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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 in 2025, 2026 and 2027. 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.

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

6 questions written for 0580 in the Trigonometry practice: 6 Core, 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):

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, 2 marks):

A right-angled triangle has hypotenuse 13 cm and one other side 5 cm. Find the length of the third side.

Show the answer and mark scheme
  • M1 13² − 5²
  • A1 12 cm

Answer: \(12\) cm

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

6 questions written for 0580 in the Trigonometry practice: 6 Core, 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 (Core, non-calculator, 2 marks):

Which ratio, sin, cos or tan, would you use to find a side in a right-angled triangle when you know

(a) an angle and the hypotenuse, and you want the side opposite the angle

(b) an angle and the side next to it (not the hypotenuse), and you want the side opposite?

Show the answer and mark scheme
  • B1 (a) sine
  • B1 (b) tangent

Answer: (a) sin (b) tan

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°.

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

  1. BC is opposite angle A: BC = 10 sin 30° = 10 × 1/2 = 5
  2. 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

14 questions written for 0580 in the Trigonometry practice: 14 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):

Without using a calculator, write down the exact value of

(a) \(\sin 30^\circ\) (b) \(\cos 60^\circ\) (c) \(\tan 45^\circ\).

Show the answer and mark scheme
  • B2 ½, ½, 1 (B1 for two correct)

Answer: (a) \(\frac{1}{2}\) (b) \(\frac{1}{2}\) (c) \(1\)

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: \(\frac{1}{2}\)

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.

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.).

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.

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

26 practice questions are written for 0580 on 6.1, 6.2, 6.3. 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.