Answer
$0,\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3},\pi,\frac{4\pi}{3},\frac{3\pi}{2},\frac{5\pi}{3}$
Work Step by Step
1. Use the Sum-to-Product Formula, we have:
$sin(2\theta)+sin(4\theta)=0\Longrightarrow
2sin(\frac{2\theta+4\theta}{2})cos(\frac{2\theta-4\theta}{2})=0\Longrightarrow sin(3\theta)cos(\theta)=0$
2. For $cos(\theta)=0$, we have $\theta=k\pi+\frac{\pi}{2}$
3. For $sin(3\theta)=0$, we have $3\theta=k\pi$, thus $\theta=\frac{k\pi}{3}$
4. Within $[0,2\pi)$, we have $\theta=0,\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3},\pi,\frac{4\pi}{3},\frac{3\pi}{2},\frac{5\pi}{3}$
