The theory of the trigonometrical functions is intimately connected with that of complex numbers-that is, of numbers of the form x+y=√1). Suppose we multiply together, by the Connexion rules of ordinary algebra, two such numbers we have with Theory (25) We observe that the real part and the real factor of the Quantities. (x1+ 191) (x2+139) = (X1X3−Y1Y2) +1(x13a + XaY1). of Complex (26) This formula was given by Euler (Nova acra, vol. x.). If we find sin E from this formula, we obtain after reduction formula given by Lexell (Acta Petrop., 1782). imaginary part of the expression on the right-hand side of this equation are similar in form to the expressions which occur in the addition formulae for the cosine and sine of the sum of two angles; in fact, if we put x= cos 0, y sin 01, x27 cos 02, yara sin 02, the above equations becomes (cos + sin 81) Xr2(cos + sin 01) = 7171⁄2(cos 01+02 +1 sin 01 +02). We may now, in accordance with the usual mode of representing complex numbers, give a geometrical interpretation of the meaning of this equation. Let P be the point whose co-ordinates referred to rectangular axes Ox, Oy are xi y; then the point P, is employed to represent the number x1+. In this mode or representation From the equations (21), (22), (23), (24) we obtain the following real numbers are measured along formulae for the spherical excess: sin' Etan R cot R, cot R, cot R 4 cos o cos c A geometrical construction has been given for E by Gudermann (in Crelle's Journ., vi. and viii.). It has been shown by Cornelius Keogh that the volume of the parallelepiped of which the radii of the sphere passing through the middle points of the sides of the triangle are edges is sin E. 16. Let ABCD be a spherical quadrilateral inscribed Properties in a small circle; let a, b, c, d denote the sides AB, BC, of Spherical CD, DA respectively, and x, y the diagonals AC, BD. It can easily be shown by joining the angular points of the quadrilateral to the pole of the circle that A+C=B+D. If we use the last expression in (23) for the radii of the circles circumscribing the triangles BAD, BCD, we have Quadri lateral Inscribed la Small Circle, sin A cos la cos i cosec y sin C cos 1b cos C cosec ly; sin A sin C whence cos o coscos la cos d This is the proposition corresponding to the relation A+C=x for a plane quadrilateral. Also we obtain in a similar manner the theorem sin y sin A cos d' sin x sin B cos analogous to the theorem for a plane quadrilateral, that the diagonals are proportional to the sines of the angles opposite to them. Also the chords AB, BC, CD, DA are equal to 2 sin ja, 2 sin 4b, 2 sin c, 2 sind respectively, and the plane quadrilateral formed by these chords is inscribed in the same circle as the spherical quadrilateral; hence by Ptolemy's theorem for a plane quadrilateral we obtain the analogous theorem for a spherical one sin x sin y=sin la sin c+sin 16 sin d.. It has been shown by Remy (in Crelle's Journ., vol. iii.) that for any quadrilateral, if a be the spherical distance between the middle points of the diagonals, cos a+cos b+cos c+cos d=4 cos x cos y cos z. This theorem is analogous to the theorem for any plane quadrilateral, that the sum of the squares of the sides is equal to the sum of the squares of the diagonals, together with twice the square on the straight line joining the middle points of the diagonals. A theorem for a right-angled spherical triangle, analogous to the Pythagorean theorem, has been given by Gudermann (in Crelle's Journ., vol. xlii.). Periodi city of Analytical Trigonometry. 