how do you find the exact value of sin^-1(-square root of 3 divided by 2)

Answer 1

better just stick with "trig"

One of the first things to do is to learn the special angles with easy trig values: 0,π/6,π/4,π/3,π/2

Knowing those, you will see that

sin π/3 = √3/2

You have -√3/2, so that means that your angle is in
QIII: π+π/3
or
QIV: 2π-π/3

Now, the inverse trig functions have principal values, and for arcsin those are -π/2 <= θ <= π/2

So, Arcsin(-√3/2) = -π/3