Answer
$0, \frac{\pi}{9}, \frac{\pi}{3}, \frac{5\pi}{9}, \frac{2\pi}{3}, \frac{7\pi}{9}, \pi, \frac{11\pi}{9}, \frac{4\pi}{3}, \frac{13\pi}{9}, \frac{5\pi}{3}, \frac{17\pi}{9}$
Work Step by Step
$\sin 3\theta-\sin 6\theta=0$
$\sin 3\theta-2\sin 3\theta\cos 3\theta=0$
$\sin 3\theta(1-2\cos 3\theta)=0$
If $\sin 3\theta=0$, then $3\theta=k\pi$, and $\theta=\frac{k\pi}{3}$. The only solutions in $[0, 2\pi)$ are $0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \frac{5\pi}{3}$.
If $1-2\cos 3\theta=0$, then:
$1=2\cos 3\theta$
$\cos 3\theta=\frac{1}{2}$
$3\theta=\frac{\pi}{3}+2k\pi, \frac{5\pi}{3}+2k\pi$
$\theta=\frac{\pi}{9}+\frac{2k\pi}{3}, \frac{5\pi}{9}+\frac{2k\pi}{3}$
The only solutions in $[0, 2\pi)$ from $\theta=\frac{\pi}{9}+\frac{2k\pi}{3}$ are $\frac{\pi}{9}, \frac{7\pi}{9}, \frac{13\pi}{9}$.
The only solutions in $[0, 2\pi)$ from $\theta=\frac{5\pi}{9}+\frac{2k\pi}{3}$ are $\frac{5\pi}{9}, \frac{11\pi}{9}, \frac{17\pi}{9}$.
