _ 2 |(^_ ^^ - _j -^j , logcos^ = log(l-^j-2|(^— 1 a. IN m^l 2 r (2r) ! rj Multiplying the logarithms on the right-hand side of these equations by the modulus '4342944819, we get the ordinary logarithms of sin — 90°, cos — 90° to the base 10 ; the formulae thus found are L (sin mjn 90°) = log m + log (2)1 - m) + log (2m + to) - 3 log m+ 9-594059885702190 - m2/»2 X -070022826605901 -to*/to*x -001117266441661 -mfijn^ X -000039229146453 - rrfijn^ x -000001729270798 - rn^o/n}" X -000000084362986 -to12/»i12 X -000000004348715 - m"/»" X -000000000231931 - mie/jiW X -000000000012659 - TO^s/w" X -000000000000702 - m^o/Ji™ X -000000000000039 j£(oosTO/re90°)= log (n-m) + log (» + m) — 2 log ii + 10-000000000000000 - m^/n^ X -101494859341892 - mVm*x -003187294065451 - rrfijn' x -000209485800017 - mS/mS X -000016848348597 -to">/?i1» X -000001480193986 -mi2/»*2 X -000000136502272 -mi*/»"x -000000012981715 - to18/w16 X -000000001261471 -mis/mW'x -000000000124567 -7ii">/m"> X -000000000012456 - m22/m22 X -000000000001258 - TO24/m24 X -000000000000128 - m28/TO26 X -000000000000013 These series were given by Euler, the decimals being given to twenty places. Examples. 301. (1) FindthevaluesofSn-%2n-S2{^n-l)-\2(,2n-l)- ^ ' 111 1 We have , sina; ,, A «2 \ ^2 1 ^4 1 2^' also ^og-^^=\os[l--^ + -^Y0~-)-'\G"l26)''2\ we have 'S,n-'^=^ir^, 2w-* = ^Tr*. Again and logcos^=log(^l-- + 24-...j=-(^2-2^)-2(^2J, therefore equating the coefficients of a? and ^, we find (2) ^Smto i!/ie m^mie series ^-^^ + g2:j-^ + 52^+ • • • ■ In the theorem (10), put ^=ixir; we thus find for the sum of the series, -T- tanh 4 irx. The sum might have been obtained directly from the expression iX for cosh irx in factors, by taking logarithms and differentiating. (3) Shew that the sum of the squares of the reciprocals of all nvmhers which are not divisible hy the square of any prime is IS/tt^. Let a, ft 7, ... denote the prime numbers 2, 3, 5, ..., then the required sum is equal to the infinite product (.-j.)-(-^)-(-a-- this is equal to -, .^^,-=^ j^-,-^ p-,- {'-a) 0-^) (^-7) or to ('4.+^-) ('4-?--) ('-?-?-•■)•■•■ 22"*" 32 "''42 "*"■■■ and this is equal to = ^'''2*'*"3*"''4i''''" or to -f — J which is equal to IS/tt^. (4) An infinite straight line is divided hy an infinite number of points into portions each of length a. Prove that if a point be taken such that y is its distance from the straight line, and x the projection on the straight line of its distance from one of the points- of division, the sum of the squares of the reciprocals of the distances of this point from all the points of division is sink — - n a ay , 27ry 27rx' ■^ cosh — - - cos INFINITE PKODUCTS 369 The series to be summed is 2 -= — -, r; , which is eauivalent to 1 " / 1 1 \ oTT, 2 I - — -.— — T— — . The sum of the series is therefore ixya \ cot "(^-^y)- cot '^^^+^^)]-, a a ] sin ■ — - a . jr(x+iy) . n{x — iy)' sin — ^^ ^ sm — ^ ^ a a which reduces to the given result. EXAMPLES ON CHAPTER XVII. 1. Prove that cos(i,rsme)=i7roos-5^1 + -2--^j \^+^^) 2. Prove that 1 + sina;: .K.+^)'{.-<^}'{i- 3. Prove that 2 S -; ^-t^ — ^ = - ""^ where i, j have all unequal {3C + 1){X-^J) — 00 — 00 \ integral values, and x is not an integer. 4. Prove that ."). Prove that (i+4.^)(i+y)(i+y) (i+-^)(i+g)(i+g) 3*^5*^7* 9* 64 V 12/ 2^g 'ix^ la? 6. Prove that 7. If express X (.»