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Sine rule, cosine rule and triangle area

Choose and use the sine rule, cosine rule, and 1/2ab sin C formula in non-right-angled triangles.

Geometry and MeasuresHigherGrades 6 to 9Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Further trigonometry

Pearson Edexcel GCSE Maths Higher geometry G22 and G23: know and apply the sine rule, cosine rule and formula for the area of a triangle.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

Basic trigonometry uses right-angled triangles. The sine rule and cosine rule are for triangles that do not have a right angle.

Start by labelling the triangle. A side and its opposite angle are a matching pair. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

Use the sine rule when you have an opposite side-angle pair and you need another side or another angle.

Use the cosine rule when you have two sides and the included angle, or when you have all three sides and need an angle.

Use area = 1/2ab sin C when you know two sides and the included angle between them.

Higher note: the hardest mark is often choosing the correct rule. The calculator work is usually straightforward once the rule is chosen.

For Edexcel marks, write the formula substitution before rounding the answer.

Key ruleSine rule needs opposite pairs. Cosine rule uses two sides and the included angle, or all three sides. Area = 1/2ab sin C.

Diagram guide

ABCabcsine rule uses opposite pairsa/sin A = b/sin B = c/sin C
Sine rule pairsThe sine rule works with side-angle pairs opposite each other.
Cbaccosine rule uses two sides and included anglec² = a² + b² - 2ab cos C
Cosine rule structureUse the cosine rule when two sides and the included angle are known, or when all three sides are known.
Cabheightarea from two sides and included angleArea = 1/2 ab sin C
Area with sineThe angle must be between the two known sides.

Worked examples

Find a side using the sine rule

In triangle ABC, angle A = 42 degrees, side a = 9 cm and angle B = 68 degrees. Find side b.

ABCabcsine rule uses opposite pairsa/sin A = b/sin B = c/sin C
Example: sine rule sidea and A are a complete opposite pair, so use the sine rule.
  1. Write the sine rule: b / sin B = a / sin A.
  2. Substitute: b / sin 68 = 9 / sin 42.
  3. b = 9 sin 68 / sin 42.
  4. b = 12.5 cm to 3 significant figures.

Answer: 12.5 cm

Find a side using the cosine rule

Two sides of a triangle are 7 cm and 10 cm. The included angle is 54 degrees. Find the third side.

Cbaccosine rule uses two sides and included anglec² = a² + b² - 2ab cos C
Example: cosine rule sideThe known angle is between the two known sides.
  1. Use c² = a² + b² - 2ab cos C.
  2. c² = 7² + 10² - 2 x 7 x 10 x cos 54.
  3. c² = 66.71...
  4. c = 8.17 cm to 3 significant figures.

Answer: 8.17 cm

Find an area using 1/2ab sin C

A triangle has sides 8 cm and 11 cm with included angle 35 degrees. Find the area.

Cabheightarea from two sides and included angleArea = 1/2 ab sin C
Example: triangle areaUse the two sides with the angle between them.
  1. Area = 1/2ab sin C.
  2. Area = 1/2 x 8 x 11 x sin 35.
  3. Area = 25.2 cm² to 3 significant figures.

Answer: 25.2 cm²

Common mistakes

  • Using right-angle SOH CAH TOA on a triangle with no right angle.
  • Using the sine rule without a complete opposite side-angle pair.
  • Using cosine rule when the angle is not the included angle.
  • Forgetting to square root after finding c².
  • Using 1/2 base x height when the perpendicular height is not known.
  • Rounding too early and losing the final accuracy mark.

Quick exercise

Try these before moving to the exam-style questions.

  1. Which rule is likely if you know two angles and one opposite side?
    ABCabcsine rule uses opposite pairsa/sin A = b/sin B = c/sin C
    Quick check: choose a ruleLook for opposite side-angle pairs.
  2. Which rule is likely if you know sides 6 cm and 9 cm and the included angle 40 degrees?
    Cbaccosine rule uses two sides and included anglec² = a² + b² - 2ab cos C
    Quick check: included angleTwo sides and included angle points to cosine rule.
  3. Find the area of a triangle with sides 5 cm and 12 cm and included angle 30 degrees.
    Cabheightarea from two sides and included angleArea = 1/2 ab sin C
    Quick check: area formulaUse 1/2ab sin C.
  4. In the cosine rule, what must you do after calculating c²?
    Cbaccosine rule uses two sides and included anglec² = a² + b² - 2ab cos C
    Quick check: final stepThe rule often gives the square of the side first.
  5. For area = 1/2ab sin C, where must angle C be?
    Cabheightarea from two sides and included angleArea = 1/2 ab sin C
    Quick check: included angleC must be between the two sides a and b.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleHigherCalculator3 marks

In triangle ABC, angle A = 50 degrees, angle B = 72 degrees and side a = 8 cm. Work out side b to 3 significant figures.

ABCabcsine rule uses opposite pairsa/sin A = b/sin B = c/sin C
Question diagram: sine ruleUse the complete pair a and A to find b.
sine rulenon-right trianglehigher trigonometry
Standard exam styleHigherCalculator4 marks

A triangle has sides 9 cm and 13 cm with included angle 61 degrees. Work out the third side to 3 significant figures.

Cbaccosine rule uses two sides and included anglec² = a² + b² - 2ab cos C
Question diagram: cosine ruleTwo sides and included angle means cosine rule.
cosine ruleincluded anglehigher geometry
ChallengeHigherCalculator4 marks

A triangular field has two sides 24 m and 31 m with included angle 118 degrees. Work out its area to the nearest square metre.

Cabheightarea from two sides and included angleArea = 1/2 ab sin C
Question diagram: area of triangleThe angle is between the two given sides.
triangle areasine formulahigher trigonometry