Integer sides and angles in a right triangle.

[MegaMoh]

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.)

 

[Deleted member 4993]

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]

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

[MATH](a,b,c)=(m^2-n^2,4mn,m^2+n^2)[/MATH] 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:

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

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

 

[MegaMoh]

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

[MATH](a,b,c)=(m^2-n^2,4mn,m^2+n^2)[/MATH] 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:

[MATH]\frac{180}{\pi}\arctan\left(\frac{m^2-n^2}{4mn}\right)=k[/MATH] 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 $$m, n \neq 0$$?

 

[MarkFL]

Shouldn't there be a condition so that $$m, n \neq 0$$?

That is implied with $m,n\in\mathbb{N}$. The natural numbers are positive integers.

 

[MegaMoh]

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]

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