The cosine rule – GCSE Maths

Higher · Grade 7 · Calculator

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Find the third side from two sides and the included angle, or find an angle from all three sides. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.

The cosine rule is $a^{2} = b^{2} + c^{2} - 2 b c cos A$ . Use it when you know two sides and the included angle (to find the third side), or all three sides (to find an angle: $cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$ ). If $cos A$ comes out negative, $A$ is obtuse — that is allowed and important.

Rearrange carefully; the $- 2 b c cos A$ term is the one people drop. Pythagoras is the cosine rule with $A = 9 0^{\circ}$ and $cos 9 0^{\circ} = 0$ . 3 s.f. for sides, 1 d.p. for angles, unrounded cosine stored if a later part needs it.

Two sides and included angle $\to$ $a^{2} = b^{2} + c^{2} - 2 b c cos A$ . Three sides $\to$ $cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$ . If $cos A$ is negative, $A$ is obtuse.

Objectives

Find a side opposite an obtuse angle of 120 degrees using cos 120° = -1/2.
Find a side opposite an acute angle of 60 degrees.
Find the cosine of an angle and determine whether the angle is acute or obtuse.

Key points

Recognise the two situations: SAS (two sides and the angle between them) and SSS (three sides).
Use $a^{2} = b^{2} + c^{2} - 2 b c cos A$ to find a side.
Use the rearranged form $cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$ to find an angle.
Handle a negative cosine, which means the angle is obtuse.

Worked examples

Example 1

Use the cosine rule: included angle 120 degrees, enclosing sides 4 and 4. Find the opposite side.

2 marks

State the cosine rule formula: c² = a² + b² − 2ab cos C.
Substitute a = 4, b = 4 and cos 120° = -1/2: c² = 16 + 16 − 32(-1/2) = 32 + 16 = 48.
Take the square root and simplify the surd: c = √48 = 4√3.

Answer: 4√3

Common slips

Slip: Write Pythagoras without the cosine term: c² = a² + b². The triangle is not right-angled, so the cosine correction term is required.
Slip: Use cos 120° = 1/2: c² = 32 − 16 = 16. Cosine of an obtuse angle is negative, so the product is added.
Slip: Leave the answer unsimplified as √48. GCSE requires surds to be written in simplest form a√b.

Example 2

Use the cosine rule: included angle 60 degrees, enclosing sides 8 and 8. Find the opposite side.

2 marks

Substitute into c² = a² + b² − 2ab cos C: c² = 8² + 8² − 2(8)(8) cos 60°.
Substitute cos 60° = 1/2: c² = 64 + 64 − 128(1/2) = 128 − 64 = 64.
Take the square root: c = √64 = 8.

Answer: 8

Common slips

Slip: Add sides directly: 8 + 8 = 16. The cosine rule relates squares of side lengths.
Slip: Multiply by √3 instead of using cos 60° = 1/2. cos 60° is 1/2 exactly.
Slip: Write c = 8√3. An isosceles triangle with an included 60° angle is equilateral, so c = 8.

Example 3

What is the cosine of the angle opposite the side 4 in a triangle with sides 2, 3 and 4?

3 marks

Rearrange the cosine rule for the angle: cos C = (a² + b² − c²) / (2ab).
Substitute a = 2, b = 3 and c = 4: cos C = (2² + 3² − 4²) / (2 × 2 × 3).
Calculate numerator and denominator: (4 + 9 − 16) / 12 = -3/12 = -1/4.

Answer: -1/4

Common slips

Slip: Rearrange with a plus sign: cos C = (a² + b² + c²) / (2ab). Subtract c² because c² was on the left of c² = a² + b² - 2ab cos C.
Slip: Substitute into denominator: 2 × 3 × 4. The denominator is 2ab, which uses the enclosing sides 2 and 3.
Slip: Drop the negative sign: 3/12 = 1/4. 4 + 9 - 16 = -3, so the cosine is negative.

Questions and answers

GCSE Maths the cosine rule questions and answers. Try each one, then open the answer. Easy questions come first.

Question 1 Easy2 marks

In triangle ABC, sides 7 and 9 enclose an angle of 47 degrees. Find the opposite side to 3 significant figures.

Show answer

Answer: 6.64

c² = 7² + 9² − 2×7×9×cos 47.
The side is 6.64.

