Angles of Elevation/Depression Revision Notes for Leaving Cert Mathematics

Angles of Elevation/Depression

Overview

Angles of elevation and depression are used to describe the angles formed by a line of sight relative to a horizontal plane. These concepts are commonly applied in trigonometry for solving problems involving distances and heights.

Angle of Elevation

The angle formed between a horizontal line and the line of sight looking upward to an object.

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Example: Viewing the top of a building from a point on the ground.

Angle of Depression

The angle formed between a horizontal line and the line of sight looking downward to an object.

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Example: Looking at a boat from the top of a cliff.

Solving Problems Involving Angles of Elevation and Depression

Right Triangle Relationships:

Both angles rely on the properties of right triangles, often involving trigonometric ratios $(\sin, \cos, \tan)$
Use $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ for calculating heights or distances.

Horizontal Reference Line:

The horizontal line acts as the baseline for measuring these angles.
Ensure diagrams are drawn accurately to visualise the problem.

Units and Modes:

Ensure the calculator is in the correct mode (degrees or radians) depending on the given angle.

Worked Examples

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Example 1: Angle of Elevation

Problem: An observer standing $50$ m away from the base of a tower observes the top of the tower at an angle of elevation of $30^\circ$ .

Find the height of the tower.

Solution:

Step 1: Use the formula

\tan \theta = \frac{\text{opposite}}{\text{adjacent}}

Step 2: Substitute values and calculate

\tan 30^\circ = \frac{\text{height}}{50}

\text{height} = 50 \times \tan 30^\circ = 50 \times 0.577 = 28.85 \, \text{m}.

Answer: The height of the tower is approximately $28.85 \, \text{m}$

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Example 2: Angle of Depression

Problem: A person at the top of a $40$ m tall cliff observes a boat at an angle of depression of $45^\circ$ .

How far is the boat from the base of the cliff?

Solution:

Step 1: Use the formula:

\tan \theta = \frac{\text{opposite}}{\text{adjacent}}

Step 2: Substitute values and calculate

\tan 45^\circ = \frac{40}{\text{distance}}

\text{distance} = \frac{40}{\tan 45^\circ} = \frac{40}{1} = 40 \, \text{m}

Answer: The boat is $40 \, \text{m}$ away from the base of the cliff.

Summary

Angle of Elevation: Measured when looking up from a horizontal plane.
Angle of Depression: Measured when looking down from a horizontal plane.
Use right triangle trigonometric ratios (⁡ $\tan, \sin, \cos$ ) to solve related problems.
Ensure proper calculator settings (degrees or radians). These angles are vital in solving practical problems involving heights, distances, and slopes.

Angles of Elevation/Depression (Leaving Cert Mathematics): Revision Notes