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It's a very clever proof. Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being us
by jonahx 4y ago
It's a very clever proof.
Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are:
1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere.
2. The law of sines is required. But the proof of this law [1] also boils down to nothing more than the ratio definition and some algebra.
So afaict no circular logic is being used, but at the same time it doesn't seem to be doing anything previously thought to be impossible, unless there was a previous belief that the law of sines could not be used in a proof, which would be a strange belief to hold. I see it simply as a creative, unexpected proof.
[1] https://www.youtube.com/watch?v=4xO8xqLyEbA https://www.youtube.com/watch?v=4xO8xqLyEbA
- deleted 4y ago[deleted]
- fnordpiglet 4y agoIs there more to trigonometry? I’m not a abstract math person, so forgive the ignorance, but my understanding was all trigonometric functions derive from ratios of angles and lengths of triangles so in the end each occurrence of a trigonometric function can be replaced by the corresponding ratios in some triangle. There are other ways to construct things, such as power series representations, etc, but even these must necessarily be replaceable by the ratio of angles and lengths of some triangle. What am I missing?
- jonahx 4y ago> What am I missing? Nearly all the nontrivial results of trigonometry do in fact rest on the pythagorean theorem. The trig identities you learned in high school, as well as more advanced results like power series, etc. These results would be inadmissible. So the “uses trigonometry” part of this story feels like an attempt to manufacture mystery and hype. Which is a shame, because the geometric series construction is imo the interesting part, and can stand on its own merits.
- hn_throwaway_99 4y agoApparently (found this while reading an article on the girls' accomplishments) somebody proved sin^2x + cos^2x = 1 without using the Pythagorean theorem in 2009: https://forumgeom.fau.edu/FG2009volume9/FG200925index.html https://forumgeom.fau.edu/FG2009volume9/FG200925index.html. I don't think the "uses trigonometry" part is hype. They do use the definitions and law of sines, they just cleverly avoid the parts of trigonometry that depend on the Pythagorean theorem.
- jonahx 4y agoI address this in my OP. If you watch the video of the proof, you will see that the "law of sines" is 1 step away from the ratio definition of sin. You just drop one altitude, apply the definition again to the similar triangles, and re-arrange. It is almost content free as a result -- I see no reason using this in a proof would have special significance. For example, the standard proof using similar triangles (https://sumantmath.wordpress.com/2020/08/16/proof-of-pythagorean-theorem-using-similar-triangle/ https://sumantmath.wordpress.com/2020/08/16/proof-of-pythago...) is implicitly using the law of sines. The hype part is the implication that impossible trig barrier was shattered by their proof.
- boycott-israel 4y agoNot at all - the hype part is that 2 teenagers derived a uniquely elegant proof that other highly-trained mathematicians had thus far failed to do so. Any other claims seem to have been added by the media, not the teenagers themselves.
- jonahx 4y agoThat is what I was referring to. I wasn’t accusing the teenagers of anything.
- onos 4y agoThis theorem is essentially the Pythagorean theorem, so … a proof of one is a proof of the other.
- pramodbiligiri 4y agoThe sine function can be defined in terms of its own behavior, using its first-order differentiation and no reference to triangles. See this detailed article on sine. https://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ https://betterexplained.com/articles/intuitive-understanding... There’s section there titled Part 2: Understanding the definitions of sine.
- chx 4y agoYou can define sine and cosine together using the functional equations S(X)C(Y)+C(X)S(Y)=S(X+Y) C(X)C(Y)−S(X)S(Y)=C(X+Y) The only solutions to this are the constant 0 functions and the sine-cosine pair.
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- dvt 4y agoThis is super cool, I've never seen it before! Do you know what this is called so I can look up a proof/theorem on it?
- SonOfLilit 4y agoI'm not familiar with this result, but this comment is phrased in the language of Ordinary Differential Equations, so I'd look for a textbook on solving systems of ODEs and expect to find a technique that can prove that this is the unique solution (at least assuming differentiability of S(x) and C(x)).
- selimthegrim 4y agoYou can also consider the pair of functional equations as implicitly using a Taylor expansion
- chx 4y agoI asked https://math.stackexchange.com/q/124887/6400 https://math.stackexchange.com/q/124887/6400 I wish I had a more modern summary of the papers mentioned in the linked paper > Tannery, Fonctions d'une Variable, 1886, p. 147. Osgood, Lehrbueh der Funktionentheorie, 1912, p. 582. Van Vleck and H'Doubler, Transactions Amer. Math. Society, vol. 17 (1916), p. 30 because we spent an entire semester at the university in one class working on these two.
- deathlight 4y agoSine and cosine can take as their input any real number including negatives and including very large positive numbers. Their outputs can also be negative numbers between negative 1 and 1 if they have real inputs. None of this necessarily makes any sense if you're considering a purely geometric naive interpretation in terms only of ratios of lengths. You have to introduce concepts like modulo the angle in a circle and analytic coordinate system for it all to square with normal naive intuition. In fact the sign and cosine can take as their inputs any and produce as their output any complex number. You have to come up with some very interesting triangles to make this makes sense. I'm sure it might be doable but they would potentially be four dimensional triangles and I haven't explored that concept very deeply.
- doublerebel 4y ago4 dimensional triangles are the same as 3 dimensional lines. They don’t exist in the 4th dimension any more than they exist in any dimension >= 3. You would need a fourth side/point in the polygon in order for it to have any position in that dimension. (It could be a triangle in dimensions 2-4 from our perspective but to the triangle it only has 3 dimensions any way you arrange it.) Or you can bend a triangle in another dimension(s), but then it’s not a triangle by the commonly accepted definition. (E.g a 270° “triangle” on a sphere)
- cabalamat 4y ago> It's a very clever proof. Which makes it a bad one, IMO. In maths, as in programming, one should go with the simplest way that works.
- iamerroragent 4y agoI disagree. 'Aesthetic' appearance in math is important in helping drive mathematical innovation and help new human beings derive pleasure from that wonderful field. Programming is kind of an applied mathematics where efficiency does matter because it's a tool, a means to an end. Not to say that people can't find aesthetics in programming, nor that they shouldn't, rather in math at least the pleasure of discovering a new way of doing/proving something is the end in of itself. It's pleasure for me to see another way to do or prove something; I can only imagine the feelings this teenager got from actually making a discovery.
- cabalamat 4y agoI find simplicity aesthetically appealing. Obviously some other people differ.
- iamerroragent 4y agoWell you're in luck since these girls' proof is pretty simple so should be pretty pleasing for you!
- colinsane 4y agoreally just an arts/hobby vs job/career disagreement. and gosh, they’re teenager students: we can’t suck all the joy out of school! but related, i wonder what a mathematics demo party would look like.
- aaronscott 4y agoFor applied maths I agree with you. But when exploring novel and innovative approaches I disagree. There is so much we can learn from finding new ways to look at the world. The initial take may be complex or clever, but it can lead to a deeper understanding that allows for later simplification of entire subject areas.
- alex_smart 4y agoThe trigonometry thing is simply a marketing gimmick for this proof. There is no more or less trigonometry in this proof than there is in Einstein's proof. In fact, you can just taken Einstein's construction and reformulated that proof in their language by using sine rule instead of similar triangles. But then the gimmick would be too obvious. Somehow the second gimmick (the infinite series construction instead of Einstein's elegant and simple construction) makes our monkey brains not notice the first gimmick.