+^a) in terms of /i (x\ and /i (a'+^a) in terms of X {x\ and thence find the limit when m is infinite of — — " , V2m+1. H. T. 24 370 EXAMPLES. CHAPTER XVII 8. If Pr denotes the products of p, ^j, =3, ... taken »• at a time, shew 9. Prove that 12 X2.32 P.! 10. Sum the series 14. 3*^3*. 5*^5*. 7* 11. Shew that the sum of the products of the fourth powers of the reciprocals of every pair of positive integers is . 12. Prove that /■ 2 2 2 \ / 1 1 1 \_^ V H-P"'"l+22"^l+32'*' ) V4 + l^'''4+.32'''4+52 + /" 8 " 13. Prove that the sum of the series \\:¥:z) "'"(2:3:4) "'■(3:4:5) "•" 14. Shew that r (TOg-l)(2^m^-l) {r^nfi-\) r=^{m'-{m-lf}{i^nfi-{m-lf} {r^m-^-{m-lf} ism— 1. 13 5 15. Shew that the sum of the series t^- — s-52 2 + ; la + aia SHa'^ 52+«"^ Jtt sech^TT^. 16. Prove that tan~ia! — tan~i Ja;+tan~i Ja;- =tan-itanh|^7r^. 17. Prove that l0gl2-21og7r=,S2 + J/S'4 + i,S'6+ +\^ir,+ - where Sr is the sum of the reciprocals of the rth powers of all numbers which are not prime. 18. The side BC of a square ABCD is produced indefinitely, and along it are measured CC^, C1C2, C2C3, each equal to BC ; if flj, 6^, be the angles BAOi, BAC^, BAC3, , shew that sin Sj sin ^2 sin ^3 ad inf. = VStt coseoh TT. 19. If 2, 3, 5, ... are all the prime numbers, shew that EXAMPLES. CHAPTER XVII 371 2^ 32 5^ 20. Express the doubly mfimte senes 2 2 (-IIm+k -— =?■ in the form of a singly infinite series of cosines of multiples of y. 21. Prove that n / (''"• + °)^+^ |. = (sinh2/3V2+cos2^V2-2cos2acos^V2 cosh j3 \/2 + cos2 2a)/4(a*+;34), where n has all integral values, positive and negative, excluding zero. 22. Prove that 1.2. 3. 4 + 5. 6. 7. 8 + 9. 10. 11.12+ = ilog2-A,r, 1,1.1. «■ 1.3.5.7 ' 9.11.13.15 ' 17.19.21.23 ' 96(2 + ^/2)" 23. li(ia;) = (l+^-^) (^ + f) i^'*'^) =^ + »^. shew that tan~i - + tan~ir+tan-i — h =tan-l -r, a b c A and hence shew that tan-xS+tan-ig+tan->|:+ Jtan-^f '"^^''"^^ ytan-^2+tanh-^^ 24. Prove that " 1 _7rv'2 sinh7ra;\/2+sin7ra;V2 1 1 ra*+a^ 4a;3 cosh 7r.»V2- cos n-a;\/2 2.»*' 71=00 1 25. Prove that 2 -. --^r^=coaec^ 6. 26. Prove that. &>+e and ~t + ^2 + (6-o)2 J 1 + 97r2+(6-c)2 J V+ 25B-H(6-c)2r e^-e' ~y-'^b^c) 1 + 4»r2+(6-c!)2 / I 167r2+(6-c)2i ■ (Euler.) 94. 9 372 EXAMPLES. CHAPTER XVII 27. If p^^ ^+^Xl__i_+li ^—+ n — m n + m 3n — m Sn+m bn—m, 6n+m ^'"{n-mf "•■ (TO+m)" ■*" {Zn-mf '^ (3re+»i)2+ • ff^^: 1 , 1 1 I {n-mf {n+mf'^i^n-mf {Zn+mf'^""" ^'^(;n-mY'^ {fi+mf'^ (^Zn-mf'^ {Zn+ntf^ ' prove that : 2m' ^"~ 2.4.# ' 2.4.6.m3' ■'^- 2. 4. 6. 8. to* ' where i=tan — . (&der.) 28. Prove that the sum of the series 1 — ^ + ^5 — 7^5+ •.••••! in' which ^^^ odd numbers not divisible by 3 are taken, is jr^/lS ^3. {Euler.) 29. Prove that the sum of the squares of the reciprocals of all numbers which are not divisible by 3 is '4jr2/27. {Euler.) 30. Prove that sinhy+sinhc ^ A y\ A gcy-yA / 2cy+.yg \ A 2(g^-y' \ sinhc V c/ V TT^ + c^/ \ 49r2+cV \ Ott^+c^/ and cosh y- cosh c _Y, _y^\ / 2cy-/ \ / 2ey+/ \ / 2ey-yg \ 1-coshc V c2/\ 47r2+cVV 4ff'+cV\ 16jr2+c2y 31. Prove that when n is odd -*^£+''°*^£ + + cot^(!^^=i(«-l)(.-2), cot*g+cot*g + + cot* i^I^=5\, (»-!)(»- 2) (»2+3»- 13). 32. Prove that the infinite product {l+x*^) (l+^) (l + ^) is lal to — ^ n (cosh77aa!+C0S7rj3a;),or -j^- -, cosh jTra; n (cosh n-aa; + Cos n-S.