Sheet № 112 · Higher only · AQA · Edexcel · OCR
Cosine Rule –
The cosine rule is the go-to formula when the sine rule cannot help — specifically when you have two sides and the included angle, or when you know all three sides and need an angle.
§Key definitions
Question:
In triangle ABC, AB = 7 cm, AC = 9 cm and angle A = 60°. Find BC to 1 decimal place.
Answer:
BC = 8.2 cm (1 d.p.).
Q1 (Foundation):
In triangle PQR, PQ = 11 cm, PR = 8 cm and angle P = 45°. Find QR to 1 d.p.
Q2 (Foundation):
A triangle has sides 5 cm, 7 cm and 9 cm. Find the angle opposite the 9 cm side to 1 d.p.
Q3 (Higher):
In triangle XYZ, XY = 14 cm, YZ = 10 cm and XZ = 18 cm. Find angle Y to 1 d.p.
§Formulas to memorise
a² = b² + c² − 2bc cos A (finding a side)
cos A = (b² + c² − a²) ÷ (2bc) (finding an angle)
For finding a side: substitute into a² = b² + c² − 2bc cos A, calculate a², then square root.
For finding an angle: substitute into cos A = (b² + c² − a²) ÷ (2bc), calculate cos A, then use cos⁻¹.
Worked example
In triangle ABC, AB = 7 cm, AC = 9 cm and angle A = 60°. Find BC to 1 decimal place.
Working: Let a = BC, b = AC = 9, c = AB = 7, angle A = 60°. a² = b² + c² − 2bc cos A a² = 9² + 7² − 2(9)(7) cos 60° a² = 81 + 49 − 126 × 0.5 a² = 130 − 63 a² = 67 a = √67
⚠ Common mistakes
- ✗Using the cosine rule when the sine rule would be simpler. If you have a complete angle-side pair, the sine rule is usually faster. Save the cosine rule for SAS and SSS situations.
- ✗Getting the sign wrong in the formula. The formula has a minus sign: a² = b² + c² minus 2bc cos A. A common error is writing plus instead, which gives the wrong answer.
- ✗Panicking when cos A is negative. A negative cosine simply means the angle is obtuse (between 90° and 180°). This is valid and expected in many exam questions.
✦ Exam tips
- →Copy the formula from the formula sheet and show the substitution clearly — this earns method marks.
- →For finding an angle, the largest angle is always opposite the longest side.
- →If your calculated value of cos A is outside the range −1 to 1, you have made an arithmetic error — go back and check.