Trig question? Calculating angles.?

Question

Ahmad and Bob want to calculate the height of a tree. They measure the angles of elevation to the top of the tree. The angle of elevation from Bob, who is standing 60 m from the base of the tree, to the top of the tree, is 40˚. The angle of elevation from Ahmad, who is standing 80 m from Bob, is 50˚. The height of the tree is 50 m.

Calculate the angle between Ahmad and Bob.

Answer 1

I've started and stopped an answer to this several times, but I think I have it now.

We know that Ahmad's angle of elevation is 50°, and that the height of the tree is 50 m. So we can calculate Ahmad's distance from the tree using tangent.

tan 50° = 50/x
x = 50/tan 50° = 41.95 feet from the tree.

If you draw a birds-eye diagram of the situation, with Bob's position 60 m along the positive x axis, then the radius representing Ahmad's position is shorter than Bob's (42 m) and arranged in either the 1st or 4th quadrant so that a line connecting Bob and Ahmad must be 80 m, forming a triangle.

So we have 3 sides (41.95, 80, and 60) but no angles...Law of cosines seems best, and might as well do it for the angle we're looking for...which is the angle opposite the side of length 80.

c² = a² + b² - 2ab cos c
80² = 41.95² + 60² - 2(41.95)(60) cos c
6400 - 1760 - 3600 = 5035 cos c
1040 = 5035 cos c
0.2065 = cos c
c = 78.08°

So the angle between them is 78.08°.