^). according as m is even or od.d, a,, |8, denoting s|n -?, cos ^ respectively, where r is an odd number. (Glatsher.) equal EXAMPLES. CHAPTER XVII 373 is equal to 1 Ji-i -J U (ooah27raa; — ooa2TrSli!), or -rr-, — ^r — 8mh7riBn(cosh27raa;-cos2jr|8«), according as w is even or odd, a, having the same meaning as in the last question. [Olaisher.) 34. Prove that l^'i + a;* T^ 22n^.^2n ^^ 32ii + a;2« ir ''-iasinh2n-aiK+,8 sin27rj3a; 1 ,j^2»-i J cosh 27ra« - cos 27r^^ 2a;*' ti, j3 having the same meaning as in the last question. {Olaisher.) 35. Shew that ax-^hy '•|" ( ax-irhy + r{a?-irb'^) eta + 6y-?-(a'-' + 6^) | is equal to CHAPTER XVIIT. CONTINUED FRACTIONS. Proof of the irrationality of nr. 302. Let /(c) denote the sum of the convergent series l.c^l.2.c(c + l) 1.2.3.c(c + l)(c + 2)^""' then /(c + 1) -/(c) = ;(^)/(c + 2); , /(c) _ a!° /(c + 2) nence ^^^_^j^ c.(c + 1) /(c + 1) ' therefore /(c + l)//(c) can be expressed as a continued fraction of the second class 1 ar'/c (c + 1) a?l(c + 1) (c + 2) a^/(c + 2) (c + 3) 1- 1- 1- 1- Let c = \, and write \x for x, the series /(c) becomes 1.2 1.2.3.4 siuic X ' tana; 1 a;= a;' or cos X, and /(c + 1) becomes hence — , .^ _ ^ ... , X 1-3-5—7— ' an expression for tan a; as a continued fi-action of the second class. 303. Lambert's proof' of the irrationality of tt depends on the continued fraction found in the last Article. Put x = \w, and if possible let l'rr = m/n, where m and ?i are integers ; we have then m m^ m^ m? n—3n—5n— 7w — " ' ' ' FnbliBhed in the memoire of the Academy of Berlin in 1761. CONTINUED FRACTIONS 375 now after a certain term, the denominators of the fractions mjn, rn^/Sn, m'/Sn, ... exceed the numerators by a number greater than unity, hence, by a well-known theorem ^ the continued frac- tion on the right-hand side of the equation has an irrational limit, and cannot therefore be equal to unity. Hence Jtt cannot be equal to a fraction m/n in which m and n are integers, and therefore tt is irrational. This result is of course included in the much wider theorem of Art. 251"*, that tt is a transcendental number. Transformation of the quotient of two hypergeometric series. 304. The fraction F{a, ^8 -1- 1, 7 -|- 1, x)IF{a, /3, 7, x), where F{a, j3,j,x) denotes the hypergeometrical series ^+1.7"^+ 1.2.7(7-H) ^+-' can be transformed into the continued fraction X fC-^tX) iCf^Su iVqoO where 0(7-^) (;5-H)(7-H-«) , ._ (a + l)(7 + l-/g) ' 7(7 + 1)' '^^ (7 + l)(7 + 2) ' "' (7 + 2)(7 + 3) , (;8 + 2)(7-H2-a) («-F»-l)(7 + w-l -/3) *^- (y + S)(ry + 4<) '•■•'^^-1 {y + 2n-2)(y+2n-l)' (l3 + n)(y + n-a) ''^~(y + 2n-l}{y + 2ny As an example of the use of this transformation, taking the series C 2 2 4 ) (^ = sin + ..■)■ , and putting a = 1, y8 = 0, 7 = ^, x = sin^ (j) in the above formula of transformation, we find 1.2 . „ ,1.2 . _3.4 . _ sin = — r: 1^ TZ r^-- The second convergent gives Snellius' formula for 0, sin \, shew that tan =——5 > =- — — - , and < :; -„ . \-¥x^ \ + x+x^' \-x+a^ 6. Prove that - S 2 -£ — is equal to the greatest integer in mln. " D=l k=0 " ' MISCKLLANEOUS EXAMPLES 379 tan~i 7. Prove that 462 (2a +6)2 + 362 is equal to tan"'- +tan-i w52 462 (2a +36)2 +362 . +tan" 462 (2a+2n- 16)2+362 , ; and hence shew that the sum of the infinite a2+»ia6 + 62' series oot-i(l2 + |) + oot-i(22+|) + cot-i(32+i)+ iscot-ij. 8. If tan^sec5 + tan£sec4= tan C, prove that tan A sec A + tan 5 sec 5 + tan C sec C+ 2 tan A tan B tan C= 0. Trace a connection between this result and the known theorem that sin A cos ^ +sin 5 cos ^+sin Ccos O- 2 sin A sin B sin 0=0, where A, B, G are the angles of a triangle. 