417. Calculate $\sin \frac{53 \pi}{6}$.
Solution. We have $\frac{53 \pi}{6}=8 \frac{5}{6}$ cycles. By omitting the whole number of periods, we get

$$
\sin \frac{53 \pi}{6}=\sin \frac{5 \pi}{6}=\sin \left(\pi-\frac{\pi}{6}\right)=\sin \frac{\pi}{6}=\frac{1}{2}
$$