Finite-order rotations are dense in SO(2)

Parametrisation. $SO(2)=\{R_\theta:\theta\in\mathbb{R}\}$ with $R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$, and $R_\theta R_\varphi=R_{\theta+\varphi}$, $R_\theta=R_\varphi$ iff $\theta-\varphi\in2\pi\mathbb{Z}$. The map $\theta\mapsto R_\theta$ is a continuous surjection $\mathbb{R}\to SO(2)$.

Finite order. $R_\theta^m=R_{m\theta}=I$ iff $m\theta\in2\pi\mathbb{Z}$. So $R_\theta$ has finite order iff $\theta/2\pi\in\mathbb{Q}$.

Density. Let $\theta\in\mathbb{R}$ and $\varepsilon>0$. Rationals are dense, so there is $q\in\mathbb{Q}$ with $|2\pi q-\theta|<\varepsilon$. Then $R_{2\pi q}$ has finite order and, since $\theta\mapsto R_\theta$ is continuous, $R_{2\pi q}$ is arbitrarily close to $R_\theta$ when $\varepsilon$ is small.

Remark. These are the roots of unity $e^{2\pi iq}$ in the unit circle, to which $SO(2)$ is isomorphic. Finite-order elements are dense but of course not all of $SO(2)$ (it also contains rotations of infinite order). Lesson. Density in a compact group often comes from torsion elements.