Answer
$\dfrac{2\pi}{3}$
Work Step by Step
Let us substitute
$g^{-1}(f (\dfrac{- 5\pi}{6}))=\cos^{-1} [\sin (\dfrac{-5 \pi}{6})]$
From the unit circle, we know that:
$\sin (\dfrac{- 5\pi}{6})=\dfrac{-1}{2}$
We know that the trigonometric function $\sin^{-1} (1)$ has the domain $[-1,1]$ and range $[-\dfrac{\pi}{2}, \dfrac{\pi}{2}]$.
Therefore, $\cos^{-1} [\sin (\dfrac{-5 \pi}{6})]=\dfrac{2\pi}{3}$; because $\sin \dfrac{- 5 \pi}{6}=\dfrac{-1}{2}$ and $\dfrac{2\pi}{3}$ lies in the range of $\sin^{-1} x$.
