Rational Parametrization of a Circle & Pythagorean Triples

Today, we discuss an interesting topic rarely taught in school: The rational parametrization of a unit circle. That is, how to find the x coordinates and y coordinates of a circle expressed as a rational function? The usual parametrization of a circle is (cos t, sin t).

unit circle

We consider the straight line (l), passing through the point A(1,0) and the point (0,h).

The gradient of this line is \displaystyle m=\frac{h-0}{0-1}=-h.

The y-intercept of this line is h.

Hence the equation of line l is \boxed{y=-hx+h}.

We know the equation of the unit circle is \boxed{x^2+y^2=1}.

By solving the two simultaneous equations (boxed), we get a quadratic formula:


Solving the above using the quadratic formula gives us \displaystyle x=\frac{h^2-1}{1+h^2}.

Using \boxed{y=-hx+h}, we get \displaystyle y=\frac{2h}{1+h^2}.

Hence, \displaystyle \boxed{ (\frac{h^2-1}{1+h^2},\frac{2h}{1+h^2})} is a parametrization of the unit circle.

We can use this to generate Pythagorean triples! Simply choose a value of h, say, h=8.

Then \displaystyle x=\frac{h^2-1}{1+h^2}=\frac{63}{65}.

\displaystyle y=\frac{2h}{1+h^2}=\frac{16}{65}.

Substituting into \boxed{x^2+y^2=1}, and multiplying by the denominator, we get the Pythagorean Triple 16^2+63^2=65^2 . Interesting?

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This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and 170 diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus. Applications include surveying and engineering problems, Platonic solids, spherical and cylindrical coordinate systems, and selected physics problems, such as projectile motion and Snell’s law. Examples over finite fields are also included.


Maths Tuition Centre: University Math Textbooks Review

Calculus Math Textbooks Review

Most students taking science related courses like Engineering or Physics need to study at least one semester of Calculus. Calculus can be a rather difficult subject, and having a good textbook to learn from is half the battle won! 🙂

We review 3 of the Top Calculus Textbooks on Amazon.com:



The Larson CALCULUS program has a long history of innovation in the calculus market. It has been widely praised by a generation of students and professors for its solid and effective pedagogy that addresses the needs of a broad range of teaching and learning styles and environments. Each title is just one component in a comprehensive calculus course program that carefully integrates and coordinates print, media, and technology products for successful teaching and learning.


Calculus, 4th edition

This book by Michael Spivak is strongly recommended for Math Majors, or for students interested in learning the theory behind calculus. Includes the theory of epsilon-delta analysis.


Thomas’ Calculus (13th Edition)
Thomas’ Calculus, Thirteenth Edition, introduces readers to the intrinsic beauty of calculus and the power of its applications. For more than half a century, this text has been revered for its clear and precise explanations, thoughtfully chosen examples, superior figures, and time-tested exercise sets. With this new edition, the exercises were refined, updated, and expanded–always with the goal of developing technical competence while furthering readers’ appreciation of the subject. Co-authors Hass and Weir have made it their passion to improve the text in keeping with the shifts in both the preparation and ambitions of today’s learners.

This text is designed for a three-semester or four-quarter calculus course (math, engineering, and science majors).

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