- #1
sharpycasio
- 16
- 0
Homework Statement
Prove the following inequality for any triangle that has sides a, b, and c.
[tex]-1<\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-\frac{b}{a}-\frac{a}{c}-\frac{c}{b}<1[/tex]
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 ([itex]sin(\theta)[/itex])
[tex]-1<(\frac{a}{b}-\frac{b}{a})+(\frac{b}{c}-\frac{c}{b})+(\frac{c}{a}-\frac{a}{c})<1[/tex]
I'm stuck. Any help? Thanks.