Statement 1 - In △ A BC if sin A + sin B + sin C ≤ 1 , then min { A + B , B + C , C + A }

Question

Statement 1 - In △ A BC if sin A + sin B + sin C ≤ 1 , then min { A + B , B + C , C + A } < 3 0 ∘ . Statement 2 - In △ A BC , 3 A + B + C ​ ≤ min { A , B , C } .

TestHub · Accepted Answer

Let A ≥ B ≥ C . Since b + c > a , sin B + sin C > sin A , so sin A + sin B + sin C > 2 sin A . Hence 2 sin A < 1 , sin A < 2 1 . As A ≥ 3 A + B + C = 6 0 ∘ , this gives A > 15 0 ∘ , so B + C < 3 0 ∘ and Statement 1 is true. Statement 2 is false: the mean is at least the minimum, 3 A + B + C ≥ min { A , B , C } .

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