When the three sides or two sides and the included angle are known:
a² = b² + c² - 2bc cos A (and cyclically)
Example ①
In △ABC, find a when b = 5, c = 7, and A = 60°.
Solution: a² = 25 + 49 - 2×5×7×(1/2) = 74 - 35 = 39 → a = √39
In △ABC, find a when b = 5, c = 7, and A = 60°.
Solution: a² = 25 + 49 - 2×5×7×(1/2) = 74 - 35 = 39 → a = √39
Example ② (Finding an Angle)
In △ABC, find cos A when a = 7, b = 5, and c = 3.
Solution: cos A = (b² + c² - a²) / 2bc = (25 + 9 - 49)/30 = -15/30 = -1/2 → A = 120°
In △ABC, find cos A when a = 7, b = 5, and c = 3.
Solution: cos A = (b² + c² - a²) / 2bc = (25 + 9 - 49)/30 = -15/30 = -1/2 → A = 120°
Key Point
- Two sides and their included angle → use the cosine rule to find the opposite side.
- Three sides → rearrange the cosine rule to find cos (each angle).
Practice Problems
- In △ABC, find cos C when a = 4, b = 3, and c = 5.
Solutions and Explanations
- Solution: cos C = 0 (C = 90°). (16+9-25)/24 = 0.