9. If m and n be any numbers, prove that n{n+l) a;2 sin^ -jl- {m+n)(m+n + l) 2! + n(n + l){n + 2)(n+3) (m+n){m+n + l){m + n + 2){m+n+3) 4! 4! .1 = (m + ncoax) 1 X m+n' 1 — {m(m. + l){m + 2) + n{n+l){n+Z)coax}j 10. ,r3 {m + n){m+n + l){m+n + 2) 3 ! Prove that 1 cos a, cos a, 1 cos (a + /3), cos /3, 1 cos (a + + 7), cos(/3 + y), cos-y, 008(0 + 18+7 + 8), COsO + 7 + 8), COS(y+S), + .... cos(a+/3), cos(a+/3 + y), cos(a+^+y + S) cos/3, cos(|3 + y), cos(;3 + y+8) cosy, 1 COS S, = 0. cos(y+8) cosS 1 11. Prove that the determinant 1, cos>4, sin^, cos{3A + X) 1, cos^, sin 5, cos(35+Z) 1, cos C, sin C, cos (ZO+X) 1, cosi), sinD, cos(3i) + Z) is equal to 2 sin (A +>S'+Z) multiplied by the product of the sines of half the differences between A, B, C, D, and also by a numerical factor, S denoting li^A+B-^C+D). 12. Prove that, if ) +sin 2S cos^ (y + <^) =0. Q 14. If tan a, tan ^, tan y are the three values of tan ^ obtained when tan 6 is given, prove that (1) cos a cos jS cos y sin (a+;3+y) + sin a sin |3 sin y cos (a+0+y)=O. (2) sin (|3+y) sin(y+a) sin (a+^) = sin 2a sin 2/3 sin 2y. 15. Shew that ^ . ,. , y+a a+0 . 2a + 33 + 3y 2 sin 03 -y) cos ^ COS -g- sin ^ i ~~ T 7+« «+^ 2a+30+3y 2 Sin (j3 - y) COS l-^r— COS — r-^ cos ^ ~ 8in2(a+^+y) + 2sin(2a + ^4-y) ~ cos2(a+^+y) + 2cos(2a+^ + y)' where the summation 2 refers to the sum formed by a cyclical interchange of the angles a, ^, y. 16. Prove that, if .2 cos - 2 cos ;rs , , 2cosfl 2 22 K=l f- 1+ 1+ 1+ ' the error made in taking the ?ith convergent to u instead of u is 2(m^-1) M - v'4 — «2 cot COS" (-2)" 17. Prove that the series 1 1.1 vfi-\ 3m2_3^ 5re2-5 has for its sum .to cx> fH^-4- 4 18. Shew that the equation tan z =00, where a is real, cannot have imaginary roots unless a,as>% and if tan~'^z mean the least positive angle whose tangent is z, shew that -1 •^^'•+^)^ =t«.n - 1 f «^nV, Tf «.. 1^ ' s" ( - 1)-- tan-1 -i^lXiZ:^ tan - 1 ] sinh ^ sec '—- 1=0 (2r+l)2-a;2 (^3 4 21. If P be any point on a circle passing through the centres of the three circles escribed to the triangle ABG, prove the relation -J— (1 + cos .4 - cos £ - cos (7) H (1 - cos .4 + cos 5 - cos C) H ^(l-cos.4-oos5+cosC) = l+cosJ.+cos5+cos C. 22. If u„ = A coa nd + £ am 7i£, where A and B are independent of n, prove geometrically the equation M„ + l-2M„COSfl+M„_l = 0. Prove that 2«sin'fl+sin70 'cos'fl-cos7d = tan e tan2 fg+'^Un^^e-^Y 23. If Oi, O2; 61, 62; N\, N2; Pi, P2 be respectively the two positions of the oircuracentre, centroid, nine-points centre, and orthocentre of a triangle in the ambiguous case, prove that 20i02 = 3(?i(?2 cosec A = 4.1^1^2 = P-^P^ sec A ; a, b, A being the given parts. 24. Lines AB'C, BC'A', CA'B' are drawn through the angular points A, B, C of a, triangle, making equal angles B with AB, BC,.GA respectively.; and lines AC"B", CB"A", BA"0" making equal angles 6 with ^(7, GB, BA respectively. Shew that the triangles A'B'C', A"B"C" are equal in all respects, the area of each being A sin^ fl (cot 6 — cot A — cot B — cot Cf. Shew also that if Tjl, Ta be the tangents to the circumcircles of these triangles from the point A, with a similar notation for the tangents from B and C, then will aTA'=cTc", hT^^aTj!', cTo'=hT^'. 