Example 1: finding the third side

• In a triangle, b = 5 cm, c = 7 cm and A = 60°. Find a.
• Solution a² = 5² + 7² − 2 × 5 × 7 × cos 60° = 25 + 49 − 70 × 0.5 = 74 − 35 = 39. So a = √39 ≈ 6.25 cm.

Example 2: finding an angle from three sides

• In a triangle, a = 7 cm, b = 8 cm and c = 9 cm. Find the angle A.
• Solution cos A = (8² + 9² − 7²) / (2 × 8 × 9) = (64 + 81 − 49) / 144 = 96 / 144 = 2/3. So A ≈ 48.2°.

Example 3: identifying the largest angle

• A triangle has sides 5 cm, 6 cm and 8 cm. Find the angle C opposite the side 8 cm and state its type.
• Solution cos C = (5² + 6² − 8²) / (2 × 5 × 6) = (25 + 36 − 64) / 60 = −3 / 60 = −0.05. So C ≈ 92.9°.

1. The numerator is negative
2. so C is obtuse. It is also the largest angle
3. because it is opposite the longest side.

Example 4: checking for a right angle

• A triangle has sides 5, 12 and 13. Is it right-angled?
• Solution The longest side is 13. The cosine of the angle opposite it is (5² + 12² − 13²) / (2 × 5 × 12) = (25 + 144 − 169) / 120 = 0. So the angle is 90°, and the triangle is right-angled.

Example 5: a real-life problem

• A car travels 8 km in one direction from a point, and another car travels 15 km in another direction from the same point.
• The angle between the two directions is 60°.
• What is the distance between the two cars?

Example 6: a side with a decimal value

• In a triangle, a = 10 cm, b = 12 cm and C = 40°. Find c, the side opposite C.
• Solution c² = 10² + 12² − 2 × 10 × 12 × cos 40° = 100 + 144 − 240 × 0.7660 ≈ 244 − 183.8 = 60.2. So c ≈ √60.2 ≈ 7.76 cm.