Bearings and directions | Year 10

01 / LEARN

The idea, step by step

Start at north

A bearing is measured clockwise from north at the starting point. East is 090°, south 180° and west 270°. Use three digits: 045°, not 45°, when a three-figure bearing is requested.

A reverse bearing differs by 180°. Add or subtract 180° to keep the answer from 000° to just below 360°. Draw north at each location.

Connect direction with components

For a distance d at a first-quadrant bearing θ, northward movement is adjacent to the bearing, so it is d cos θ. Eastward movement is d sin θ.

In other quadrants, determine north/south and east/west first. A reference-angle triangle gives positive lengths; signed components must match the correct compass directions.

A common trap

Measuring a bearing from east.

Instead: Bearings start at north. A triangle reference angle may start elsewhere, so label both carefully.

02 / WORKED EXAMPLES

See the thinking

Example 1

Combine movements

A hiker moves 6 km east then 8 km north. Find the direct distance and bearing from the start.

$d = \sqrt{6^{2} + 8^{2}} = 10$ km.
$tan θ = 6 / 8$ because the bearing starts at north.

10 km on bearing 036.9°

Example 2

A south-east journey

A boat travels 120 m on bearing 150°. Find eastward and southward movements to 0.1 m and its return bearing.

The direction is $30^{\circ}$ east of south.
East $= 120 sin 30^{\circ} = 60$ m.
South $= 120 cos 30^{\circ}$ .
Reverse bearing = 330°.

60.0 m east; 103.9 m south; 330°

Check your understanding

Find the reverse bearing of 065°.

Show answer

245°

065° + 180° = 245°.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the reverse bearing of 045°.
2 marks
A walker goes 9 km north and 12 km east. Find direct distance and bearing to 0.1°.
3 marks
Find the northward/eastward movements of a 100 m journey on bearing 030°, to 0.1 m.
3 marks
The bearing of B from A is 230°. Find the bearing of A from B.
2 marks
Paths leave one point on bearings 035° and 110°. Find the smaller included angle.
2 marks
A student says bearing 300° travels east. Correct the direction and explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the reverse bearing of 045°.
2 marks
Add 180°.
225°
A walker goes 9 km north and 12 km east. Find direct distance and bearing to 0.1°.
3 marks
$d = \sqrt{9^{2} + 12^{2}} = 15$ km.
$θ = {tan}^{- 1} (12 / 9) = 53.130^{\dots \circ}$ .
15 km on bearing 053.1°
Find the northward/eastward movements of a 100 m journey on bearing 030°, to 0.1 m.
3 marks
North $= 100 cos 30^{\circ} = 86.602 \dots$ m.
East $= 100 sin 30^{\circ} = 50$ m.
86.6 m north; 50.0 m east
The bearing of B from A is 230°. Find the bearing of A from B.
2 marks
Subtract 180° and use three digits.
050°
Paths leave one point on bearings 035° and 110°. Find the smaller included angle.
2 marks
$110^{\circ} - 35^{\circ} = 75^{\circ}$ .
75°
A student says bearing 300° travels east. Correct the direction and explain.
2 marks
300° lies between west (270°) and north (360°).
Its eastward component is negative.
North-west, with a westward component.

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian school course scope

Right-triangle applications include AC9M10M03. The approved chapter also develops non-right-triangle methods, exact values and trigonometric functions within one normal Year 10 course.

Official curriculum source ↗

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