Compute the given expression: $\;$ $\sin \left[3 \tan^{-1} \left(\sqrt{3}\right) + 2 \cos^{-1} \left(\dfrac{1}{2}\right)\right]$
Given expression: $\;$ $\sin \left[3 \tan^{-1} \left(\sqrt{3}\right) + 2 \cos^{-1} \left(\dfrac{1}{2}\right)\right]$
$= \sin \left[3 \times \dfrac{\pi}{3} + 2 \times \dfrac{\pi}{3}\right]$
$= \sin \left[\pi + \dfrac{2 \pi}{3}\right]$
$= - \sin \left[\dfrac{2 \pi}{3}\right]$
$= - \sin \left[\pi - \dfrac{\pi}{3}\right]$
$= - \sin \left[\dfrac{\pi}{3}\right]$
$= \dfrac{- \sqrt{3}}{2}$