Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The focus of this section is on understanding and applying the concepts of angles of elevation and depression, along with incorporating bearings in trigonometry. These concepts are essential for solving real-world problems, including navigation, surveying, and architecture.

Angles of Elevation and Depression

Introduction to Angles of Elevation and Depression

• Angle of Elevation: The angle above the horizontal line from the observer to some point of interest.
• Angle of Depression: The angle below the horizontal line from the observer to some point of interest.

Image courtesy of Cue Math

Key Concepts

• Both angles are measured from the horizontal.
• These angles are used in solving problems involving distances and heights that are not directly measurable.
• Bearings are used to indicate direction. They are measured in degrees, clockwise from the north.

Calculating Angles of Elevation and Depression

Basic Formulae

• $\text{sin } \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$

• $\text{cos } \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$

• $\text{tan } \theta = \dfrac{\text{opposite}}{\text{adjacent}}$

These trigonometric ratios are pivotal in calculating angles and distances in problems involving elevation and depression.

Incorporating Bearings in Trigonometry

Bearings provide a method of describing direction. In trigonometry, bearings are used to define the direction of one point from another, measured in degrees from the north line (0° to 360°).

Understanding Bearings

• True Bearing: The direction to a point is measured in degrees clockwise from the north direction.
• Bearings are usually given in three digits, for example, 045°.

Application in Problem Solving

When solving trigonometry problems, bearings help to establish the angles involved in the scenario, providing a framework for applying the trigonometric ratios.

Worked Examples

Example 1: Calculating Angle of Elevation

Question: From a point A on the ground, the angle of elevation to the top of a tower B is 30°. If the distance from A to B is 100 meters, find the height of the tower.

Solution:

1. Identify Known Values:

2. Apply Trigonometric Ratio:

3. Calculate Height:

Example 2: Calculating Angle of Depression

Question: From the top of a lighthouse 60m high, the angle of depression to a boat at sea is 15°. Calculate the distance of the boat from the base of the lighthouse.

Solution:

1. Identify Known Values:

2. Apply Trigonometric Ratio:

3. Calculate Distance:

Example 3: Incorporating Bearings

Question: A plane flies from A to B, 300 km away, on a bearing of 045°. It then turns and flies 400 km to C on a bearing of 130°. Calculate the bearing from C to A.

Solution:

This problem involves using the Law of Cosines and Sine Rule to find the angles and distances between points. It requires a detailed understanding of geometry and trigonometry, beyond the scope of simple trigonometric ratios but illustrates the application of bearings in navigation.

• Step 1: Draw a diagram to represent the problem.
• Step 2: Identify known values and what needs to be calculated.
• Step 3: Choose the appropriate trigonometric ratio.
• Step 4: Apply the ratio to find the answer.
• Step 5: If bearings are involved, adjust calculations to incorporate the direction.

Tips for Success

• Always draw a sketch of the problem.
• Remember that angles of elevation and depression are measured from the horizontal.
• Bearings are measured clockwise from north.
• Practice with a variety of problems to build confidence.