Can rotations in a plane be described in terms of plane polar coordinates?

Question

It makes sense to talk about rotation through an angle $\theta$ in a plane in terms of Cartesian coordinates, and it can also be quantified as the matrix transformation $$\begin{pmatrix}x^\prime\\ y^\prime\end{pmatrix}=\begin{pmatrix}\cos\phi & \sin\phi\\-\sin\phi & \cos\phi\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}.$$

Can one describe a rotation in a plane in terms of plane polar coordinates? If yes, how? Sorry if it doesn't make sense.

Answer 1

$$\begin{pmatrix}\rho\\ \theta^\prime\end{pmatrix}=\begin{pmatrix}\rho\\ \theta_0\end{pmatrix} +\begin{pmatrix}0\\ \theta\end{pmatrix}$$

Where $\rho$ is the length of the given vector, $\theta_0$ is the angle of the given vector, $\theta$ is the angle of rotation, and $\theta^\prime$ is the angle of the given vector after rotation.