570 Expan- Considerthe identity og(l- sion of Expand both sides of this equation in powers of a 1 , and equate the sines and coefficients of x n , we then get cosines of . . , n(n - 3) multiple n n - n -*- arcsiu TRIGONOMETRY = sin0 [ANALYTICAL. , ,, powers of + ( - 1 ) sines and . (p + q) n r p r q r + ... cosines of If we write this series in the reverse order, we have when n is even, and < n-5 n-l
- -i) 2
when vi is odd. If in these three formulae we put jj=e tfl , q=e~ l , we obtain the following series for cos n8 : 2 cos nd = (2 cos 0)" - ?i(2 cos g)n-2 + ^ TO ~ 3 cos 0)- 4 - ... - when 71 is any positive integer ; 7i 2 , 7i 2 (7t 2 -2 2 ) ? i a (?i 2 -2 2 )(i 2 -4 2 ) .- ( - 1 ) 2 cos n0 - 1 - ps- cos 2 + i-r-j - cos 4 -- * - j-^ - cos 6 I * * o + ... + (-l)22- 1 cos0 when n is an even positive integer ; ~ n(t 2 - 1 2 ) 7 t (7i 3 - I 2 )(n 2 - 3 2 ) (-1) 2 cos0=7icos?i0- - 3 - - (8) ...+(-1) 2 2- 1 cos"0 ..................... (9) when ?i is odd. If in the same three formulae we put p = e t6 , q -e~ i8 , we obtain the following four formulae : - 1) 2 2 cos 7i0 = (2 sin 0)" - w(2 sin 0)- 2 2 sin 0)"- 4 - ... (-1) 2 2 sin 7i0 = the same series (n odd) n 2 . , n 2 (7i 2 -2 2 ) . . i 2 (7i 2 -2 2 )(n 2 -4 2 ) . cos 710 = 1 - nr- siu 2 + y-j - - y sm 4 -- S - r - sin 6 !__ I * | o + ... + 2"- 1 sin"0 (71 even) ............... (12) ; , 7l(7l 2 -! 2 ) . , (t 2 -l 2 )(7l 2 -3 2 ) . sm 710 = n sm - -^-5 - sin 3 + - r-^ - ? sm 5 - ... LJL LA n-l + (-1) 2 2"- 1 sin" ( odd) ............... (13). P~1 Next consider the identity - - - - - _ l-px l-qx l-(p + q)x+pqa? Expand both sides of this equation in powers of x, and equate the coefficients of x"~ l , then we obtain the equation If, as before, we write this in the reverse order, we have the series IJL when n is even, and n - 1 _. n - 1 when ?i is odd. If we put^=e 9 , q = e~ l9 , we obtain the formulae i-r-l)(?t - r-2). . . (71 - 2?-), where n is any positive integer ; (-1) siu7i0=sin0i 71COS0- ) 7( M 2 -2 2 ) Jl 2 -4 2 ) y COS 3 0+ - r^ - - ; COS S - ... + (-l) 2 (2COS0)"- 1 (71 even) ............ (15); + (-l) 2 (2 cos 0)"- 1 (TI odd) (16). If we put in the same three formulae ^=6^, q= - e ~ l6 , we obtain the series n-2 (-1) , , ., + (-1) ~ "1 ... J . ?teven)(17) ; (-1) 2 cos 710 = the same series (n odd) ...... (18); a /.i /, ( 2 - 2-) . , a 7i(7i 2 -2-)(t--4 2 ) . ... sin H0 = cos } )i sin - v , . / sm 3 + 3 - X sm 5 + I o I 4 - ... + (-1)2 (2 sin 0)~ l }(TI even) ............ (19); + (2 sin 0)"- 1 } (71 odd; ............... (20). We have thus obtained formulae for cos ?i0 and sin 710 both in ascending and in descending powers of cos 6 and sin 6. Viete ob tained formula? 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 scctiones. James Bernoulli found formulae equivalent to (12) and (13) (Mem. de I Academic des Sciences, 1702), and trans formed these series into a form equivalent to (10) and (11). John Bernoulli published in the Acta eruditorum for 1701, among other formula already found by Viete, one equivalent to (17). These formulas have been extended to cases in which n is fractional, nega tive, or irrational ; sec a paper by D. F. Gregory in Camb. Math. Journ., vol. iv., in which the series for cos 7i0, sin0 in ascending powers of cos 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 acia, vol. ix., by Lagrange in his Calcul d?s fonctioiis (1806), and by Poinsot in Rccherchcs sur Vatuilyse des sec tions angulaircs (1825). The general definition of Napierian logarithms is that, if 6*+^ Theory = a + ib, then x + iy= log (a + ib). Now we know that e x+t y= e*cos y y= ic c siny; hence e? cos y=a, e* sin y=b, or e x =(a? arc tan - mir, where m is an integer. If b=Q, then m must be even or odd according as a is positive or negative ; hence log,, (a + i&)=log e ( 2 +& 2 )i+t (arc tan -2ra7r)
rithnis - or log e (a + ib) = log e (a 2 + 6 2 )* + 1 (arc tan - 2?i + ir), (L 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 27rt ; in particular, the most general form of the logarithm of a real positive quantity is obtained by adding positive or negative multiples of 27n to the arithmetical logarithm. On this subject, see De Morgan s Trigonometry and Double Algebra, chap, iv., and a paper by Prof. Cayley in vol. ii. of Proc. London Math. Soc. We may suppose the exponential values of the sine and cosine Hyper- extended to the case of complex arguments ; thus we accept bolic e i(x+iy) + e -i(x+iy) e t(x+iy)_ e -i(x+ty) trigono- and-- as the definitions of the me try. 2 2t functions cos (x + iy), sin (x + iy) respectively. If x=Q, we have cosiy= and sin iy=.~($-e ?/ ). The quantities s are called the hyperbolic cosine and sine of y and arc written cosh y, sinh y ; thus cosh j/ = cos ly, sinh y= i sin ty. The functions cosh y, sinh y are connected with the rectangular hyperbola
in a manner analogous to that in which the cosine and sine are
