Cambridge IGCSE Maths (0580) · Revision notes · Topic 6 of 9

Cambridge IGCSE Maths 0580 Trigonometry Notes

Revision notes for topic 6, Trigonometry, of the Cambridge IGCSE Mathematics (0580) syllabus for exams from 2025 to 2030 (Cambridge says the 2028–2030 syllabus has no significant changes that affect teaching). Each of the 6 sub-topics lists what the syllabus asks, marked Core or Extended, with a short explanation and a worked example. Cover the answer and try each example first.

6 sub-topics
2 Core and Extended
4 Extended only
1 with notes written for 0580

Core Core students (Papers 1 and 3) learn the Core statements. Extended Extended students (Papers 2 and 4) learn everything: the Core statements, the Extended additions and the Extended-only sub-topics. Papers 1 and 2 are non-calculator.

6.1 Pythagoras' theorem

Core C6.1 Extended E6.1

Know and use Pythagoras' theorem in two dimensions

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

Pythagoras: in a right-angled triangle, a² + b² = c² (c = hypotenuse).

Example: Right-triangle legs 7 cm and 24 cm — hypotenuse?

Answer: √(49 + 576) = √625 = 25 cm.

0580 practice questions

8 questions written for 0580 in the Trigonometry practice: 8 Core, 4 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

The two shorter sides of a right-angled triangle are 6 cm and 8 cm. Find the length of the hypotenuse.

Show the answer and mark scheme

M1 6² + 8²
A1 10 cm

Answer: $10$ cm

Practice question (Core, non-calculator, 4 marks):

A triangle has sides of length 20 cm, 21 cm and 29 cm.

(a) Show that the triangle is right-angled.

(b) Work out the area of the triangle.

Show the answer and mark scheme

M1 (a) 20² + 21² = 400 + 441 = 841
A1 841 = 29², so the triangle is right-angled
M1 (b) ½ × 20 × 21
A1 210 cm²

Answer: (a) $20^{2} + 21^{2} = 29^{2}$ , so right-angled (b) $210$ cm²

0580 questions: Pythagoras' Theorem questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Pythagoras' Theorem
Practise (4MA1 questions, same mathematics): Pythagoras' Theorem questions · Right-Angled Trigonometry questions

6.2 Right-angled triangles

Core C6.2 Extended E6.2

Sine, cosine and tangent in right-angled triangles to find sides and angles
Problems in two dimensions, including bearings

Extended also: Angles of elevation and depression.

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

SOH CAH TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj — used in right-angled triangles.

Example: Hypotenuse 14 cm, angle 30°. Opposite side?

Answer: Opp = 14 × sin 30° = 7 cm.

0580 practice questions

8 questions written for 0580 in the Trigonometry practice: 7 Core and 1 Extended, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

In a right-angled triangle, angle $A$ is opposite a side of 3 cm. The other two sides are 4 cm (next to $A$ ) and 5 cm (the hypotenuse). Write down, as fractions, $\sin A$ , $\cos A$ and $\tan A$ .

Show the answer and mark scheme

B2 sin A = 3/5, cos A = 4/5, tan A = 3/4 (B1 for two correct)

Answer: $\sin A = \frac{3}{5}, \cos A = \frac{4}{5}, \tan A = \frac{3}{4}$

Practice question (Extended, calculator, 4 marks):

From the top of a vertical cliff 64 m high, the angle of depression of a boat is 17°.

(a) Work out the distance of the boat from the foot of the cliff.

(b) The boat then moves 50 m directly towards the cliff. Work out the new angle of elevation of the top of the cliff from the boat.

Show the answer and mark scheme

M1 (a) 64 ÷ tan 17°
A1 209 m (209.3…)
M1 (b) tan⁻¹(64 ÷ (209.3… − 50))
A1 21.9°

Answer: (a) $209$ m (b) ${21.9}^{\circ}$

0580 questions: Right-Angled Trigonometry questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Right-Angled Trig (SOH CAH TOA)
Practise (4MA1 questions, same mathematics): Right-Angled Trigonometry questions · Sine and Cosine Rules questions

6.3 Exact trigonometric values

Extended only E6.3

Extended: Exact values of sin and cos of 0°, 30°, 45°, 60° and 90°, and of tan of 0°, 30°, 45° and 60°.

New for 0580: not in Edexcel 4MA1: 4MA1 does not ask for exact trigonometric values; on 0580 they are needed on the non-calculator Paper 2.

Notes for 0580

Learn these for the non-calculator paper: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3; sin 45° = cos 45° = √2/2 (which is 1/√2), tan 45° = 1; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3; sin 0° = 0, cos 0° = 1, tan 0° = 0; sin 90° = 1, cos 90° = 0 (tan 90° is not defined).

