In △ABC, if R is the radius of the circumcircle:
a/sin A = b/sin B = c/sin C = 2R
Example ①
In △ABC, find the radius R of the circumcircle when a = 5 and A = 30°.
Solution: a/sin A = 2R → 5/(1/2) = 2R → R = 5
In △ABC, find the radius R of the circumcircle when a = 5 and A = 30°.
Solution: a/sin A = 2R → 5/(1/2) = 2R → R = 5
Example ②
In △ABC, find b when A = 45°, B = 60°, and a = √6.
Solution: b/sin 60° = a/sin 45° → b = √6 × (√3/2)/(√2/2) = √6 × √3/√2 = 3
In △ABC, find b when A = 45°, B = 60°, and a = √6.
Solution: b/sin 60° = a/sin 45° → b = √6 × (√3/2)/(√2/2) = √6 × √3/√2 = 3
Key Point
- Sine rule: side/sin(opposite angle) is constant = 2R.
- Use it with a “side and opposite angle” pair.
Practice Problems
- In △ABC, find the radius R of the circumcircle when B = 30° and b = 3.
Solutions and Explanations
- Solution: R = 3. b/sin B = 2R → 3/(1/2) = 2R → R = 3.