AIME I 2026 (Problem 5)

A plane contains points

A

and

B

with

AB=1

. Point

A

is rotated in the plane counterclockwise through an acute angle

\theta

around point

B

to point

A'

. Then

B

is rotated in the plane clockwise through angle

\theta

around point

A'

to point

B'

. Suppose

A B'=\frac43

. The value of

\cos\theta

can be written as

\frac mn

, where

m

and

n

are relatively prime positive integers. Find

m+n