The angles of elevation theta and alpha to an airplane from the airport control tower and from an observation post 2 miles away are being continuously monitored. If theta is 42 degrees when alpha is 81 degrees, how far is the plane from the observation post?

Answer 1

You need to clarify the diagram. Label the airplane A, the tower T and the post P. The plane is at altitude h, with AP=t and AT=p. Point Q is directly below the plane. The distance PQ=q and thus TQ = 2-q.

If the plane is between the tower and the post, then

h/q = tanα
h/(2-q) = tanθ

Eliminate h and plug in the angles, and to get the desired distance t,

q tan 81° = (2-q)tan42°
6.31q = 0.90(2-q)
q = 0.249
so, h = 0.249*6.31 = 1.571

t^2 = .249^2 + 1.571^2
t = 1.59 miles

If the plane is not between the two measurement locations, then you need to set up different triangles.