Answer
$210^{\circ}$ and $330^{\circ}$
Work Step by Step
$\csc\theta=-2$
$\sin\theta=\frac{1}{\csc\theta}=-\frac{1}{2}$
The value of $\sin \theta$ is negative, so $\theta$ may lie in either quadrant III or IV.
We know that $\sin 30^{\circ}=\frac{1}{2}$.
Recall: $-\sin x= \sin(180^{\circ}+x)$ and $-\sin x= \sin(360^{\circ}-x)$
$\implies -\sin 30^{\circ}=-\frac{1}{2}=\sin(180^{\circ}+30^{\circ})$
$=\sin 210^{\circ}$
$210^{\circ}$ which is in the third quadrant is one value of $\theta$.
Now $-\sin 30^{\circ}=-\frac{1}{2}= \sin (360^{\circ}-30^{\circ})$
$=\sin 330^{\circ}$
$330^{\circ}$ is in the fourth quadrant.
The values of $\theta$ are $ 210^{\circ}$ and $330^{\circ}$.
