Answer
$-\frac{\pi}{3}$
Work Step by Step
$\csc^{-1}{-\frac{2\sqrt3}{3}}=-\frac{\pi}{3}$, because $\csc{-\frac{\pi}{3}}=-\frac{2\sqrt3}{3}$ and $-\frac{\pi}{3}$ is in the range of $\csc^{-1}{x}$, which is $[-\frac{\pi}{2},\frac{\pi}{2}]$ \ $\left\{0\right\}.$
