Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

The Sine & Cosine Rules

In one sentence. Trigonometry escapes the right angle: solving ANY triangle, plus the ½ab sin C area formula. Higher Tier.

Trigonometry — one of 4 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Regiomontanus. Learn The Sine & Cosine Rules, teach it, then defend it against Regiomontanus.

Where this sits in the specification

Specification 4MA1, 4.8 Trigonometry — sine rule, cosine rule, area formula (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 4.8 — Trigonometry — sine rule, cosine rule, area formula (Higher)

1 · Understand it

What it is

For any triangle with sides a, b, c opposite angles A, B, C: the sine rule a/sin A = b/sin B links opposite pairs; the cosine rule c² = a² + b² − 2ab cos C links three sides and the included angle. Area = ½ab sin C uses two sides and the angle BETWEEN them.

Why it works

The sine rule works because every side, divided by the sine of its opposite angle, gives the same number (the circumscribed circle's diameter) — pairs are exchangeable. The cosine rule is Pythagoras with a correction term: when C = 90°, cos C = 0 and it collapses to c² = a² + b² exactly; the −2ab cos C measures how far from right-angled the triangle is. The area formula is ½ × base × height with the height rewritten as b sin C.

When a question is asking for it

Choose by what you're given: an opposite PAIR in the mix → sine rule. Three sides, or two sides with the INCLUDED angle → cosine rule. Two sides + included angle and 'find the area' → ½ab sin C. Beware the obtuse case with the sine rule: sin⁻¹ returns the acute answer, but 180° minus it can also be valid.

How to hold it in your head

The cosine rule is 'Pythagoras plus a penalty': the wider the angle, the smaller cos C gets (turning negative past 90°), and the longer the far side grows. At exactly 90° there's no penalty and it IS Pythagoras — one formula containing the other.

2 · The mistakes that cost marks

  • Using the cosine rule with a NON-included angle. The angle in c² = a² + b² − 2ab cos C must sit between a and b.
  • Sine rule pairing a side with a non-opposite angle. The rule only links OPPOSITE pairs.
  • Order of operations: a² + b² − 2ab cos C is NOT (a² + b² − 2ab) cos C. The cosine term is computed and subtracted whole.
  • Taking sin⁻¹'s answer as the only one. If the triangle could be obtuse, 180° − answer is the other candidate.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

In triangle ABC, angle A = 40°, angle B = 75° and side a = 8 cm. Find side b.

  1. Sine rule with the two opposite pairs: b/sin 75° = 8/sin 40°.
  2. b = 8 sin 75° / sin 40° = 8 × 0.9659/0.6428.
  3. = 12.0 cm (1 d.p.).

Answer: 12.0 cm

A triangle has sides 8 cm and 11 cm with an included angle of 47°. Find the third side.

  1. Cosine rule: c² = 8² + 11² − 2(8)(11) cos 47°.
  2. = 64 + 121 − 176 × 0.6820 = 185 − 120.03 = 64.97.
  3. c = √64.97 = 8.06 cm (2 d.p.).

Answer: 8.06 cm

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — In triangle ABC, angle A = 40°, angle B = 75° and the side a (opposite A) is 8 cm. Use the sine rule to work out side b, correct to 1 decimal place. (answer in cm)
  2. Grade 7 — A triangle has sides of 8 cm and 11 cm meeting at an angle of 47°. Work out its AREA, correct to 1 decimal place. (answer in cm²)
  3. Grade 8 — A triangle has sides 8 cm and 11 cm with an included angle of 47°. Use the cosine rule to work out the third side, correct to 2 decimal places. (answer in cm)
  4. Grade 9 — A triangle has sides 5 cm, 7 cm and 8 cm. Work out the size of the angle opposite the 8 cm side, correct to 1 decimal place. (answer in °)
  5. 9+ — In triangle ABC, angle A = 35°, side a = 10 cm and side b = 16 cm. Angle B is OBTUSE. Work out angle B, correct to 1 decimal place. (answer in °)

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on The Sine & Cosine Rules, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how The Sine & Cosine Rules works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue The Sine & Cosine Rules against Regiomontanus — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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