Sine Rule, Cosine Rule and the Area of a Triangle — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the three formulas used for triangles that are not right-angled: the sine rule, the cosine rule, and the area formula ½ab sin C. By the end of this guide you should be able to decide which of the three a question needs, substitute into it correctly, and rearrange it to find either a missing side or a missing angle.

You should also be able to use the standard labelling convention, recognise the ambiguous case of the sine rule, and know when a right-angled triangle makes all of this unnecessary.

The organising idea is the choice between the two rules, and it rests on one thing: does the triangle give you a matching pair? A matching pair means a side together with the angle directly opposite it. If you have one, use the sine rule. If you do not, use the cosine rule. Everything else in this topic is substitution and rearrangement; the choice is where the thinking happens, and it takes about five seconds once you know what to look for.

Key terms and definitions

Standard labelling — the convention that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The lower-case letter always names the side facing the matching capital.

Matching pair — a side and the angle opposite it, both known. The sine rule needs one to get started.

Included angle — the angle sitting between two named sides. The cosine rule and the area formula both need one.

Sine rule — a/sin A = b/sin B = c/sin C.

Cosine rule — a² = b² + c² − 2bc cos A.

Ambiguous case — a situation in which the sine rule gives two possible angles, because an obtuse angle and its acute partner have the same sine.

Core concepts

Labelling the triangle

Before choosing anything, label the triangle in the standard way: each side takes the lower-case version of the letter at the opposite vertex.

This is not decoration. Both rules are written in terms of that pairing, and substituting a side against the wrong angle is the commonest way of getting a wrong answer from a correctly chosen formula.

Choosing between the two rules

Look for a matching pair — a side whose opposite angle you also know.

If there is one, use the sine rule. This covers two situations: two angles and any side (find another side), or two sides and an angle opposite one of them (find another angle).

If there is no matching pair, use the cosine rule. This also covers two situations: two sides and the angle between them (find the third side), or all three sides (find any angle).

A quick way to remember the cosine rule's cases: it is the rule for when the known angle sits between the known sides, or when there is no angle at all.

The sine rule

The sine rule states that a/sin A = b/sin B = c/sin C. Only two of the three fractions are needed at a time.

To find a side, use it in the form written above, with the unknown side on top: a = b × sin A ÷ sin B.

To find an angle, turn every fraction upside down first, so that sin A/a = sin B/b. That keeps the unknown on top and avoids an awkward rearrangement.

For a triangle with a = 8, A = 40° and b = 10, finding B gives sin B = 10 × sin 40° ÷ 8 = 0.8035, so B = 53.5° to one decimal place.

The ambiguous case

Because sin 130° and sin 50° are equal, an angle found from the sine rule may have two possible values: the acute one the calculator gives, and its obtuse partner, found by subtracting from 180°.

So an answer of 53.5° also admits 126.5°. Both are genuine unless the question rules one out — often because the angles would then add to more than 180°, or because the diagram or wording describes an acute-angled triangle.

Check whether both fit. Where a question says "find the obtuse angle", it is telling you directly that the second value is the one wanted.

The cosine rule

The cosine rule states that a² = b² + c² − 2bc cos A, where A is the angle between sides b and c, and a is the side opposite it.

To find a side, substitute and work through carefully. For b = 7, c = 9 and A = 50°: a² = 49 + 81 − 2 × 7 × 9 × cos 50° = 130 − 126 × 0.6428 = 130 − 80.99 = 49.01, so a = 7.0 to one decimal place.

The subtraction is one calculation, not two: the whole of 2bc cos A is taken away from b² + c². Working left to right as 49 + 81 − 126 and then multiplying by cos 50° is a frequent and serious error.

To find an angle, rearrange to cos A = (b² + c² − a²) ÷ 2bc, then apply cos⁻¹. For sides 5, 6 and 8, the angle opposite the 8 is found from cos A = (25 + 36 − 64) ÷ 60 = −0.05, giving A = 92.9°.

A negative value for cos A is not an error. It simply means the angle is obtuse, and the inverse function handles it correctly — which is also why the cosine rule has no ambiguous case.

The area formula

The area of a triangle is ½ab sin C, where C is the angle between the two sides a and b.

The angle must be the included one. Using a side and an angle that do not enclose the region gives a meaningless answer, and this is the only real difficulty with the formula.

