Answer
$\theta = 210^{\circ}$
$\theta = 330^{\circ}$
Work Step by Step
$csc ~\theta = -2$
$\frac{r}{y} = -2$
Since $r$ is not negative, we can let $r = 2$ and $y = -1$.
We can find the angle $\alpha$ below the x-axis.
$sin(\alpha) = \frac{1}{2}$
We know in a $30^{\circ}-60^{\circ}$ triangle, the side opposite the $30^{\circ}$ angle is 1 while the hypotenuse is 2. Therefore, $\alpha = 30^{\circ}$
Since $\alpha$ is the angle below the x-axis, then $\theta = 180^{\circ}+\alpha$ or $\theta = 360^{\circ}-\alpha$
$\theta = 210^{\circ}$ or $\theta = 330^{\circ}$
