Sine and cosine rules · Grade 12
In Grade 12 you learn: the cosine rule.
The sine and cosine rules extend trigonometry to triangles without a right angle. Label the sides a, b, c opposite angles A, B, C. Then choose the rule that fits the information you have.
Key facts
- Sine rule: a/(sin A) = b/(sin B) = c/(sin C) (use it when you know a side and its opposite angle).
- Cosine rule for a side: a² = b² + c² − 2bc cos A (two sides and the angle between them).
- Cosine rule for an angle: cos A = (b² + c² − a²)/(2bc) (all three sides).
- Sine rule for an angle can have two answers: θ and 180° − θ.
Worked examples
Example 1
Two sides of a triangle are 8 cm and 10 cm with an angle of 70° between them. Find the third side x, to 1 d.p.
Answer: x = 10.5 cm
Example 2
In triangle ABC, angle A = 40°, angle B = 75° and a = 9 cm. Find b, to 1 d.p.
Answer: b = 13.5 cm