Answer
$-30^o$
Work Step by Step
The value of $\theta$ must be in the interval $[-90^o, 0^o) \cup (0^o, 90^o] $.
$\theta=\csc^{-1}{\left(-2\right)}$ means that $\csc{\theta} = -2$.
Since cosecant is the reciprocal of sine, then $\csc{\theta} = -2$ means $\sin{\theta}=-0.5$.
Note that $\sin{30^o} = 0.5$.
Since $\sin(-x)=-\sin{x}$, then
$\sin{(-30^o)}=-\sin{30^o} = -0.5$
Thus, $\csc^{-1}{\left(-2\right)}=-30^o$