For sides 4 and 5 with an included angle of 30°: area = ½ × 4 × 5 × sin 30° = 10 × 0.5 = 5 square units.

When the included angle is 90°, sin 90° = 1 and the formula reduces to ½ × base × height, which is the familiar result — a useful sign that the formula is behaving sensibly.

When the triangle is right-angled

None of these rules is needed for a right-angled triangle. Pythagoras' theorem and SOH CAH TOA are quicker and exact.

It is worth knowing that the cosine rule contains Pythagoras' theorem as a special case: when A = 90°, cos A = 0, and a² = b² + c² − 0 is exactly the theorem.

Worked examples

Example 1: The sine rule, finding a side

In triangle ABC, angle A = 35°, angle B = 65° and side a = 9 cm. Find side b.

There is a matching pair — side a with angle A — so the sine rule applies.

a/sin A = b/sin B, so b = 9 × sin 65° ÷ sin 35°.

That gives b = 9 × 0.9063 ÷ 0.5736 = 14.2 cm to one decimal place.

Sense check: angle B is larger than angle A, so side b must be longer than side a. ✓

Example 2: The cosine rule, finding a side

A triangle has sides of 6 cm and 10 cm with an angle of 120° between them. Find the third side.

The known angle lies between the two known sides, so there is no matching pair and the cosine rule is needed.

a² = 6² + 10² − 2 × 6 × 10 × cos 120° = 36 + 100 − 120 × (−0.5).

Because cos 120° is negative, subtracting it adds: a² = 136 + 60 = 196.

So a = √196 = 14 cm.

Sense check: the angle is obtuse, so the side opposite it should be longer than either given side. ✓

Example 3: The area formula

Find the area of a triangle with sides 7 cm and 12 cm and an included angle of 40°.

The two sides and the angle between them are exactly what the formula needs.

Area = ½ × 7 × 12 × sin 40° = 42 × 0.6428 = 27.0 cm² to one decimal place.

Note the units: an area is measured in square centimetres.

Common mistakes and how to avoid them

Choosing the wrong rule. Look for a matching pair. One exists, use the sine rule; none exists, use the cosine rule.

Pairing a side with the wrong angle. Label the triangle in the standard way before substituting.

Mishandling the cosine rule's subtraction. The whole of 2bc cos A is subtracted from b² + c². Work out that product completely before taking it away.

Forgetting to take the square root. The cosine rule gives a², not a.

Using a non-included angle in the area formula. The angle must lie between the two sides used.

Missing the ambiguous case. An angle from the sine rule may also have an obtuse partner, found by subtracting from 180°.

Being thrown by a negative cosine. It simply means the angle is obtuse. Keep the sign through the calculation.

Rounding partway through. Keep full accuracy until the final answer.

Exam technique for "Sine Rule, Cosine Rule and Area"

Label the triangle before anything else, and write down which pairs you have. The choice of rule then makes itself.

Write the formula out, then write it again with the numbers substituted. Both lines carry method marks, and they are available even when the calculator work goes wrong.

Check the calculator is in degrees by confirming that sin 30° gives 0.5.

Sense-check with the rule that the largest angle faces the longest side. If your answer puts a short side opposite a large angle, something has gone wrong.

For an angle found by the sine rule, ask whether the obtuse partner also fits. If the question says "obtuse", it wants that second value.

Give areas in square units and lengths in ordinary units, and follow the rounding instruction exactly.

Quick revision summary

Label the triangle so that side a faces angle A, and so on. Then ask one question: is there a matching pair — a side with its opposite angle?

Yes → sine rule: a/sin A = b/sin B. Invert every fraction when finding an angle, so the unknown stays on top.

No → cosine rule: a² = b² + c² − 2bc cos A, where A sits between b and c. Rearranged for an angle, cos A = (b² + c² − a²) ÷ 2bc. A negative result just means an obtuse angle.

Area = ½ab sin C, where C is the angle between sides a and b. At C = 90° this reduces to ½ × base × height.

Beware the ambiguous case: an angle from the sine rule may also be 180° minus the calculator's value, and both can be valid.

Subtract the whole of 2bc cos A in one step, and remember to square-root a² at the end.

For a right-angled triangle, use Pythagoras' theorem and SOH CAH TOA instead — the cosine rule reduces to Pythagoras when the angle is 90°.