Sine Rule –

The sine rule lets you work with non-right-angled triangles when you have a matching angle-side pair. It is one of the two major Higher-tier trigonometry rules and appears regularly on AQA, Edexcel and OCR papers.

What Is the Sine Rule?

The sine rule is a formula that relates each side of a triangle to the sine of its opposite angle. It works for any triangle, not just right-angled ones. In a triangle where the sides are a, b, c and the opposite angles are A, B, C, the rule states that the ratio of each side to the sine of its opposite angle is constant.

You use the sine rule when you have at least one complete angle-side pair (a side and its opposite angle) plus one other piece of information — either another angle or another side. It comes in two forms: one for finding sides and one for finding angles. The formulas are provided on the AQA and Edexcel formula sheets, but you still need to know when and how to apply them.

The sine rule also has an ambiguous case: when you are given two sides and a non-included angle (SSA), there may be two possible triangles. At GCSE this rarely comes up, but awareness of it is expected at the top end of Higher.

Key Formulas

Step-by-Step Method

1. Label the triangle so that side a is opposite angle A, side b is opposite angle B, etc.
2. Identify the complete pair (a side and its opposite angle both known).
3. Decide whether you need a side or an angle.
4. For a side: use a/sin A = b/sin B. Substitute and solve for the unknown side.
5. For an angle: use sin A/a = sin B/b. Substitute, solve for sin of the unknown angle, then use sin⁻¹.
6. Round as instructed.

Worked Example 1 — Foundation Level

Question: In triangle ABC, angle A = 50°, angle B = 70° and side a = 10 cm. Find side b to 1 decimal place.

Working: Use the sine rule for finding a side: a/sin A = b/sin B 10/sin 50° = b/sin 70° 10/0.7660 = b/0.9397 13.055 = b/0.9397 b = 13.055 × 0.9397

Answer: b = 12.3 cm (1 d.p.).

Worked Example 2 — Higher Level

Question: In triangle PQR, side p = 14 cm, side q = 9 cm and angle P = 80°. Find angle Q to 1 decimal place.

Working: Use the sine rule for finding an angle: sin P/p = sin Q/q sin 80°/14 = sin Q/9 0.9848/14 = sin Q/9 0.07034 = sin Q/9 sin Q = 9 × 0.07034 = 0.6331 Q = sin⁻¹(0.6331)

Answer: Q = 39.3° (1 d.p.).

Worked Example 3 — Exam Style

Question: Two ships leave a port. Ship A sails on a bearing of 060° for 15 km. Ship B sails on a bearing of 120° for 20 km. The angle between the two paths at the port is 60°. The angle at Ship B is 42°. Find the distance between the two ships to 1 decimal place.

Working: Angle at Ship A = 180° − 60° − 42° = 78°. Let the distance between the ships be c (opposite the 60° port angle), and side b = 15 km is opposite angle B = 42°. c/sin 60° = 15/sin 42° c/0.8660 = 15/0.6691 c/0.8660 = 22.418 c = 22.418 × 0.8660

Answer: The distance between the ships is 19.4 km (1 d.p.).

Common Mistakes

• Mismatching sides and angles. Side a must be opposite angle A. If you pair a side with the wrong angle, the answer is wrong.
• Forgetting to use sin⁻¹ when finding an angle. If sin Q = 0.6331, the angle is sin⁻¹(0.6331), not 0.6331.
• Using the sine rule when the cosine rule is needed. The sine rule requires a complete pair. If you have SAS (two sides and the included angle) or SSS (three sides), you need the cosine rule instead.

Exam Tips

• Copy the formula from the formula sheet and write it into your working — this earns a method mark.
• Always check: do I have a complete angle-side pair? If yes, use the sine rule. If no, consider the cosine rule.
• For the ambiguous case, if sin⁻¹ gives you an acute angle but the diagram suggests an obtuse angle, the second solution is 180° minus the acute angle.

Practice Questions

Q1 (Foundation): In triangle ABC, angle A = 45°, angle C = 65° and side a = 8 cm. Find side c to 1 d.p.

Q2 (Foundation): In triangle DEF, angle D = 55°, side d = 12 cm and side e = 10 cm. Find angle E to 1 d.p.

Q3 (Higher): In triangle XYZ, angle X = 38°, angle Y = 72° and side y = 25 cm. Find side x to 1 d.p.

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Related Topics

• Trigonometry Sine and Cosine Rules
• Cosine Rule
• Trigonometry SOHCAHTOA

Summary

• The sine rule is a/sin A = b/sin B = c/sin C and works for any triangle with a complete angle-side pair.
• Use it to find a missing side (when you know two angles and one side) or a missing angle (when you know two sides and a non-included angle).
• Always match each side with its opposite angle — mislabelling is the most common error.
• The formula is on the exam formula sheet, so the key skill is recognising when to use it and substituting correctly.