I am reading about unitary matrices in Horn and Johnson's Matrix Analysis. On page 68, the exercise asks, letting $T(\theta)=\begin{pmatrix}\cos\theta & \sin\theta \\-\sin\theta & \cos\theta \\ \end{pmatrix}$ where $\theta$ is a real parameter:
If $U\in M_{2}(\mathbb{R})$ is a unitary matrix, show that $U$ is real orthogonal iff $U=T(\theta)$ or $U=\begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}T(\theta)$
Since $U$ is unitary, it we know that it is an isometry, i.e. if $x\in \mathbb{R}$, $Ux=y \implies \|Ux\| = \|y\|$. But what does does this have to do with sine and cosine functions?