Angles II

Grade 12 Math - Angles of Elevation and Depression

Lesson Objectives

Understand and identify angles of elevation and depression.
Apply trigonometric ratios to solve elevation and depression problems.
Interpret real-world scenarios using angles and trigonometry.

Lesson Introduction

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

An angle of depression is the angle between the horizontal and the line of sight when looking down at an object.

We often use right-angled triangles and trigonometric ratios like sine, cosine, and tangent to solve such problems.

Core Lesson Content

Using Trigonometric Ratios

Given a right triangle:

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

Worked Examples

Example 1: A person 1.5 m tall sees the top of a tower at an angle of elevation of

30^\circ

. The tower is 50 m away. Find the height of the tower.

\tan(30^\circ) = \frac{h - 1.5}{50}

→

h = 50 \cdot \tan(30^\circ) + 1.5 \approx 28.4 \text{ m}

Example 2: A boat is 100 m from the base of a cliff. The angle of elevation to the top of the cliff is

45^\circ

. Find the height of the cliff.

\tan(45^\circ) = \frac{h}{100} \Rightarrow h = 100

Example 3: From a lighthouse 120 m above sea level, the angle of depression to a boat is

25^\circ

. How far is the boat from the base of the lighthouse?

\tan(25^\circ) = \frac{120}{x} \Rightarrow x = \frac{120}{\tan(25^\circ)} \approx 257.4 \text{ m}

Example 4: A kite is flying at a height of 80 m. The string makes an angle of

60^\circ

with the ground. Find the length of the string.

\sin(60^\circ) = \frac{80}{\text{string}} \Rightarrow \text{string} = \frac{80}{\sin(60^\circ)} \approx 92.4 \text{ m}

Example 5: A man on a hill 200 m high observes a car at an angle of depression of

35^\circ

. Find the horizontal distance from the car to the hill.

\tan(35^\circ) = \frac{200}{x} \Rightarrow x = \frac{200}{\tan(35^\circ)} \approx 285.7 \text{ m}

Example 6: A balloon is flying directly above a point on the ground. The angle of elevation from a point 300 m away is

40^\circ

. Find the height.

\tan(40^\circ) = \frac{h}{300} \Rightarrow h = 300 \cdot \tan(40^\circ) \approx 251.2 \text{ m}

Example 7: A drone is observed from two points on a straight road 400 m apart. The angles of elevation are

30^\circ

and

60^\circ

. Find the height. (Harder problem; requires trigonometric system of equations.)

Example 8: From the top of a pole 15 m high, the angle of depression to a point on the ground is

50^\circ

. Find the distance from the point to the base.

\tan(50^\circ) = \frac{15}{x} \Rightarrow x = \frac{15}{\tan(50^\circ)} \approx 12.6 \text{ m}

Example 9: The top of a building is viewed at an angle of elevation of

53^\circ

from a point 30 m away. The observer is 1.6 m tall. Find the building height.

h = 30 \cdot \tan(53^\circ) + 1.6 \approx 41.1 \text{ m}

Example 10: A cable from the top of a tower makes an angle of

60^\circ

with the ground and touches the ground 25 m from the base. Find the tower height.

\tan(60^\circ) = \frac{h}{25} \Rightarrow h = 25 \cdot \tan(60^\circ) \approx 43.3 \text{ m}

Exercises

[WAEC] The angle of elevation to a tree is $40^\circ$ from 20 m away. Find the tree height. [Past Question]
A lighthouse is 30 m tall. The angle of depression to a boat is $25^\circ$ . How far is the boat from the base?
[NECO] A ladder leans against a wall at $65^\circ$ . The base is 5 m from the wall. Find the height reached. [Past Question]
The angle of elevation to a mountain top is $28^\circ$ from 500 m away. Find the mountain height.
[WAEC] From a building 60 m tall, the angle of depression to a point on the ground is $37^\circ$ . Find the distance to the point. [Past Question]
A drone hovers at an angle of elevation $45^\circ$ from 150 m away. Find the drone's height.
[NECO] The string of a kite is 40 m long and makes an angle of $35^\circ$ with the ground. Find the height. [Past Question]
A building is viewed at $55^\circ$ angle of elevation from 25 m. Observer is 1.5 m tall. Find the building height.
From a tower, the angle of depression to a ship is $20^\circ$ . Tower height is 40 m. Find distance to ship.
A hot air balloon rises at an angle of $70^\circ$ from ground level. After 100 m along the path, how high is it?

Conclusion/Recap

Understanding angles of elevation and depression allows us to solve practical problems involving height and distance using trigonometry. Accurate diagram drawing and careful use of trigonometric ratios are key in mastering these concepts. Practice regularly to build confidence and fluency.