The easiest way I know how to explain this is with a picture so here it is: http://pbrd.co/19RxqqV
For simplicity the origin is 0,0 and the angles are easy to work with but they could potentially be anything. The only things I really consistently know for this type of problem is the rotation of the circle, the origin and the radius and then of course the new degree/rotation of the circle.
This question, I think is similar but not really the exact same thing: Finding the coordinates on the edge of a circle
Thanks! I hope my question is clear enough.