A plane contains points A and B with AB=1. Point A is rotated in the plane counterclockwise through an acute angle θ around point B to point A′. Then B is rotated in the plane clockwise through angle θ around point A′ to point B′. Suppose AB′=34. The value of cosθ can be written as nm, where m and n are relatively prime positive integers. Find m+n.
Note that A′B forms an angle of θ with the line AB, so when we do our second clockwise rotation, we undo the effects of our counterclockwise rotation. Therefore, A′B′ is simply parallel to AB, and since it is the same length, ABB′A′ is a parallelogram.
Let A=(1,0) and B=(0,0), so A′=(cosθ,sinθ). Since we have a parallelogram, B′=(cosθ−1,sinθ).
