The sine and cosine rules work in any triangle, not just right-angled ones. Use the sine rule, a/sin A = b/sin B = c/sin C, when you know a side and its opposite angle. Use the cosine rule, a² = b² + c² − 2bc cos A, when you know two sides and the angle between them, or all three sides. The area of any triangle is ½ab sin C.
Rules for any triangle
- Sine rule: a/sin A = b/sin B = c/sin C (or sin A/a = sin B/b for angles)
- Cosine rule: a² = b² + c² − 2bc cos A
- Cosine rule for an angle: cos A = (b² + c² − a²)/(2bc)
- Area = ½ab sin C
Labelling the triangle
Label the angles with capital letters A, B and C and the sides opposite them with the matching small letters a, b and c. Side a is opposite angle A. Careful labelling decides which values go where in the formulae.
The sine rule
Use the sine rule when you know a matching pair: a side and the angle opposite it. To find a side, use a/sin A = b/sin B; to find an angle, use the version with sines on top, sin A/a = sin B/b. With A = 35°, B = 75° and a = 6 cm, b = 6 sin 75° ÷ sin 35° = 10.10 cm.
The cosine rule
When you know two sides and the angle between them, there is no matching pair, so use the cosine rule to find the third side. When you know all three sides, rearrange it to find an angle: cos A = (b² + c² − a²)/(2bc). A negative cosine means the angle is obtuse, which the calculator handles automatically.
Area of a triangle
With two sides and the angle between them, the area is ½ab sin C, with no perpendicular height needed. For sides of 7 cm and 11 cm with an included angle of 35°, the area is ½ × 7 × 11 × sin 35° = 22.1 cm².
Choosing the right rule
Ask what you know. A side and its opposite angle: sine rule. Two sides and the included angle: cosine rule for the third side, or ½ab sin C for the area. Three sides: cosine rule for an angle. If the triangle is right-angled, SOH CAH TOA or Pythagoras is quicker.
Worked examples
Example 1. In triangle ABC, A = 35°, B = 75° and a = 6 cm. Find b, to 2 decimal places.
- Sine rule: b/sin 75° = 6/sin 35°.
- b = 6 sin 75° ÷ sin 35° = 10.104…
Answer: 10.10 cm.
Example 2. In triangle ABC, a = 9 cm, b = 7 cm and A = 70°. Find angle B, to 1 decimal place.
- sin B/7 = sin 70°/9, so sin B = 7 sin 70° ÷ 9 = 0.7309…
- B = sin⁻¹(0.7309…)
Answer: 47.0°.
Example 3. In triangle ABC, b = 6 cm, c = 10 cm and A = 48°. Find a, to 1 decimal place.
- a² = 6² + 10² − 2 × 6 × 10 × cos 48° = 136 − 80.29… = 55.70…
- a = √55.70…
Answer: 7.5 cm.
Example 4. A triangle has sides 5 cm, 7 cm and 10 cm. Find its largest angle, to 1 decimal place.
- The largest angle is opposite the 10 cm side.
- cos θ = (5² + 7² − 10²)/(2 × 5 × 7) = −26/70.
- θ = cos⁻¹(−0.3714…)
Answer: 111.8°.
Example 5. Two sides of a triangle are 7 cm and 11 cm, with an included angle of 35°. Find the area, to 1 decimal place.
- Area = ½ × 7 × 11 × sin 35° = 22.08…
Answer: 22.1 cm².
Common mistakes
- Using the sine rule without a matching pair. You need a side and its opposite angle.
- Mislabelling sides. Side a must be opposite angle A.
- Calculating b² + c² − 2bc and then multiplying by cos A. Work out 2bc cos A as one term, then subtract it.
- Forgetting to square-root. The cosine rule gives a²; take the square root for a.
- Using SOH CAH TOA in a non-right-angled triangle. Only right-angled triangles use SOH CAH TOA.
How the sine and cosine rules are examined
Calculator papers at Higher tier test the sine rule, the cosine rule and ½ab sin C, in the UK GCSE and the International GCSE alike. Questions ask you to:
- find a missing side or angle with the sine rule
- find a side or angle with the cosine rule
- find the area of a triangle using ½ab sin C
- solve a bearings or real-world problem using several rules
The trigonometry worksheet includes Higher sine and cosine rule questions with answers. The geometry engine gives more practice.
These rules extend right-angled trigonometry. At A-Level, see sine and cosine rules.
Frequently asked questions
When do you use the sine rule?
When you know a side and its opposite angle, plus one more side or angle.
When do you use the cosine rule?
When you know two sides and the angle between them, or all three sides.
What is the area formula for any triangle?
Area = ½ab sin C, using two sides and the angle between them.