Find all polynomials $p(x)$ satisfying condition

Question

Find all polynomials $p(x)$ such that $\sin{p(x)}$ is periodic.

I think that the condition for the solution is $\deg{p(x)}\le 1$, but i can't find a formal way to prove it.

Answer 1

You need to meet

$$p(x+T)=p(x)+2k\pi$$ or

$$p(x+T)-p(x)=2k\pi.$$

But for the LHS to be continuous, $k$ must be constant, hence $p(x)$ must indeed be linear.