Geometric problem in spherical coordinates

Geometric problem in spherical coordinates

I have the following problem.
I wanted to find a model for a line in spherical coordinates!
At first I had a line that had a constant z-coordinate $z_c$. so i just
said: whenever $$z_c= r \cos(\theta)$$ a point is on that line.
Now I had to rotate this line by an angle beta. But now I do not know
anymore how to find a similar expression for this line, like I had
earlier. Does anybody know? ( I should stress that the origin of the
spherical coordinate system is in the CENTER of the sphere) Image that
describes my problem.
Maybe my picture gives you a wrong idea, since the center of the rotation
is the center of the sphere. so i should have better drawn the angle
between a new and the old z-axis.