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A 3-4-5 triangle is a triangle with sides of the smallest integers.
I am wondering why it forms a right triangle with 36.87° and 53.13°
Do 36.87 and 53.13 relate to π or have some kind of ratio in some ways?
Can we express 35 and 53 in some nicer forms? Or 345 just happen to form these angles which have no relationship with other areas of maths?

I understand 37 and 53 are approximations. My question is if 37 and 53 form some relationship, for example, golden ratio or something like that.

wada
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    By definition, the smaller acute angle of a 3-4-5 triangle is $\cos^{-1}\left(\frac{4}{5}\right)$ – Charles Hudgins Sep 20 '20 at 03:44
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    Those aren't the exact angles. They are $\arcsin(3/5)$ and $\arcsin(4/5)$ respectively. It is easy to prove for example that they are not rational multiples of $\pi$ (or the 90 degree angle if you prefer that). – Jyrki Lahtonen Sep 20 '20 at 03:45
  • $\arcsin(3/5) = 36.869,897{\dots}^\circ$ and $\arcsin(4/5) = 53.130,102{\dots}^\circ$. – Eric Towers Sep 20 '20 at 03:55
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    "can we express the angles in a 3-4-5 triangle in some nicer forms?" With square roots and such? No. With inverse trig functions? Yes, though not in any way that is fundamentally different than just saying "the angles are the angles in a 3-4-5 triangle." – JMoravitz Sep 20 '20 at 04:05

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The only relationship to $\pi$ is that the internal angles must add up to $180^\circ$ or $\pi$ radians. Given that the right angle is $90^\circ$, the other two must add up to $90^\circ$.

poetasis
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