Triangle Inequality Proving: Use Sine Law & Find Solution

[sharpycasio]

Homework Statement

Prove the following inequality for any triangle that has sides a, b, and c.

$$-1<\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-\frac{b}{a}-\frac{a}{c}-\frac{c}{b}<1$$

Homework Equations

The Attempt at a Solution

I think we have to use sine or cosine at a certain point because the bounds of the inequality are the same as the bounds of the two functions' ranges. Perhaps the Sine Law since that applies to all triangles? Tried rearranging it, pairing up the reciprocals. Maybe the fractions represent ratios ($sin(\theta)$)

$$-1<(\frac{a}{b}-\frac{b}{a})+(\frac{b}{c}-\frac{c}{b})+(\frac{c}{a}-\frac{a}{c})<1$$

I'm stuck. Please help. Thanks.

 

[LCKurtz]

This is a duplicate of this thread:

https://www.physicsforums.com/showthread.php?t=642712

 

[sharpycasio]

I am sorry for reposting the same question. It's just that I've been working on it for hours and I really have to solve it for tomorrow. My apologies.