THE SINE SERIES 379 



f "cos 2 n0 d0 - I [ "d6 + i f "cos 2n6 dQ 



J-w -J- ^J- 



f* 1 f* 1 f* 



I cos 2 wO d0 = I dQ H I i 



* 'I * ' I 



Jo Jo Jo 



cos 2w0 



These results may be used as a means by which a periodic 

 function can be expressed as a series of sines or cosines of multiple 

 angles. 



188. The Sine Series. The sine series is expressed in the form 

 y = f( x ) = A t sin x + A 2 sin 2,r f A 3 sin 3# + . . . A n sin nx + . . . 

 where A,, A 2 , A 3 , etc., are constant coefficients, and any integration 

 which might be necessary must be taken between the limits 

 x = and x = n. 



If we multiply throughout by sin nx and integrate each term 

 with respect to x between the limits and IT, 



Then I y sin nx dx 

 Jo 



Aj I sm#sinn#da: + A 2 | sin 2x sin nx dx + . . . A^ I sin 2 nx dx+. . . 

 Jo Jo Jo 



and all the integrals on the right-hand side vanish except 



I 

 Jo 



sin 2 nx dx, which becomes A^ 







1 f 



Hence **& = \ y sinnx dx 



* Jo 



2 f* 



and A_ = I v sin nx dx 



rcJo 



If y is known in terms of x, the integral can be determined, and 

 by giving n the values, 1, 2, 3, etc., the values of A lt A 2 , A 3 can 

 be found. 



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

 that when x = TC, y = c. 



c c 



Then m = - and y = -x. 



K n 



Then y = A, sin x + A 8 sin 2x + A 3 sin Sx + . . . A,, sin nx + . . . 



