How do you convert $r^{2} = cos 2 θ$ $r^2=cos2 theta$ to rectangular form?

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Answer 1 · 2016-09-27T08:05:08

We know the relations

$x = r cos θ and y = r sin θ$ $x=rcostheta and y =rsintheta$

where r and $θ$ $theta$ are the polar coordinate of a point having rectangular coordinate $(x, y)$ $(x,y)$
So $r^{2} = x^{2} + y^{2}$ $r^2=x^2+y^2$

Now the given relation is

$r^{2} = cos 2 θ = {cos}^{2} θ - {sin}^{2} θ$ $r^2=cos2theta=cos^2theta-sin^2theta$

$\Rightarrow r^{4} = r^{2} {cos}^{2} θ - r^{2} {sin}^{2} θ$ $=>r^4=r^2cos^2theta-r^2sin^2theta$

$\Rightarrow {(x^{2} + y^{2})}^{2} = x^{2} - y^{2}$ $=>(x^2+y^2)^2=x^2-y^2$

This is the rectangular form of the given polar equation.

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