How do I calculate the angle between the tangent to an outer circumference and a line passing through a specific point on the inner circumference?

Question

I would like to calculate the angles indicated by the arrows in my sketch. All I know is the segment AB between the two circumferences and the angles Theta and Theta'. The two circles are concentric, so I also know the distance between B and the centre. Is there a quick clever procedure? I can't think of one.

Answer 1

Apply sine rule to triangle $AOC$: $$ {\sin\beta\over r}={\sin\alpha\over r+d}. $$