Measure bearings clockwise from North and choose the quadrant
Use North as the zero-bearing reference · Find displacement length and bearing from perpendicular components · Calculate a reverse bearing with the correct wrap
Section Paper 3 · Historical Core · calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Right-triangle ratios
- Compass directions and full turns
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Use North as the zero-bearing reference
- Find displacement length and bearing from perpendicular components
- Calculate a reverse bearing with the correct wrap
North, clockwise, three figures
A bearing is measured clockwise from North, normally from 000° up to but not including 360°. For 6 km east and 8 km north, displacement is 10 km and the angle east of North is tan⁻¹(6/8). Round and write 037° when a nearest-degree three-figure bearing is requested. Do not measure this angle from the horizontal eastward leg.
Sketch the quadrant
For northwest, subtract the acute west-of-North angle from 360°. For southeast, subtract the acute east-of-South angle from 180°. A direct tan⁻¹ ratio alone does not determine the quadrant. Under parallel North references, a reverse bearing differs by 180°, reducing the result modulo 360°. A known northeast distance D at bearing β has north component D cosβ and east component D sinβ.
Sources for this lesson
Pause and explain
Does tan⁻¹ of a positive component ratio alone identify the full bearing?
Worked example
Find displacement and nearest-degree bearing for 6 km east and 8 km north.
Show the worked solution
- Distance=√(6²+8²)=10 km.
- Bearing angle=tan⁻¹(6/8)≈36.8699°, clockwise from North.
Answer 10 km; bearing 037°.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A journey ends 3 km east and 4 km north of its start. Find direct distance and nearest-degree three-figure bearing.
Give me a hint
Use the angle east of North.
Compare my reasoning
- Distance=√(9+16)=5 km.
- Bearing=tan⁻¹(3/4)≈36.8699°, rounded to 037°.
5 km; bearing 037°.
Look for these in your work
- Use perpendicular components
- Name the North reference
A point is 5 km west and 12 km north of the start. Find distance and nearest-degree bearing.
Give me a hint
The direction is northwest.
Compare my reasoning
- Distance=√(25+144)=13 km.
- Bearing=360°−tan⁻¹(5/12)≈337.3801°.
13 km; bearing 337°.
Look for these in your work
- Use the westward sign
- Wrap clockwise from North
A point is 6 km east and 8 km south of the start. Find the outward and reverse bearings to the nearest degree.
Give me a hint
Work in the southeast quadrant, then reverse by 180°.
Compare my reasoning
- Outward bearing=180°−tan⁻¹(6/8)≈143.1301°.
- Reverse bearing≈323.1301°.
Outward 143°; reverse 323°.
Look for these in your work
- Choose the correct quadrant
- Apply the reverse-bearing rule
Common mistakes
- The bearing begins at North, not East.
- Use the displacement signs to select the quadrant.
What can you explain now?
Does tan⁻¹ of a positive component ratio alone identify the full bearing?
Compare with the explanation
No; the quadrant must also be established from the directions.
An acute ratio angle can correspond to different compass quadrants.
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Make it stick
Does tan⁻¹ of a positive component ratio alone identify the full bearing?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.