Answer
$0,\frac{2\pi}{3},\pi,\frac{4\pi}{3}$
Work Step by Step
1. $sin(\theta)+sin(2\theta)=0 \Longrightarrow sin(\theta)+2sin(\theta)cos(\theta)=0 \Longrightarrow sin(\theta)=0,\ or\ cos(\theta)=-\frac{1}{2}$,
2. For $sin(\theta)=0$, we have $\theta=k\pi$
3. For $cos(\theta)=-\frac{1}{2}$, we have $\theta=2k\pi+\frac{2\pi}{3}$ or $\theta=2k\pi+\frac{4\pi}{3}$
4. Within $[0,2\pi)$, we have $\theta=0,\frac{2\pi}{3},\pi,\frac{4\pi}{3}$
