Why is the determinant of a $2\times2$ rotation matrix equal to $1$?

Question

I've seen Why is the determinant of a rotation matrix equal to 1?

I do get that no matter what value you put for theta you will end up getting $1$, but I would like to explain it using the cos and sin graphs (do some reasoning based on graphs).

I think there might be a way to proof this using the graphs.

Thanks.

Answer 1

The general form of a $2 \times 2$ rotation matrix is:

$\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}$

By definition, the determinant of this matrix is $\cos^2{\theta} + \sin^2{\theta}$. This is equal to $1$ by the Pythagorean trigonometric identity.