M1 — c² = 7² + 9² − 2×7×9×cos 47.
A1 — Answer 6.64

Question 2 Easy2 marks

Use the cosine rule with included angle 120 degrees, enclosing sides 6 and 6. What is the opposite side?

Show answer

Answer: 6√3

cos 120 = −1/2.
The opposite side is 6√3.

M1 — cos 120 = −1/2.
A1 — Answer 6√3

Question 3 Medium2 marks

Is the angle opposite the side 4 in a triangle with sides 2, 3 and 4 acute or obtuse?

Show answer

Answer: obtuse

4² is greater than 2² + 3².
The angle is obtuse.

M1 — 4² is greater than 2² + 3².
A1 — Answer obtuse

Question 4 Medium2 marks

Find the cosine of the angle opposite the side 6 in a triangle with sides 4, 5 and 6.

Show answer

Answer: 1/8

cos C = (4² + 5² − 6²) / (2×4×5).
The cosine is 1/8.

M1 — cos C = (4² + 5² − 6²) / (2×4×5).
A1 — Answer 1/8

Question 5 Hard2 marks

In triangle ABC, sides 5 and 7 enclose an angle of 120 degrees. Find the opposite side to 3 significant figures.

Show answer

Answer: 10.4

cos 120 is negative, so the third side is longer than for an acute angle.
The side is 10.4.

M1 — cos 120 is negative, so the third side is longer than for an acute angle.
A1 — Answer 10.4

Question 6 Hard2 marks

Find the cosine of the angle opposite the side 5 in a triangle with sides 3, 4 and 5.

Show answer

Answer: 0

cos C = (3² + 4² − 5²) / (2×3×4).
The cosine is 0.

M1 — cos C = (3² + 4² − 5²) / (2×3×4).
A1 — Answer 0

Question 7 Exam-style4 marks

In triangle ABC, the sides are 6, 7 and 8. Find the angle opposite 8 to the nearest degree.

Show answer

Answer: 76

cos C = (6² + 7² − 8²) / (2×6×7).
Round the angle to the nearest degree.

M1 — cos C = (6² + 7² − 8²) / (2×6×7).
M1 — Round the angle to the nearest degree.
A1 — Round the angle to the nearest degree.
A1 — Answer 76

Common mistakes

Most lost marks on this topic are one slip.

Using a positive cosine for 120 degrees
cos 120° is -1/2 because 120° is an obtuse angle
Using Pythagoras when the triangle is not right-angled
Include the -2ab cos C correction term for non-right triangles
Assuming a negative cosine means an acute angle
A negative cosine corresponds to an obtuse angle between 90° and 180°

Spot the slip

An angle of 120 degrees is obtuse. What is the value of cos 120°?

-1/2
1/2
-√3/2

Show the slip

-1/2 For obtuse angles (90° < θ < 180°), cosine is negative. cos 120° = -cos 60° = -1/2.

Exam tips

A calculator is useful here. Show the calculation you typed, not only the final line.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the substituted formula in full, then the value of $a^{2}$ , then $a$ — three lines, three marks in a typical 3-mark question.
In bearings questions, draw a north line at the starting point and mark the angle between the two paths clearly; that angle is usually the first mark.

Printable worksheet

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Questions students ask

Is The cosine rule Foundation or Higher?

The cosine rule is Higher only, aimed at grade 7.

What is a common slip on The cosine rule?

Using a positive cosine for 120 degrees. cos 120° is -1/2 because 120° is an obtuse angle.

Where can I get The cosine rule questions and answers?

The questions on this page are free. Open Show answer under each one, or download the worksheet PDF. Start free to practise this topic in the app.

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AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.

The cosine rule – GCSE Maths

Objectives

Key points

Worked examples

Example 1

Example 2

Example 3

Questions and answers

Question 1 Easy2 marks

Question 2 Easy2 marks

Question 3 Medium2 marks

Question 4 Medium2 marks

Question 5 Hard2 marks

Question 6 Hard2 marks

Question 7 Exam-style4 marks

Common mistakes

Spot the slip

Exam tips

Printable worksheet

Questions students ask

Is The cosine rule Foundation or Higher?

What is a common slip on The cosine rule?

Where can I get The cosine rule questions and answers?

Related topics

More on The cosine rule

Practise The cosine rule free