Find Height of Bridge?

Question

Sightings to the same point at water level directly under the bridge are taken from each side of the 880-foot-long bridge.
The angle of depression on the left side under the bridge is 69.2 degrees; the angle of depression on the right side under the bridge is 65.5 degrees. How high is the bridge?

Answer 1

Call the Height H.
Let the distance be D from the left hand side along the bridge to the point directly above the point the sightings are aimed at
The left angle L=69.2 degrees
The right angle R=65.5 degrees

tan(L) = H/D and tan(R) = H/(880 - D)

Now eliminate the D between these equations and get

D = H/tan(L) and tan(R) = H/(880 - H/tan(L))

880tan(R) - H(tan(R)/tan(L)) = H

880tan(R) = H(1 + tan(R)/tan(L))

So H = 880tan(R)/(1 + tan(R)/tan(L))

tan(L) = 2.6325 , tan(R) = 2.1943 , tan(R)/tan(L) = 0.8335

H = 880(2.1943)/(1 + 0.8335) = 1053 feet