Evaluate: $\;$ $\sin \left[3 \sin^{-1} \left(\dfrac{1}{3}\right)\right]$
Let $\sin^{-1} \left(\dfrac{1}{3}\right) = \theta \;\;\; \text{ where } \theta \in \left[- \dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$
Then, $\sin \left(\theta\right) = \dfrac{1}{3}$
$\begin{aligned}
\therefore \; \sin \left[3 \sin^{-1} \left(\dfrac{1}{3}\right)\right] & = \sin \left(3 \theta\right) \\\\
& = 3 \sin \theta - 4 \sin^3 \theta \\\\
& = 3 \times \dfrac{1}{3} - 4 \times \left(\dfrac{1}{3}\right)^3 \\\\
& = 1 - \dfrac{4}{27} \\\\
& = \dfrac{23}{27}
\end{aligned}$