Skip to content
IGCSEMath.com.my
Extended Only Trigonometry

Cosine Rule for IGCSE Maths

Using the cosine rule to find sides and angles when you have SAS or SSS. This subtopic is part of Trigonometry in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding cosi

What You Need to Know

Using the cosine rule to find sides and angles when you have SAS or SSS. This subtopic is part of Trigonometry in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding cosine rule is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Cosine Rule

The Cosine Rule states a² = b² + c² − 2bc cos A. It is used when SOH CAH TOA and the Sine Rule do not apply: specifically when you know two sides and the included angle (SAS, to find the third side) or all three sides (SSS, to find any angle). In IGCSE Extended Paper 4, it often appears in multi-step problems involving bearings, 3D shapes, or combined with the area formula. Rearranging for the angle gives cos A = (b² + c² − a²) / 2bc.

Step-by-Step Method

  1. 1

    Identify the case

    Two sides + included angle → find the third side. All three sides → find an angle. Label the triangle: the side you're finding (or solving for the angle opposite) is a, and the other two sides are b and c.

  2. 2

    Write the formula

    For a missing side: a² = b² + c² − 2bc cos A. For a missing angle: cos A = (b² + c² − a²) / 2bc.

  3. 3

    Substitute values

    Replace b, c, and A (or a, b, c) with the given numbers. Be careful with the sign: if A > 90°, cos A is negative, making 2bc cos A negative, so a² > b² + c².

  4. 4

    Calculate step by step

    Work out b², c², 2bc cos A separately before combining. Show each arithmetic step to earn method marks.

  5. 5

    Find the square root / inverse cosine

    For a side: take the positive square root. For an angle: use cos⁻¹. Round to the required accuracy.

Worked Example

Question

Triangle ABC has AB = 7 cm, AC = 10 cm and angle BAC = 120°. Find BC, correct to 3 significant figures.

Solution

Step 1: We have two sides (7 and 10) and the included angle (120°). Use the Cosine Rule to find BC. Step 2: Label: a = BC, b = AC = 10, c = AB = 7, A = 120°. Step 3: a² = b² + c² − 2bc cos A a² = 10² + 7² − 2(10)(7) cos 120° a² = 100 + 49 − 140 × (−0.5) a² = 149 + 70 a² = 219 Step 4: a = √219 = 14.799... ≈ 14.8 cm Answer: BC = 14.8 cm (3 s.f.)

Exam Tips for Cosine Rule

  • The Cosine Rule is only needed for SAS or SSS — if the triangle has a right angle or you have AAS/ASA, use a simpler method.
  • cos 90° = 0, so the Cosine Rule reduces to Pythagoras for right-angled triangles — this is a useful check.
  • When the angle is obtuse (> 90°), cos A is negative. A negative result inside the brackets is correct — don't panic.
  • Always find the side first if possible; finding the angle requires the rearranged formula which is slightly harder to remember under pressure.

Practice Questions

Q1: In triangle PQR, PQ = 9 cm, PR = 12 cm and angle QPR = 55°. Find QR, correct to 3 significant figures.

Show hint

Use a² = 9² + 12² − 2(9)(12)cos 55°. Work out each term separately.

Q2: A triangle has sides of length 5 cm, 7 cm and 10 cm. Find the largest angle, correct to 1 decimal place.

Show hint

The largest angle is opposite the longest side (10 cm). Use cos A = (5² + 7² − 10²)/(2 × 5 × 7).

Q3: Two ships leave port at the same time. Ship A travels 15 km on a bearing of 040° and ship B travels 22 km on a bearing of 115°. Find the distance between them.

Show hint

The angle between the two directions is 115° − 40° = 75°. Use the Cosine Rule with sides 15, 22 and included angle 75°.

Frequently Asked Questions

What is cosine rule in IGCSE Maths?

Using the cosine rule to find sides and angles when you have SAS or SSS.

Is cosine rule in the Core or Extended syllabus?

Cosine Rule is part of the Extended only syllabus for IGCSE Mathematics 0580.

How do I revise cosine rule effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on cosine rule rather than trying to cover everything at once.

Need Help with Cosine Rule?

Book a free trial class and get personalised support from Teacher Rig.

Book Free Trial WhatsApp Us

More on Trigonometry

Related Subtopics

Explore More

Master Cosine Rule with Expert Help

Book a free 60-minute trial class with Teacher Rig. Get personalised guidance on Trigonometry and every other IGCSE Maths topic.

Book Free Trial WhatsApp Us