Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want the Cosine Rule — finding missing sides and angles when SSS or SAS is given — to become a reliable source of marks instead of a formula they only half-remember.
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 handles the triangle cases that the Sine Rule cannot — when you know two sides and the included angle (SAS), or all three sides (SSS). In Cambridge IGCSE Mathematics (0580/0607), a² = b² + c² − 2bc cos A is essential for navigation, geometry and multi-step problems. This guide explains exactly what the Cosine Rule covers, how to handle the question types that actually appear, and where to practise each skill.

Key takeaways

• Cosine Rule for a side: a² = b² + c² − 2bc cos A, where A is the angle between sides b and c.
• Cosine Rule for an angle: cos A = (b² + c² − a²) / (2bc) — use when three sides are known (SSS).
• Choose Cosine Rule for SAS and SSS; use Sine Rule for AAS and SSA.
• Always square-root at the end when finding a side; use cos⁻¹ when finding an angle.

What is the Cosine Rule in Cambridge IGCSE Maths?

The Cosine Rule generalises Pythagoras to any triangle. When angle A is 90°, cos A = 0 and the rule becomes a² = b² + c². For non-right triangles it finds a missing side or angle from SAS or SSS information. In Cambridge IGCSE Mathematics it appears in surveying problems, vector-style journeys and questions that combine with the Sine Rule and Area Rule.

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 → find side	a² = b² + c² − 2bc cos A	Two sides and included angle
SSS → find angle	cos A = (b² + c² − a²)/(2bc)	Three sides given
Labelling	a is opposite angle A	Mark diagram before substituting
Multi-step	Cosine then Sine Rule	Find one side, then an angle

How to use the Cosine Rule — step by step

The safest method works for every Cosine Rule question.

1. Label the triangle: side a opposite angle A, etc.
2. Identify SAS (find a side) or SSS (find an angle).
3. Write the formula — full version for a side; rearranged for an angle.
4. Substitute carefully — squaring each term before combining.
5. Solve — square-root for a side; cos⁻¹ for an angle.
6. Check — angle sum 180°; longest side opposite 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.

Finding a side vs finding an angle: which form of the rule?

Students lose marks by using the wrong form or the wrong angle. Use what is given to decide.

Situation	Formula to use	Typical signal
SAS — two sides + included angle	a² = b² + c² − 2bc cos A	Angle between the two known sides
SSS — three sides	cos A = (b² + c² − a²)/(2bc)	No angles given initially
After finding a side	Switch to Sine Rule for remaining angles	Multi-step navigation
Right angle	Pythagoras may be faster	Angle marked 90°

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

Most lost marks come from substituting the wrong angle or forgetting to square-root. 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	Full method with formula	”Work out the length of BC.”
Show that	Prove a given side length	”Show that AC = 7.5 cm.”
Find angle …	Rearranged cosine formula	”Find angle BAC correct to 1 d.p.”
Give your answer correct to …	Round as instructed	”Give your answer correct to 3 significant figures.”
Obtuse angle	cos⁻¹ may need supplement	”Angle ABC is obtuse.”

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 = 7 cm, AC = 9 cm and angle A = 120°. Work out the length of BC.” BC² = 7² + 9² − 2(7)(9) cos 120° = 49 + 81 − 126(−0.5) = 130 + 63 = 193 → BC = √193 ≈ 13.9 cm. Mark-scheme reward: included angle A used with sides AB and AC.
2. “Triangle PQR has sides PQ = 5 cm, PR = 8 cm and QR = 10 cm. Work out angle P.” cos P = (5² + 8² − 10²)/(2×5×8) = (25 + 64 − 100)/80 = −11/80 → P = cos⁻¹(−0.1375) ≈ 97.9°. Reward: correct rearrangement for cos P.
3. “Show that the length of the third side is 6.2 cm correct to 1 d.p., given two sides 4 cm and 9 cm with an included angle of 35°.” x² = 16 + 81 − 72 cos 35° ≈ 97 − 58.98 = 38.02 → x ≈ 6.17 ≈ 6.2 cm. Reward: full substitution shown on “Show that”.

When you can recognise the wording instantly, compare with Sine Rule and take the Cosine Rule quiz to lock the method in.

How the Cosine Rule connects to the rest of Trigonometry

The Cosine Rule complements the Sine Rule — many exam questions need both in sequence. Once all sides are known, the Area Rule finds triangle area. Bearing journeys with two legs often need Cosine Rule for the closing side. The Cambridge IGCSE Maths resource hub links all subtopics.

Common mistakes students make

• Using an angle that is not between the two known sides in SAS.
• Forgetting to square-root when finding a side.
• Sign errors when cos A is negative (obtuse angles).
• Using Cosine Rule when Sine Rule would be shorter (AAS case).
• Rounding too early before the final square-root or cos⁻¹.

When you need more support

If SAS and SSS questions keep tripping you up, work through 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 longer than Sine Rule — marks are lost on labelling, sign errors and forgetting to square-root.

When do I use Cosine Rule instead of Sine Rule? Use Cosine Rule for SAS (find a side) and SSS (find an angle). Use Sine Rule for AAS and SSA.

Is Cosine Rule related to Pythagoras? Yes — when angle A = 90°, cos A = 0 and the rule becomes a² = b² + c².

How do I revise the Cosine Rule effectively? Read the subtopic notes, practise both forms (side and angle), then take the Cosine Rule quiz. Pair each session with one Sine Rule question to reinforce when to switch.

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