Answer
$\theta=\frac{k\pi}{2}$
Work Step by Step
$\sin\theta+\sin 3\theta=0$
Use the Sum-to-Product Formula $\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$:
$2\sin\frac{\theta+3\theta}{2}\cos\frac{\theta-3\theta}{2}=0$
$\sin\frac{4\theta}{2}\cos\frac{-2\theta}{2}=0$
$\sin 2\theta\cos(-\theta)=0$
If $\sin 2\theta=0$, then $2\theta=k\pi$, and $\theta=\frac{k\pi}{2}$.
If $\cos(-\theta)=0$, then $-\theta=\frac{\pi}{2}+k\pi$, and $\theta=-\frac{\pi}{2}-k\pi$. This is equivalent to $\theta=\frac{\pi}{2}+k\pi$. However, note that this is already included in the above solution, $\theta=\frac{k\pi}{2}$.
