I need to prove that the special orthogonal group $\displaystyle SO(2,\mathbb{R}) is$ isomorphic to the circle group $\displaystyle S^1$.

I was considering using the case in which

$\displaystyle \varphi :A= \begin{bmatrix}

cos \alpha & -sin \alpha \\

sin \alpha & cos \alpha

\end{bmatrix} \mapsto (cos \alpha, sin \alpha)$