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).

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

The gradient of this line is .

The y-intercept of this line is .

Hence the equation of line l is .

We know the equation of the unit circle is .

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

Solving the above using the quadratic formula gives us .

Using , we get .

Hence, is a parametrization of the unit circle.

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

Then .

.

Substituting into , and multiplying by the denominator, we get the Pythagorean Triple . Interesting?

Featured book:

Divine Proportions: Rational Trigonometry to Universal Geometry

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.

### Like this:

Like Loading...

*Related*

Pingback: Video on Simplices and Simplicial Complexes | Singapore Maths Tuition

Reblogged this on Project ENGAGE.

LikeLike