I've had to do this before where the shaft was connected to a continuously rotatable pot. That's a normal pot with two wipers 1/2 turn apart.
I found that the pot was reasonably linear, but not good enough for what we were trying to do. I added a calibration procedure and ended up doing a piece-wise linear lookup. If I remember right for that pot and the accuracy we wanted, a calibration point every 20 degrees or so seemed to be good enough. Any one pot stayed pretty consistant once calibrated. I'm sure they would wear more over time and require re-calibration, but initially a single calibration adjusted the system well enough to ship. We did specify calibration at regular intervals when other maintanence was already scheduled to be performed.
Long term pots are probably not a good measure of angle unless you recalibrate regularly after a cerain amount of use. The wiper rubs against the slider, which eventually wears off material and changes the resistance.
If you need long term consistancy, get a rotary encoder.