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Convert the rectangular equation to polar form

Convert the rectangular equation to polar form-example-1

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The coordinate change from rectangular to polar is


\begin{cases}x=\rho\cos(\theta)\\y=\rho\sin(\theta)\end{cases}

So, you simply have to substitute each occurrence of x and y with their expression:


(x^2+y^2)^2 = x^2-y^2 \iff (\rho^2\cos^2(\theta)+\rho^2\sin^2(\theta))^2 = \rho^2\cos^2(\theta)-\rho^2\sin^2(\theta)

Using the fundamental trigonometric equation
\cos(x)^2+\sin(x)^2=1 we have


(\rho^2)^2 = \rho^2(\cos^2(\theta)-\sin^2(\theta)) \iff \rho^4-\rho^2(\cos^2(\theta)-\sin^2(\theta))=0

We can use
\cos(2x)=\cos^2(x)-\sin^2(x) to get


\rho^4-\rho^2(\cos(2\theta))=0 \iff \rho^2(\rho^2-\cos(2\theta))=0

User Darrel Hoffman
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