25. Sum the series where the value ?i=0 is omitted, and 51, q are positive integers to be increased without limit. 26. Shew that, if a = 27r/l 7, the quantities coso+cos32a + c083*a+cos3^a, and cos3a+cos 3*a-(-cos 3^a-t-cos3'a are the roots of the equation z^-\-\z=l, and explain how the process thus indicated can be continued to obtain the value of cos a. A, B, C, D, E, F, O, H, K are nine consecutive vertices of a regular polygon of seventeen sides inscribed in a circle whose centre is ; a, /3, -y, 8 are the 382 MISCELLANEOUS EXAMPLES projections upon OA of the middle points of the chords BE, GK, DP, OE respectively ; shew that the common chord of the two circles on a^ and yS as diameters passes through 0, and is of length \0A. 27. If a, /3, y, 8 be the distances of the nine-points centre from those of the inscribed and escribed circles of a triangle ABC, shew that ^+y+8-lla"'"y-|-8 + a-llj3"^8+a+/3-lly"''a+j3-|-y-118^ ' and that a2+;32-|-/+82=iJ='(13-8cos4 cos^cos C), where R is the radius of the circumcircle. 28. Prove that tan^-|-4 sin-r^=Vll. 29. Prove that if /be the centre of the inscribed circle of a triangle ABC, and Z, M, iVthe centres of the escribed circles, the circles inscribed in the triangles IMN, INL, ILM touch the circle ABC, and the tangents of the angles of the triangle formed by the three points of contact are respectively equal to 2 cos \ A +COS \B -l-cos \ C— sin \B — sin \ C— 2 1— cos^.B-cos^C+sin^5+sin JC and two similar expressions. 30. Shew that if x be not an integer, the series %x-\-m,-'rn '(a;+m)2(a;+m)2' in which m, and n receive in every possible way unequal values, zero or integers lying between / and — /, vanishes when / increases indefinitely. 31. Shew that sin™ 6 cos" B can be expanded in the form ^» co° ^"'"'■"^ ^"^'*' cos ^™'''""^) ^■'■^2 cos ^"^ ■'"''" ^^ ^■'"'^• when m, and n are positive integers. Shew also that except in the case of the last terms of the series, when both tn. and n are even. 32. The circumference of a cii'cle whose centre is is divided into n equal parts at the points Pi, P^, P^, P„, and § is any internal point. Prove that tanPi§0+tanP2§0 + + tan P„§0=to tan P'§'0, where P' is a point on the circle such that QOP'=n . QOPi, and g' is a point on QO such that (if the ordinates QR, Q'R' cut the circle in B, R') QOR'=n.QOR. MISCELLANEOtrS EXAMPLES 383 33. Prove that, if mi, mg m, are the integers less than and prime to m, and itpi, Pi, are the different prime factors of m, nsin^ + — !^) 1 \ ™ / sin md . n sm . n sin - Pi Vi PxPiPsPi „,„ . me „ . m6 2'nsm — .nsin Pi PiPiPs 34. Prove that the sum of the products sin^a sin J ( a + -^ j sin ?• ( o + -^ J for all positive integral values of ^, q, r which are such that ^^-g' +»•=«, when i >3, is zero unless « is a multiple of 3, and is - Jsin«a, when s is a multiple of 3. 35. Prove that tan. = |{l-5 + ^-A,.+ }, where .J; = tan 25. H-Lcb a^A. /J-U^. Klc/^woM^ - /"roc. ^o^^„« Ai.M. Ac. ^^. ^' ( ' ^ ^"J f- ^^^ f*- t*-i (Kins T'f t« * L^ I S CAMBBIDOE: PKINTED by JOHN clay, M.A. at the nNIVEBSITT PBESS.