abc is a right-angled trangle. All its angles and sides are known.

c is also the center of a circle in the same plane as b and p (but not a).

p is a point moving around on the perifery of the circle (all other points are fixed).

radius cp is known.

Given an angle v (=pac), how do determine the position of p, for example in terms of angle w (=pcb)?

Given is also which half ("right/left" as cut by bc) of the circle p is on.

I notice that for every v, there are two possible p with different w.

There must be some simple function w = f(v), right? Even if there are multiple solutions for each v.

I hope the illustrations above clearify this 3D situation!

I've tried law of sines on triangle acp because cp, ac and v are known.

This gives two solutions for p's position (in the end, for w).

One where angle apc is obtruse and one where it is acute. That's fine, but I fail to make it add up.

Are maybe the two solutions of the law of sines invalid because of the three dimensions involved here?

It assumes that ap here can have two lengths, and thus angle apc also can have two values. But I don't just change the length of ap, I move p to another position in another plane than the triangle in question!