Cosine Rule

1Summary 重點摘要

Learn one universal formula that works for any triangle, not just right-angled. I'll teach you step by step.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Why the cosine rule is neededThe ratios sin, cos and tan work only inside right-angled triangles, where a 90° angle exists. Yet most real triangles are not right-angled — a triangular field may have no right angle at all, and the same holds for triangles met at sea or in land surveying. The cosine rule applies to any triangle, with no right angle required, filling exactly this gap. It is Pythagoras' theorem upgraded for general use: with it, every triangle's angles and sides become computable.
2. The formula and how to remember itThe cosine rule states c² = a² + b² − 2ab cos C, where c is the wanted side, a and b the other two, and C the angle between a and b. To remember: the square of the wanted side equals the sum of the squares of the other two sides, minus twice their product times the cosine of the included angle. The minus sign in the middle and the 2ab factor are the usual traps; write the formula out three times from memory, and note that 'included angle' means the angle between the two known sides — using the opposite angle instead invalidates the formula entirely.
3. Case one: two sides and the included angle (SAS)Given two sides and the angle between them (SAS), the rule yields the third side directly. With a = 3, b = 4 and included angle C = 60°, substitute to get c² = 3² + 4² − 2 × 3 × 4 × cos 60°, leaving a single unknown c to compute. If the given angle is not the included one, the formula does not apply directly and senior methods such as the sine rule are needed; at junior level, questions normally supply the included angle, so substitution goes straight through.
4. Case two: three sides, find an angle (SSS)When all three sides are known (SSS) and an angle is wanted, rearrange the formula to cos C = (a² + b² − c²) ÷ (2ab). With sides 3, 4, 5 and the largest angle C facing the side of length 5: cos C = (9 + 16 − 25) ÷ 24 = 0, so C = 90° — confirming that 3-4-5 is a right-angled triangle. In the numerator, the square of the side facing the wanted angle is subtracted from the sum of the other two squares; for a different angle, swap the sides accordingly.
5. A full worked calculationExample: a = 3, b = 4, included angle C = 60°, find c. First write the formula c² = a² + b² − 2ab cos C; second, substitute the values, c² = 9 + 16 − 2 × 3 × 4 × cos 60°; third, with cos 60° = 0.5, c² = 25 − 24 × 0.5 = 25 − 12 = 13; fourth, take the root, c = √13 ≈ 3.6. Note the order — square before you take the root — and keep extra digits through the working, rounding to two decimal places only at the end, to limit accumulated error.
6. The link with PythagorasPythagoras' theorem is a special case of the cosine rule: when C = 90°, cos 90° = 0 and the formula collapses to c² = a² + b² − 0 = a² + b², exactly Pythagoras. The rule is thus more general: in an obtuse triangle cos C is negative, making c² larger than a² + b², which matches the fact that the longest side faces the largest angle. A useful check: your computed longest side must face the largest angle; any mismatch signals a slip in the working and calls for a re-check.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Works for any triangle, no need right angle
• c² = a² + b² − 2ab cos C
• SAS find side, SSS find angle
• Pythagoras is its special case
• Remember to square root after calculating