Yes that’s Jason Zimba’s proof from the paper On the Possibility of Trigonometric Proofs of the
Pythagorean Theorem. This proof is indeed a trigonometric proof of the Pythagorean Theorem and it’s from 2009.
I am using ‘independent’ to mean does not depend on.
Just an amusing comment of the question. I tried to find a proof of the PT using the law of sines. I hadn’t seen the diagram with the infinite decomposition. Anyway, I started with the law of sines, drew a diagram including a circle, saw how it proved the PT by a proof that is surely known but from which all the sines had disappeared.
Another comment. It is stated above that the double angle formula can be proved without the PT. But I’ll bet that the general sum formula cannot be. For if it could it would yield an immediate proof of the PT. For if you have
sin(x+y) = sin x * cos y +cos x * sin y, then when x + y = 90 deg, this gives immediately 1 = sin x * cos y + sin y * cos x. But x and y are complimentary, so cos y = sin x and sin y = cos x, so 1 = sin^2 x + cos^2 x .
Why not? Consider the diagram in the above PDF, for one thing. I did suggest this may or may not be considered veering close to simply proving the Pythagorean theorem, as you say.
You would lose that bet.
For completeness this is pretty close to the proof I was thinking of when I wrote this post.
Unlikely. What most likely happpened was that the special session organizers, who may or may not have any role in the AMS beyond possibly being AMS members, read their abstract and thought it sounded sufficiently interesting and worthwhile to warrant adding their talk to the program.
I’ve given talks at lots of AMS meetings, both national and local, and session organizers have never asked to see any materials beyond an abstract. Certainly the AMS itself as an organization never steps in to vet speakers’ work before allowing their talks to be scheduled.
Maybe session organizers for undergraduate speakers subject submissions to more scrutiny, but I doubt it. They know the undergraduates’ faculty advisors are doubtless supervising their efforts to some extent and that the talk content isn’t likely to be flat-out wrong.
And it’s not even unheard-of for the occasional talk at such a conference to actually be flat-out wrong, for that matter. Conference talks are much less rigorously screened than peer-reviewed publications. They’re intended as an earlier step on the road to peer-reviewed publication, though of course it’s always nice if your findings are obviously publication-worthy first crack out of the box.
This also is not true. I mean, plenty of individual research findings that were first presented at regional or national AMS meetings do subsequently get published, but such meetings don’t typically have official “conference proceedings” volumes the way that more subject-specific small conferences tend to do.
This video seems to cover their entire proof in more step-by-step detail than the other video:
Just re-discovered this thread.
I just think how remarkable these highschoolers’ achievement is.
They managed to pull off something tenured university professors and Fields Medal winners couldn’t.
Even the likes of Newton, Euler, and Gauss didn’t succeed.
While these kids deserve all due credit for their work, what are you getting at with this hyperbolic disparagement of Newton, Euler, and Gauss? Is it a quote from some news clip? If they are suggesting that any one of Newton, Euler, or Gauss could not come up with a dozen original proofs of Pythagoras’s theorem then that is simply bullshit they made up. (I am reminded of a story about Gauss along the lines of, frustrated that he could not prove the quadratic reciprocity theorem, basically locked himself in his study for a week, at the end of which he emerged with 8 different proofs. Then again, Euler and Gauss never completely proved the Prime Number theorem…)
More to the point is how “amateur” mathematicians are treated. I recall Martin Gardner described one oddball who was working in the post office because his mother thought that would make a more reliable career? At least these days it is easier than ever to access books and course materials if one has the time and energy for self-study. One thing the OP kids did get absolutely right is that, beyond merely being competent or brilliant in one’s field, self-promotion, up to and including talking to journalists, is a good move these days.
It’s pretty easy to prove the sine of sum formula directly from Euler’s formula (eix = cos x + i sin x):
ei(a+b) = eia eib = (cos a + i sin a)(cos b + i sin b) =
(cos a cos b - sin a sin b) + i (sin a cos b + cos a sin b)
That’s Euler’s formula applied twice and then some simple algebra.
But ei(a+b) is also equal to cos(a+b) + i sin(a+b), again by Euler’s formula.
Equating the imaginary parts of these two expressions,
sin(a+b) = sin a cos b + cos a sin b
QED
Yeah, I’m not sure where @EnolaStraight is getting that either. Even if these teens did genuinely accomplish something that has never been done before (and isn’t the jury still out on that?), it’s not as though this is something that Newton, Euler, and Gauss had been trying to do.
As I said way earlier in the thread:
If the task is defined as just “come up with a proof of the Pythagorean Theorem using trigonometry”, we have a few examples of that in this thread, and with all due respect to @Hari_Seldon , @Lance_Turbo , and @DPRK , none of them are the equal to Newton, Euler, or Gauss. What these girls did would not have been remarkable, had it come from a professional mathematician.
