A 2 by 2 matrix is given with irrational elements and we can derive that the top left element is equal to $\cos(θ)$, and the bottom left element is equal to $\sin(θ)$.
How could we find the value of $θ$ and therefore the angle that the rotation does. I think by using the signs of the trigonometric functions in each quadrant.
As shown in this answer, a nice formula for getting $\theta$ from $\sin(\theta)$ and $\cos(\theta)$ that takes care of quadrants is $$ \theta=2\tan^{-1}\left(\frac{\sin(\theta)}{1+\cos(\theta)}\right)\tag1 $$ The only ambiguity might be when $\cos(\theta)=-1$, where we get $2\tan^{-1}\left(\frac00\right)$. In this case, $\theta$ could be $\pm\pi$.
