Study guides › Level 9 · Trigonometry

The law of cosines

Level 9 · Trigonometry · for questions like “Two sides of a triangle are 3 and 8, with 60° between them. How long is the third side?”

💡 The idea

For any triangle, not just right ones:

c² = a² + b² − 2ab × cos C

C is the angle between sides a and b.

✏️ Worked examples

1. Sides 3 and 8, 60° between them: c² = 9 + 64 − 2 × 3 × 8 × 1/2 = 73 − 24 = 49, so c = 7.

2. Sides 3 and 5, 120° between them: cos 120° = −1/2, so c² = 9 + 25 + 15 = 49 and c = 7.

⚠️ Watch out

• cos 120° is negative, so the last part gets added.
• At 90° the last part is 0, and it's just Pythagoras.

🎯 Try these

1. Sides 5 and 8, 60° between them
2. Sides 7 and 8, 120° between them
3. Sides 6 and 16, 60° between them

Answers: 1) 7 2) 13 3) 14

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.