We will now consider some practical applications of trigonometry in the calculation of angles of elevation and angles of depression.

**What is angle of elevation?**

The **angle of elevation **is the angle between a
horizontal line from the observer and the line of sight to an object that is above the horizontal line.

In the diagram below, *AB* is the horizontal line. *θ* is the angle of elevation
from the observer at * A* to the object at * C*.

**What is angle of depression?**

The **angle of depression **is the angle between a horizontal line
from the observer and the line of sight to an object that is below the horizontal line.

In the diagram below, *PQ* is the horizontal line. *θ* is the angle of depression from
the observer at * P* to the object at * R*.

Find angles of depression and angles of elevation, and the relationship between them.

Angles of elevation and depression are equal.

* Example: *

In the diagram below, *AB* and *CD* are two vertical poles on horizontal ground.
Draw in the angle of elevation of *D* from *B* and the angle of depression of *C*
from *B*.

* Solution:*

Step 1: Draw a sketch of the situation.

Step 2: Mark in the given angle of elevation or depression.

Step 3: Use trigonometry to find the required missing length

* Example: *

Two poles on horizontal ground are 60 m apart. The shorter pole is 3 m high. The angle of depression of the top of the shorter pole from the top of the longer pole is 20˚. Sketch a diagram to represent the situation.

* Solution: *

**Step 1**: Draw two vertical lines to represent
the shorter pole and the longer pole.

** Step 2**: Draw a line from the top of the longer pole to the top
of the shorter pole. (This is the line of sight).

** Step 3**: Draw a
horizontal line to the top of the pole and mark in the angle of depression.

* Example: *

A man who is 2 m tall stands on horizontal ground 30 m from a tree. The angle of elevation of the top of the tree from his eyes is 28˚. Estimate the height of the tree.

*Solution:*

Let the height of the tree be *h*. Sketch a diagram to represent the situation.

tan 28˚ =

*h* – 2 = 30 tan 28˚

*h *= (30 ´ 0.5317) + 2 ← tan 28˚ = 0.5317

= 17.951

The height of the tree is approximatelyExamples:

1. An observer standing on top of a vertical 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 at the top of Building B is 24°, and the angle of depression of the base of Building B is 34°. How tall is each building?

Applications of Trig Functions: Solving for unknown distances.

Example:

How far away is a boat from the lighthouse if the lighthouse is 120° tall and the angle of depression to boat is 15°?

Example:

A boat is 500 meters from the base of a cliff. Jackie, who is sitting in the boat, notices that the angle of elevation to the top of the cliff is 32°15'. How high is the cliff? (Give your answer to the nearest meter).

Examples:

1) From a boat on the lake, the angle of elevation to the top of the cliff is 24°22'. If the base of the cliff is 747 feet from the boat, how high is the cliff (to the nearest foot)?

2) From a boat on the river below a dam, the angle of elevation to the top of the dam is 24°8'. If the dam is 2039 feet above the level of the river, how far is the boat from the base of the dam (to the nearest foot)?

3) If a man is just about to ski down a steep mountain. He estimates the angle of depression from where he is now to the flag at the bottom of the course to be 24°. He knows that he is 800 feet higher than the base of the course. How long is the path that he will ski? (Round to the nearest foot)

4a) A man at ground level measures the angle of elevation to the top of the building to be 67°. If at this point, he is 15 feet away from the building, what is the height of the building?

4b) The same man now stands atop a building. He measures the angle of elevation to the building across the street to be 27° and the angle of depression (to the base of the building across the street) to be 31°. If the two buildings are 50 feet apart, how tall is the taller building?

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