382 PRACTICAL MATHEMATICS 



. me 8c 1 



when n = 3 sin = 1 and A.> = -- - 



2 7i 2 3 2 



W7C 



when n = 4 sin = and A 4 = 



.me 8c 1 



when n = 5 sin = 1 and A 5 = 



Then y = ^ \ sin a; - -5 sin 3# + -5 sin 5# - -5 sin 7 + . . .1 

 TU" IP 3^ 5^ 7 2 J 



working in terms of c. 



4m r 1 1 1 1 



or ?/ = J sm a; - -5 sm 3# + ^ sin 5,r - 2 sin 7#+ . . . j- 



189. T/ze Cosine Series. The cosine series is expressed in the 

 form y = /(#) = 60+6! cos x + B 2 cos 2# + . . . B n cos nx + . . . 

 where B , B a , B 2 , B 3 , etc., are constant coefficients and any 

 integration which might be necessary must be taken between the 

 limits x = and x = TC. It should be noticed that the cosine 

 series differs from the sine series in having an initial constant 

 term B . 



In working with this series two operations are necessary, one 

 operation to find the initial term B and the other to find the 

 general coefficient B n . 



If we integrate throughout with respect to x between the limits 

 and TC. 



Then I y dx 



Jo 



= B I cte+Bj I cos x (ir + B 2 I cos 2a? dx + . . . E n \ cos nx dx+. . . 

 Jo Jo Jo Jo 



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



B I dx, which becomes 7rB . 

 Jo 



Hence 7cB = I y dx 



Jo 



= -I y dx 



^Jo 



and B 



If we multiply throughout by cos nx and integrate each ten 

 with respect to x between the limits and TT, 



Then I cos nx dx 



I y 



Jo 



B I cos nx dx + ~Bi \ cos x cos nx dx+. . , B n I 

 Jo Jo Jo 



cos nx 



