What sounds like the plot of a Netflix film genuinely happened in the United States: two sixth-form students uncovered a genuinely new route to the Pythagorean theorem, catching the attention of mathematicians. Their method relies on trigonometry alone and deliberately avoids the familiar geometric rearrangements and algebraic shortcuts that usually appear in classrooms.
Why a 2,000-year-old theorem can still feel new
The Pythagorean theorem is part of mathematics’ basic vocabulary. Most pupils meet it early on as the statement that, in any right-angled triangle, the side lengths satisfy:
a² + b² = c², describing the relationship between the sides of every right-angled triangle.
In other words, the sum of the squares of the two shorter sides (the legs) equals the square of the hypotenuse-the longest side, opposite the right angle. From surveying and architecture to computer graphics and navigation, the theorem sits quietly inside countless practical tools.
Over the centuries, mathematicians have produced hundreds of proofs: by dissecting areas, using similar triangles, applying algebra, and even drawing on ideas from calculus. Yet one approach has long been treated as close to “off limits”: a purely trigonometric derivation-one that works only with angle functions such as sine and cosine, without smuggling in the theorem through the back door.
Jackson and Johnson’s trigonometry-first proof of the Pythagorean theorem
That is exactly the challenge Ne’Kiya Jackson and Calcea Johnson, two students from Louisiana, set themselves. Over several years-largely outside ordinary lessons-they tackled a question that usually belongs to research-level discussion:
Can you prove the Pythagorean theorem using only trigonometry without falling into circular reasoning?
The problem is subtle. In many textbooks, sine and cosine are introduced using the Pythagorean theorem (often via the unit circle or distances derived from it). If you later use sine and cosine to “prove” the theorem, the argument can quietly assume what it claims to establish.
Their strategy was to build angles and proportions carefully first, and only then develop trigonometric relationships-without presupposing Pythagoras.
To do that, they went back to foundational Euclidean ideas: angle properties, proportional reasoning, and the geometry of similar shapes. From these building blocks they constructed right-angled triangles and related figures arranged so that side-to-angle relationships could be justified step by step.
Trigonometry without Pythagoras: is it actually possible?
A key move in their work is how sine and cosine are handled. Rather than treating them as ready-made abbreviations for “opposite over hypotenuse” and “adjacent over hypotenuse” in a right-angled triangle, Jackson and Johnson treat them as ratio quantities that arise from their geometric set-up and proportional structure.
From those definitions they derived relationships between side lengths in right-angled triangles. This naturally leads to a central trigonometric identity:
sin²(x) + cos²(x) = 1 forms the backbone of their approach, functioning in place of a direct appeal to the Pythagorean theorem.
Crucially, in their framework this identity is not simply the Pythagorean theorem wearing new labels. Instead, it is obtained from angle properties and proportional arguments framed independently of the classical statement. From there, they work methodically until they reach the ancient destination once more: a² + b² = c².
Why mathematicians don’t dismiss this as a gimmick
In mathematics, the endpoint matters-but so does the route. A new proof can:
- clarify logical dependencies and expose hidden assumptions,
- open doors to generalisations,
- make familiar concepts feel unfamiliar in productive ways,
- reshape how topics are taught over the long term.
Many researchers saw Jackson and Johnson’s contribution as exactly that: a fresh lens on an idea so well known that most people stop questioning how it is built.
From sixth form to the professional stage
After four years of sustained work, Jackson and Johnson presented their results in 2023 at the annual meeting of the Mathematical Association of America in Atlanta. That is a venue where university academics, established researchers, and doctoral candidates typically speak-not school students.
Their talk drew attention because it treated a supposedly impossible idea as something worth doing carefully.
The mathematics held up under scrutiny. Specialists queried the structure, tested for weak points, and examined whether any hidden circular step had slipped in. That process ultimately led to a peer-reviewed paper published in the highly regarded American Mathematical Monthly-an exceptional achievement for two young women just leaving school.
Their academic paths have since diverged in direction but not in spirit: Jackson is studying pharmacy at Xavier University of Louisiana, while Johnson is studying environmental engineering at Louisiana State University-both fields where quantitative thinking remains central.
