problem in finding polar angle in spherical coordinate system

http://img704.imageshack.us/img704/8079/test1yc.th.gif

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!