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