17. Analytical trigonometry is that branch of mathematical analysis in which the analytical properties of the trigonometrical functions are investigated. These functions derive their importance in analysis from the fact that they are the simFunctions. plest singly periodic functions, and are therefore adapted to the representation of undulating magnitude. The sine, cosine, secant and cosecant have the single real period 27; i.e. each is unaltered in value by the addition of 2 to the variable. The tangent and cotangent have the period. The sine, tangent, cosecant and cotangent belong to the class of odd functions; that is, they change sign when the sign of the variable is changed. The cosine and secant are even functions, since they remain unaltered when the sign of the variable is reversed. the axis of x and imaginary ones along the axis of y, additions being performed according to the parallelogram law. The points A, A represent the numbers 1, the points a, a, the numbers 1. Let P, represent the expression A, FIG. 8. ty and P the expression (x1+132) (x2+432). The quanti ties 71, 01, 72, are the polar coordinates of Pi and P, respectively, referred to O as origin and Ox as initial line; the above equation shows that 7 7 and 0+ are the polar co-ordinates of P; hence OA: OP OP: OP and the angle POP is equal to the angle POA. Thus we have the following geometrical construction for the determination of the point P. On OP, draw a triangle similar to the triangle OAP, so that the sides OP, OP are homologous to the sides OA, OP1, and so that the angle POP, is positive; then the vertex P represents the product of the numbers represented by Pi, P. If+y were to be divided by x+y, the triangle OP'P, would be drawn on the negative side of P2, similar to the triangle OAP, and having the sides OP', OP, homologous to OA, OP1, and P' would represent the quotient. Theorem. 18. If we extend the above to n complex numbers by continual repetition of a similar operation, we have(cos + sin) (cos + sin 2)... (cos on + sin 0.) De Molvre's =cos (01 = 02+ . . . + On) + sin (01 +02+.. +O2) If 01 = 02 = ... =0,01, this equation becomes (cos + sin @)^ #cos no+ sin no; this shows that cose + sin is a value of (cos ne+ɩ sin ne). If now we change into 0/n, we see that cos 0/+ sin o/n is a value of (cos + sin 0); raising each of these quantities to any positive integral power m, cos me/n+ sin men is one value of (cos + sin 0). Also I cos (mo/n) + sin (—mê/n) = cos mo/n+ sin meni hence the expression of the left-hand side is one value of (cos + sin @). We have thus De Moivre's theorem that cos ko+ sin ko is always one value of (cos + sin 0), where k is any rational number. This theorem can be extended to the case in which k is irrational, if we postulate that a value of (cos + sin )* denotes the limit of a sequence of corresponding values of (cos + sin 0)*,. where ki, k...k.... is a sequence of rational numbers of which k is the limit, and further observe that as cos ke+sin ko is the limit of cos k + sin k. The principal object of De Moivre's theorem is to enable us to find all the values of an expression of the form (a+b), where m and n are positive integers prime to each other. Then Roots If a=r cos e, br sin e, we require the values of (cos + sin ). One value is immediately furof a Complex nished by the theorem; but we observe that since the Quantity, expression cos 0+ sin is unaltered by adding any multiple of 2 to 6, the n/mth power of rm (cos m.0+25/n+ sin m.0+25/n) is a+b, if s is any integer; hence this expression is one of the values required. Suppose that for two values s and s2 of the values of this expression are the same; then we must have m0+25,/nm0+25/n; a multiple of 27, or 3-5 must be a multiple of n. Therefore, if we give s the values o. 1, 2,..n-1 successively, we shall get n different values of (a+b), and these will be repeated if we give s other values; hence all the values of (a+b) are obtained by giving_s the values o, 1, 2,...n-I Also x-2xy cos no + in the expression (cos m.0+25T/n + sin m.0+25x/n), where = (a+b) and arc tan b/a. We now return to the geometrical representation of the complex numbers. If the points B1, B2, B3,...B represent the expres FIG. 9. B, sion x+y, (x+1y)2, (x+1y)2, ..