9.sin x + sin 3x + sin 5x = 0
Given that,
sin x + sin 3 x + sin 5 x = 0 ( sin x + sin 5 x ) + sin 3 x = 0 ( ∵ sin x + siny = 2 sin x + y 2 cos x − y 2 ) 2 sin 3 x cos 2 x + sin 3 x = 0 ( ∵ cos ( − x ) = cos x ) sin 3 x ( 2 cos 2 x + 1 ) = 0
Hence,
sin 3 x = 0
Or
2 cos 2 x + 1 = 0 cos 2 x = − 1 2
General solution for sin 3 x = 0 ,
3 x = n π x = n π 3 ( n ∈ z )
General solution for cos 2 x = − 1 2 cos 2 x = cos 2 y (1)
Here,
cos 2 x = − 1 2 (2)
From equations (1) and (2),
cos 2 y = − 1 2 cos ( 2 y ) = cos ( 2 π 3 ) 2 y = 2 π 3
General solution is given by,
2 x = 2 n π ± 2 y ( n ∈ z )
For, 2 y = 2 π 3
2 x = 2 n π ± 2 3 π x = n π ± 1 3 π
Where, n ∈ z
Thus, the general solutions are,
For sin 3 x = 0 , x = n π 3
For cos 2 x = − 1 2 , x = n π ± π 3 where, n ∈ z
Given that,
Hence,
Or
General solution for
General solution for
Here,
From equations (1) and (2),
General solution is given by,
For,
Where,
Thus, the general solutions are,
For
For
