114 



111 these expressions, n is any integer. Since: 

 sin^ x=y2 (1 — cos 2x), and cos 2x sin nx^ji sin (2-{-n)x—y2 sin (2 — n)x, 



An== {2 1 it) I sin^ X sin nx dx 



Jo 



= (l/x) I smnxdx—(l/Tr) cos 2^:' sin ?7,a; (/ic 

 Jo Jo 



= (1/t) I smnxdx—(ll2Tr) I sin (2-{'n)x dx 

 Jo Jo 



+ (l/27r) I sin {2—n)x dx 



_ cos waj'l'^ cos (n-|-2)x~j"" cos {n—2)x~\' (118) 



~ ^r^V 2(n + 2)7r Jo"^ 2(n-2)7r Jo 



The vahies of cos nx, cos {n-]-2)x and cos (n— 2)a:; are +1 when 

 x=0. If n is odd their value is —1 when x=t; but if n is even their 

 value is +1 when x='ir. Therefore, for values of x between and t, 

 the value of An is, when n is odd : 



^ __1__J 1 _ 8 , (119) 



" WTT (71 + 2)t {n-~2)T n{n^ — 4:)ir 



but when n is even, An=0. 

 Substituting successive odd values of n: 



Ai = 8/(3t), A.^-S/ildir), A,= -8/(105it), ^7=-8/(3157r), etc. 



Por values of x between and tt, therefore: 



sin2a; = (8/37r)(sin a--l/5 sin Sx—l/Zo sin 5a-- 1/105 sin 7a: . . . ) (120) 



Similarly, for values of x between x and 2x: 



. _ cos nxT'' cos (n+2)a::"|2'' cos (n— 2)xT' non 



"^'^- ^^J. + 2(n+2)T J. + 2(n-2)7r J. ^^^^^ 



Since, when n is odd, the functions cos nx, cos (n-\-2)x and cos (n— 2)x 

 have a value of —1 when x=ir and of +1 when x=2ir, the value of 

 An becomes, between these limits. 



But when n is even the values of these cosine functions is -fl both 

 when a;=7r and when a;=27r, and the coefficient is zero. 

 For values of x between tt and 27r, therefore; 



