Euclidean plan/Proper isometry/Rotation/Fact

Let

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

be a proper linear isometry.

Then ${\displaystyle {}\varphi }$ is a rotation,

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

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

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