(x+y) respectively, the triangles OA B1, OBB2, OBB are all similar. Let (x+y)=a+b, then the converse problem of finding the nth root of a+b is equivalent B to the geometrical problem of describing such a series of triangles that 04 is the first side of the first triangle and OB the second side of the nth. Now it is obvious that this geometrical problem has more solutions than one, since any number of complete revolutions round O may be made in travel =(x*—ya cos no +ɩ sin në)(x”—3′′ cos në —ɩ sin në) Airy and Adams have given proofs of this theorem which do not In equation (6) put y=1/x, take logarithms, and then (aa — b2) (a2n — 2a′′b* cos no+b2) Σ 1 s=0 a2-2ab cos 0+2+b2 ling from B1 to B. The first for the sum of the series solution is that in which the vertical angle of each triangle is BOA/n; the second is that in which each is (B2OA+2)/", in this case one complete revolution being made round 0; the third has (BOA+47)/n for the vertical angle of each triangle; and so on. There are n sets of triangles which satisfy the required conditions. For simplicity we will take the case of the determination of the values of (cos + sin @). Suppose B to represent the expression cos + sin e. If the angle AOP, is 10, PP represent the root cos 10+ sin 10; the angle AOB is filled up by the angles of the three similar triangles AOP, POP, POB. Also, if P2, P, be such that the angles POP, POP, are }, } respectively, the two sets of triangles AOP, POP, POB and AOP,, Pop, POB satisfy the conditions of similarity and of having OA, OB for the bounding sides; thus P1, P. represent the roots cos (0+2)+ sin (0+2), cos (+4) + sin (0+4) respectively. If B coincides with A, the problem is reduced to that of finding the three cube roots of unity. One will be represented by A and the others by the two angular points of an equilateral triangle, with A as one angular point, inscribed in the circle. P Pa The problem of determining the values of the nth roots of unity is equivalent to the geometrical problem of inscribing a regular The nth polygon of n sides in a circle. Gauss has shown in his Disquisitiones arithmeticae that this can always be done Roots of by the compass and ruler only when n is a prime of the Unity. form 2+1. The determination of the nth root of 25T Example of Series. It 20. Denoting the complex number x+iy by z, let us consider the series 1+2+2/2+...+/n!+... This series converges uniformly and absolutely for all values of z whose moduli do not exceed an arbitrarily chosen positive The Exnumber R. Consequently the function E(s), defined Ponential as the limiting sum of the above series, is continuous in every finite domain. The two series representing E(s) and E(z), when multiplied together give the series represented by E(&+2). In accordance with a known theorem, since the series for E(21) E(2) are absolutely convergent, we have E(31) XE(2) = È(+22). From this fundamental relation, we deduce at once that (E(2) = E(ns), where n is any positive integer. The number E(1), the sum of the convergent series 1+1+1/2+1/3!..., is usually denoted by e; its value can be shown to be 2-718281828459.... is known to be a transcendental number, i.e. it cannot be the root of any algebraical equation with rational coefficients; this was first established by Hermite. Writing 3-1, we have E(n) =e", where n is a positive integer. If z has as a value a positive fraction pla we find that E(p/q)) = E(p)=e"; hence E(p/q) is the real positive value of e. Again E(-p/g)XE(p/q)=E(0)=1, hence E-p/g) is the real positive value of epi It has been thus shown that for any real and rational number x, the value of E(x) is the principal value of e. This result can be extended to irrational values of x, if we assume that e* is for such a value of x defined as the limit of the sequence ez, ez,. where x, x is a sequence of rational numbers of which x is the limit, since E(x), E(x)..., then converges to E(x). Next consider (1+z/m)", where m is a positive integer. We have by the binomial theorem, any complex number requires in addition, for its geometrical (1+)=1++(-) + + (1−1) (1–2) ....... solution, the division of an angle into n equal parts. 