@Semimclassical: I tried to solve it this way before reading the questions/answers mentioned, but it doesn't seem to give right results. If anyone can see something obviously wrong, I'd be happy to learn where I misstepped:
α is an angle in the XY plane (angle from the X axis), β is an angle between the XY plane and the Z axis. Find the corresponding unit vector v.
We know that:
x^2 + y^2 + z^2 = 1
Let f be the length of the projection of v into the XY plane. Then:
sqrt(f) = sqrt(x^2 + y^2)