Question

Convert this parametric equations : { x=rho*cos[phi] ; y =rho*sin[phi]
and {rho= +sqrt (x^2+y^2) ; phi= arc tan [y/x] ,
to the cartesian form @Math @Algebra

Answer 1

\[\left\{ \rho=+ \sqrt{x^{2}+y ^{2}} ; \phi = arc \tan [y/x]\right\}\]\[\left\{ x=\rho*Cos[\phi] ; y=\rho*Sin[\phi] \right\}\]
Convert to cartesian equations

Answer 2

Just sub in the expressions for rho and phi into the expression for x and y.
Remember that \[\sin(\arctan x)={x \over \sqrt{x^2+1}}\]and \[\cos(\arctan x)={1\over \sqrt{x^2+1}}\]

Answer 3

I believe what you have is just the equation of a circle with radius rho.\[x^2+y^2=\rho^2\]