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