Question

Find the value of $\sin \frac{31\pi }{3}$.

Answer 1

$\sin \frac{31\pi }{3}=\sin (10\pi +\frac{\pi }{3})=\sin \left(5\right(2\pi )+\frac{\pi }{3})$

Values of $\sin x$ repeats after an interval of $2\pi$.
Hence ignoring $5\times \left(2\pi \right)$, we get
$\sin \left(5\right(2\pi )+\frac{\pi }{3})=\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$

Therefore, the value of $\sin \frac{31\pi }{3}$ is $\frac{\sqrt{3}}{2}$.