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