Two rays with common endpoint O form a 30 degree angle. Point A lies on one ray, point B on the other ray, and AB=1. What is the maximum possible length of OB?

Answer 1

If we call the lengths OA and OB x and y, then we have, using the law of cosines,

x^2+y^2-√3 xy = 1

That is just an ellipse with its extreme values at x=2 or y=2.

So, the maximum value of OB is 2.