They come from two triangles: half an equilateral triangle of side 2 (sides 1, √3 and 2, angles 30°, 60° and 90°) and a right-angled isosceles triangle with shorter sides 1 (hypotenuse √2, angles 45°, 45° and 90°).

Example (Extended, non-calculator): Without a calculator, find the exact value of sin 60° × tan 30°.

sin 60° = √3/2 and tan 30° = 1/√3
(√3/2) × (1/√3) = 1/2

Answer: 1/2

Example (Extended, non-calculator): In triangle ABC, angle B = 90°, angle A = 30° and AC = 10 cm. Find the exact lengths of BC and AB.

BC is opposite angle A: BC = 10 sin 30° = 10 × 1/2 = 5
AB is adjacent to angle A: AB = 10 cos 30° = 10 × √3/2 = 5√3

Answer: BC = 5 cm and AB = 5√3 cm

0580 practice questions

15 questions written for 0580 in the Trigonometry practice: 15 Extended, all non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Extended, non-calculator, 2 marks):

Use exact values, not a calculator, to show that $2 \sin 60^{\circ} \times \cos 30^{\circ}$ is a rational number, and write down its value.

Show the answer and mark scheme

M1 sin 60° = cos 30° = √3/2
A1 3/2 oe

Answer: $\frac{3}{2}$

Practice question (Extended, non-calculator, 3 marks):

Without using a calculator, find the exact value of $(\cos 30^{\circ})^{2} - (\sin 30^{\circ})^{2}$ .

Show the answer and mark scheme

B1 cos 30° = √3/2 and sin 30° = ½
M1 ¾ − ¼
A1 ½

Answer: $\frac{1}{2}$

0580 questions: Exact Trigonometric Values questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Right-Angled Trig (SOH CAH TOA)

6.4 Trigonometric functions

Extended only E6.4

Extended: Graphs of y = sin x, y = cos x and y = tan x for 0° ≤ x ≤ 360°; Solve trigonometric equations for 0° ≤ x ≤ 360°.

Same mathematics as Edexcel 4MA1.

0580 practice questions

8 questions written for 0580 in the Trigonometry practice: 8 Extended, 4 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Extended, non-calculator, 2 marks):

Without using a calculator, solve $\sin x = - 0.5$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Show the answer and mark scheme

B1 210°
B1 330°

Answer: $x = 210^{\circ}$ , $330^{\circ}$

Practice question (Extended, calculator, 4 marks):

Solve $3 - 8 \sin x = 5$ for $0^{\circ} \leq x \leq 360^{\circ}$ . Give your answers correct to 1 decimal place.

Show the answer and mark scheme

M1 −8 sin x = 2, so sin x = −0.25
M1 sin⁻¹(0.25) = 14.47…, used in 180 + 14.5 or 360 − 14.5
A1 194.5°
A1 345.5°

Answer: $x = {194.5}^{\circ}$ , ${345.5}^{\circ}$

0580 questions: Trigonometric Graphs and Equations questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Graphs, Sequences and Calculus (all notes)
Practise (4MA1 questions, same mathematics): Trigonometric Graphs questions

6.5 Non-right-angled triangles

Extended only E6.5

Extended: The sine rule and the cosine rule, including obtuse angles and the ambiguous case; Area of a triangle = ½ab sin C.

Same mathematics as Edexcel 4MA1.

Revise it

For triangles without a right angle: sine rule a/sin A = b/sin B, cosine rule a² = b² + c² − 2bc cos A, area = ½ab sin C.

Example: Find the area of a triangle with sides 7 cm and 9 cm and included angle 40°.

Answer: ½ × 7 × 9 × sin 40° = 20.2 cm² (3 s.f.).

0580 practice questions

8 questions written for 0580 in the Trigonometry practice: 8 Extended, all calculator (Papers 3 and 4), each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Extended, calculator, 2 marks):

In triangle $A B C$ , $C A = 8$ cm, $C B = 11$ cm and angle $A C B = 40^{\circ}$ .

Work out the area of the triangle.

Show the answer and mark scheme

M1 ½ × 8 × 11 × sin 40°
A1 28.3 cm²

Answer: $28.3$ cm²

Practice question (Extended, calculator, 4 marks):

A ship sails 12 km from port $P$ on a bearing of 080° to a buoy $B$ . It then sails 9 km on a bearing of 150° to a lighthouse $L$ .

(a) Show that angle $P B L = 110^{\circ}$ .

(b) Work out the distance $P L$ .

