The sine rule, a / sin A = b / sin B = c / sin C, and the cosine rule, a² = b² + c² − 2bc cos A, extend trigonometry to triangles that are not right-angled. Use the cosine rule with two sides and the included angle, or with all three sides; use the sine rule when you know a side and its opposite angle. The area of any triangle is ½ab sin C.
The sine rule, cosine rule and area formula
- Sine rule: a / sin A = b / sin B = c / sin C
- Sine rule for angles: sin A / a = sin B / b = sin C / c
- Cosine rule for a side: a² = b² + c² − 2bc cos A
- Cosine rule for an angle: cos A = (b² + c² − a²) / 2bc
- Area = ½ab sin C
In every formula, side a is opposite angle A, side b opposite angle B, and side c opposite angle C.
When do you use the cosine rule?
Use the cosine rule when you know two sides and the angle between them and want the third side, or when you know all three sides and want an angle. In a² = b² + c² − 2bc cos A, the angle A sits between the two known sides b and c, and a is the side opposite it.
Work out a² completely before taking the square root, and make sure your calculator is in degrees.
Finding an angle with the cosine rule
Rearranged, the cosine rule gives cos A = (b² + c² − a²) / 2bc. If that fraction is negative, the angle is obtuse, and the cosine rule tells you so directly with no ambiguity.
The largest angle is always opposite the longest side, so if a triangle has an obtuse angle, that is where it will be.
The sine rule and the obtuse case
The sine rule links pairs of opposite sides and angles. Put the sides on top when finding a side, and the sines on top when finding an angle.
When you find an angle, sin⁻¹ only gives the acute value. Because sin θ = sin(180° − θ), the obtuse alternative 180° − θ has the same sine. Information in the question, or the fact that the largest angle faces the longest side, tells you which to take.
Area of a triangle, and working backwards
Area = ½ab sin C uses two sides and the angle between them. Working backwards from a given area, sin C = 2 × area / ab. Since sine is positive for both acute and obtuse angles, that gives two possible values of C unless the context rules one out.
Multi-step triangle problems
Exam problems usually chain the rules: the cosine rule for a side, then the sine rule for an angle, then perhaps an area or a bearing. A reliable habit is to find the largest angle first with the cosine rule, which settles any obtuse angle before you use the sine rule for the others.
Carry full calculator accuracy from one step to the next. Rounding a side to 1 d.p. and then using it in the sine rule can shift the final angle enough to lose the accuracy mark.
Worked examples
Example 1. In triangle ABC, b = 7, c = 5 and A = 60°. Find a, giving your answer as an exact surd.
- Cosine rule: a² = 7² + 5² − 2(7)(5) cos 60°.
- Evaluate: a² = 49 + 25 − 70 × 0.5 = 39.
- Square root: a = √39 ≈ 6.24.
Answer: a = √39.
Marks: M1 for the cosine rule, A1 for √39.
Example 2. A triangle has sides 4, 5 and 6. Find its largest angle, to 1 d.p.
- The largest angle, A, is opposite the longest side, 6.
- cos A = (5² + 4² − 6²) / (2 × 5 × 4) = 5/40 = 0.125.
- A = cos⁻¹(0.125) ≈ 82.8°. The cosine is positive, so the angle is acute.
Answer: 82.8°.
Marks: M1 for the rearranged cosine rule, A1 for 82.8°.
Example 3. In triangle ABC, A = 70°, B = 50° and b = 12. Find a to 1 d.p.
- Sine rule: a / sin 70° = 12 / sin 50°.
- Rearrange: a = 12 sin 70° / sin 50°.
- Evaluate: a ≈ 14.7.
Answer: a ≈ 14.7.
Marks: M1 for the sine rule, A1 for 14.7.
Example 4. In triangle ABC, AB = 8, AC = 11 and angle BAC = 70°. Find BC, then angle ABC, each to 1 d.p.
- Cosine rule: BC² = 8² + 11² − 2(8)(11) cos 70° = 124.80…, so BC = 11.17… ≈ 11.2.
- Sine rule, keeping BC unrounded: sin ABC / 11 = sin 70° / 11.17…
- sin ABC = 11 sin 70° / 11.17… = 0.9253…, so angle ABC ≈ 67.7°.
Answer: BC ≈ 11.2, angle ABC ≈ 67.7°.
Marks: M1 for the cosine rule, M1 for the sine rule, A1 for both answers.
Common mistakes
- Using the cosine rule with the wrong angle. In a² = b² + c² − 2bc cos A, the angle must be the one between sides b and c.
- Taking only the acute angle from sin⁻¹. When the sine rule gives an angle, check whether 180° minus that angle is the one you need.
- Rounding between steps. Use the unrounded side in the next calculation. Early rounding is the most common reason for losing the final accuracy mark.
- Square-rooting too early. Find a² fully, including the − 2bc cos A term, before taking the square root.
- Calculator in radians. These questions use degrees unless stated otherwise. A wrong mode gives a wrong answer with no warning.
How the sine and cosine rules are examined
In Edexcel International A-Level, the sine and cosine rules are Pure 1 content (paper WMA11), in the trigonometric ratios chapter, and they belong to the AS (Year 1) Pure content of Edexcel A-Level (9MA0). Common question types:
- find a side or an angle in a triangle, often with an obtuse angle to handle
- find the area of a triangle, or an angle from a given area
- multi-step problems that combine both rules
- bearings and navigation problems
Practise with the Pure 1 trigonometric ratios exam questions and their mark schemes, or the trigonometric ratios worksheet. For unlimited mixed questions, use the trigonometry revision engine.
These rules reappear in radians, arcs and sectors, where chords and composite shapes need the cosine rule.
Frequently asked questions
How do I know whether to use the sine rule or the cosine rule?
If you know a side and its opposite angle, use the sine rule. If you know two sides and the angle between them, or all three sides, use the cosine rule.
What is the ambiguous case of the sine rule?
When finding an angle with the sine rule, both θ and 180° − θ have the same sine, so two triangles can fit the information. The question, or the longest-side rule, tells you which one applies.
Can the cosine rule give an obtuse angle?
Yes. If (b² + c² − a²) / 2bc is negative, cos⁻¹ gives an obtuse angle directly, with no ambiguity.