19. We are now in a position to factorize an expression of Factoriza- the form x-(a+b). Using the values which we tions. have obtained above for (a+b), we have Also sin 25+ 1x +1 ](n even). (4) -2x COS (n 2m Since the series for E(3) converges, s can be fixed so that for all values of m>s the modulus of */(s + 1)! + +/m! is less than an arbitrarily chosen number e. Also the modulus of 1+0 12/1+...+0m2TM3 /(m −2)! is less than that of 1+1|=|/1! +1=1/2! +..... or of e mod hence mod R,<+(1/2m). mod (2)<e, if m be chosen sufficiently great. It follows that (5) lim-(1+z/m)= E(z), where is any complex number. Το evaluate E(s), write 1+x/m=p cos . y/mp sin, then E(2) = lim (p(cos mo+i sin mo)), by De Moivre's theorem. | when n is odd. These formulae enable us to express any positive Since p" = - (1 + ;) " { 1 +m(√ m + x/√m)}3 } *", we have limm-op =. limo+; less than √m+x/√m, then lim-o lies integral power of the sine or cosine in terms of sines or cosines of multiples of the argument. There are corresponding formulae when n is not a positive integer. Consider the identity log(1 − px)+log(1−qx) = log(1-p+qx+pqx2). Expand both sides of this equation in powers of x, and equate the coefficients of x", we then get between 1 and lim-{+} + ", or between 1 and ev; hence "+g"=(p+q)"−n(p+q)~~2pq sincer can be taken arbitrarily large, the limit is 1. The limit of Values of Trigonometrical Functions. sin Expansion of Sines and + "(n = 3) (p+q)~•p2g2+... Cosines of +(− 1)′n (n − r — 1) (n −?— 2) .. 2! Arc. +E(-iy)), sin y=E(iy)-E(-iy); and using If we write this series in the reverse order, we have Exponential the series defined by E(iy) and E(-iy), we find that cos y 1-y3/2!+y4! +y/5! where y is any real number. These are the well-known expansions of cos y, sin y in powers of the circular measure y. Where z is a complex number, the symbole may be defined to be such that its principal value is E(); thus the principal values of e, ev are E(iy), E(—iy). The above expressions for cos y, sin y may then be written cos y = (+), sin yi(e). These are known as the exponential values of the cosine and sine. It can be shown that the symbol e as defined here satisfies the usual laws of combination for exponents. metrical Functions. or = 22. The two functions cos z, sin z may be defined for all complex or real values of 2 by means of the equations cos y=E(2) + E(-2)), sin (†){ E(z) — E( − 2)}, where E(2) represents Analytical the sum-function of 1+ + 22/2! + .. + 8"/n! + Definitions For real values of 2 this is in accordance with the of Trigono ordinary definitions, as appears from the series obtained above for cos y, sin y. The fundamental properties of cos 2, sin z can be deduced from this definition. Thus Cos i sin = E(2), COs 2-i sin = E(- is); therefore cos's+sin'z-E(iz) E(-iz)=1. Again cos (+) is given by E(iz + iz) + E(− iz — iz2) } = } | E(iz,) E (iz2) + E(— iz) E (−iz)} E(iz)+E ( − iz,) } { E (iz2) +E ( − iz) } + {{E (iz,) — E (— iz,) {E(iz)-E(-i), whence we have cos (+2)= cos & cos sin sin . Similarly, we find that sin (+2)=sin cos + cos sin . Again the equation E(z)=1 has no real roots except 2=0, for e>1, if z is real and >o. Also E(z)=1 has no complex root a+iß, for a-iẞ would then also be a root, and E(2a) = E(a+18) E(a-18)=1, which is impossible unless a=0. The roots of E(z)=1 are therefore purely imaginary (except z=0); the smallest numerically we denote by 2 in, so that E(2ix)=1. We have then E(2ixr) = {E(2ix)=1, if r is any integer; therefore 2ir is a root. It can be shown that no root lies between 2irr and 2(+1); and thus that all the roots are given by 2=21x7. Since E(y+21x)=E(s)E(2ix)= E(z), we see that E() is periodic, of period 21. It follows that cos 2, sin z are periodic, of periods 27. The number here introduced may be identified with the ratio of the circumference to the diameter of a circle by considering the case of real values of z. 23. Consider the binomial theorem Expansion of Powers (a+b)"=a"+na"-1b+”(n − 1) a^~^b2+... 