Find the distance between the points whose polar coordinates are $\left(a, \dfrac{\pi}{2}\right)$ and $\left(3a, \dfrac{\pi}{6}\right)$.
The given polar coordinates are $\;$ $A\left(r_1, \theta_1\right) = \left(a, \dfrac{\pi}{2}\right)$, $\;\;$ $B\left(r_2, \theta_2\right) = \left(3a, \dfrac{\pi}{6}\right)$
$\begin{aligned}
\text{Distance } AB & = \sqrt{r_1^2 + r_2^2 -2 r_1 r_2 \cos \left(\theta_1 - \theta_2\right)} \\\\
& = \sqrt{a^2 + \left(3a\right)^2 - 2 \times a \times 3a \cos \left(\dfrac{\pi}{2} - \dfrac{\pi}{6}\right)} \\\\
& = \sqrt{10a^2 - 6a^2 \cos \left(\dfrac{\pi}{3}\right)} \\\\
& = \sqrt{10a^2 - 3a^2} \\\\
& = a \sqrt{7} \; \text{units}
\end{aligned}$