The cosine rule, a² = b² + c² − 2bc cos A, finds the side opposite a known angle when you know the other two sides. It is like Pythagoras with a correction term.
With BC = 18 cm, AC = 17 cm and ∠ACB = 110°, AB² = 18² + 17² − 2 × 18 × 17 × cos 110° = 822.3, so AB = 28.7 cm. To find an angle from three sides, rearrange: cos A = (b² + c² − a²)/2bc.
Worked example
BC = 5 cm, AC = 12 cm and ∠ACB = 88°. Find AB to 1 d.p.
- AB² = 5² + 12² − 2 × 5 × 12 × cos 88°.
- AB² = 169 − 4.19 = 164.81, so AB = 12.84…
Answer:12.8 cm
Key words
- cosine rule
- a² = b² + c² − 2bc cos A
- included angle
- the angle between the two known sides
- obtuse angle
- between 90° and 180°; its cosine is negative
Practise this
New questions every time.
Common Core-referenced: HSG-SRT.D.10, HSG-SRT.D.11