What an odd question. Anyway, you do not seem to have a useful idea. Let's see if I can make some sense of it...

I believe the intent is to choose ANY angle, not a specific one, but ALL of them in the given range. Perhaps examples would be useful.

P(X<0) = 1/2 and this corresponds to

P(X<-1) = 1/4 and this corresponds to

P(X<1) = 3/4 and this corresponds to

Essentially, what I am trying to define is a mapping from a uniform distribution on [-pi/2,pi/2] to the entire x-axis. Sounds like it might be related to a tangent, perhaps.