Bearings in Trigonometry – Year 9 Mathematics

1. What Are Bearings?

Bearings are a way of describing direction using angles measured clockwise from the north direction. Bearings are always measured as three-digit numbers, from 000° to 360°.

2. Types of Bearings

• True Bearings: Measured clockwise from the north direction. For example, east is 090°, south is 180°, and northwest is 315°.
• Compass Bearings: Use directions such as N30°E, which means 30° east of north.

3. How to Read Bearings

1. Start at the north direction (000°).
2. Measure the angle clockwise until you reach the direction of the second point.
3. Always write the angle as a three-digit number (e.g. 045°, 120°, 270°).

4. Bearings in Diagrams

Bearings are often used in diagrams to show the direction from one point to another. The diagram usually includes a north arrow and two points connected by a line. A protractor is used to measure the bearing angle from north.

5. Using Trigonometry with Bearings

When solving problems involving bearings, trigonometry is often used. If you have a triangle formed by two points and their relative direction, you can use:

• Sine rule or cosine rule to find sides or angles.
• SOHCAHTOA for right-angled triangles.

Example: A ship sails 10 km on a bearing of 045°. How far north and how far east does it travel?

• North component: 10 × cos(45°) = 7.07 km
• East component: 10 × sin(45°) = 7.07 km

6. Common Problem Types

• Finding a bearing given a triangle and angle.
• Finding a distance between two points given bearings and one side.
• Using trigonometric ratios to solve navigational problems.

7. Practice Questions

1. Convert the following directions to true bearings: (a) East, (b) South-West, (c) North-East.
2. A plane flies 300 km on a bearing of 120°. How far east and how far south does it travel?
3. From point A, a boat sails 5 km east, then 5 km north. Find the bearing from the starting point to the final position.

8. Solutions

1. (a) 090°
   (b) 225°
   (c) 045°
2. East = 300 × sin(60°) = 259.81 km
   South = 300 × cos(60°) = 150 km
3. Triangle formed is right-angled.
   Use tanθ = opposite/adjacent = 5/5 = 1.
   θ = 45°, so bearing is 045°