R mod Z2 pi/Isomorphism theorem/Example

The group homomorphism

$\mathbb {R} \longrightarrow S^{1},t\longmapsto {\begin{pmatrix}\cos t\\\sin t\end{pmatrix}},$

is surjective. Due to the periodicity of the trigonometric functions, the kernel equals ${}\mathbb {Z} 2\pi$. By the isomorphism theorem, there exists a canonical isomorphism

${}\mathbb {R} /\mathbb {Z} 2\pi \cong S^{1}\,.$