Solution

For certain values, we can use the Unit Circle to determine the exact value of the inverse trig function. Unless otherwise indicated, you should always assume the desired answer is exact, and therefore, requires the Unit Circle to determine. Each of the answers below are determined by looking at the coordinate on the Unit Circle that relates to the given inverse function. For example, the first question is \(\sin^{-1}(-1)\), so I will look at the \(y\) coordinates on the Unit Circle, where the \(y\) value is \(-1\). Remember, however, that you will need to use the \(\left[\frac{-\pi}{{2}},\frac{\pi}{{2}}\right]\) semi-circle version of the Unit Circle to get the correct value. You can find the semi-circles here: Reference: Inverse Trig Semicircles

1. \(\sin^{-1}\left(-1\right)=-\frac{\pi}{{2}}\)
2. \(\cos^{-1}\left(-1\right)=\pi\)
3. \(\tan^{-1}\left(-1\right)=-\frac{\pi}{{4}}\)
4. \(\cos^{-1}\left(\frac{\sqrt{{3}}}{{2}}\right)=\frac{\pi}{{6}}\)