Trig Equations and identity

Question

A radio tower is located 250 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 41° and that the angle of depression to the bottom of the tower is 36°. How tall is the tower?

Answer 1

To do this problem the major triangle has angle 77° = "γ".

There are 2 right triangles. The top one has α=41°, β=49°; γ=90°; side b=250.

bottom 36° 54° 90 250.

a/sinα=b/sinβ=c because sin 90°=1.

Top: b/sinβ=250/(sin49°)=250/.755=331.253=a/sinα=a/(sin41)=a/.656; so .656*331.253=217.322'=up side "a".

Bottom: " =250/(sin54°) =309.017=a/(sin36) so a=.588*309.017= 181.635'=lower side "a".

+=398.957' height.

Answer 2

If you draw this out you'll see that the distance from the window to the tower is 250'. Using tan(θ) = opposite/adjacent ⇒ adjacent*tan(θ) = opposite. For the first angle of 41, we get 250*tan(41) = opposite1. Similarly, 250*tan(36) = opposite2. Then add the two opposites to get the height of the tower.