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Stage 5.1-3

9.08 Bearings

Lesson

Introduction

In surveying and air navigation, bearings are used to help identify the location of an object.

A compass showing the directions north, east, south, and west and smaller divisions. Ask your teacher for more information.

The four main directions of a compass are known as cardinal directions. They are north (\text{N}), east (\text{E}), south (\text{S}), and west (\text{W}). In between each of these cardinal direction are intermediate directions of northeast (\text{NE}), southeast (\text{SE}), southwest (\text{SW}), and northwest (\text{NW}) that are 45\degree from the two cardinal directions closest. There are smaller divisions shown on the compass as well.

Three-figure (true) bearings

A three-figure bearing are:

  • Measured from north (\text{N})

  • Measured in a clockwise direction

  • Written using three figures

A \text{T} is often but not always used to indicate a true bearing. If the angle measure is less than 100\degree it would be written as 040\degree or 040\degree \text{T}.

To use true bearing to describe the location of a plane at point B from the airport at point A:

  • Place the centre of a compass on the starting point, in this case the airport.

  • Starting at North, rotate clockwise until we get to the line AB.

  • Write angle as the true bearing of point B.

A compass with A as the centre and line A B at an angle of 127 degrees clockwise from the north direction.

The true bearing of B from A is 127\degree or 127\degree \text{T}.

A compass with O as the centre and line O P at an angle of 55 degrees clockwise from the north direction.

The diagram describes the bearing of P from O. Rotating clockwise from North, we get an angle of 55\degree.

Since this measure is less than three digits, we put a 0 in front of it so the true bearing of P is 055\degree. Consider the true bearing of O from P. Since angle of elevation is equal to angle of depression and we are starting at P the true bearing would be 180+55=235\degree.

Examples

Example 1

Consider the point A.

A compass with O as the centre and line O A at an angle of 35 degrees clockwise from the north direction.

Determine the true bearing of point A from O.

A
55\degree \text { T}
B
035\degree \text { T}
C
055\degree \text { T}
D
35\degree \text { T}
E
325\degree \text { T}
F
125\degree \text { T}
Worked Solution
Create a strategy

Measure the bearing clockwise from North, and write it using three digits.

Apply the idea

The angle between the North direction and the line OA measured clockwise is 35\degree . So the true bearing is in option B, 035\degree \text{T}.

Example 2

Consider the point A.

A compass with O as the origin and line OA rotated counterclockwise from north with an angle of 42.

Find the true bearing of point A from O.

Worked Solution
Create a strategy

Subtract the given angle from 360 \degree.

Apply the idea

The angle between the North direction and the line OA measured clockwise is 360-42=318\degree . So the true bearing is 318\degree \text{T}.

Example 3

Find the true bearing of point A from point B.

Three compasses with O, B, and A as the centres connected to make a triangle. Ask your teacher for more information.
Worked Solution
Create a strategy

Consider the compass directions from point B.

Apply the idea

Point A is directly West from point B. We know that for every quadrant measures 90\degree, so we will move three quadrants clockwise until we reach West.

\text{True bearing}=3\times 90 = 270\degree \text{T}

Idea summary

True bearings are:

  • Measured from north (\text{N})

  • Measured in a clockwise direction

  • Written using three figures

  • Usually written with a \text{T} at the end

Compass bearings

A compass bearing describes the location of a point using:

  • The starting direction of either north or south;

  • The acute angle needed to rotate

  • The direction to rotate, east or west.

The image describes how to write an angle using the starting and rotation direction. Ask your teacher for more information.
A compass with A at the centre and an angle of 53 degrees between the south direction and line AB measured anti clockwise.

To describe the position of point B from A we would say: "Starting at South, I then rotate 53\degree towards East."

We can write this mathematically as: S\,53\degree E

Exploration

The bearing needed or used completely depends on which position comes first. Have a look at the applet below. It quickly shows you how the angle changes depending on if we are measuring the bearing of A from B or B from A.

Loading interactive...

If A is between north and east, the compass bearing of A from B is measured clockwise from the north.

If A is between south and east, the compass bearing of A from B is measured anticlockwise from the south.

If A is between south and west, the compass bearing of A from B is measured clockwise from the south.

If A is between north and west, the compass bearing of A from B is measured anticlockwise from the north.

Examples

Example 4

Consider the point A.

A compass with O at the centre and line O A rotated clockwise from north with an angle of 35.
a

For the compass bearing of point A from O, which option shows the correct letters that should be written either side of the angle?

A
\text{N$\,⬚\,$E}
B
\text{N$\,⬚\,$W}
C
\text{S$\,⬚\,$E}
D
\text{S$\,⬚\,$W}
Worked Solution
Create a strategy

Consider whether point A is closer to the north or south directions.

Apply the idea

Since A is closer to north, we start at North, rotate clockwise towards the East direction. So the letters that should be written is in option A, \text{N}\,⬚\,\text{E}.

b

Complete the compass bearing of point A from O.

\text{N$\,⬚\,$E}

Worked Solution
Create a strategy

Write the angle measure shown in the figure.

Apply the idea

\text{N$\,35 \degree $E}

Idea summary
The image describes how to write an angle using the starting and rotation direction. Ask your teacher for more information.

A compass bearing describes the location of a point using:

  • The starting direction of either north or south

  • The acute angle needed to rotate

  • The direction to rotate, east or west.

Outcomes

MA5.2-13MG

applies trigonometry to solve problems, including problems involving bearings

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