Special orthogonal group in two dimensions ($\mathrm{SO}(2)\cong U(1)$)

Every real orthogonal two-by-two matrix of determinant one has the unique formIndeed the first column is a unit vector, the perpendicular second column has one of two signs, and positive determinant selects the displayed one. The map $z\mapsto R(z)$ from the circle group is bijective and satisfies $R(zw)=R(z)R(w)$ by direct multiplication, proving a group isomorphism. Writing $z=e^{i\theta }$ parametrizes these matrices as planar rotations, with the angle defined modulo $2\pi$.