Integer sides and angles in a right triangle.

MegaMoh

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As simple as the title says it. Is it possible to have a right triangle with 3 integer sides and 3 integer angles(30 degrees, 11 degrees, 79 degrees, etc.)
 

Subhotosh Khan

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As simple as the title says it. Is it possible to have a right triangle with 3 integer sides and 3 integer angles(30 degrees, 11 degrees, 79 degrees, etc.)
I am assuming you want to measure angles in degrees - not in radians!

What will be the unit of measurements for the length of the sides - angstroms, cm, inches, light-years?
 

MarkFL

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I would begin with a well-known method for generating Pythagorean triples:

\(\displaystyle (a,b,c)=(m^2-n^2,4mn,m^2+n^2)\) where \(m,n\in\mathbb{N}\) and \(n<m\)

And so, if you are using degrees to measure the angles, it will suffice to find an ordered pair \((m,n)\) such that:

\(\displaystyle \frac{180}{\pi}\arctan\left(\frac{m^2-n^2}{4mn}\right)=k\) where \(k\in\mathbb{N}\) and \(k<90\).

I would likely use a computer to search for a solution.
 

MegaMoh

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I would begin with a well-known method for generating Pythagorean triples:

\(\displaystyle (a,b,c)=(m^2-n^2,4mn,m^2+n^2)\) where \(m,n\in\mathbb{N}\) and \(n<m\)

And so, if you are using degrees to measure the angles, it will suffice to find an ordered pair \((m,n)\) such that:

\(\displaystyle \frac{180}{\pi}\arctan\left(\frac{m^2-n^2}{4mn}\right)=k\) where \(k\in\mathbb{N}\) and \(k<90\).

I would likely use a computer to search for a solution.
Shouldn't there be a condition so that \(\displaystyle m, n \neq 0\)?
 

MarkFL

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Shouldn't there be a condition so that \(\displaystyle m, n \neq 0\)?
That is implied with \(m,n\in\mathbb{N}\). The natural numbers are positive integers. :)
 

MegaMoh

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That is implied with \(m,n\in\mathbb{N}\). The natural numbers are positive integers. :)
Oh, sorry. I thought it was \(\mathbb{R}\) somehow 😱
 

MegaMoh

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So isn't there some other fancy number theory way to get all solutions for this question without code?
 
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