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. * q* 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. * q* is the angle of depression from
the observer at * P* to the object at * R*.

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

* 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 approximately **17.95 m**.

The following videos show how to solve word problems using tangent and the angle of elevation.

Example:

Neil sees a rocket at an angle of elevation of 11 degrees.
If Neil is located 5 miles from the rocket's launchpad, how high is the rocket? Round your answer to the
nearest hundredth.

Angles of Elevation and Depression

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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