Sine rule, cosine rule and area of a triangle
Trigonometry in triangles without a right angle.
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Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 6.2.1Use the sine and cosine rules and the formula ½ab sin C for area.Supplement
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Sine rule, cosine rule and area of a triangle
When there is no right angle, SOHCAHTOA does not apply. Two rules cover every case.
Labelling
Side a is opposite angle A, side b opposite B, side c opposite C. Getting this right is half the battle.
The sine rule
a/sin A = b/sin B = c/sin C Use when you have a matching pair — a side and its opposite angle — plus one more piece of information. Invert it when finding an angle: sin A/a = sin B/b.
The cosine rule
a² = b² + c² − 2bc cos A Use when the sine rule cannot start, namely:
- Three sides given (finding any angle), or
- Two sides and the included angle (finding the third side).
Rearranged for an angle: cos A = (b² + c² − a²)/(2bc).
Choosing between them
Ask one question: do I have a side and its opposite angle? If yes, sine rule. If no, cosine rule. This single test resolves nearly every question.
Area of a triangle
Area = ½ab sin C — two sides and the included angle between them. If the angle is not between the two sides, it does not work.
The ambiguous case
When the sine rule gives an angle, remember sin θ = sin(180 − θ), so an obtuse solution may also be valid. If the diagram or context shows an obtuse angle, use 180 − θ.
Think of it like this
The two rules divide the work like two tools for one job: the sine rule needs a matched pair to get started, and when you have no matched pair, the cosine rule is the crowbar that makes the first opening.
Worked examples
Method, step by step
A triangle has sides 7 cm and 9 cm with an included angle of 40°. Find the third side, to 3 significant figures.
- 1Two sides and the included angle means the cosine rule.
- 2a² = 7² + 9² − 2(7)(9)cos 40°.
- 3= 49 + 81 − 126 × 0.766 = 130 − 96.5 = 33.5.
- 4a = √33.5 ≈ 5.79 cm.
5.79 cm
Find the area of a triangle with sides 8 cm and 5 cm and an included angle of 30°.
- 1Use Area = ½ab sin C with a = 8, b = 5, C = 30°.
- 2= ½ × 8 × 5 × sin 30°.
- 3= 20 × 0.5 = 10 cm².
10 cm²
Common misconceptions
- Using the sine rule with three sides given. There is no matching angle to start from, so the cosine rule is required.
- Using `½ab sin C` with an angle that is not between the two sides. The angle must be included.
- Forgetting the ambiguous case, so an obtuse triangle is reported as acute.
In the exam
- Label the diagram with a, b, c and A, B, C before choosing a rule — mismatched labelling is the main source of error.
- If the cosine rule produces a negative cosine, the angle is obtuse. That is correct, not a mistake.