Launching your learning experience…
Detailed notes on Trigonometry for Cambridge IGCSE Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.
Compass directions written as three-figure angles measured CLOCKWISE from NORTH. Cambridge expects 0°≤ bearing <360° — three digits even for small angles.
Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).
Three-figure angle, measured clockwise from north.
A bearing is a direction expressed as a three-figure angle measured clockwise from north.
Compass cardinal points.
Why three figures? Bearings always use three digits — write 060°, not 60°. This avoids confusion with non-bearing angles.
Worked. A point lies due east of an observer. The bearing is 090°.
Worked. A point lies 30° west of north. The bearing is 360°−30°=330°.
The phrase "bearing of B FROM A" — start at A, draw a north line, measure the angle clockwise to AB.
"Bearing of B from A" means: stand at A, face north, then turn clockwise until you face B.
Step-by-step.
Worked. A is at the origin, B is at (4,3) (i.e. east-then-north of A). Find the bearing of B from A.
Tip. Drawing the north arrow at the STARTING point — and ONLY at the starting point — is the key visual.
The back bearing of B from A is the bearing of A from B. Add 180° if the original is <180°, otherwise subtract 180°.
If the bearing of B from A is θ, the bearing of A from B is the back bearing: back bearing={θ+180°θ−180°if θ<180°if θ≥180°
The two opposite directions are exactly 180° apart, but you stay in the 0 to 360 range.
Worked. Bearing of B from A is 075°. Find the bearing of A from B.
Worked. Bearing of C from D is 230°. Find the bearing of D from C.
| Bearing of B from A | Back bearing (A from B) |
|---|---|
| 040° | 220° |
| 075° | 255° |
| 150° | 330° |
| 230° | 050° |
| 310° | 130° |
Find the angles inside the triangle, then apply right-angled trig or the sine/cosine rule.
Cambridge's hardest bearing questions combine a journey (multiple legs, each with a bearing and distance) with trigonometry to find a missing distance or angle.
Worked. A ship sails on a bearing of 060° for 40km, then on a bearing of 150° for 30km. How far is it from the start, and on what bearing?
Step 1. Draw the journey. Note that the two legs differ in bearing by 150−60=90°, so the angle between the two legs at the turning point is 90° (the journey makes a right angle).
Step 2. The displacement from start to end is the hypotenuse of a right-angled triangle:
Step 3. Find the bearing. The angle inside the triangle at the start, opposite the second leg (length 30), is θ=tan−1(30/40)≈36.87°. The bearing from start to end is 060°+36.87°≈097°.
General template.
Verbatim phrases and definitions Cambridge mark schemes credit.
Bearings appear most years on Paper 4 as a 4-6 mark question, often combining bearings with trigonometry (sine/cosine rule) to find distances or angles in a journey. Examiner reports flag two errors: (i) missing leading zeros in three-figure bearings, (ii) measuring anticlockwise from north (or from the wrong axis).
Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E8.2); 0580/42 Oct/Nov 2024 — Q14 (bearings + cosine rule); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.
Step-by-step solutions to past-paper-style questions on bearing, written exactly the way a tutor would explain them at the board.
Almost every bearing exam question is one of these shapes. Learn to spot each one and you will always know how to start.
Recognise it by
A diagram (or a verbal description of one) showing a point relative to north, with a write down / state the bearing instruction — you read or interpret an angle rather than calculate.
How to approach it
Measure the angle clockwise from the north line at the starting point, then write it as a three-digit value between 000° and 360°.
Common trap
Measuring anticlockwise (the standard maths convention) or omitting leading zeros. Examiner reports flag both — bearings go clockwise and always use three digits.
Recognise it by
A bearing is given and a back bearing is asked for, or two bearings from the same point need combining into an angle — a single arithmetic step.
How to approach it
For a back bearing add 180° if the original is below 180°, otherwise subtract 180°. For an angle between two bearings from one point, subtract them.
Common trap
Adding when you should subtract (or vice versa) and leaving the range 000°-360°. Examiner reports flag the 180° direction error and missing leading zeros.
Recognise it by
A navigation context — a ship or hiker travelling several legs on stated bearings — with no triangle drawn, asking for a distance and/or a final bearing.
How to approach it
Draw a north line at each point, use back-bearings to find the interior angles, then apply Pythagoras, the sine rule or the cosine rule. Add the calculated angle to the original bearing for the final answer.
Common trap
Adding the calculated angle to the back-bearing instead of the forward bearing, or misjudging the quadrant. Examiner reports stress sketching the north lines before computing.
Question
From point A, point B lies 40° east of north. Write the bearing of B from A.
Step-by-step solution
Step 1
Bearings are measured CLOCKWISE from north and given as 3 digits.
Bearing=040°
Answer
040°
Examiner tip
Always state bearings as THREE digits (e.g. 040° not 40°).
Question
The bearing of B from A is 075°. Find the bearing of A from B.
Step-by-step solution
Step 1
Add 180° if the original bearing is less than 180°.
075°+180°=255°
Answer
255°
Question
A ship sails 20 km from A to B on a bearing of 060°. Find how far east of A the ship is now.
Step-by-step solution
Step 1
East = adjacent leg if we measure 60° from north (so it's the side opposite the angle from north).
Step 2
Use sin60°=20east.
east=20sin60°≈17.3km
Answer
≈17.3km
Question
From A, the bearing of B is 080°. From A, the bearing of C is 140°. Find ∠BAC.
Step-by-step solution
Step 1
Both bearings share the same north line at A, so subtract.
