A range problem

05102010, 07:25 AM
Post: #1




A range problem
Let \(0<\alpha \le \beta \le \gamma, \quad \alpha+\beta+\gamma=\pi\). Find the range of the value \(\displaystyle{ \min \left\{\frac{\sin\beta}{\sin\alpha},\frac{\sin \gamma}{\sin \beta}\right\}}\)


05182010, 03:13 PM
Post: #2




RE: A range problem
By the law of sines, problem is equivalent to find \(\min \left\{b/a,c/b\right \}\)
where \(a, b, c\) are the lengths of a triangle with \(1 = a \le b \le c\), By the necessary condition of the lengths of a triangle, we have the constrain that \(b \le c < b+1\) thus \(c = b + \lambda\) for some \(\lambda \in [0,1)\) and \(f(a,b,c) = \min \left\{b/a,c/b\right \} = \min\left\{b, 1+ \lambda/b\right \}\) Notice that for fixed \(\lambda\), \(1+ \lambda/b\) is decreasing thus when \(b = 1+\lambda /b\), \(f(a,b,b+\lambda)\) reaches the \(\lambda\)maximum \(b = \frac{1+\sqrt{1+4\lambda}}{2}\) Therefore the range we are looking for is \([1, \frac{1+\sqrt{5}}{2})\) 

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