ANALYTICAL.] TRIGONOMETRY 571 connected with the circle. We may easily show from the definitions that cos 2 (a; + iy) + sin*(x + iy) = 1, cosh 2 y - sinh - y 1 ; cos(ar + iy) = cos x cosh y - 1 sin x sinh y, sin(x + iy) = sin x cosh y + 1 cos x sinh y, cosh(a + |3) = cosh a cosh /3 + sinh a sinh /3, sinh(a + /3) = sinh a cosh - cosh a sinh /3. These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert. Expan- If we equate the coefficients of 71 on both sides of equation (13), sionofan we set . . , . . * a , _ . ... . 1 aaro 1 . 3 sm 5 1.3.5 sm 7 an g e m 6 = sm6 + - 5- + ;r-7 iT~ + o t ~ ^~ + (21); powers of 2 3 2.4 5
- its sine. Q must Ue ^tween the va i ues ?T
2.4.6 7 This equation may also be written in the form when x lies between 1. By equating the coefficients of n 2 on both sides of equation (12) we get 2sin 4 2.4 sin0 2.4.6sin 8 which may also be written in the form / arcsma . =a j! ,2 ** 2^4 x 6 2_ L 4J5a 32 3.53 3.5.74 when x is between 1. Differentiating this equation with regard to x, we get arc sin a; 2 , 2.4 2.4.6 , if we put arc sin x arc tan y, this equation becomes >, l 2 1 + s. ! Gregory s I series. Series for calcula tion of TT. (23). This equation was given with two proofs by Euler in the Nova acta We have o^S-j = x+ ^- + -=- + -= + ... ; put iy for x, the left side then becomes {log (1 + iy) - log (1 - iy)} or larctanyimTT ; I/O yd |/7 The series is convergent if y lies between 1 ; if we suppose arc tan y restricted to values between -, we have 7/"^ V^ arc tan y=y- "t + ^ - (24), which is Gregory s series. Various series derived from (24) have been employed to calculate the value of TT. At the end of the 17th century ir was calculated to 72 places of decimals by Abraham Sharp, by means of the series obtained by putting arc tan y=, V=j= in (24). The cal culation is to be found in Sherwin s Mathematical Tables (1742). About the same time Machin employed the series obtained from the equation 4 arc tan ^ - arc tan HOQ = I to ca l cu l ate T to 100 de cimal places. Long afterwards Euler employed the series obtained from 7 = arc tan- + arctan-, which, however, gives less rapidly con verging series (Introd., Anal, infin., vol. L). Lagny employed the 1 7T formula arc tan /== ~ to calculate ?r to 127 places ; the result was communicated to the Paris Academy in 1719. Vega calculated IT to 140 decimal places by means of the series obtained from the equation - = 5 arc tan = + 2 arc tan . The formula ^ = arc tan - + 1 4 j / 10 4 2 arc tan = + arc tan - was used by Base to calculate IT to 200 decimal 11 places. Rutherford used the equation TT= 4 arc tan = - arc tan =- + 1 5 70 If in (23) we put y-- and ^, we have o / 56 a rapidly convergent series for ir which was first given by Hutton in Phil. Trans, for 1776, and afterwards by Euler in Nova acta for 1793. Euler gives an equation deduced in the same manner from t q the identity ir = 20 arc tan = + 8 arc tan ^. The calculation of TT has been carried out to 707 places of decimals ; see Proc. Roy. Soc., xxi. and xxii. ; also SQUARING THE CIRCLE (vol. xxii. p. 435 sq.). We shall now obtain expressions for sin x and cos x as infinite Factoriz- products of rational factors. We have sina;=2sin s sin =2 3 sin 7 sin & A 444 4 proceeding continually in this way with each factor, we obtain . , . x . x + ir . a: + 2jr x + n-lir sma;=2 n - 1 sin - sm sin ... sin ^ n n n n where n is any positive integral power of 2. Now ation of sine and cosine. . x + rv . sm sm and -nr x + rir . rir-x . nr . .x =sm sm =sm 2 sin 2 -, I n n n n . x + &nir x sm = = cos -. 7i n Hence the above may be written sina:=2 n - 1 sin -(sin 2 - - sin 2 - Vsin 2 _ sin 2? . .
n Ti/V n nj
(lew a; x sin 2 sin 2 - I cos -. ?i n/ n where k=
- 1. Let x be indefinitely small, then we have
hence sm a: = . 2"" . * . ,27r ,/br 1 = - sm- - sin 2 ... sm- ; n n n n sin 2 x/n in a! =nsin^ C os^l- s 4H^Vl- si t a;/ rV./l- n n sin 2 7r/7i/ siu-2ir/nj sl We may write this . x xf si sma;=?isin- cos-( 1 r n 7i si where R denotes the product / sin 2 ? / (l- =-}(!-
sin 2 / V
N n / x and m is any fixed integer independent of n. It is necessary, when we make n infinite, to determine the limiting value of the quantity R ; then, since the limit of ^^ is s ~, and that of sin THTT/Ti . n sin - cos - mr y n > U1 "ty. we have sin a; Now R is less than unity, since sin - is less than sin n s n
- also by an elementary algebraical proposition R is greater
than 1 -sin 2 |(cosec 2 1 ^~ + ... +cosec 2 ^ and cosec ^<^,if < ; R is therefore greater than orthan 4 Im 7W + 1 m+l 7/1 + 2 By? __ k-1 k or than 1 - . Hence R=l - , where is some proper fraction whence When TO is indefinitely increased this becomes The expression for cos x in factors may be found in a similar manner by means of the equation cos x = 2 sin ^ cos ~ , or may be deduced thus 4 sin 2a; 2 sin x Pil- n=+<a 2a; .(26). n=-<n 2?i + l?r> If we change x into tx, we have the formulae for sinh x, cosh x as infinite products v=xP (l+_ ) C osha;=P (T+ |. 7i=0 n-ir*J n=Q 271 + 1 -IT-/ In the formula for sin a; as an infinite product put x^, we then get 1 =
- if we stop after 2?t factors in the numer
- . ator and denominator, we obtain the approximate equation j.ri j^.^-iji (2re+1)
~2 2 2 .4 2 .6 2 ... (2ra) 2 -^ n + i >
