inequality with polynomials

Question

What are the polynomials $p(x)$ such that $|\sin(p(x))| \leq \frac{1}{2} \forall x \in \mathbb{R}$ ? I think that the only polinomials are of the type $p(x) = k$. That because any polinomial of deg >=1 tends to + or - infinity (or both) and so it will assume a value of the type $\pi/2 +k\pi$ that makes $|sin(p(x)) = 1|$. Do you think is good answer or should I motivate it better?