Posted on 04/21/2023 2:30:05 PM PDT by nickcarraway
At an American Mathematical Society meeting, high school students presented a proof of the Pythagorean theorem that used trigonometry—an approach that some once considered impossible
Two high school students have proved the Pythagorean theorem in a way that one early 20th-century mathematician thought was impossible: using trigonometry.
Calcea Johnson and Ne’Kiya Jackson, both at St. Mary’s Academy in New Orleans, announced their achievement last month at an American Mathematical Society meeting. “It’s an unparalleled feeling, honestly, because there’s just nothing like it, being able to do something that ... people don’t think that young people can do,” Johnson told WWL-TV, a New Orleans CBS affiliate.
If verified, Johnson and Jackson’s proof would contradict mathematician and educator Elisha Loomis, who stated in his 1927 book The Pythagorean Proposition that no trigonometric proof of the Pythagorean theorem could be correct. Their work joins a handful of other trigonometric proofs that were added to the mathematical archives over the years. Each sidestepped “circular logic” to prove the pivotal theorem. So what exactly is a trigonometric proof of the Pythagorean theorem, and why was Loomis so closed off to the idea?
The Pythagorean theorem provides an equation to calculate the longer side of a right triangle by summing the squares of the other two sides. It is often phrased as a2 + b2 = c2. In this equation, a, b and c represent the lengths of the three sides of a right triangle, a triangle with a 90-degree angle between two of its sides. The quantity c is the length of the longest side, called the hypotenuse. Though the theorem is named for the ancient Greek philosopher Pythagoras, some historians believe it was known in Babylon around 1,000 years earlier.
The theorem “connects algebra and geometry,” says Stuart Anderson, a professor emeritus of mathematics at Texas A&M University–Commerce. “The statement a2 + b2 = c2, that’s an algebraic statement. But the figure that it comes from is a geometric one.”
Meanwhile trigonometry focuses on functions that depend on angles. These functions, such as the sine and cosine, are defined using right triangles. Imagine a right triangle with one side that lies flat against a table and another that shoots straight up from where it meets the first side at a right angle. The hypotenuse will reach diagonally between these two sides.
Now measure the angle between the hypotenuse and the table. Mathematicians define the sine of this angle as the height of the vertical side divided by the length of the hypotenuse. The cosine of this angle is the length of the horizontal side divided by the hypotenuse. The Pythagorean theorem is therefore equivalent to the equation sin2 x + cos2 x = 1. “A lot of the basic trig ‘identities’ are nothing more than Pythagoras’ theorem,” explains Anderson, referring to equations that describe relationships among different trigonometric functions.
Loomis believed that if you used these functions in a proof of the Pythagorean theorem, you would have assumed the theorem to begin with—a circular argument and thus an unforgivable mathematical error.
But that’s not always true. In their talk at the American Mathematical Society meeting, Jackson and Johnson said a trigonometric identity called the law of sines didn’t depend on the Pythagorean theorem and that they could use it to prove the theorem.
Anderson hopes that Jackson and Johnson’s proof will raise interest in mathematics among students. “It kind of makes me wish I still had a class so I could talk about it,” he says.
The other trigonometric proofs of the theorem that have appeared in the past include a few that are described on mathematician Alexander Bogomolny’s website. One of these was crafted by Jason Zimba, then a physicist and mathematician at Bennington College, and published in Forum Geometricorum in 2009. This proof used a trigonometric identity that allows you to calculate the cosine and sine of an angle x – y without using the Pythagorean theorem—if you know the cosines and sines of x and y on their own.
On October 26, 2009, Bogomolny added Zimba’s proof to his website, writing “Elisha Loomis, myself and no doubt many others believed and still believe that no trigonometric proof of the Pythagorean theorem is possible.... I happily admit to being in the wrong.” Over time, Bogomolny added more trigonometric proofs to the site: one such proof could be written in just four lines.
The saga shows how even the simplest mathematics can surprise us. “Mathematicians, I think, have learned to not make a bold claim that something is impossible because we’ve been embarrassed over the years too many times by doing that,” Anderson says.
The American Mathematical Society has encouraged the New Orleans students to submit their proof for publication in a peer-reviewed scientific journal.
Leila Sloman is a math writer based in Princeton, New Jersey. She is a contributor to Quanta Magazine, and she creates and edits outreach content for the American Mathematical Society. She holds a Ph.D. in mathematics from Stanford University.
Recent Articles by Leila Sloman
Really?
Philosophy is the study of questions to which there are no answers. Mathematics is the study of answers to which there are no questions.
That’s pretty cool!
Color me skeptical. I “could be” convinced, but ...
"/>
‘Color me’ skeptical is right. History is being rewritten to make certain folks feel good about themselves. I think this is just more of that.
well done!!
Reading this just makes me want to pull this guys underwear over his head.
Shouldn’t it be easy for mathematicians be able to validate the proof quickly?
The thing I loved is that there’s no mention of ethnicity. It’s just two smart girls proving something that was once thought impossible. About damn time.
From Wikipedia
“The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods - possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.”
And these high school kids are the first ones to do this using Trigonometry in all the proceeding 2,593 years?
Color me skeptical.
And it has not been submitted for peer review yet?
Could be we’re celebrating a bit early.
Then again, in todays hyper charged politically correct academia community, will anyone dare tell them they’re wrong?
It’s not just validating the proof. It must be shown to be noncircular. For example, a proof that uses on the familiar trig identity sin^2(x)+cos^2(x)=1 would be a circular proof.
I’ll wait until their proof is peer reviewed. I run into kids using circular logic in their arguments on an almost daily basis and that might be only how these girls “proved” this. If they didn’t, then good for them.
Shouldn’t they know by now?
>>>And these high school kids are the first ones to do this using Trigonometry in all the proceeding 2,593 years? Color me skeptical.<<<
Everyone in the loop knows the trig squ... gal, who is named SOHCAHTOA.
(She used to be on the butter boxes until she got cancelled. Circle of life I suppose.)
SOH means Slight Of Hand and Sense Of Humor, but in math it's a sin when people want to know Y.
Glad I could help.
If this holds up they will be listed alongside President Garfield as original authors of proofs of the Pythagorean Theorem.
You Go Girls!!
They had to know their stuff, because it was before electronic calculators and computers. They made many of their calculations with slide rules, a huge blackboard (or whiteboard) and a ladder.
Pictures and descriptions of the main characters are at the link -- scroll down.
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.