# Angles Of Elevation And Depression

In these lessons, we will study how to find the angles of elevation and depression using trigonometry.

### What Is Angle Of Elevation And Depression?

The angle of elevation is the angle between a horizontal line and the line joining the observer’s eye to some object above the horizontal line.

The angle of depression is the angle between a horizontal line and the line joining the observer’s eye to some object beneath the horizontal line.

In real world situations, we often discuss the angles of elevation and depression. The angles of elevation and depression is used often in word problems, especially those involving a person’s line of sight as they look up at an object.

This video will explain what is the angle of elevation and what is the angle of depression.
It will also give some examples of how to use the angles of elevation and angles of depression.

Angle of Elevation/Depression Story Problems

Examples:
The angle of elevation from point A to the top of a cliff is 38 degrees. If point A is 80 feet from the base of the cliff, how high is the cliff? Let x be the height of the cliff.

1. From the top of the tower, the angle of depression to a stake on the ground is 72 degrees. The top of the tower is 80 feet above the ground. How far is the stake from the foot of the tower?
2. A tree 40 feet high casts a shadow 58 feet long. Find the measure of elevation of the sun.
3. A ladder leaning against a house makes an angle of 60 degrees with the ground. The foot of the ladder is 7 feet from the foundation of the house. How long is the ladder?
4. A balloon on a 40-foot string makes an angle of 50 degrees with the ground. How high above the ground is the balloon if the hand of the person holding the balloon is 6 feet above the ground?

Angles of Elevation and Depression

This video goes through three word problems that require trigonometry to calculate side lengths or angle measures in right triangles.

Examples:

1. A salvage ship uses sonar to determine that the angle of depression to the wreck on the ocean floor is 13.25 degrees. The depth chart shows that the ocean floor is 40 meters below the surface. How far must the diver lowered from the salvage ship walk along the ocean floor to reach the wreck?
2. Find the angle of elevation of the sun when a 7.6 m flag pole casts a 18.2 m shadow.
3. Olivia is in a lighthouse on a cliff. The top of the cliff is 110 feet above the water and Olivia is in a place in the lighthouse 85 feet above the top of the cliff. She observes two sailboats due east of the lighthouse. The angle of depression to the two boats are 33 degrees and 57 degrees. Find the distance between the two boats.

How to solve application problems using angles of depression and elevation?

Examples:

1. A homeowner is to construct a ramp to his front door to make it wheelchair-accessible. How long is the ramp if the door is 4ft above the ground level and the angle of elevation is 20°
2. For a laser light show at an amusement park, the laser beam directed from the top of a 30 ft building is to reflect from an object that is 100 ft away from a point directly below the location of the laser. What is the angle of depression from the laser to the reflecting object?

Angle of Elevation/Angle of Depression Problems

Examples:

1. An observer standing on the top of a vertical cliff spots a house in the adjacent valley at an angle of depression of 12°. The cliff is 60m tall. How far is the house from the base of the cliff?
2. Buildings A and B are across the street from each other, 35m apart. From a point on the roof of Building A, the angle of elevation of the roof of Building B is 24°, and the angle of depression of the base of Building B is 34°. How tall is each building?

Angle of elevation part 1

Examples:
The passengers on a ship, 3700 m from the base of a 1800 m high cliff are able to see the Byron Bay lighthouse on top of a cliff. Find the angle of elevation from the ship to the top of the cliff.

Angle of elevation part 2

Determining the Angle of Elevation

How to determine the angle of elevation?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 