Set of orthogonal $2\times 2$ matrices I need to do a proof for a paper, and I need to make sure of something. Is it true that the full set of orthogonal $2\times 2$ matrices can be represented by: $$\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\\ \sin(\theta) & \cos(\theta) \end{bmatrix}$$ or are there other possible rotations not accounted by this representation? Many thanks!

All rotations of the plane have this form, but the set of orthogonal matrices also contains axial symmetries, who have determinant $-1$. You need to add the matrices of the form $$\begin{bmatrix} \cos(\theta) & \sin(\theta) \\\ \sin(\theta) &-\cos(\theta) \end{bmatrix}.$$ For a proof that any orthogonal matrice has one of these forms, see this question.