6.6 The Angle of Elevation and the Angle of Depression

6.6 The Angle of Elevation and the Angle of Depression

The angle between the horizontal line $AD$ (eye level) and a line from the eye $A$ to the top of building ($B$) is called an angle of elevation.

[Diagram showing angle of elevation and angle of depression]

The angle between the horizontal line $AD$ (eye level) and the line from the eye '$A$' to the foot of the building ($C$) is called the angle of depression.

Example 19

The angle of elevation of the top of a pole $40$ m high is $60^\circ$ when seen from a point on the ground level. Find the distance of the point from the foot of the pole.

Solution: In the triangle $ABC$, we have

$m\overline{BC} = 40 \text{ m}$
$m\angle A = 60^\circ$

Let $m\overline{AB} = x$ (the point $B$ is the foot of the pole $BC$)

In right angled triangle $ABC$,

Hence, distance of the point from the foot of the pole $= 23.09$ m

Example 20

From the top of a lookout tower, the angle of depression of a building has on the ground level of $45^\circ$. How far is a man on the ground from the tower, if the height of the tower is $30$ m?

Solution: In the triangle $ABC, AB$ is the tower and point $C$ is the position of man. We have

$m\overline{AB} = 30 \text{ m}$
$m\angle CAD = m\angle C = 45^\circ$
$m\overline{BC} = x \text{ m} = ?$

Let $x$ be the base of right angled triangle $ABC$,

Hence, man is $30$ m far from the tower.