∠BAC=140°−80°=60°
Answer
60°
Question
The bearing of Y from X is 235°. Find the bearing of X from Y.
Step-by-step solution
Step 1
Since the original bearing exceeds 180°, subtract 180° rather than add.
235°−180°=055°
Answer
055°
Examiner tip
Always state the answer with three digits. Writing 55° instead of 055° commonly loses one mark in 0580 paper 2 questions.
Question
A ship sails 40 km from A on a bearing of 060° to B. From B it then sails 30 km on a bearing of 150° to C. Find the bearing of C from A and the distance AC.
Step-by-step solution
Step 1
Interior angle at B: the back-bearing from B to A is 060°+180°=240°. So ∠ABC=240°−150°=90°.
Step 2
Triangle ABC is right-angled at B. Apply Pythagoras to find AC.
AC=402+302=2500=50km
Step 3
Find ∠BAC using tan: opposite is BC=30, adjacent is AB=40.
tan(∠BAC)=4030=0.75
Step 4
Apply tan−1.
∠BAC=tan−1(0.75)≈36.87°
Step 5
The bearing of C from A is the bearing of B from A plus ∠BAC.
Bearing=060°+36.87°≈096.9°
Answer
AC=50 km; bearing of C from A≈097°
Examiner tip
The 2023 examiner report repeatedly warns students to add the calculated angle to the original bearing (not the back-bearing). Sketch the north lines at each point before computing.
Question
From a harbour H, a yacht sails on a bearing of 090° for 7 km to P, then turns and sails on a bearing of 180° for 24 km to Q. Find the distance HQ and the bearing of Q from H.
Step-by-step solution
Step 1
Bearings 090° and 180° are perpendicular. Triangle HPQ is right-angled at P.
HQ=72+242=625=25km
Step 2
Find the angle at H between north and HQ. The east-component is 7 and the south-component is 24.
tanθ=247⇒θ≈16.26°
Step 3
This angle is measured east of south, so the bearing from H is 180°−16.26°=163.74°.
Answer
HQ=25 km; bearing ≈164°
Examiner tip
Watch the quadrant: the yacht is south-east of the harbour, so its bearing must be between 090° and 180°.
Question
Town B is 58 km from town A on a bearing of 068°. Town C is 42 km from A on a bearing of 156°. Find the distance BC and the bearing of C from B.
Step-by-step solution
Step 1
Angle at A inside triangle ABC: ∠BAC=156°−68°=88°.
Step 2
Apply the cosine rule.
BC2=582+422−2(58)(42)cos88°≈3364+1764−4872(0.0349)≈5128−170.0=4958
Step 3
Square root.
BC≈4958≈70.41km
Step 4
Apply the sine rule for ∠ABC: 42sinB=70.41sin88°.
sinB=70.4142sin88°≈0.5963⇒∠B≈36.6°
Step 5
Back-bearing of A from B=068°+180°=248°. The bearing of C from B lies clockwise from this by ∠ABC.
Bearing=248°−36.6°≈211.4°
Answer
BC≈70.4 km; bearing of C from B≈211°
Examiner tip
Decide whether to add or subtract ∠ABC from the back-bearing based on the diagram orientation. Sketching it before computing avoids the most common sign error.
Question
On a map, a hiker walks from point X to point Y on a bearing of 325°. After 6 km she turns and walks 4 km due east to reach a campsite Z. State the bearing of Z from Y, and find how far north of X the campsite is.
Step-by-step solution
Step 1
Due east corresponds to a bearing of 090°.
Step 2
Northward component from X to Y: 325° is 35° west of north, so the northward leg is 6cos35°≈4.915 km.
y1=6cos35°≈4.915
Step 3
From Y to Z the displacement is purely east, so it contributes nothing to the north component.
ytotal≈4.915km
Answer
Bearing of Z from Y is 090°; campsite is ≈4.92 km north of X.
Examiner tip
Resolve each leg into north/east components separately and add. The east leg from Y to Z doesn't change the north-distance from X.
Question
The bearing of Q from P is 128°. Find the bearing of P from Q.
Step-by-step solution
Step 1
128° is less than 180°, so add 180°.
back bearing=128°+180°=308°
Answer
308°
Examiner tip
Apply the rule consistently: if the original bearing is less than 180°, ADD 180°; if it is at least 180°, SUBTRACT 180°.
The formulae you need to memorise for bearing on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.
Measured clockwise from NORTH, given as 3 digits 000°–360°
When to use
Always.
back=θ±180°(+180° if θ<180°)
When to use
To reverse a bearing.
Definitions to memorise and the exact keywords mark schemes credit for bearing answers — sharpened from recent examiner reports for the 2026 0580 sitting.
An angle measured clockwise from north, written as a 3-digit number between 000° and 360°.
The bearing in the reverse direction. Differs from the original by exactly 180°.
A vertical line drawn at each point in a bearings diagram, used as the reference direction.
The traps other students keep falling into on bearing questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.
0580/42 — recurring
Why it happens
Defaulting to maths-style positive direction.
How to avoid it
Bearings: always CLOCKWISE from north. Different from standard angle convention.
Why it happens
Treating it like a normal angle.
How to avoid it
Pad with zeros: 30°→030°, 5°→005°.
Why it happens
Each point has its own north line.
How to avoid it
Bearing of B from A uses the north line at A, not at B.
Why it happens
Forgetting the 0–360 range.
How to avoid it
If the bearing is less than 180°, ADD 180°. If it's more, SUBTRACT 180°.
The things students keep getting wrong in this sub-topic, answered.