Uhhh I haven't taken physics AP but don't you just do law of cosines?
Assume |A| = |B| = 1 (problem doesn't state otherwise, so why not)
Then, by Law of Cosines:
|A+B| = sqrt(1 + 1 - 2 cos θ
= sqrt(2 - 2 cos θ
|A-B| = sqrt(1 + 1 - 2 cos (180-θ) = sqrt(2 + 2 cos θ
|A+B| = n*|A-B|
sqrt(2 - 2 cos θ = n*(sqrt(2 + 2 cos θ)
2 - 2 cos θ = 2n^2 + 2n^2 cos θ
2 - 2n^2 = (2 + 2n^2) cos θ
cos θ = (1 - n^2)/(1 + n^2)
Probably able to be simplified, but I can't be bothered.
Assume |A| = |B| = 1 (problem doesn't state otherwise, so why not)
Then, by Law of Cosines:
|A+B| = sqrt(1 + 1 - 2 cos θ
= sqrt(2 - 2 cos θ|A-B| = sqrt(1 + 1 - 2 cos (180-θ
) = sqrt(2 + 2 cos θ|A+B| = n*|A-B|
sqrt(2 - 2 cos θ
= n*(sqrt(2 + 2 cos θ)2 - 2 cos θ = 2n^2 + 2n^2 cos θ
2 - 2n^2 = (2 + 2n^2) cos θ
cos θ = (1 - n^2)/(1 + n^2)
Probably able to be simplified, but I can't be bothered.