How do you convert $r = 1 + 0.5 cos 2 (θ)$ $r=1+0.5cos2(theta)$ into cartesian form?

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Answer 1 · 2016-09-19T09:50:39

$\frac{x^{2}}{\sqrt{x^{2} + y^{2}}} - \frac{x^{2}}{x^{2} + y^{2}} = \frac{1}{2}$ $x^2/sqrt(x^2+y^2)-x^2/(x^2+y^2)=1/2$

The pass equations are

$((x=rcostheta),(y=rsintheta))$

and also keeping in mind that $cos2theta =1-2sin^2theta$

$r = 1+(1-2sin^2theta)/2 = 1+1/2-sin^2theta = 1/2+cos^2theta$

then

$x/costheta = 1/2 + cos^2theta$

but $costheta = x/sqrt(x^2+y^2)$

so

$x^2/sqrt(x^2+y^2)-x^2/(x^2+y^2)=1/2$

How do you convert r=1+0.5cos2(theta)r=1+0.5cos2(θ)r=1+0.5cos2(theta) into cartesian form?