Answer
The solution set of the equation is $$\{90^\circ, 210^\circ,330^\circ\}$$
Work Step by Step
$$\sin3\theta=-1$$ 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: $$\sin3\theta=-1$$
Over interval $[0^\circ,1080^\circ)$, there are 3 values whose sine equals $-1$, which are $\{270^\circ, 630^\circ,990^\circ\}$
Therefore, $$3\theta=\{270^\circ, 630^\circ,990^\circ\}$$
It follows that $$\theta=\{90^\circ, 210^\circ,330^\circ\}$$
This is the solution set of the equation.
