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 A range problem
05-10-2010, 07:25 AM
Post: #1
 elim Moderator     Posts: 578 Joined: Feb 2010 Reputation: 0
A range problem
Let $$0<\alpha \le \beta \le \gamma, \quad \alpha+\beta+\gamma=\pi$$. Find the range of the value $$\displaystyle{ \min \left\{\frac{\sin\beta}{\sin\alpha},\frac{\sin \gamma}{\sin \beta}\right\}}$$
05-18-2010, 03:13 PM
Post: #2
 elim Moderator     Posts: 578 Joined: Feb 2010 Reputation: 0
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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