2! In Series of Putting a = e1®, b=e, we obtain (2 cos 0)"=2 cos ne+n2 cos n — 20 + n(n- (-1) 4! 2 sin no-the same series (n odd); (11) sin' +21 sin " (n even); (12) sin non sin @ 3! n(n2 — 12) (n3 — 32) sin' 5! (13) of Sines and Cosines Sines and Cosines of Multiple Arc. and when n is even it is When n is odd the last term is 22 2 cos(n-21)0+... +(−1) 7′2′′1 sin" (n odd). 2 cos (n-4)0-... + (n − 3) (n − 4) (p + q)~~* p2q —..... 2 (− 1), (n − r — 1) (n − r−2).... (n− -27) If we put a=e, be ̃o, we obtain the formula (— 1) 1" (2 sin 0)” = 2 cos no − 2n cos (n − 2)0 +” (n − 1) 1.2 7! +(− 1 ) n ( n − 1)... ( } n + 1) If, as before, we write this in the reverse order, we have the series +(−)÷n(n−1). . }(n+3), sin @ 3! + (n2 sine are connected with the circle. We may easily show from the definitions that cos(x+y)+sin(x+y)=1, cos(x+y)=cos x cosh y-i sin x sinh y, These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert. 26. If we equate the coefficients of n on both sides of equation (13), this process requiring, however, a justification of its validity, we get + 2! (n − 3) -3) (n = 4) ( } (14) Expansion 0 = sin 0 +12 I sin3013 sin 0 13.5 sin2 0. ·+.. ...; (21) of an Angie 3 2.4 5 2.4.6 7 in Powers must lie between the values }, This equation of its Sine. may also be written in the form 1 x 1.3 x 1.3.5 arc sin x=x+3 2.4 5 2.4. By equating the coefficients of 2 on both sides of equation (12) we get sin' 0 2.4 sin 0 2.4.6 sin+.... (22) 3-5-7 4 +.. *=sin3 0+3 2 3.5 3 which may also be written in the form 2 x4 ・+2.4 x 2.4.6x8 (arc sin x)=x+3+3.5 3 3.5.7 4 (-1) sinne = cos @sin"-10—(n−2)sin"-10+ (~— 3) (n —4) sin "~60—... + ( − 1), (n − − 1 ) (n − − 2) ... (n≈ — 2r) sin**18+.. sin”-10-+-...] (% even); (17) | to x, we get •{" sin 0_"(n3—23) sin10+ "(x2—2a) (x2—4°) sin10+ sin no-cos en sin e-: 3! 4! (19) (20) We have thus obtained formulae for cos no and sin no both in ascending and in descending powers of cos e and sin e. Vieta obtained formulae for chords of multiple arcs in powers of chords of the simple or complementary arcs equivalent to the formulae (13) and (19) above. These are contained in his work Theoremata ad angulares sectiones. Jacques Bernoulli found formulae equivalent to (12) and (13) (Mém. de l' Académie des Sciences, 1702), and transformed these series into a form equivalent to (10) and (11). Jean Bernoulli published in the Acta eruditorum for 1701, among other formulae already found by Vieta, one equivalent to (17). These formulae have been extended to cases in which n is fractional, negative or irrational; see a paper by D. F. Gregory in Camb. Math. Journ. vol. iv., in which the series for cos no, sín ne in ascending powers of cos 0 and sin are extended to the case of a fractional value of n. These series have been considered by Euler in a memoir in the Nova acta, vol. ix., by Lagrange in his Calcul des fonctions (1806), and by Poinsot in Recherches sur l'analyse des sections angulaires (1825). 