Compute the given expression: $\;$ $\sin^{-1} \left[\sin \left(\dfrac{33 \pi}{7}\right)\right] + \cos^{-1} \left[\cos \left(\dfrac{46 \pi}{7}\right)\right]$
Given expression: $\;$ $\sin^{-1} \left[\sin \left(\dfrac{33 \pi}{7}\right)\right] + \cos^{-1} \left[\cos \left(\dfrac{46 \pi}{7}\right)\right]$
$= \sin^{-1} \left[\sin \left(5 \pi - \dfrac{2 \pi}{7}\right)\right] + \cos^{-1} \left[\cos \left(7 \pi - \dfrac{3 \pi}{7}\right)\right]$
$= \sin^{-1} \left[\sin \left(\dfrac{2 \pi}{7}\right)\right] + \cos^{-1} \left[-\cos \left(\dfrac{3 \pi}{7}\right)\right]$
$= \dfrac{2 \pi}{7} + \pi - \cos^{-1} \left[\cos \left(\dfrac{3 \pi}{7}\right)\right]$
$= \dfrac{2 \pi}{7} + \pi - \dfrac{3 \pi}{7} = \dfrac{6 \pi}{7}$
Formulas:
$\sin^{-1} \left(\sin x\right) = x \;\; \forall \;\; x \in \left[\dfrac{- \pi}{2}, \dfrac{\pi}{2}\right]$
$\cos^{-1} \left(\cos x\right) = x \;\; \forall \;\; x \in \left[0, \pi\right]$
$\cos^{-1} \left(-x\right) = \pi - \cos^{-1} x \;\; \forall \;\; x \in \left[-1, 1\right]$