More than one proof: a toolkit of trigonometric derivations
Their contribution is not limited to a single argument. In their work, they outline several distinct trig-based proofs of the Pythagorean theorem, along with a method for generating further variants.
| Variant | Core idea | What makes it distinctive |
|---|---|---|
| Main proof | Building sine and cosine from angles and proportions | Avoids any direct reliance on the Pythagorean theorem |
| Follow-on proof A | Rearranging the identity sin² + cos² = 1 | Reaches Pythagoras in only a few steps |
| Follow-on proof B | Working with similar right-angled triangles | Blends trigonometric structure with geometric reasoning |
| Generating procedure | A general construction scheme | Produces five additional Pythagorean-theorem proofs |
Taken together, this “bundle” matters because it suggests the result is not a one-off trick. It looks more like a systematic framework-a set of components that can be recombined to build multiple valid routes to the same theorem.
What it could change in classrooms, universities, and applied work
In teaching, this kind of approach can make trigonometry feel less like a bag of formulas and more like a logically assembled system. For example, teachers could:
- present trigonometry as a structured theory rather than a collection of rules,
- highlight how tightly angles and length ratios are linked,
- discuss what makes a proof robust-and where circular reasoning can hide,
- set project work on alternative proofs of classical results.
At university level, disentangling trigonometric arguments from the Pythagorean theorem can also make it easier to move into broader settings, such as non-Euclidean or curved geometries, or into topics where careful foundations matter for later generalisation.
In areas such as computer graphics, robotics, and navigation algorithms, geometry and trigonometry repeatedly become critical design levers.
A cleaner logical base can help practitioners validate numerical methods, spot sources of error, and refine algorithms. Trigonometric functions also appear routinely in machine learning-particularly when handling spatial data, rotations, periodic signals, and certain neural-network representations.
A wider point: how peer review and communication shape mathematics
Another overlooked aspect of this story is that the result did not become “real” simply because it was clever. It became part of the mathematical record because it was communicated clearly, questioned publicly, and then tested through peer review. That path-presentation, critique, revision, publication-is how mathematics protects itself against subtle gaps, especially in areas where an argument might accidentally rely on the very statement it aims to prove.
This is also why writing matters in mathematics: the difference between a promising idea and a publishable proof is often the precision with which assumptions are stated and steps are justified. Jackson and Johnson’s work stands out not only for the concept, but for the discipline of the exposition.
Why this resonates with so many young people
Many teenagers experience mathematics as a finished building: everything already settled, every theorem boxed and labelled. Jackson and Johnson’s story unsettles that picture. It shows that even a classroom classic like the Pythagorean theorem can still prompt genuine questions.
They have emphasised that curiosity and persistence were decisive. They worked for years without any guarantee the outcome would be worthy of publication. That cycle-trying, failing, revising, and trying again-is ordinary in research, yet it rarely becomes visible in school lessons.
For girls in particular, the message can be powerful: mathematics and science are not private clubs for a tiny group of “geniuses”. They are fields where sustained effort, careful thinking, and the courage to question assumptions can genuinely pay off-whether you later study engineering, medicine, computing, or the social sciences.
Key terms that become more interesting once you look closely
Their work brings several familiar school terms back into focus:
- Right-angled triangle: a triangle containing a 90° angle; this is where the Pythagorean theorem appears in its standard form.
- Hypotenuse: the longest side in a right-angled triangle, always opposite the right angle.
- Trigonometric functions: sine, cosine, and tangent connect angles to ratios of lengths; they sit at the heart of analysing waves and oscillations.
- Circular reasoning: a logical loop in which a “proof” quietly assumes what it is supposed to demonstrate.
Avoiding circularity is not only a mathematical concern. The same skill applies when checking safety cases in engineering, evaluating evidence in scientific studies, or spotting self-supporting arguments in law and public policy.
How to recreate the basic idea in practice
Even without the full journal paper, motivated pupils or students can explore the underlying principle through hands-on experimentation:
- Draw or construct several right-angled triangles that share the same acute angle but have different side lengths.
- Measure the sides and record ratios such as “opposite over hypotenuse”.
- Compare those ratios across triangles-the values stay strikingly consistent for a fixed angle.
- Use that consistency to motivate your own definitions of sine and cosine for the angle, without invoking the Pythagorean theorem.
Small investigations like this-done in a classroom, with simple instruments, or using dynamic geometry software-show how mathematical knowledge can grow from observation, pattern recognition, and progressively tighter justification. Jackson and Johnson followed that research-like route and arrived at a surprising conclusion: even the most familiar theorem can look different when you rebuild it from the ground up.
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