High-schoolers come up with 'impossible' trigonometry proofs for Pythagorean theorem

Two Louisiana high school students, Ne'Kiya Jackson and Calcea Johnson, have achieved a remarkable feat by proving the Pythagorean theorem using trigonometry, an accomplishment long considered impossible due to circular reasoning. Their proofs have been officially published in the "American Mathematical Monthly," where they presented new proofs that have been peer-reviewed. The journal's editor-in-chief, Della Dumbaugh wrote that, "the Monthly feels honored and is pleased to publish the work of these two students on its pages".

Their journey began with a math contest at St. Mary's Academy in New Orleans in December 2022. The teacher added a bonus question: to provide a new proof for the Pythagorean theorem using trigonometry, a task previously thought to be technically implausible because it challenges foundational principles of mathematics, creating a potential circular reasoning problem. Motivated by a $500 reward, the students took on the challenge, surprising their teachers. Jackson and Johnson wrote, "$500 in prize money motivated us to take on this task".

"The task turned out to be much more difficult than we had initially imagined, and we spent many late nights trying and failing to create a proof," they stated. They sacrificed their free time for the project for several months, even working on it during their vacations. Despite the challenges, they showed determination through the trial and error period. "There were many moments when we both wanted to give up on the project, but we decided to persevere and finish what we had started," they wrote. Jackson and Johnson told CBS News that they balled up sheets of paper and kept throwing them away for months.

After much trial and error, they presented their findings at the 2023 American Mathematical Society annual conference, where the response was very positive. Their high school encouraged them to present their proof on a larger platform. They were surprised when their work was showcased at the conference. "I would have never expected that this would go so far, so quickly, because it just really took off," Johnson stated to LSU. The recognition highlighted their achievement as high school students and African-American women. They even received significant recognitions, including the keys to the city and congratulations from former First Lady Michelle Obama.

In their new study, Jackson and Johnson described four new ways to prove the Pythagorean theorem using trigonometry, contributing to the hundreds of other proofs available from various mathematical disciplines. They discovered a total of ten proofs, including five new proofs and an alternative method that provides five more proofs. One of the proofs is one that had been previously presented; nine of the proofs are completely new. They used the Law of Sines in their proof, avoiding circular reasoning, which has long been considered unworkable due to the assumption that the theorem is true in trigonometric formulas.

Jackson and Johnson wrote, "There may be no other topic in mathematics that causes more confusion and uncertainty among high school students than trigonometry". They explained that the conflation of two different forms of trigonometry means that "trying to make sense of trigonometry can be like trying to make sense of a picture where two different images have been printed on top of each other". By teasing the two methods apart, researchers can discover "a large collection of new proofs of the Pythagorean theorem," they added.

Their work has been recognized as a rare achievement for young mathematicians, as they are only the third and fourth people known to have proven the Pythagorean theorem using trigonometry without resorting to circular reasoning (claim 5). The two other people were professional mathematicians. "At such a young age to publish an article—that's really amazing," Johnson agreed. Jackson expressed pride in their achievements, stating, "I didn't think it would go this far. I was pretty surprised to be published".

Both students have expressed pride in representing young women and diversity in the mathematical field. "I am very proud that we can both have such a positive impact," Johnson said. They have shown that young women are capable of significant achievements and "let other young women know that they can do anything they want to do". Jackson and Johnson see themselves as pioneers.

Currently, Jackson is studying pharmacy at Xavier University of Louisiana, and Johnson is studying environmental engineering at Louisiana State University. They are very happy with the results of their work. Despite their initial lack of experience with academic publishing tools like LaTeX, the journal was prepared to publish their work and guided the authors through the peer-review process. The peer-review process led to some improvements in the study.

Their accomplishment adds to the rich history of proofs of the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (c²) is equal to the sum of the squares of the lengths of the other two sides (a² + b²) (claim 6). Mathematical truth is timeless; once a mathematical theorem is proven, its correctness can be assumed for all eternity. People continue to find and publish new proofs of the Pythagorean theorem, including famous mathematicians like Euclid, Leonardo da Vinci, James A. Garfield, and Albert Einstein.

"We didn't think it would go this far, so quickly, because it just really took off," Johnson reflected.

Sources: Live Science, Blue News, Der Standard, Interesting Engineering, Scienze Notizie, Die Welt

This article was written in collaboration with generative AI company Alchemiq