Find the general solution of each of the following equations:

To Find: General solution.

Given: $\sin x+\sin 3 x+\sin 5 x=0 \Rightarrow \sin 3 x+2 \sin 3 x \cos 2 x=0 \Rightarrow \sin 3 x(1+2 \cos 2 x)=$ 0

[NOTE: $\sin C+\sin D=2 \sin (C+D) / 2 \times \cos (C-D) / 2]$

$\Rightarrow \sin 3 x=0$ or $\cos 2 x=\frac{-1}{2}=\cos \left(\frac{2 \pi}{3}\right)$

Formula used: $\sin \theta=0 \Rightarrow \theta=n \pi, n \in I, \cos \theta=\cos \alpha \Rightarrow \theta=2 k \pi \pm \alpha, k \in I$

$\Rightarrow 3 x=n \pi$ or $2 x=2 k \pi \pm \frac{2 \pi}{3} \Rightarrow x=\frac{n \pi}{3}$ or $x=k \pi \pm \frac{\pi}{3}$ where $n, k \in I$

So general solution is $x=\frac{n \pi}{3}$ or $x=k \pi \pm \frac{\pi}{3}$ where $n, k, \in \mid$