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Is it possible to prove the Pythagorean theorem using trigonometry in a non-circular way? I would bet that it is impossible. However, to prove it is impossible, we need rigorous definitions of a "circular proof" and also of "uses trigonometry". I know there are some results from computability theory that state that there is no "natural" proof of P=NP, so it is conceivable that there are rigorous definitions of "circular proof" and "proof that uses trigonometry". The reason I am asking this is because of the recent viral story that two high school students came up with a non-circular trigonometric proof of the Pythagorean theorem. I believe that they are still using circular reasoning, despite what the news says, but of course, to judge whether it is a non-circular proof, we would need rigorous definitions.

Hanul Jeon
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user107952
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    As you are aware, the key issue is to define trigonometry. Why not start with your personal definition? – lulu Sep 21 '24 at 21:56
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    "The reason I am asking this is because of the recent viral story that two high school students came up with a non-circular trigonometric proof of the Pythagorean theorem. I believe that they are still using circular reasoning" -- Then ... wouldn't it be more prudent to state why you think so, so other people can talk about that, instead of asking for a proof of something that (if taken at face value) may well be false? Sort of an "XY" problem. – PrincessEev Sep 21 '24 at 21:56
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    I suppose you can define sine and cosine via power series, and prove what you need from there, but it will probably be a lot of work to establish their geometric meaning from those definitions. – Thomas Andrews Sep 21 '24 at 22:00
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    Or, if you have a particular problem with the, seriously well publicized, proof you are talking about, why not focus on that? – lulu Sep 21 '24 at 22:01
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    I found the proof given compelling and I didn't see any circular reasoning. They explain the definitions they're using clearly in the paper. – CyclotomicField Sep 21 '24 at 22:15
  • This is a good general question, posed for the wrong specific reasons, with little hope for an accepted answer. From a proof theorist's perspective: there is no universal notion of "uses trigonometry" or "uses technique X" for proofs. Such a notion would need to be both (1) technically engaging (2) tractable in that one would be able to prove anything about it, and (3) also accepted by mathematicians as a fitting formalization of their "uses technique X" questions, which are now primarily psycho-social, rather than proof-theoretic. It is really hard to come up with a notion satisfying (1)-(3). – Z. A. K. Sep 21 '24 at 22:21
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    W.r.t. the article you put the cart before the horse! To convince referees they made a mistake, engage with that specific proof and present a reason strong enough for them to reverse their opinion and label it circular. This would have to be something direct, e.g. pointing out an explicit reasoning step where most would use Pythagoras and it's unclear how else it could be done. I doubt you'll find such a step, but in any case technical impossibility results from proof theory won't come into play: you need to show only that this proof fails, not that all possible trig-based proofs are circular. – Z. A. K. Sep 21 '24 at 22:40
  • What is a "circular proof" ? Regarding the term "natural" in the natural proofs in complexity theory it has a precise meaning which only seems "natural" to the few hundreds of experts in that field on this planet... – plm Sep 21 '24 at 23:00
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    The question of whether the student's proof is circular is completely independent of how you define "proof that uses trigonometry". On the other hand, any questions about the definition of "circular proof" would concern how you determine whether an error in logic is "circular" or is an example of some other fallacy. That distinction is irrelevant to whether the proof is an actual valid proof. – David K Sep 22 '24 at 00:17
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    Related: "Is this a circular proof of Pythagorean Theorem?" discusses what is essentially a standard ratio proof, attaching names "$\sin\theta$" and "$\cos\theta$" to those ratios. My answer asserts that such use of trig is not "circular", as it does not invoke (explicitly or implicitly) $\sin^2\theta+\cos^2\theta=1$. See also my comments to this (closed) question, which attribute confusion about "using trig" to a misreading of Elisha Loomis. – Blue Sep 22 '24 at 05:03
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    Trigonometry could be considered the study of similar triangles. The key result about similar triangles is that the ratios of the lengths of corresponding sides are the same. That is the result which allows us to define the trigometric functions and develop their properties, including the law of sines. It is also the key result on which this proof is based. It could be recast to not even mention trig functions. It you define "Trigonometric proof" as meaning based on similarity of triangles, then this is a trigonometric proof, but then so also are plenty of others. – Paul Sinclair Sep 23 '24 at 11:51

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