I was reading a document illustrating Euler angles representation of rotations. A Rotation that depends on 3 angles $\alpha$ around z axis, $\beta$ around y axis, and $\gamma$ around x axis is described as follows:
The author tried to evaluate a value for the angles given an arbitrary matrix. He first tried to evaluate the value of $\beta$ and stated that: $\cos^2 (\beta) = r_{11}^2 + r_{21}^2$ , as long as $\cos (\beta) \neq{0}$ .
I can't understand the need to impose this constraint on $\beta$ (given my basic knowledge of trigonometry). Is it because $\cos(90^{\circ})$ is zero and zero cannot be common factor?
Also, The author divided the possible values of the other angles $\alpha$ and $\gamma$ depending on whether $\beta$ is equal $\pm{90}$ degrees or not. Can you please refer to some other resources on this point?
When $\beta=\frac{\pi}{2}$ then the last rotation is parallel to the first rotation. Each successive rotation applies to the local intermediate axes. An orthogonal rotation about y rotates the z axis into the x axis, and the x axis onto the -z axis. So the subsequent rotation about x acts in the same direction as the first rotation about z. Hence the rotations cannot be decomposed from one another.