Rotating frames with euler angles

I need to rotate a starting frame (classic xyz cartesian frame) multiple times, using Euler angles, and at the end I need to calculate the angle between the initial and the final Z axys. Specifically, I need to perform these clockwise rotations (of a certain degree value), each time the next rotation has to be performed on the newly generated coordinate system:

1. rotation of angle theta1 about the Y axis
2. rotation of angle theta2 about the Z axis
3. rotation of angle theta3 about the Y axis
4. rotation of angle theta4 about the Z axis
5. rotation of angle theta5 about the Y axis
6. calculate the angle between the initial and the final Z axis

I'm pretty new to this and despite my effort I didn't understand how to correctly proceed with this task. If this issue has already been answered elsewhere, please point me there. Many thanks!