The $\frac{n}{3} - \big{[}{\frac{n}{3}}\big{]}$ part only takes three values. $0$, $\frac{1}{3}$ and $\frac{2}{3}$. If you want to prove this set n to be equal to $3k$, $3k+1$ and $3k+2$ where $k$ is a non negative integer and see what this gives you.
Then just find which is larger. $\sin(0)$, $\sin(\tfrac{\pi}{3})$ or $\sin(\tfrac{2\pi}{3})$.