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.

55\degree \text { T}

035\degree \text { T}

055\degree \text { T}

35\degree \text { T}

325\degree \text { T}

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.

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

\text{N$\,⬚\,$E}

\text{N$\,⬚\,$W}

\text{S$\,⬚\,$E}

\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}.

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

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.

6.05 Bearings

Introduction

Ideas

Three-figure (true) bearings

Examples

Example 1

Example 2

Example 3

Compass bearings

Exploration

Examples

Example 4

What is Mathspace

About Mathspace