Find the general solution of $\csc x = -2$.

(N/A) Given $\csc x = -2$.
We know that $\csc \frac{\pi}{6} = 2$.
Since $\csc x$ is negative in the third and fourth quadrants,we have:
$\csc(\pi + \frac{\pi}{6}) = -\csc \frac{\pi}{6} = -2$ and $\csc(2\pi - \frac{\pi}{6}) = -\csc \frac{\pi}{6} = -2$.
Thus,$\csc \frac{7\pi}{6} = -2$ and $\csc \frac{11\pi}{6} = -2$.
The principal solutions are $x = \frac{7\pi}{6}$ and $x = \frac{11\pi}{6}$.
To find the general solution,we use $\csc x = \csc \frac{7\pi}{6}$,which implies $\sin x = \sin \frac{7\pi}{6}$.
The general solution for $\sin x = \sin \alpha$ is $x = n\pi + (-1)^n \alpha$,where $n \in \mathbb{Z}$.
Substituting $\alpha = \frac{7\pi}{6}$,the general solution is $x = n\pi + (-1)^n \frac{7\pi}{6}$,where $n \in \mathbb{Z}$.