Angle of elevation and depression

Angles of Elevation and Depression

Angle of Elevation of an Object as seen by an Observer is the angle between the horizontal and the line from the Object to the Observer's eye (the line of sight). See the figure below for better understanding.

The angle of Elevation of the Object from the Observer is $\alpha^\circ$.

Angle of depression of an Object which is below the level of Observer is the angle between the horizontal and the Observer's line of sight. To have the angle of depression, an Object must be below the Observer's level. Consider an illustration below:

The angle of depression of the Object from the Observer is $\beta^\circ$.

Problems involving Angles of Elevation and Angles of Depression

Example 5

From the top of a vertical cliff 40 m high, the angle of a depression of an object that is level with the base of the cliff is $35^\circ$. How far is the Object from the base of the cliff?

Solution

We can represent the given information in diagram as here below:

Angle of depression = $35^\circ$

Angle ASO = Angle BOS (alternate Angles)

Consider Triangle ASO

$\tan 35^\circ = \frac{AO}{AS} = \frac{40 \text{ m}}{AS}$

$AS \times \tan 35^\circ = 40 \text{ m}$

$AS = \frac{40 \text{ m}}{\tan 35^\circ} = \frac{40 \text{ m}}{0.7} = 57.14 \text{ m}$

Therefore, the Object is 57.14 m from the base of the cliff.