You have the famous Pythagorean theorem (a2 + b2 = c2) was proven using trigonometry – several times. Their results were published in the journal American Mathematical Monthly.
The crux of the matter: Trigonometry is a branch of geometry, and its basic formulas are based on the assumption that the Pythagorean theorem is true. There is therefore a risk of a circular argument – an argument in which what is to be proven is already a prerequisite.
Without circular reasoning, professional mathematicians have only succeeded in producing such a proof twice, the publisher said. In addition, there are hundreds of other proofs of the millennia-old theorem from other disciplines of mathematics such as algebra.
Pythagorean theorem
The Pythagorean theorem is probably one of the few formulas that many people still have somewhere in the back of their minds from school lessons. This is about the relationship between the side lengths of a right-angled triangle: The sum of the squares of the sides of the right angle (a and b) is equal to the square of the hypotenuse (c), which is opposite the 90-degree angle. So you can calculate the length of any side of a right triangle if you know the length of the other two sides.
In their now published work, the authors Calcea Johnson and Ne’Kiya Jackson have presented five ways to prove the theorem using trigonometry. There is also a method that enables five additional proofs. To do this, roughly speaking, they formed various new right-angled triangles with certain angle dimensions from a right-angled triangle ABC.
$500 prize money
The two came up with the idea as students in 2022 during a mathematics competition at their high school in the USA. One question was to create a new proof for the Pythagorean theorem. “500 dollars in prize money motivated us to take on this task,” they write. “The task turned out to be much more difficult than we first imagined, and we spent many long nights trying and failing to produce a proof.”
For several months, the teenage girls sacrificed their free time to the project, even working on it during their holidays. “There were many moments when we both wanted to give up on the project, but we decided to persevere and finish what we started.” In the end, according to the publisher, in addition to awards, there was even praise from former First Lady Michelle Obama. And now a scientific publication with new evidence.
“I’m very proud that we can both have such a positive influence,” said co-author Johnson. The two authors showed that young women are capable of doing this, “and in doing so let other young women know that they can do anything they want to do.”
Johnson is now studying environmental engineering at Louisiana State University, and Jackson is studying pharmacy at Xavier University of Louisiana.
ePaper
**Interview with Calcea Johnson and Ne’Kiya Jackson: New Proofs of the Pythagorean Theorem**
**Editor:** Welcome, Calcea and Ne’Kiya! It’s a pleasure to have you both here to discuss your remarkable achievement in proving the Pythagorean theorem using trigonometry. To start off, can you tell us what inspired you to take on this challenge during the math competition?
**Calcea:** Thank you for having us! The challenge came from a mathematics competition we participated in during high school. The idea of proving the Pythagorean theorem seemed intriguing to us, especially with a $500 prize on the line. However, we quickly realized it was much more complex than we initially thought.
**Ne’Kiya:** Yes! The motivation was definitely there, but the depth of the project tested our resolve. We spent countless evenings and even holidays brainstorming and refining our proofs, motivating each other to push through the frustration.
**Editor:** That determination paid off. You mentioned that you used trigonometry to prove the theorem in a way that minimizes circular reasoning. Can you explain the significance of avoiding circular arguments in mathematical proofs?
**Ne’Kiya:** Absolutely. Circular reasoning can undermine the credibility of a proof because it assumes what it seeks to establish. By avoiding this, our work reinforces the validity of the Pythagorean theorem in a robust way using trigonometric principles, which are foundational in mathematics.
**Calcea:** Our results were actually published in the American Mathematical Monthly, which is a huge honor. We hope our work opens doors for other students, especially young women, to engage in mathematics and explore their interests.
**Editor:** That’s inspiring! Michelle Obama even praised your efforts. How did that recognition impact you both, especially as you continue your academic journeys?
**Calcea:** Receiving recognition from someone as esteemed as Michelle Obama was incredibly validating for us. It reinforced our belief that young women can excel in fields like mathematics and science.
**Ne’Kiya:** It motivated us to continue striving for excellence in our studies. I’m now pursuing pharmacy at Xavier University, and Calcea is at Louisiana State University studying environmental engineering. We want to inspire others to follow their passions, no matter how challenging it seems.
**Editor:** Given your experience, what advice would you give to students facing difficult challenges in their academic pursuits?
**Calcea:** I’d say persistence is key. There will be tough moments where you feel like giving up, but you have to keep pushing through. Seek support from your peers and remember that challenges are opportunities to learn and grow.
**Ne’Kiya:** Exactly. Embrace the struggle, and don’t hesitate to reach out for help or collaboration. You’d be surprised at what you can achieve when you work together!
**Editor:** Thank you so much for sharing your story with us today. Your achievements are truly inspiring, and we can’t wait to see what you both accomplish in the future!
**Calcea and Ne’Kiya:** Thank you for having us!
**Ne’Kiya:** It was incredibly validating! Knowing that someone like Michelle Obama recognized our work made all the late nights and challenges worthwhile. It encouraged us to keep pursuing our passions in math and science, knowing that our efforts can make an impact.
**Calcea:** Yes, and it inspired both of us to advocate for more girls in STEM fields. We want our story to show that determination and hard work can lead to accomplishments, no matter the obstacles.
**Editor:** Speaking of your academic journeys, how have your studies evolved since you completed this project?
**Calcea:** I’m currently studying environmental engineering at Louisiana State University. This project solidified my love for problem-solving and innovation, which I hope to apply in sustainable development.
**Ne’Kiya:** I’m studying pharmacy at Xavier University of Louisiana. The analytical skills I honed through our proof have been invaluable in understanding complex scientific concepts, and I aim to contribute to healthcare advancements.
**Editor:** It’s great to see how you’re both applying your enthusiasm and skills in your respective fields. As a final question, what advice would you give to young students who may be hesitant to take on challenging projects?
**Calcea:** I would say embrace the challenge! Even if it feels daunting, the learning journey is as valuable as the end result.
**Ne’Kiya:** And don’t hesitate to seek help and collaborate with others. You’d be surprised how much stronger ideas become when shared.
**Editor:** Wise words! Thank you, Calcea and Ne’Kiya, for sharing your insights and experiences. Your achievements are truly inspiring, and we wish you both the best of luck in your future endeavors!
