Cosine Rule, Knowledge Organiser

Key Terms

• SAS: Two sides and the INCLUDED angle between them
• SSS: All three sides known, no angles
• Included angle: The angle that sits BETWEEN the two known sides
• Cosine rule: Generalised version of Pythagoras for any triangle

Must-Know Facts

• Use the cosine rule when: SAS (2 sides + included angle) or SSS (all 3 sides)
• Angle A is BETWEEN sides b and c in the formula
• When A = 90°, cos 90° = 0, so it simplifies to Pythagoras
• After finding cos A, use cos⁻¹ to get the angle
• Always check: the largest angle is opposite the longest side

Key Formulas

• Find side: a² = b² + c² − 2bc cos A
• Find angle: cos A = (b² + c² − a²) ÷ 2bc
• Use when: SAS or SSS

Common Mistakes

• Wrong angle pairing: In a² = b² + c² − 2bc cos A, angle A must be opposite side a
• Sign error with −2bc cos A: If angle A is obtuse, cos A is negative, so the term becomes positive: let the calculator handle this
• Finding angle: wrong rearrangement: Use cos A = (b² + c² − a²) ÷ 2bc, then cos⁻¹ to find A
• Using cosine rule when sine rule is simpler: If you have an angle-opposite-side pair, sine rule is usually easier

Worked Examples

Find side a when b = 7cm, c = 9cm, and A = 60°. Solution a² = b² + c² - 2bc cos A a² = 49 + 81 - 2(7)(9)cos 60° a² = 130 - 126(0.5) = 130 - 63 = 67 a = √67 = 8.2 cm (1 d.p.)