Find the exact value of $\sin^{-1}\left (-\frac{\sqrt{3}}{2}\right)$ and the like. 
a. $\sin^{-1} \left(-\frac{\sqrt{3}}{2}\right)$
b. $\cot^{-1} \left(-1\right)$
c. $\sec^{-1} \left(\frac{2}{\sqrt{3}}\right)$
d. $\cos^{-1} \left(-\frac{\sqrt{2}}{2}\right)$
e. $\tan^{-1} \left(-\sqrt{3}\right)$

How do you do these questions without a calculator? You don't have to do them all—I just need to know how to do at least one of them! I appreciate any advice and/or solutions.

There are a few values of $\sin$ or $\cos$ that you should know, e.g. 
$\sin(\pi/3) = \sqrt{3}/2$.  Using the fact that $\sin$ is an odd function, 
$\sin(-\pi/3) = -\sqrt{3}/2$.  So $\sin^{-1}(-\sqrt{3}/2) = -\pi/3$.
