Question

Convert the rectangular equation to polar form

Answer 1

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`