Answer
$0,\frac{\pi}{2},\pi,\frac{3\pi}{2}, \frac{\pi}{3},\frac{2\pi}{3}, \frac{4\pi}{3},\frac{5\pi}{3}$
Work Step by Step
1. $sin(2\theta)+sin(4\theta)=0 \Longrightarrow sin(2\theta)+2sin(2\theta)cos(2\theta)=0 \Longrightarrow sin2\theta=0,\ or\ cos2\theta=-\frac{1}{2}$
2. For $sin2\theta=0$, we have $2\theta=k\pi$, thus $\theta=\frac{k\pi}{2}$.
3. For $cos2\theta=-\frac{1}{2}$, we have $2\theta=2k\pi+\frac{2\pi}{3}$ or $2k\pi+\frac{4\pi}{3}$, thus $\theta=k\pi+\frac{\pi}{3}$ or $k\pi+\frac{2\pi}{3}$,
4. Within $[0,2\pi)$, we have $\theta=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}, \frac{\pi}{3},\frac{2\pi}{3}, \frac{4\pi}{3},\frac{5\pi}{3}$
