Verify that $x\mapsto (\cos(x),\sin(x))$ from the real line to the unit circle is an open map.

Hint. Open arc can be broken up into the union of "small" open arcs. Such small that each one can easily be represented in your way.

Let $I$ be an open set of $\Bbb R$. Then the set $O=\{ r(\cos x, \sin x) \ , \ r>0 ; x\in I \}$ is an open subset of $\Bbb R^2$ , we have

$p(I)=O\cap S^1$ open subset of $S^1$.

EDIT: $O$ is open: Let $f:\Bbb R^2\to \Bbb R^2$ defined by $f(r,x)=r(\cos x,\sin x)$ it is clear that $f$ is continuous , since $\Bbb R_+^*\times I$ is an open of $\Bbb R^2$ (product of two open) then $O=f^{-1}(\Bbb R_+^*\times I)$ is open of $\Bbb R^2$.