How to converting rectangular equations into polar equations

Three formulas for converting between rectangular and polar equations

To convert rectangular equations to polar equations, we’ll use the following conversion formulas.

$x=r\cos{\theta}$

$y=r\sin{\theta}$

$r^2=x^2+y^2$

Example

Convert the rectangular equation to a polar equation.

$x^2-3x+y^2+2y=0$

Use $r^2=x^2+y^2$ .

$x^2+y^2-3x+2y=0$

$r^2-3x+2y=0$

Use $x=r\cos{\theta}$ and $y=r\sin{\theta}$ .

$r^2-3r\cos{\theta}+2r\sin{\theta}=0$

$r^2=3r\cos{\theta}-2r\sin{\theta}$

$r=3\cos{\theta}-2\sin{\theta}$

Once we’ve eliminated all $x$ and $y$ variables, and replaced them with $r$ and $\theta$ variables, we’re done with the conversion.