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?

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