Uniform Convergence of $(\sin x)^n$

Hint: find the pointwise limit of this sequence of functions on $[0,\pi]$. If a sequence is uniformly convergent, then its limit is a continuous function. Is it continuous in this case?

It is not uniformly convergent: At $\pi/2$ we have $\sin(\pi/2)=1$ and thus $$ \sin(\pi/2)^n\to 1 $$ What happens everywhere else where $\sin(x)<1$?