## problem in finding polar angle in spherical coordinate system

As the aboved image, polar angles in spherical coordinate system are not the same in order to plot projected points, situated on the X-Z plane, in the straight line.

Known variables of the problem are as followed:
1)length of all black segment the same
2)length of all blue segment the same
3)all r values are given
4)theta' given (theta' is the polar angle for point which occur on the Y-Z plane.) In the other way, theta is the angle between the positive Y axis and the red segment.

Please help in finding theta'' , theta''' and so on (may be relative to theta') depending on given facts. thanks in advance!