833. How many different values does the expression $\sin \frac{k \pi}{3}$ take when $k$ takes all integer values from 0 to $+\infty$?
833. Let $q$ be the integer quotient of the division of $k$ by $3$, and $r$ the remainder, which lies between -2 and 2, then $k=3 q+r$.

We have $\sin \frac{k \pi}{3}=\sin \left(q \pi+\frac{r \pi}{3}\right)=\sin \left( \pm \frac{r \pi}{3}\right)$. Everything reduces to the number of different values of the seven members of the series

$$
\sin \left(-\frac{2 \pi}{3}\right), \sin \left(-\frac{\pi}{3}\right), \sin 0^{\circ}, \sin \frac{\pi}{3}, \sin \frac{2 \pi}{3}
$$

In this series, the first two and the last two members are equal; as a result, we have 3 different values, two of which differ only in sign.