OxfordAQA International GCSE Mathematics: Sine rule, cosine rule and area of a triangle (9260) – Revision Notes

These revision notes are for the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). They cover statement G20 in section 3.3.2, Mensuration and calculation. G20 appears only in the Extension content column, so everything here is Extension only. For full explanations and longer worked examples, use the study guide.

Test yourself with the practice questions. Course pages: the OxfordAQA IGCSE Mathematics hub and the printable checklist. Not sure what to revise first? Take the free 10-minute diagnostics.

The set-up

Triangle ABC has angles A, B, C and sides a, b, c. Side a is opposite angle A, b is opposite B and c is opposite C. Right-angled triangles (G19) use Pythagoras and sin, cos, tan instead.

Formulae

Rule	Formula	Use it to find
Sine rule	a / sin A = b / sin B = c / sin C	A side or angle when you have a matching side and angle pair
Sine rule (angle form)	sin A / a = sin B / b = sin C / c	An angle
Cosine rule	a² = b² + c² − 2bc cos A	A side, given two sides and the angle between them
Cosine rule (angle form)	cos A = (b² + c² − a²) / (2bc)	An angle, given all three sides
Area	Area = ½ab sin C	Area, a side or an angle

The specification’s appendix lists these among the formulae you are expected to know. Learn them so you can write them down without a prompt.

Choosing the rule

Given	Wanted	Rule
Two angles and any side	Another side	Sine rule
Two sides and a non-included angle	An angle	Sine rule
Two sides and the included angle	The third side	Cosine rule
Three sides	Any angle	Cosine rule
Two sides and the included angle	Area	½ab sin C

Two angles in a triangle give you the third by subtraction from 180°. Do that first if it gives you a matching pair.

Method boxes

Sine rule for a length

1. Label the triangle with capital letters for angles and small letters for opposite sides.
2. Pick the two fractions that hold the known pair and the unknown.
3. Put the unknown side on top: b = a × sin B / sin A.
4. Evaluate in one go, then round.

Sine rule for an angle

1. Put the unknown sine on top: sin B = b × sin A / a.
2. Take the inverse sine to get the acute angle.
3. Ask whether 180° minus that angle also fits. It does only if the sum of the two angles stays below 180°.

Cosine rule for a side

1. Check the known angle is between the two known sides.
2. Write a² = b² + c² − 2bc cos A and substitute.
3. Work out the whole product 2bc cos A before subtracting.
4. Take the square root.

Cosine rule for an angle

1. The angle you want is opposite the side that is subtracted: cos A = (b² + c² − a²) / (2bc).
2. Evaluate the fraction, then use inverse cosine.
3. A negative value means an obtuse angle.

Worked reminders

Cosine rule (side). Two sides are 5.7 cm and 8.4 cm with an included angle of 79°. Find the third side.

x² = 5.7² + 8.4² − 2 × 5.7 × 8.4 × cos 79°
x² = 103.05 − 18.27...
x² = 84.78... , x = 9.2075...
x = 9.21 cm (3 s.f.)

Sine rule (angle). In a triangle, a = 12.8 cm, b = 15.4 cm and angle B = 61°. Find angle A.

sin A / 12.8 = sin 61° / 15.4
sin A = 12.8 × sin 61° / 15.4 = 0.7270...
A = 46.6° (1 d.p.)

Side a is shorter than side b, so angle A is smaller than angle B and the obtuse value is not possible.

Area. Two sides are 7.9 cm and 11.6 cm with an included angle of 105°.

Area = ½ × 7.9 × 11.6 × sin 105° = 44.2587...
Area = 44.3 cm² (3 s.f.)

Cosine rule (angle). A triangle has sides 6, 9 and 10. Find the angle opposite the side of length 6.

cos θ = (9² + 10² − 6²) / (2 × 9 × 10) = 145 / 180 = 0.8056...
θ = 36.3° (1 d.p.)

Must-know distinctions

• Sine rule or cosine rule? The sine rule needs a side and its opposite angle as a matched pair. If you have no matched pair, use the cosine rule.
• Included or not? In ½ab sin C and in the cosine rule for a side, the angle is between the two sides used.
• Acute or obtuse? Inverse sine gives an acute angle only. Inverse cosine gives acute or obtuse correctly.
• Smallest angle. The angle opposite the shortest side is always acute, so the sine rule is safe for it.
• Exact or rounded? Keep the full calculator value between steps, since the specification expects no rounding in the middle of a calculation.
• Area from an angle. If sin C = x, then C and 180° − C both give the same area. Check the question for “acute” or “obtuse”.