But remember that, first of all, they were high schoolers, and second, the level of literature review they were realistically able to do claimed (incorrectly) that it was impossible. A lot of students, on reading that, would have given up on trying… but these students, instead of taking it as a sign to give up, not only took it as a challenge, but succeeded at that challenge. And that is definitely worth celebrating.
Agree on all counts.
A guy who taught high school math for the most part made an offhand comment in his quirky book that a trigonometric proof of the Pythagorean Theorem was impossible. I’m not sure any real mathematician ever took that statement seriously much less tried and failed to find such a proof.
But the work of Ne’Kiya D Jackson and Calcea Rujean Johnson is still impressive. As far as I can tell this is a new proof of the Pythagorean Theorem. That’s not nothing and I’m not sure I’ve ever met someone who has done that, and I know a lot of mathematicians having earned a math PhD myself.
And as someone who has beaten my head against a problem not knowing if there was a solution, I know the joy of the last piece falling into place and finally cracking that nut. I am happy for them for getting to experience that.
I can hardly describe the feeling of taking a step back, looking at your work, and being able to honestly say, “I’m the only person in the world who knows this.”
Ah.
I had been operating on the notion that this WAS something of an unproven math problem along the lines of the Reimann Hypothesis or, earlier, Fermat’s Last Theorem.
Nevertheless…
I stand by my praise for these junior mathematicians:
I WILL CELEBRATE ACOMPLISHMENT AND ACHIEVEMENT AS I SEE FIT!!!
(so there 
)
Wow! Beautiful. Then the claim that there is no purely trigonometrical proof of the PT is simply wrong.
That’s what I think at this point.
Both these young women and Jason Zimba from upthread cited Loomis for the claim and none of these four are mathematicians. Loomis didn’t prove his claim or even attempt to do so.
I have no reason to believe that any serious mathematician ever considered this a problem worth working on. I’m also pretty certain many mathematicians doing manipulating of trig functions could have proofs of this nature essentially fall out of their work, but none would even dream of publishing it.
@Hari_Seldon, you’ve been a research mathematician for years and came up with a trig proof of the PT in this thread with almost no effort. Are you going to submit it to a journal? Of course not.
My sense is that the students did something interesting and new, and that it got wildly oversold as solving some impossible, millennium-old problem (certainly browsing the resulting headlines shows that).
Scientific American has a nice summary that doesn’t undermine the new/interesting part, but doesn’t oversell it either.
2 High School Students Prove Pythagorean Theorem. Here’s What That Means - Scientific American
Perhaps not, but per the article at least one mathematician thought it was interesting enough to capture examples on his website. So, it’s not 100% trivial either.
I may have overstated the interesting part. It’s not something a serious research mathematician would pursue. There is however an area of mathematics that I don’t often think about for which this would be interesting. Math education.
This is the exact kind of thing people who are interested in pedagogy enjoy. It increases awareness and interest in math because it is accessible to recreational mathematicians and students.
Thanks to the excellent post of @Lance_Turbo I now have
a trigonometric proof of the Pythagorean theorem
that involves neither infinite series nor complex
exponentials. And to answer another poster, no I
will not be trying to publish it. Maybe in a HS
math magazine (there is one in Canada).
I would also add that I agree completely with
@Chronos on the girl’s achievements. Not only did
they show commendable skepticism about the "no
trigonometric proof"assertion, but they also found
one.
The argument by describing a diagram. If you are
interested in following it, you have to start by
drawing it. I don’t know how to include it in a
post. If I could, it would be a lot shorter.
Begin with a vertical line segment of lenght 1
unit, endpoints labeled A at the bottom and B
at the top. Draw horizontal rays (half lines) to
the right starting from A and B. Given an
acute angle \alpha, let \beta be its
complement. Draw a ray towards the right up from
A that makes an angle \alpha with the
horizontal line through A and another, starting
at B, towards the right down that makes an angle
\alpha with AB. These will meet at a point
C. Draw a line through C parallel to AB and
suppose it meets the upper horizontal line at D
and the lower one at E. Since \angle
(CBD)=\beta and \angle(BDE) is a right angle,
it follows that \angle(BCD)=\alpha. Since
\angle(AEC) is a right angle,
\angle(ACE)=\beta and so \angle(ACB) is a
right angle. At this point you have a rectangle
of height 1 and unknown width (it is actually
\sin\alpha\cos\alpha, but this is not needed)
divided into three similar right triangles.
Denote by x,y,u,v the lengths of CD,EC,BC,AC,
respectively. From \triangle(EAC) we see that
\sin\alpha=y/v, while from \triangle(ABC) that
\sin\alpha=v. Comparing them yields
y=v^2=\sin^2\alpha. From \triangle(ABC) we
see that \cos\alpha=u, while from
\triangle(BCD), we see that \cos\alpha=x/u.
Comparing them yields x=u^2=\cos^2\alpha. Then
\sin^2\alpha+\cos^2\alpha=y+x=1. This completes
the argument.