Euclidean plane/Improper isometry/Axis reflection/Fact/Proof

Proof

We consider

$\varphi \circ {\begin{pmatrix}1&0\\0&-1\end{pmatrix}},$

which is by the multiplication theorem for the determinant a proper isometry. Because of fact, there exists a uniquely determined angle ${}\alpha \in [0,\pi [$ such that

${}\varphi \circ {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}={\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}\,.$

Therefore,

${}\varphi ={\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}\circ {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}={\begin{pmatrix}\cos \alpha &\sin \alpha \\\sin \alpha &-\cos \alpha \end{pmatrix}}\,.$

To fact