388 



PRACTICAL MATHEMATICS 



n 



, / vi 



- (e* cos WTC e~* cos ( WTC) } 



1 



n cos nn 



and 



A = 



Tcn 



cos 



When w is odd, cos nn = 1 and A,, = 



n(e" 



and 



. 

 , A 3 - 



; , A 6 = 



2TC 10TC - 26TC 



When n is even, cos me = 1 and A n = -r-z -^ 



and 



A 2 = 



i, A 4 =- 



Then n = 



e~ 



17rc 



cos x ~ cos 



i, A 6 = 



37 n 



1 



T7T cos 

 10 



--(- 



I \ /I 2 . 



T= cos 4# . .) 4- V - sin a. 1 - sin 



17 / \2 5 



3 . 4 . 



+ sin Sx - sin 40 . 



191. Hitherto we have been working with and TC, and 27c, 

 and TC and TC as the limits for x, but the work is not restricte 

 to these limits; it is possible to work with any limits. Taking 

 the function y = /(0) and expressing it as a sine series betwee 

 and TC. 



Then y =/(6) = A l sin + A 2 sin 20 + . . . A n sin 



If is replaced by x in such a way that when = 0, x = 



TC'7 1 



and when = TC, x = c, then we have the relation = , whi 



c 



renders it possible to work in terms of between the limit 

 and TC. 



Example. Expand the function y = x z as a sine series workii 

 between the limits x = and x = c. 



Then y = A x sin + A 2 sin 20 + A 3 sin 30 + . . . A n sin n0+ . . 

 taken between the limits and TC. 



sn 



c- o *t 2 



Since = , then = x 2 = - 



2 



TC 



and 



C 2Q2 



= 5- = A, sin + A 2 sin 20 + . . . A 



TC 2 



working between the limits and TC, 



