7.1 Extracting Euler angles from a rotation matrix

We discussed different angle set conventions in class. One common convention is the set of X-Y-Z Euler Angles. With this convention, the B frame is first rotated about  by angle , then rotated about  by angle , and then finally rotated about  by angle . This angle set has the following form.

Using the examples from class as a primer, and using the atan2 function, please derive the formulas necessary to extract the values of angles , , and  from the above rotation matrix. For simplicity, you may assume that no angles equal either 0° or 90°. Please express your answers in terms of elements of the rotation matrix (, , etc.) and clearly state any assumptions that you make.