Triangle Inequality Proving: Use Sine Law & Find Solution

In summary, the conversation revolves around proving an inequality for any triangle with sides a, b, and c. The equation involves fractions and the suggestion is to use sine or cosine functions. The discussion also touches on the use of the Sine Law and rearranging the equation. The individual is stuck and seeking help to solve the problem by tomorrow.
  • #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. Please help. Thanks.
 
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  • #3
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.
 

What is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, the shortest distance between two points is a straight line, so the shortest distance between two sides of a triangle must be greater than the third side.

How do you prove the Triangle Inequality Theorem?

There are a few different ways to prove the Triangle Inequality Theorem, but one common method is to use the Sine Law. This law states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant for all three sides of a triangle. By using this law, you can show that the sum of the lengths of two sides is always greater than the third side.

What is the Sine Law?

The Sine Law is a mathematical rule that relates the lengths of the sides of a triangle to the sine of the angles opposite those sides. It states that the ratio of the length of a side to the sine of the opposite angle is constant for all three sides of a triangle. This law is often used in proving the Triangle Inequality Theorem.

How do you use the Sine Law to prove the Triangle Inequality Theorem?

To use the Sine Law to prove the Triangle Inequality Theorem, you first need to draw a triangle and label its sides and angles. Then, you can use the law to set up equations involving the ratios of the sides and the sines of the angles. By manipulating these equations, you can show that the sum of the lengths of two sides is always greater than the length of the third side.

Why is the Triangle Inequality Theorem important?

The Triangle Inequality Theorem is important because it is a fundamental rule in geometry that helps us understand the relationships between the sides and angles of a triangle. It is also used in many real-world applications, such as in navigation, architecture, and engineering. Additionally, understanding the Triangle Inequality Theorem can help us solve more complex problems involving triangles and other geometric shapes.

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