380 



Then 



Now 



PRACTICAL MATHEMATICS 



f* f* 1 



y sin nx dx = A M I sin 2 nx dx = -T 



Jo Jo a 



?/ sin we cfo = - I x sin ur da? 



Jo rcjo 



cos nx sm nx 



1 : 



n n 



= ( 7T COS W7T) 

 7TO 



= - when n is odd 

 n 



- when n is even 

 n 



Hence A., = when n is odd 



icn 



2c i, 

 = when n is even 



TOT 



2c . 2c 2c 2c . 



Thus w = sin x sm 2# H sm 3x sm 4a; + . . . 



TC 2rc STC 4;r 



2cr i i i "i 



- { sm x - sin 2x + - sm Sa? - sm 4x + . . . r 

 TC I 2 3 4 



working in terms of c. 

 or i/ = 2m | sin # - sin 2a? + - sin 3x - sin 4>x + . . . \ 



working in terms of m. 



Example. Expand the function y = mx as a sine series, knowing 

 that when x = 0, y = ; when x = -, y = c, and when a? = TT, g/ = 0. 



Y 



1t 



FIG. 125 



