What began as a high-school project has now turned into a surprising mathematical milestone: a fresh proof of Pythagoras’ theorem using only trigonometry, something generations of mathematicians had long assumed could not be done without running in circles.
Ancient theorem, modern challenge
Pythagoras’ theorem is one of the first serious formulas many pupils meet in school. In a right-angled triangle, the square of the longest side, the hypotenuse, matches the sum of the squares of the two shorter sides.
Written as an equation, it becomes:
a² + b² = c², where c is the hypotenuse and a and b are the other sides.
This relationship has been known for more than 2,000 years. It underpins everything from carpentry and architecture to satellite navigation and 3D graphics. Because of its central role, mathematicians have produced hundreds of different proofs, using geometry, algebra, and even calculus.
Yet one type of proof was widely considered off-limits: a proof based solely on trigonometry.
Why a trig proof was thought impossible
Trigonometric functions such as sine and cosine are usually defined using right-angled triangles. In school textbooks, sin(x) is introduced as a ratio involving the hypotenuse, which itself is defined using Pythagoras’ theorem.
Use trigonometry to prove Pythagoras, and you risk using Pythagoras to define trigonometry in the first place.
This circularity made most experts shrug at the idea. A “pure” trigonometric proof would have to avoid smuggling Pythagoras into the definitions. It would need more basic geometric facts as a starting point: angles, similar triangles, and proportions, but not the length formula everyone already knows.
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Two high-school students break the deadlock
In 2022, two American high-school students from Louisiana, Ne’Kiya Jackson and Calcea Johnson, decided to tackle this challenge. Working over several years, they developed a new way to reach Pythagoras’ theorem using trigonometry without relying on it.
Their work focused on rebuilding the house from the ground up. Instead of assuming that trigonometric functions were tightly bound to Pythagoras, they defined them using more elementary geometry.
Building triangles from first principles
Jackson and Johnson started by constructing families of right-angled triangles and related geometric figures. They relied on classical ideas:
- properties of angles in triangles
- similarity of triangles (same shape, different size)
- ratios between corresponding sides
- basic proportional reasoning
Within these carefully chosen configurations, they defined sine and cosine not as mysterious functions from a calculator, but as specific ratios between sides that arise directly from angle relationships and similarity, not from Pythagoras.
The key move: define sine and cosine using similarity and proportions, then prove their main identity from scratch.
From sin and cos to Pythagoras
Once they had trigonometric functions standing on this new foundation, Jackson and Johnson examined the classic identity that links them:
sin²(x) + cos²(x) = 1
Usually, this identity is proved using Pythagoras. The students reversed that logic. They used geometry and proportions to justify the identity, then turned it into a direct route back to the relationship between the sides of a right-angled triangle.
By following a chain of calculations, they showed that, for the triangles they constructed, the squared lengths of the two shorter sides must add up exactly to the squared length of the hypotenuse.
Crucially, nowhere in their argument did they assume Pythagoras as a starting point. That removed the circular reasoning that had long blocked a purely trigonometric proof.
Recognition from the mathematical community
After working on the project for about four years, Jackson and Johnson presented their findings at the annual conference of the Mathematical Association of America, held in Atlanta in March 2023.
Two teenagers took the stage at a major maths conference and argued they had done what many thought could not be done.
Their presentation drew immediate attention. Mathematicians in the audience examined the logic, checked the steps, and found the work credible. Their paper was later accepted for publication in the journal American Mathematical Monthly, a respected venue for mathematical research and expository work.
The published work goes further than a single clever argument. The students proposed several distinct proofs, and one of their constructions turns out to generate five separate demonstrations of Pythagoras’ theorem.
What makes this breakthrough different
Mathematicians already knew Pythagoras’ theorem was true; the novelty here lies in the method. The new proof:
- uses only trigonometry built from basic geometry
- avoids referring back to Pythagoras at any stage
- opens paths to fresh ways of defining trig functions
- shows that long-established results can still be approached in original ways
That approach has drawn interest not just for its ingenuity, but also for its potential use in teaching and research.
Possible impacts on future maths and technology
Rebuilding a central theorem through a new lens can have subtle but wide consequences. In pure mathematics, alternative proofs often reveal hidden connections between fields. This trig-based route to Pythagoras may help researchers rethink how geometry and trigonometry fit together.
In applied mathematics, different formulations of the same fact can lead to more robust algorithms. Geometry and trigonometry underpin computer graphics, signal processing, robotics and navigation. A clearer understanding of trigonometric identities, tied tightly to geometric reasoning, can influence how software handles angles, distances and error estimates.
New proofs sometimes become new tools, shaping how engineers and scientists design and analyse systems.
Some researchers have already suggested that refined geometric and trigonometric frameworks might benefit machine learning and optimisation, where distances in high-dimensional spaces play a crucial role. Any progress that tightens the links between formulas and geometry can improve numerical stability and interpretation.
Role models for the next generation of scientists
Jackson and Johnson say they hope their work encourages other young people to view mathematics not just as a fixed collection of rules, but as an unfinished project. Both have since gone on to university in Louisiana, one studying environmental engineering, the other pharmacy.
Their message is deliberately simple: passion and persistence can make room for new voices, even in fields dominated by long histories and established experts.
For many teenagers, seeing peers contribute to published research shifts maths from abstract homework to something living and reachable.
Key ideas behind Pythagoras and trigonometry
For readers who have not touched a maths textbook for years, a few core terms sit at the heart of this story:
| Term | Meaning | Why it matters here |
|---|---|---|
| Right-angled triangle | A triangle with one 90-degree angle | The setting where Pythagoras’ theorem applies |
| Hypotenuse | The side opposite the right angle, the longest side | The side whose length is determined by a² + b² = c² |
| Similarity | Two shapes with the same angles and proportional sides | Used to define trig functions without Pythagoras |
| Sine and cosine | Ratios that link angles and side lengths | Core tools in the students’ new proof |
| Identity | An equation true for all allowed values | sin²(x) + cos²(x) = 1 is the bridge to Pythagoras |
How teachers and students can use this story
For teachers, this episode offers a concrete way to show pupils that mathematics is still being refined. Presenting Pythagoras’ theorem alongside a discussion of different proofs can help students grasp that reasoning, not memorisation, sits at the centre of the subject.
Students can experiment with small-scale versions of the same idea. For instance, they might:
- try to prove a familiar result in two or three different ways
- build sets of similar triangles and track how side ratios behave
- reconstruct sin and cos as geometric ratios rather than as calculator keys
Exercises like these train the habit that drove Jackson and Johnson’s success: questioning where definitions come from and asking whether a different route is possible.
Why alternative proofs matter beyond the classroom
Multiple proofs of the same statement can look redundant, yet they carry distinct benefits. Some are visual and intuitive, useful for explanation. Others adapt well to computation and algorithms. A few reveal deeper structural links, guiding future research.
A trig-based route to Pythagoras gives mathematicians another angle on one of their oldest theorems. It also sends a quiet but powerful message: even the most familiar cornerstones of science are open to fresh thinking, and those breakthroughs do not always come from the places people expect.
Originally posted 2026-02-19 07:19:33.