Question

angle of depression
when looking down to an object, the angle of depression is formed by an observers line of sight and a horizontal line.
the angle of depression is congruent to the angle of elevation because they are alternate interior angles.
examples
directions: draw and label a diagram, then solve for the missing part. round to the nearest tenth.

1. a lifeguard is sitting on a platform, looking down at a swimmer in the water. if the lifeguards line of sight is 8 feet above the ground and the angle of depression to the swimmer is 18°, how far away is the swimmer from the lifeguard?
2. a pilot in a helicopter spots a landing pad below. if the angle of depression is 73° and the horizontal distance to the pad is 1200 feet, what is the altitude of the helicopter?
3. building a is 480 feet tall and building b is 654 feet tall. if the angle of depression from the top of building b to the top of building a is 42°, how far apart are the buildings?
4. zack is standing at the top of a lookout tower and spots a water fountain below. if the lookout tower is 75 feet tall and the angle of depression is 28°, what is the horizontal distance between zack and the water fountain?

Explanation:

Step1: Recall tangent - ratio formula

In a right - triangle formed by the angle of depression, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Solve problem 5

The height (opposite side) is 8 feet and the angle of depression $\theta = 18^{\circ}$. Let the distance from the lifeguard to the swimmer be $d$. Then $\tan(18^{\circ})=\frac{8}{d}$, so $d=\frac{8}{\tan(18^{\circ})}\approx\frac{8}{0.3249}\approx24.6$ feet.

Step3: Solve problem 6

The angle of depression $\theta = 73^{\circ}$ and the horizontal distance (adjacent side) is 1200 feet. Let the altitude (opposite side) be $h$. Then $\tan(73^{\circ})=\frac{h}{1200}$, so $h = 1200\times\tan(73^{\circ})\approx1200\times3.2709\approx3925.1$ feet.

Step4: Solve problem 7

The height difference between the two buildings is $654 - 480=174$ feet. The angle of depression $\theta = 42^{\circ}$. Let the distance between the buildings be $x$. Then $\tan(42^{\circ})=\frac{174}{x}$, so $x=\frac{174}{\tan(42^{\circ})}\approx\frac{174}{0.9004}\approx193.3$ feet.

Step5: Solve problem 8

The height of the tower (opposite side) is 75 feet and the angle of depression $\theta = 28^{\circ}$. Let the horizontal distance be $y$. Then $\tan(28^{\circ})=\frac{75}{y}$, so $y=\frac{75}{\tan(28^{\circ})}\approx\frac{75}{0.5317}\approx141.0$ feet.

Answer:

1. Approximately 24.6 feet
2. Approximately 3925.1 feet
3. Approximately 193.3 feet
4. Approximately 141.0 feet