# trigonometry

posted by jack

the angle of elevation of the top of the tower from the foot of a flagpole is twice the angle of elevation of the top of the flagpole from the foot of the tower. at the point midway between the tower and the flagpole, the angles of elevation to their tops are complimentary. if the tower and the flagpole are 120 feet apart, find the height of the flagpole

1. Steve

If we call
h = height of flagpole
t = height of tower
The four angles are x,y,a,b in the order named, then we have

h/120 = tanx
t/120 = tany
x = 2y
a+b = pi/2
that means that sin(a+b) = 1

That's a lot of equations to work with, but we know that

tanx = tan 2y = 2tany/(1-tan^2y)
sina cosb + cosa sinb = 1

so, that means

h/120 = 2(t/120)/(1-(t/120)^2)
(h/√(h^2+60^2))(60/√(t^2+60^2))+ (60/√(h^2+60^2))(t/√(t^2+60^2))=1

That looks horrible, but it simplifies quite a bit, and you wind up with

the flagpole is 90 ft

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