1. What Is an Angle of Depression?
The angle of depression is the angle formed when an observer looks downwards from a higher position to an object below eye level.
This angle is measured from the observer's horizontal line of sight to the line joining the observer's eye to the object below.
2. Understanding the Diagram (Explained in Words)
Imagine you are standing on a balcony and looking down at a car on the road. First, draw a horizontal line from your eye level.
From your eye, draw another line joining to the car below. The angle between the horizontal line and this downward line of sight is the angle of depression.
This naturally forms a right triangle when the vertical height of the balcony and the horizontal distance to the car are considered.
3. Relation Between Angle of Depression and Angle of Elevation
The horizontal line of sight is parallel to the ground. Therefore, using alternate interior angles:
\( \text{Angle of depression = Angle of elevation} \)
So, in most problems we convert the angle of depression into an angle of elevation for the triangle formed on the ground.
4. Applying Trigonometric Ratios
The right triangle formed allows us to apply trigonometric ratios. Usually, the vertical height and horizontal distance form the opposite and adjacent sides respectively, so we commonly use:
\( \tan \theta = \dfrac{\text{vertical height}}{\text{horizontal distance}} \)
Depending on the given values, sin and cos can also be used.
5. Example to Understand Angle of Depression
Example: A lifeguard spots a swimmer in the sea at an angle of depression of \(20^\circ\). If the lifeguard tower is 10 m high, find the horizontal distance to the swimmer.
\( \tan 20^\circ = \dfrac{10}{d} \)
\( d = \dfrac{10}{\tan 20^\circ} \)
This gives the distance from the base of the tower to the swimmer.