Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want the Cosine Rule — finding missing sides and angles when Sine Rule does not apply — to become a reliable source of marks instead of a formula they confuse with Pythagoras.
What query it owns: how to understand and revise the Cosine Rule in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the Cosine Rule revision-guide angle, while Tutopiya’s Cosine Rule subtopic page owns the learning resource and the free Cosine Rule quiz owns the practice.

The Cosine Rule is essential in the Trigonometry unit of Cambridge IGCSE Mathematics (0580/0607). Whenever a triangle is not right-angled and you know two sides with the included angle (SAS), or all three sides (SSS), examiners expect you to apply a² = b² + c² − 2bc cos A with clear working. This guide explains when to use the rule, how it extends Pythagoras, and where to practise each skill.

Key takeaways

• Cosine Rule (find a side): a² = b² + c² − 2bc cos A — A is the angle between sides b and c.
• Cosine Rule (find an angle): cos A = (b² + c² − a²)/(2bc) — rearrange when all three sides are known.
• Use it for SAS (two sides + included angle) or SSS (three sides, find an angle).
• Always state the formula, show substitution and give the final answer with correct units or degrees.

What is the Cosine Rule in Cambridge IGCSE Maths?

The Cosine Rule relates the three sides of any triangle to the cosine of one of its angles. In Cambridge IGCSE Mathematics it generalises Pythagoras Theorem to non-right-angled triangles. It is used to find a third side when two sides and the included angle are known, or to find an angle when all three sides are given.

You can read the full explanation, worked examples and notes on Tutopiya’s Cosine Rule subtopic page before you attempt questions.

The core ideas you must master

These four ideas appear again and again. Learn what each one means and the exam phrasing that signals it.

Idea	What it means	How the exam uses it
SAS	Two sides + included angle → find third side	”AB = 7, AC = 9, angle A = 52°“
SSS	Three sides → find an angle	”Find angle B” with all sides given
Included angle	Angle between the two known sides	Must be A in a² = b² + c² − 2bc cos A
Pythagoras link	When A = 90°, cos 90° = 0 → a² = b² + c²	Right angle is a special case

How to use the Cosine Rule — step by step

The safest method works for finding a side or an angle.

1. Label the triangle — identify which side or angle you need.
2. For a missing side: use a² = b² + c² − 2bc cos A with A the included angle.
3. Substitute the known values; calculate a² then square-root for a.
4. For a missing angle: use cos A = (b² + c² − a²)/(2bc).
5. Apply cos⁻¹ to find the angle in degrees.
6. Check: the longest side is opposite the largest angle.

Once you have worked through a few, test yourself with the free Cosine Rule quiz — it tells you fast whether the method has actually stuck.

Cosine Rule vs Sine Rule: which does the question want?

Students lose marks by using Sine Rule for SSS problems or by using a non-included angle in the Cosine Rule. Use the given information to decide.

Situation	What to do	Typical signal words
Two sides + included angle	Cosine Rule (find side)	“included angle”, angle between two known sides
Three sides, find an angle	Cosine Rule (find angle)	All three lengths given
Two angles + one side	Sine Rule	Two angles stated
Two sides + opposite angle	Sine Rule	Angle opposite one of the sides

Cosine Rule in past-paper wording: command words that matter

Most lost marks come from misreading the command word or using the wrong angle in the formula. These are the command words you will see and what each one demands.

Command word / phrase	What the question wants	Typical stem
Calculate / Work out	Find a side or angle with full method	”Work out the length of BC.”
Show that	Prove a given result — method earns marks	”Show that AC = 11.3 cm correct to 3 s.f.”
Write down	State a value; minimal working (usually 1 mark)	“Write down the size of angle A.”
Give your answer correct to …	Round as instructed	”Give your answer correct to 1 decimal place.”
Find the largest angle	Use Cosine Rule; largest angle opposite longest side	”Find the largest angle of the triangle.”

Worked exam-style stems (how to answer the wording)

Practising the wording — not just the formula — is what method marks reward. Here is how three real-style stems are answered.

1. “In triangle ABC, AB = 8 cm, AC = 11 cm and angle A = 47°. Calculate the length of BC.” BC² = 8² + 11² − 2(8)(11) cos 47° = 64 + 121 − 119.7 = 65.3 → BC = 8.08 cm (3 s.f.). Mark-scheme reward: correct formula with included angle A.
2. “Triangle PQR has PQ = 6 cm, QR = 9 cm and PR = 11 cm. Show that angle Q = 84.3° correct to 1 d.p.” cos Q = (6² + 9² − 11²)/(2×6×9) = (36 + 81 − 121)/108 = −4/108 → Q = cos⁻¹(−0.0370) = 84.3°. Reward: full substitution — the answer alone scores nothing on “Show that”.
3. “A triangular plot has sides 120 m, 150 m and 180 m. Work out the size of the largest angle.” Largest angle opposite 180 m side: cos A = (120² + 150² − 180²)/(2×120×150) = 6300/36000 = 0.175 → A = 79.9°. Reward: correct side paired as opposite the required angle.

When you can recognise the wording instantly, work the full set on the Trigonometry topical past paper questions and the Cosine Rule quiz to lock the method in.

How the Cosine Rule connects to the rest of Trigonometry

The Cosine Rule pairs with the Sine Rule and Area Rule for non-right-angled triangles, and generalises Pythagoras Theorem when the angle is 90°. When you are ready to mix topics, the Cambridge IGCSE Maths resource hub lets you move straight from a weak subtopic into the next.

Common mistakes students make

• Using a non-included angle in a² = b² + c² − 2bc cos A.
• Applying Sine Rule when three sides are given (use Cosine Rule for angles).
• Forgetting to square-root after finding a².
• Sign errors when cos A is negative (obtuse angle).
• Calculator in radians instead of degrees.

When you need more support

If Cosine Rule questions keep tripping you up — especially SSS angle problems — work through the Trigonometry topical past paper questions and the Cosine Rule quiz to pinpoint the exact gap, then get focused help from a Cambridge IGCSE Maths tutor to fix it quickly.

Frequently asked questions

Is the Cosine Rule hard in Cambridge IGCSE Maths? The formula is an extension of Pythagoras. Marks are lost when students use a non-included angle or pick Sine Rule for SSS problems.

When do I use the Cosine Rule instead of the Sine Rule? For SAS (two sides and the included angle) or SSS (three sides, find an angle). Use Sine Rule for AAS or ASA.

How is Cosine Rule related to Pythagoras? When the included angle is 90°, cos 90° = 0 and the rule becomes a² = b² + c².

How do I revise the Cosine Rule effectively? Read the subtopic notes, mark the included angle on every diagram, then take the Cosine Rule quiz. Revisit any SSS problems you got wrong before moving on.

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