Labelling
- Label the sides a, b and c, and the angles opposite them A, B and C. Side a is opposite angle A.
- These rules work in any triangle, not just right-angled ones.
The sine rule
- a/sin A = b/sin B = c/sin C. Use it when you know a side and the angle opposite it.
- If A = 40°, B = 65° and a = 8 cm, then b = 8 × sin 65° ÷ sin 40° = 11.3 cm.
- To find an angle, flip it: sin A/a = sin B/b.
The cosine rule
- a² = b² + c² − 2bc cos A. Use it when you know two sides and the angle between them, or all three sides.
- If b = 7 cm, c = 9 cm and A = 50°, then a² = 49 + 81 − 126 × cos 50° = 49.01, so a = 7.0 cm.
- To find an angle: cos A = (b² + c² − a²) ÷ 2bc.
Area of a triangle
- Area = ½ab sin C, using two sides and the angle between them.
- Sides of 6 cm and 10 cm with an angle of 30° between them: area = ½ × 6 × 10 × sin 30° = 15 cm².
Key terms
- Sine rule
- a/sin A = b/sin B = c/sin C.
- Cosine rule
- a² = b² + c² − 2bc cos A.
- Included angle
- The angle between two given sides.
- Opposite side
- The side facing an angle; side a is opposite angle A.
- Area formula
- Area = ½ab sin C, for any triangle.