Sine Rule vs Cosine Rule: When to Use Each (GCSE Higher)
Quick answer: SOH CAH TOA only works in right-angled triangles. For any other triangle, use the sine rule or the cosine rule. Use the sine rule when you know a side and its opposite angle (a matching pair). Use the cosine rule when you know two sides and the angle between them, or all three sides. Both are GCSE Higher tier content, and you need to know them.
Is SOH CAH TOA only for right-angled triangles?
Yes. SOH CAH TOA uses the hypotenuse, which only exists in a right-angled triangle. For non-right-angled triangles, you need the sine and cosine rules.
OCR's GCSE formulae list puts the sine rule, cosine rule and area of a triangle (½ab sin C) under "Higher tier students should also know…" (OCR — GCSE formulae). Learn them; don't rely on them being given.
The formulas
Label the triangle so that side a is opposite angle A, side b opposite B, and side c opposite C.
Sine rule (finding a side): a / sin A = b / sin B = c / sin C
Sine rule (finding an angle): sin A / a = sin B / b = sin C / c
Cosine rule (finding a side): a² = b² + c² − 2bc cos A
Cosine rule (finding an angle): cos A = (b² + c² − a²) / 2bc
Area of a triangle: Area = ½ab sin C
The one-question test
Ask: "Do I know a side AND the angle opposite it?"
A side and its opposite angle, plus one more side or angle: sine rule
Two sides and the angle between them: cosine rule (to find the third side)
All three sides: cosine rule (to find an angle)
Two sides and the angle between them, and you want the area: ½ab sin C
If you have a matching pair, it's the sine rule. If not, it's the cosine rule.
Four worked examples
1. Sine rule: finding a side
In triangle ABC, A = 40°, B = 65° and a = 8 cm. Find b.
We know a and A (a matching pair), so use the sine rule.
b = 8 × sin 65° / sin 40° = 8 × 0.9063 / 0.6428 ≈ 11.3 cm.
2. Sine rule: finding an angle
a = 9 cm, b = 12 cm and A = 35°. Find B.
sin B = 12 × sin 35° / 9 = 12 × 0.5736 / 9 ≈ 0.7648, so B ≈ 49.9°.
3. Cosine rule: finding a side
b = 7 cm, c = 10 cm and the angle between them, A = 50°. Find a.
No matching pair, but we have two sides and the included angle, so use the cosine rule.
a² = 7² + 10² − 2 × 7 × 10 × cos 50° = 49 + 100 − 140 × 0.6428 ≈ 59.0
a ≈ 7.68 cm.
4. Cosine rule: finding an angle
a = 6 cm, b = 8 cm and c = 11 cm. Find the largest angle.
The largest angle is opposite the longest side, so find C.
cos C = (6² + 8² − 11²) / (2 × 6 × 8) = (36 + 64 − 121) / 96 = −21/96 ≈ −0.2188
C ≈ 102.6°. A negative cosine means an obtuse angle, which makes sense.
Bonus: area
Two sides of 7 cm and 10 cm with a 50° angle between them.
Area = ½ × 7 × 10 × sin 50° = 35 × 0.7660 ≈ 26.8 cm².
The ambiguous case (sine rule for angles)
When you use the sine rule to find an angle, there can sometimes be two possible triangles, because sin θ = sin (180° − θ). In example 2, B could be 49.9° or 130.1°. Here, 130.1° + 35° = 165.1°, which still leaves room for angle C, so both triangles are possible. Exam questions usually tell you if the angle is obtuse; if they don't, check whether the second angle fits.
Exam tips
Label the triangle first. Match each side with the angle opposite it.
Check your calculator is in degrees.
Keep full values until the final answer to avoid rounding errors.
Sense-check. The longest side is opposite the largest angle.
Tips are MMS editorial guidance.
Common mistakes
Using SOH CAH TOA on a non-right triangle. It only works with a right angle.
Forgetting to square-root after the cosine rule for a side.
Mislabelling sides and angles. a must be opposite A.
Rounding too early. Small errors grow through multi-step questions.
When extra support might help
Support is worth considering when a student keeps choosing the wrong rule, when calculator slips are costing marks, or when trigonometry is the weak spot before mocks.
A tutor can't guarantee a grade, but can practise the decision and the method until it sticks. Math Make Smart offers online maths tutoring in the UK and GCSE preparation, and you can book a free trial lesson. Start with the basics in our GCSE trigonometry guide.
Frequently asked questions
Is SOH CAH TOA only for right-angled triangles?
Yes. Use the sine or cosine rule for other triangles.
When do I use the sine rule?
When you know a side and the angle opposite it.
When do I use the cosine rule?
When you know two sides and the angle between them, or all three sides.
Are the sine and cosine rules on the Foundation paper?
They're Higher tier content.