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

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*.

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*.

How to define an angle of elevation or an angle of depression.

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:*

* 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 video shows 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

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 metre).