Answer
The solution set is $$\{0^\circ, 60^\circ,120^\circ, 180^\circ, 240^\circ, 300^\circ\}$$
Work Step by Step
$$\sin3\theta=0$$ over interval $[0^\circ,360^\circ)$
1) Find corresponding interval for $3\theta$
The interval for $\theta$ is $[0^\circ,360^\circ)$, which can also be written as the inequality:
$$0^\circ\le\theta\lt360^\circ$$
Therefore, for $3\theta$, the inequality would be
$$0^\circ\le3\theta\lt1080^\circ$$
Thus, the corresponding interval for $3\theta$ is $[0^\circ,1080^\circ)$.
2) Now we examine the equation: $$\sin3\theta=0$$
Over interval $[0^\circ,1080^\circ)$, there are 6 values whose sine equals $0$, which are $\{0^\circ, 180^\circ, 360^\circ,540^\circ, 720^\circ, 900^\circ\}$
Therefore, $$3\theta=\{0^\circ, 180^\circ, 360^\circ,540^\circ, 720^\circ, 900^\circ\}$$
It follows that $$\theta=\{0^\circ, 60^\circ,120^\circ, 180^\circ, 240^\circ, 300^\circ\}$$
This is the solution set of the equation.
