How do you calculate the inverse sin(-7/3)?

There is no such thing. (There is no real number whose sine is $- \frac{7}{3}$.)
$y = \arcsin \left(x\right)$ if and only if $\sin \left(y\right) = x$ and $- \frac{\pi}{2} \le y \le \frac{\pi}{2}$
For any real $y$, we have $- 1 \le \sin y \le 1$.
Since $- \frac{7}{3} < - 1$, there is no real $y$ with $\sin \left(y\right) = - \frac{7}{3}$.