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This is about the $3,4,5$ Pythagorean triangle.

Question in the title. I think the answer is "no", but if not, then why not? What about the $5, 12, 13$ triangle, or even non-Pythagorean triangles like $1, 4, \sqrt {17} $ ? Can you write the angles in these triangles in a "nice way"?

ViHdzP
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Adam Rubinson
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    Well, Niven's Theorem rules out a lot of "nice" answers. For the second part of your question, of course every angle is an angle in a right triangle if you don't require integrality (or rationality) of the sides. – lulu Nov 19 '19 at 00:25
  • Searching up nice values of trigonometric functions on Math SE gives plenty of questions related to yours. – Toby Mak Nov 19 '19 at 00:29
  • Well I didn't know Niven's Theorem was a thing, but that doesn't really answer my question. However, I should be clearer about what I mean by "exact value". I think I mean "can be written as the product of a finite amount of surds", in the same way that $cos 18 ^\circ $ can be written as the product of a finite amount of surds. – Adam Rubinson Nov 19 '19 at 00:32
  • The Lindemann Weirstrass Theorem tells us that if $x$ is algebraic, and non-zero, $\cos (x)$ is transcendental. Since you want $\cos (x)=\frac 45\in \mathbb Q$ $x$ must be transcendental. Note: I doubt that's really what you meant though. Usually in these situations people would ask about $\frac x{\pi}$, as in Niven. I would tend to assume that this $x$ would also be transcendental in your case, but that's less evident (at least to me). – lulu Nov 19 '19 at 00:47
  • lulu- For me, if x in degrees or radians is a "nice" (algebraic I think?) number, then that would be nice. Wouldn't it be nicer than writing 36.86989... $^\circ$ ? To me, that's gross (because it's not "exact"). – Adam Rubinson Nov 19 '19 at 00:56
  • @AdamRubinson Sure, but as the theorems I linked to illustrate, it's asking a lot to require that both $x$ and $\cos(x)$ be "nice". The L-W theorem I linked to, for instance, says that, other than at $x=0$, they can't both be algebraic. But otherwise. – lulu Nov 19 '19 at 00:59
  • @WillJagy I'm not sure what the intent is here... Do you know, by the way, any theorems about the transcendence of $\frac x{\pi}$ given the algebraicity of $\cos (x)$? I'd assume that, as in the posted question here, $\cos x = \frac 45$ or some other rational must imply that $\frac x{\pi}$ is transcendental, but I can't instantly prove it. – lulu Nov 19 '19 at 01:01
  • " I'm not sure what the intent is here... " Honestly, the original question was purely out of curiosity. But isn't that what a lot of maths is about anyway (especially pure maths)? lol. "it's asking a lot to require that both x and cos(x) be "nice"." Maybe I'll crawl back into my little maths cave and just use 36.86989... $^\circ $ for now. But I don't forget questions I ask and I never know when the answers/comments here could be of interest in the future... – Adam Rubinson Nov 19 '19 at 01:06
  • I wasn't criticizing the question! It's a perfectly sensible question as it stands, though I don't think it has a terribly enlightening answer. Most special values of transcendental functions are transcendental. The people commenting here are debating possible variants of your question which may or may not have pleasant answers. Of course, the variants we come up with may or may not have anything to do with what you were thinking about. – lulu Nov 19 '19 at 01:09
  • I'm not saying you were. I'm just reading up on algebraic and transcendental numbers for the first time actually. Very interesting. Algebraic numbers are countable? It's 1am though, so I should really go to bed – Adam Rubinson Nov 19 '19 at 01:18
  • Yes, the algebraic numbers are countable. – lulu Nov 19 '19 at 01:27
  • What's interesting to me is that when I studied linear independence and orthogonal/orthonormal bases with regards to inner product spaces, we didn't use algebraic numbers as an example. We always used (vector) subspaces of R^3 from what I can remember. I wonder if you can do something similar for "algebraic subspaces of R"- split them up into orthogonal bases? I dunno. I need bed – Adam Rubinson Nov 19 '19 at 01:32
  • A235509. – g.kov Nov 19 '19 at 03:01

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