Sense checks

Run these on every answer before you move on.

• The longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.
• A triangle can have only one obtuse angle, and its angles add up to 180°.
• If the included angle is obtuse, the third side from the cosine rule is longer than both of the other sides.
• If the included angle is 90°, the cosine rule reduces to Pythagoras, and ½ab sin C reduces to ½ab.
• An area from ½ab sin C can never be larger than ½ab, because sin C is at most 1.

Two-step reminder

Cosine rule then sine rule. In triangle ABC, AB = 5.9 cm, AC = 8.6 cm and angle A = 61°. Find BC and then angle B.

BC² = 5.9² + 8.6² − 2 × 5.9 × 8.6 × cos 61°
BC² = 108.77 − 49.19... = 59.57...
BC = 7.71... cm
sin B / 8.6 = sin 61° / 7.71...
sin B = 0.9745... , B = 77.0° or 103.0°

The sine rule gives two candidates here, and the angle sum alone cannot remove 103.0°. Use the cosine rule with all three sides instead:

cos B = (5.9² + 7.71...² − 8.6²) / (2 × 5.9 × 7.71...) = 0.2242...
B = 77.0°

The cosine of an obtuse angle is negative, so a positive value confirms that B is acute.

When a question gives you all three sides after the first step, switch to the cosine rule for the next angle. It gives acute and obtuse angles correctly.

Quick self-test

Give lengths to 3 significant figures and angles to 1 decimal place.

1. Two sides of a triangle are 8 cm and 15 cm with an included angle of 60°. Find the third side.
2. Find the area of a triangle with sides 12 cm and 9 cm and included angle 30°.
3. In triangle ABC, a = 11 cm, angle A = 58° and angle B = 47°. Find b.
4. A triangle has sides 7 cm, 8 cm and 12 cm. Find the largest angle.
5. In triangle ABC, a = 13 cm, b = 9 cm and angle A = 62°. Find angle B.
6. A triangle has area 30 cm². Two sides are 6 cm and x cm, with an included angle of 90°. Find x.
7. A triangle has area 27 cm² and two sides 6 cm and 10 cm. Find the two possible included angles.
8. Two sides of a triangle are 6 cm and 11 cm with an included angle of 115°. Find the third side.
9. Which rule do you use first when you know two angles and one side, and want a second side?
10. Which rule do you use first when you know all three sides and want an angle?

Answers

1. 13.0 cm. Cosine rule: 64 + 225 − 240 cos 60° = 169, and √169 = 13.
2. 27.0 cm². ½ × 12 × 9 × sin 30° = 27.
3. 9.49 cm. b = 11 × sin 47° / sin 58°.
4. 106.1°. The largest angle is opposite 12: cos = (49 + 64 − 144) / 112 = −0.2768.
5. 37.7°. sin B = 9 × sin 62° / 13 = 0.6113. The obtuse value is not possible because b is shorter than a.
6. x = 10 cm. 30 = ½ × 6 × x × sin 90°, so 30 = 3x.
7. 64.2° and 115.8°. sin C = 54 / 60 = 0.9.
8. 14.6 cm. Cosine rule: 36 + 121 − 132 cos 115° = 212.79, and the square root is 14.59.
9. The sine rule.
10. The cosine rule, in its rearranged form.

Where marks are usually lost

• Placing a side next to an angle that is not opposite it in the sine rule.
• Using the cosine rule or ½ab sin C when the angle is not the included one.
• Typing 2bc cos A as (b² + c² − 2bc) × cos A on a calculator.
• Stopping at a² and forgetting the square root.
• Dropping the second angle when the question could have two answers.
• Rounding early, so the final digit is out.
• Leaving out units, or writing cm instead of cm² for an area.
• Starting a multi-step question with a rule that does not fit the information given, then getting stuck. Write down what you know first.
• Writing only a final answer when a question says “show that”. Show the substitution and the unrounded value.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), specification Version 5.1 (for exams May/June 2018 onwards), published by OxfordAQA. Statement G20, section 3.3.2 Mensuration and calculation.