First was in 2022, when Calcea Johnson and Ne’Kiya Jackson wowed their teachers. At that moment, the two young women found five new ways to prove the Pythagorean theorem. They achieved this using trigonometry and their efforts have just been rewarded with an article that has just been published in the journal American Mathematical Monthly. But there is more to this story.
What does the Pythagorean theorem say?
Let’s remember. The famous theorem explains the geometric relationship that exists between the three sides of a right triangle. He states that in this type of triangles, The sum of the squares of the legs is equal to the square of the hypotenuse.
It is represented by the formula a² + b² = c², ‘a’, ‘b’ and ‘c’ are the lengths of the sides.
Over the years, many mathematicians have proven the theorem using algebra and geometry. But explaining it through trigonometry, the branch of mathematics that studies triangles from angles, was considered almost impossible. The belief derives from the fact that any proof of this type resorts to the mathematical expression sin² a + cos² a = 1which is nothing more than a reformulation of the Pythagorean theorem itself. Therefore, any attempt in that direction would inevitably lead to a vicious circle known as ‘circular reasoning’.
It is likely that the discovery will also serve to improve our knowledge in other areas of the discipline.
5 + 5 = 10
The two students proved the Pythagorean theorem without resorting to circular reasoning, a feat that had only been achieved on two previous occasions. Their finding is based on a trigonometric result: the Law of Sines, with which they revealed that the test is independent of the equation sin² a + cos² a = 1. Calcea and Ne’Kiya found five new ways to explain the theorem using trigonometry and another new approach by which they discovered five more, for a total of ten proofs.
“I was very surprised by the post,” said Ne’Kiya Jackson, who is currently studying pharmacy at Xavier University of Louisiana. “I didn’t think it would come to this. Getting a paper published at such a young age is really mind-blowing,” adds Calcea Johnson, an environmental engineering student at Louisiana State University’s Roger Hadfield Ogden Honors College. Although they presented their work at a meeting of the American Mathematical Society in 2023, it was not authorized until today, October 28.
Why is trigonometry so difficult?
The two authors maintain that One of the reasons why trigonometry creates so much confusion in students is because there are two completely different versions of this branch and are defined using the same terms. This means that trying to make sense of this mathematical study can be like trying to outline a painting in which there are two different printed images on top of each other.
Jackson and Johnson advise separating the two versions and focusing on just one; Only in this way can new proofs of the Pythagorean theorem be found: “I am so proud that we are such a positive influence in showing that young women and girls of color can do these things, and instilling in other young women the idea that they are capable of doing anything they set their minds to.” Johnson concludes.
Article originally published in WIRED Italy. Adapted by Alondra Flores.
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