Angle of Elevation and Depression

Lesson Objectives

By the end of this lesson, students should be able to:

Define angle of elevation and angle of depression.
Interpret angle problems using right-angle triangles.
Solve real-life problems involving elevation and depression using trigonometric ratios.
Apply trigonometric ratios (sine, cosine, tangent) in calculating heights and distances.

Introduction

Have you ever looked up at the top of a tree or a building and wondered how high it is? Or looked down from a tall tower at a car on the road? The angle your eyes make with the ground while looking up or down is what we call the angle of elevation or angle of depression. In this lesson, we’ll learn how to use these angles in real-life calculations.

Core Lesson Content

The angle of elevation is the angle formed between the horizontal and the line of sight when looking up at an object.
The angle of depression is the angle formed between the horizontal and the line of sight when looking down at an object.

These problems are usually modeled using right-angled triangles. We apply trigonometric ratios:

$\sin(\theta) = \frac{opposite}{hypotenuse}$
$\cos(\theta) = \frac{adjacent}{hypotenuse}$
$\tan(\theta) = \frac{opposite}{adjacent}$

Worked Example

Example 1 (Basic): A ladder leans against a wall making an angle of

60^\circ

with the ground. If the ladder is 10 m long, how high does it reach up the wall?
Use

\sin(60^\circ) = \frac{opposite}{hypotenuse}

\Rightarrow \sin(60^\circ) = \frac{h}{10}

\Rightarrow h = 10 \times \sin(60^\circ)

= 10 \times \frac{\sqrt{3}}{2}

\approx 8.66 \, \text{m}

The ladder reaches approximately 8.66 m up the wall.

Example 2 (Moderate): From a point 50 m away from the base of a tower, the angle of elevation to the top of the tower is

45^\circ

. Find the height of the tower.

\tan(45^\circ) = \frac{h}{50}

\Rightarrow h = 50 \times \tan(45^\circ) = 50 \times 1 = 50

The height of the tower is 50 m.

Example 3 (Challenging): A ship is 100 m away from a cliff. The angle of elevation to the top of the cliff is

60^\circ

. Find the height of the cliff.

h = 100 \times \tan(60^\circ)

= 100 \times \sqrt{3}

\approx 173.21 \, \text{m}

Height of the cliff ≈ 173.21 m

Example 4 (Difficult): From a tower 50 m high, the angle of depression to a car on the ground is

30^\circ

. How far is the car from the base of the tower?
Let the distance be

x

\tan(30^\circ) = \frac{50}{x}

\Rightarrow x = \frac{50}{\tan(30^\circ)} = 50 \times \sqrt{3}

\approx 86.60 \, \text{m}

Example 5 (Real-world): A drone flying at 120 m above the ground observes a person on the ground with an angle of depression of

40^\circ

. How far is the person horizontally from the drone?

\tan(40^\circ) = \frac{120}{x}

\Rightarrow x = \frac{120}{\tan(40^\circ)}

\approx \frac{120}{0.8391} \approx 143.02 \, \text{m}

Exercises

$\tan(45^\circ) = \frac{h}{20}$ , find $h$ .
A kite is flying at a height of 60 m. If the string makes an angle of $30^\circ$ with the horizontal, find the length of the string.
$\sin(60^\circ) = \frac{h}{15}$ , find $h$ .
[WAEC] A building is 70 m tall. From a point on the ground, the angle of elevation to the top is $45^\circ$ . Find the distance of the point from the base. (Past Question)
A tree casts a shadow 12 m long when the sun’s angle of elevation is $45^\circ$ . Find the height of the tree.
[NECO] From a helicopter 300 m above sea level, a rescue boat is spotted at an angle of depression of $25^\circ$ . Find the horizontal distance to the boat. (Past Question)
An observer is 50 m away from a building and sees the top at an angle of $60^\circ$ . Find the building's height.
[JAMB] A boy looks up at the top of a pole with an angle of elevation of $30^\circ$ . If the pole is 10 m tall, how far is the boy from the pole? (Past Question)
A ladder 10 m long makes an angle of $60^\circ$ with the ground. How high does it reach up the wall?
Find the angle of elevation to the top of a tower 100 m high from a point 173.2 m away.

Conclusion / Recap

Angle of elevation and depression help solve height and distance problems in real life. With right-angle triangles and basic trigonometry, we can find unknown heights or distances using $\sin, \cos, \tan$ . In the next lesson, we’ll apply these concepts to navigation and indirect measurement problems.