Euclidean plan/Proper isometry/Rotation/Fact

Let

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

be a proper linear isometry.

Then ${}\varphi$ is a rotation,

and the describing matrix with respect to the standard basis has the form

${}D(\theta )={\begin{pmatrix}\operatorname {cos} \,\theta &-\operatorname {sin} \,\theta \\\operatorname {sin} \,\theta &\operatorname {cos} \,\theta \end{pmatrix}}\,$

with a uniquely determined rotation angle ${}\theta \in [0,2\pi [$.

Proof, Write another proof