24. The general definition of Napierian logarithms is that, if ex+y=a+b, then x+y=log (a+b). Now we know that Theory of ex+y=excos y+ex sin y; hence ex cos y=a, ex sin y =b, or ex=(a+b), y=arc tan b/amx, where m Logarithms. is an integer. If bo, then m must be even or odd according as a is positive or negative; hence or log. (a+b)=log. (a2+b2)}+ (arc tan b/a #2nx) log. (a+b)=log. (a2+b2)+(arc tan b/a =2n+x), according as a is positive or negative. Thus the logarithm of any complex or real quantity is a multiple-valued function, the differ. ence between successive values being 2; in particular, Hyperbolic the most general form of the logarithm of a real positive Trigono- quantity is obtained by adding positive or nega metry. tive multiples of 2 to the arithmetical logarithm. On this subject, see De Morgan's Trigonometry and Double Algebra, ch. iv., and a paper by Professor Cayley in vol. ii. of Proc. London Math. Soc. = 25. We have from the definitions given in § 21, cos y= (ex+e-y) and sin yi(ex-e-y). The expressions, (ey+e-y), (ex-e-y) are said to define the hyperbolic cosine and sine of y and are written cosh y, sinh y; thus cosh y=cos y, sinh y sin ty. The functions cosh y, sinh y ted with the rectangular hyperbola in a manner anal which the cosine and when x is between 1. Differentiating this equation with regard if we put arc sin x=arc tan ̧y, this equation becomes 2 32 2.4 arc tan y=++++33(5)+.... (23) 1 the value of *. At the end of the 17th century was calculated Various series derived from (24) have been employed to calculate to 72 places of decimals by Abraham Sharp, by means Series for of the series obtained by putting arc tan y=x/6, Calculation y=1/3 in (24). The calculation is to be found in of s. Sherwin's Mathematical Tables (1742). About the same time J. Machin employed the series obtained from the equation 4 arc tan-arc tan to calculate to 100 decimal places. Long afterwards Euler employed the series obtained from arc tan arc tan, which, however, gives less rapidly converging series (Introd., Anal. infin. vol. i.). T. F. de Lagny employed the formula arc tan 1/√3/6 to calculate to 127 places; the result was communicated to the Paris Academy in 1719. G. Vega calculated ☛ to 140 decimal places by means of the series obtained from the equation 5 arc tan +2 arc tan The formula arc tan +are tan +arc tan was used by J. M. Z. Dase to calculate to 200 decimal places. W. Rutherford used the equation 4 arc tan arc tan arc tan. If in (23) we put y and, we have Sin mx/n ·(1-; sin x/n sin' kan It is necessary, when value of the quantity and that of sin x is x The modulus of R-1 is less than lim R. 药品 where p mod. sin x/n. Now >1+Ap, if A is positive; hence mod. (R-1) is less than exp. (cosec2 m+1x/n + ... + cosec kr/n)-1, or than exp. pn1/(m + 1)2+. +1/1, or than exp. p2n2/4m2 } — 1. Now psin a/n.cosh' B/n+cos a/n. sinh* Bán, if x =a+iB or p = sin an+sinh B/n. Hence lim p2na2+ẞ2, limpn=mod. x. It follows that mod. (R-1) is between o and exp. {(mod. 4)2/ xm2) −1, and the latter may be made arbitrarily small by taking m large enough. It has now been shown that sin x=x(1-x2x2)(1 − x2/22x2) (1 − x3/m2x2) (1+), where mod. decreases indefinitely as m is increased indefinitely. When m is indefinitely increased this becomes x+x cos (x+y) manner the series tan x= 2 7-2x and thence into factors we should obtain in a similar +2x 3π-2x 3x+2x for sin x and cos x by taking logarithms and then differentiatThese four formulae may also be derived from the product formulae ing. Glaisher has proved them by resolving the expressions for cos x/sin x and 1/sin x ... as products into partial fractions (see Quart. Journ. Math., vol. xvii.). The series for cot x may also be obtained by a continued use of the equation cot x=cot x+ cot (x+) (see a paper by Dr Schröter in Schlömilch's Zeitschrift, vol. xiii.). Various series for may be derived from the series (27), (28), (29), (30), and from the series obtained by differentiating them one or more times. For example, in the formulae (27) and (28), by putting x=/n we get I n-1 I == 3√3(1 Series for * derived from Series for Cot and Cosec. '(1-1). (25) put x=7, and we get 2=9{ ++++.... |