How do you convert $r= cos^2(theta/2)$ into cartesian form?

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Answer 1 · 2016-06-30T23:31:45

$2(x^2 + y^2) = x + sqrt{x^2 + y^2}$

With $r= cos^2(theta/2)$ , we can get that $theta/2$ back into a $theta$

as we have the double angle formula

$cos 2Q = 2 cos^2 Q -1$

so

$cos^2 Q = (cos 2Q +1)/2$

here that means that

$r = (cos theta +1)/2 implies 2 r = cos theta +1$

now we have $x = r cos theta$ so $cos theta = x/r$

meaning that

$2 r = x/r +1$

$2 r^2 = x + r$

in cartesian, that is very very ugly

$2(x^2 + y^2) = x + sqrt{x^2 + y^2}$

How do you convert r= cos^2(theta/2) r= cos^2(theta/2) into cartesian form?