math help needed

Question

A radio tower is located 425 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 34∘ and that the angle of depression to the bottom of the tower is 22∘. How tall is the tower?

Answer 1

One of the main trigonometric identities, for solving geometrical values, requires having a Side and two other Angles. Therefore, you contain Side-Angle-Angle. For this problem, you have a tower-to-building distance of 425 feet.

Now, imagine a point that has 2 lines being drawn out of it. The upper line is going up at an angle of 34 degrees. The lower line is going down at an angle of 22 degrees.

A horizontal line with length = 425. Adjacent angle with 34 and 22 degrees. The trig. function "tangent" can be used to relate the opposing triangle side length (or a section of the building height).

tan(θ) = opposite / adjacent

This step needs to be done twice, because the opposite triangle length is a portion of the Tower height.

tan(θ1) = height1 / distance

height1 = distance * tan(θ1)

height1 = 425 * tan(34 deg) = 297.6 feet

tan(θ2) = height2 / distance

height2 = distance * tan(θ2)

height2 = 425 * tan(22 deg) = 171.7 feet

total Tower Height = height1 + height2 = 297.6 + 171.7

Total Tower Height = 469.3 feet