Show the answer and mark scheme

M1 (a) angle between the bearings at B: 180 − (150 − 80)
A1 110° (shown)
M1 (b) PL² = 12² + 9² − 2 × 12 × 9 × cos 110°
A1 17.3 km

Answer: (a) $110^{\circ}$ (shown) (b) $17.3$ km

0580 questions: Sine and Cosine Rules questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Sine & Cosine Rules
Practise (4MA1 questions, same mathematics): Right-Angled Trigonometry questions · Sine and Cosine Rules questions

6.6 Pythagoras' theorem and trigonometry in 3D

Extended only E6.6

Extended: Pythagoras and trigonometry in three dimensions, including the angle between a line and a plane.

Same mathematics as Edexcel 4MA1.

0580 practice questions

8 questions written for 0580 in the Trigonometry practice: 8 Extended, 1 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Extended, calculator, 2 marks):

A cuboid measures 6 cm by 4 cm by 3 cm.

Work out the length of a diagonal joining opposite corners of the cuboid.

Show the answer and mark scheme

M1 √(6² + 4² + 3²)
A1 7.81 cm (√61)

Answer: $7.81$ cm

Practice question (Extended, calculator, 4 marks):

A pyramid has a rectangular base 16 cm by 12 cm. Its vertex is 10 cm vertically above the centre of the base.

(a) Work out the angle between a sloping edge and the base.

(b) Work out the length of a sloping edge.

Show the answer and mark scheme

M1 half the base diagonal ½√(16² + 12²) = 10
M1 (a) tan θ = 10/10
A1 45°
A1 (b) √(10² + 10²) = 14.1 cm

Answer: (a) $45^{\circ}$ (b) $14.1$ cm

0580 questions: 3D Pythagoras and Trigonometry questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Pythagoras' Theorem · Right-Angled Trig (SOH CAH TOA)
Practise (4MA1 questions, same mathematics): Pythagoras' Theorem questions · Sine and Cosine Rules questions

Practise trigonometry

55 practice questions are written for 0580 on every sub-topic of this topic. For the 5 sub-topics that are the same mathematics in Edexcel 4MA1, the practice also has our 4MA1 Higher questions, which are Extended level. Core students: use the Core filter on the practice page; every sub-topic with Core content has questions written for 0580 Core. Practice is part of the IGCSE plan, with a free preview; these notes are free.

Trigonometry practice for 0580: the 55 questions written for 0580, tagged Core or Extended and non-calculator or calculator, with the Edexcel 4MA1 Higher questions for the sub-topics that are the same mathematics
Student hub: the Edexcel 4MA1 units (Unit 4, Unit 5), each question marked
Cambridge IGCSE Maths 0580 past papers: the 2025 specimen papers and past papers, with a timer
Formulas for trigonometry on the one-page 0580 formula sheet (free PDF), showing which are on the paper's formula list
Cambridge 0580 trigonometry questions by topic, Core and Extended, with worked answers

More on this topic

Knowledge organiser: Trigonometry on one page (printable)

Knowledge organiser: Trigonometry on one page (key facts, formulas and mistakes to avoid; printable PDF)

Questions about trigonometry

What is in Cambridge IGCSE 0580 trigonometry?

Topic 6 has 6 sub-topics: 6.1 Pythagoras' theorem; 6.2 Right-angled triangles; 6.3 Exact trigonometric values; 6.4 Trigonometric functions; 6.5 Non-right-angled triangles; 6.6 Pythagoras' theorem and trigonometry in 3D.

Which trigonometry sub-topics are Extended only?

6.3 Exact trigonometric values; 6.4 Trigonometric functions; 6.5 Non-right-angled triangles; 6.6 Pythagoras' theorem and trigonometry in 3D. Core students (Papers 1 and 3) do not need them.

What do students get wrong in trigonometry?

Bearings: the wrong angle, or measured from the wrong line. Our 0580 Examiner Insights page explains each one with a fix.

Are these notes free?

Yes. Every note and worked example on these pages is free to read, with no sign-in.

Join free: practise trigonometry with marking →

Cambridge IGCSE Maths 0580 Trigonometry Notes

6.1 Pythagoras' theorem

Revise it

0580 practice questions

6.2 Right-angled triangles

Revise it

0580 practice questions

6.3 Exact trigonometric values

Notes for 0580

0580 practice questions

6.4 Trigonometric functions

0580 practice questions

6.5 Non-right-angled triangles

Revise it

0580 practice questions

6.6 Pythagoras' theorem and trigonometry in 3D

0580 practice questions

Practise trigonometry

Try it now

Questions about trigonometry