Find the value of

To find: Value of $\sin \frac{31 \pi}{3}$

$\sin \frac{31 \pi}{3}=\sin \left(10 \pi+\frac{1}{3} \pi\right)$

$=\sin \left(5 \times(2 \pi)+\frac{1}{3} \pi\right)$

Value of $\sin x$ repeats after an interval of $2 \pi$, hence ignoring $5 \times(2 \pi)$

$=\sin \left(\frac{1}{3} \pi\right)$

$=\sin \left(\frac{1}{3} \times 180^{\circ}\right)$

$=\sin 60^{\circ}$

$=\frac{\sqrt{3}}{2}\left[\because \sin 60^{\circ}=\frac{\sqrt{3}}{2}\right]$