GCSE MATHS • HIGHER TIER

Cosine Rule Made Simple

Find missing sides and angles in non-right-angled triangles using the Cosine Rule.

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Cosine Rule in Four Moves

1

Label

Label the triangle carefully so side a is opposite angle A.

2

Choose

Decide whether you are finding a missing side or a missing angle.

3

Substitute

Put the known values into the correct form of the Cosine Rule.

4

Calculate

Use your calculator accurately and check that the answer is sensible.

What You’ll Learn

Understand what the Cosine Rule is used for.

Recognise when the Cosine Rule is appropriate.

Use the side form of the Cosine Rule.

Use the angle form of the Cosine Rule.

Label non-right-angled triangles correctly.

Choose between the Cosine Rule and Sine Rule.

Use inverse cosine and degree mode correctly.

The Cosine Rule Formula

To find a side:

a² = b² + c² − 2bc cos(A)

To find an angle:

cos(A) = (b² + c² − a²) ÷ 2bc

Worked Proof: Cosine Rule

Step-by-Step Method

  • 1. Draw and label the triangle.
  • 2. Decide whether you are finding a side or an angle.
  • 3. Choose the correct version of the formula.
  • 4. Substitute carefully.
  • 5. Use brackets correctly.
  • 6. Calculate accurately.
  • 7. Check the answer is sensible.

More Worked Examples

1

Finding a Side
b = 12 cm
c = 15 cm
A = 40°

a² = 12² + 15² − 2(12)(15)cos40

a² ≈ 93.2
a ≈ 9.65 cm
Answer: 9.65 cm

2

cos(A) = (9² + 12² − 7²) ÷ (2×9×12)

cos(A) ≈ 0.815
A ≈ 35.4°
Answer: 35.4°

Finding an Angle
a = 7 cm
b = 9 cm
c = 12 cm

3

Finding an Angle
a = 10 cm
b = 13 cm
c = 15 cm

cos(A) = (13² + 15² − 10²) ÷ (2×13×15)

A ≈ 41.8°
Answer: 41.8°

4

Find the longest side when:
b = 14 cm
c = 18 cm
A = 80

Using the Cosine Rule:

a ≈ 23.1 cm
Answer: 23.1 cm
  • Common GCSE Mistakes
  • Using the Sine Rule instead.
  • Incorrectly labelling sides.
  • Missing brackets.
  • Leaving the calculator out of degree mode.
  • Making arithmetic errors.
  • Forgetting to square values.
  • Exam Tips
  • Label the side opposite A as a before substituting.
  • Use Cosine Rule for two sides plus the included angle.
  • Use the angle form when all three sides are known.
  • Keep brackets around the product 2bc.
  • Check the calculator is in degree mode.
  • Use cos⁻¹ when finding an angle.

Skills Used in Cosine Rule

Trigonometry

Non-Right-Angled Triangles

Cosine Rule

Algebraic Substitution

Calculator Skills

Geometry

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Frequently Asked Questions

It is used to find missing sides and angles in non-right-angled triangles.

Use it when two sides and the included angle are known.

Use it when all three side lengths are known.

The side opposite angle A is labelled a.

Use the Sine Rule when you have a matching side-angle pair.

Use degree mode and cos⁻¹ when finding an angle.