Sine and cosine rule calculator
Cosine rule: side
7.08
The side is 7.0835…, 7.08 to 3 significant figures.
Working
- You know two sides and the angle between them, so use the cosine rule: a² = b² + c² − 2bc cos A.
- a² = 8² + 11² − 2 × 8 × 11 × cos 40° = 185 − 134.8238… = 50.1762….
- a = √50.1762… = 7.08 (3 s.f.).
- Area = ½bc sin A = ½ × 8 × 11 × sin 40° = 28.3 (3 s.f.).
Your result
7.08. The side is 7.0835…, 7.08 to 3 significant figures. The triangle's area is 28.3.
The cosine rule, step by step
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Your mock plan, in three taps
By the cosine rule, a² = b² + c² − 2bc cos A, so sides of 8 cm and 11 cm with 40° between them give a third side of 7.08 cm (3 s.f.).
Questions people ask
When do you use the cosine rule?
When you know two sides and the angle between them and want the third side, or when you know all three sides and want an angle.
When do you use the sine rule?
When you know a side and the angle opposite it, plus one more side or angle. It pairs each side with its opposite angle.
What is the ambiguous case of the sine rule?
When you find an angle with the sine rule, both B and 180° − B have the same sine. If the obtuse one still leaves room in the triangle, two triangles fit, and the diagram decides which.
How do you find the area of a triangle without a right angle?
Area = ½ab sin C, using two sides and the angle between them. With 8 cm, 11 cm and 40°: ½ × 8 × 11 × sin 40° = 28.3 cm².
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How we work this out
What it does
Find a side or an angle in any triangle with the sine rule or the cosine rule, including the ambiguous case.
Any triangle, angles in degrees.
Method
- Cosine rule for a side: a² = b² + c² − 2bc cos A. For an angle: cos A = (b² + c² − a²) ÷ 2bc.
- Sine rule for a side: b = a sin B ÷ sin A. For an angle: sin B = b sin A ÷ a, and 180° − B may also fit.
- Area = ½ab sin C.