I think this will work: imagine there's a square containing your larger circle. If the large circle has radius r, then the square has corners at (r,r), (-r,r), (r,-r), (-r,-r). The polygon can be made up of the origin, the two endpoints of the large arc, and some of the corners of the square.

For example, if your two angles are 30 and 60, then the polygon is:
(0,0), (x30,y30), (r,-r), (x60,y60).

If the angles are in different quarters of the circle, then you need to include more of the corners from the square. ex: if the angles are 30 and 120:
(0,0), (x30,y30), (r,-r), (-r,-r), (x120,y120)

Or angles of 120 and 290:
(0,0), (x120,y120), (-r,-r), (-r,r), (r,r), (x290,y290)