Application and Problem Solving using Trigonometry?

Question

A forest ranger 6 feet tall is standing on top of an obsevation tower 50 feet high. He spotted a deer at an angle of depression of 28 degrees.
a) How far is the deer from the base of the tower?
b) How far is the deer from the forest ranger?

Guys if you help me please explain how you got the answer.
Thank You

Answer 1

The tower (plus ranger), the ground, and the line of sight form a right triangle.

The angle of depression (28 degrees) is also the angle of ascension of the line of sight from the deer back up to the ranger. (They are alternate interior angles of the ground and a line parallel to the ground from the ranger's line of sight, crossed by the line of sight down to the deer.)

Given one angle, and one leg of the triangle (the height of the tower plus the ranger), you can compute the other leg (distance from the tower base to the deer), and the hypotenuse (distance from the ranger to the deer).

From the 28-degree angle, the opposite side is the height of the tower and ranger, the adjacent side is the distance along the ground from the deer to the base of the tower, and the hypotenuse is the line of sight from the ranger to the deer.

tan(angle) = opposite/adjacent
tan(28 degrees) = 56 feet / distance
0.53171 = 56 feet / distance
distance = 56 / 0.53171
distance = 105.32

The deer is about 105 feet from the base of the tower.

sin(angle) = opposite/hypotenuse
sin(28) = 56 feet / distance
0.4695 = 56 / distance
distance = 56 / 0.4695
distance = 119.28

The deer is about 119 feet from the ranger.

Answer 2

Draw the picture. Label the three (20º, 90º, and 70º) angles. Label the 56foot side.

You have the opposite side length. You need the lateral distance, and the slant distance. Use the trigonometric relations.

If you're trying to do these problems without drawing the simple diagrams, then you are making the problems unnecessarily difficult.