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Extended Only Trigonometry

Sine Rule for IGCSE Maths

Using the sine rule to find sides and angles in non-right-angled triangles. This subtopic is part of Trigonometry in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding s

What You Need to Know

Using the sine rule to find sides and angles in non-right-angled triangles. This subtopic is part of Trigonometry in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding sine rule is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Sine Rule

The Sine Rule states a/sin A = b/sin B = c/sin C (where a, b, c are sides and A, B, C are the opposite angles). It applies to any triangle — not just right-angled. In IGCSE Extended Paper 4, it is used to find a missing side when you know two angles and one side (AAS or ASA), or to find a missing angle when you know two sides and a non-included angle (SSA). The ambiguous case (two possible triangles) is occasionally tested at Extended level.

Step-by-Step Method

  1. 1

    Check which case applies

    You need two angles and one side (finding a side) OR two sides and a non-included angle (finding an angle). If you have two sides and the included angle, use the Cosine Rule instead.

  2. 2

    Label the triangle

    Label sides a, b, c and their opposite angles A, B, C consistently. You only need one pair plus one other element.

  3. 3

    Write the relevant pair

    Write a/sin A = b/sin B using the known pair and the unknown. Cross-multiply only — do not use the third ratio unless needed.

  4. 4

    Solve

    Finding a side: a = b × sin A / sin B. Finding an angle: sin A = a × sin B / b, then A = sin⁻¹(...).

  5. 5

    Find remaining angles/sides if needed

    Remember angles in a triangle sum to 180°. Find the third angle by subtraction, then use the Sine Rule again if a third side is needed.

Worked Example

Question

In triangle ABC, angle A = 48°, angle B = 67° and side AB = 13 cm. Find the length of side BC, correct to 3 significant figures.

Solution

Step 1: Find angle C. Angle C = 180° − 48° − 67° = 65° Step 2: Label sides. Side BC is opposite angle A (48°). Side AB (= c = 13 cm) is opposite angle C (65°). Step 3: Write the Sine Rule. BC/sin 48° = 13/sin 65° Step 4: Solve for BC. BC = 13 × sin 48° / sin 65° BC = 13 × 0.7431 / 0.9063 BC = 10.657 cm Answer: BC = 10.7 cm (3 s.f.)

Exam Tips for Sine Rule

  • The Sine Rule is for non-right-angled triangles — if there is a right angle, SOH CAH TOA is simpler and just as valid.
  • When finding an angle with the Sine Rule, always check whether the obtuse solution is valid — if the other known angle plus the obtuse answer exceeds 180°, it is impossible.
  • Write the formula with the unknown in the numerator straight away: rather than a/sin A = b/sin B, flip it to sin A/a = sin B/b when finding an angle.
  • Show the full substitution line before calculating — this line earns the method mark in the mark scheme.

Practice Questions

Q1: In triangle XYZ, angle X = 53°, angle Y = 72° and XY = 18 cm. Find YZ.

Show hint

Find angle Z first (180° − 53° − 72°). YZ is opposite angle X. XY is opposite angle Z.

Q2: In triangle PQR, PQ = 11 cm, QR = 15 cm and angle QPR = 34°. Find angle PQR.

Show hint

Use sin(PQR)/PQ = sin(QPR)/QR. Rearrange to sin(PQR) = 11 × sin34°/15.

Q3: A triangular field has two angles of 55° and 75°. The side between those two angles is 220 m. Find the longest side of the field.

Show hint

The longest side is opposite the largest angle. Find the third angle (50°). The longest side is opposite 75°.

Frequently Asked Questions

What is sine rule in IGCSE Maths?

Using the sine rule to find sides and angles in non-right-angled triangles.

Is sine rule in the Core or Extended syllabus?

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

How do I revise sine 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 sine rule rather than trying to cover everything at once.

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