Is 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?
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.
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.
I 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-pythago...) is implicitly using the law of sines.
The hype part is the implication that impossible trig barrier was shattered by their proof.
Sine 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.
4 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)
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.
> because we spent an entire semester at the university in one class working on these two.
Yeah, the Math Overflow answers are a bit sparse, and I think the Euler formulation (while clever) is a bit of a red herring and might be circular. Trying to slowly go through that 1917 paper, thanks for linking!
I'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)).
I did not know that one! It's a more complex version of the well-known functional definition of the exponential function, i.e. the unique continuous function satisfying
