Question

How do you convert `r=3theta - csctheta` to Cartesian form?

Answer 1

`sqrt(x^2+y^2)=3arctan(y/x)-sqrt(x^2+y^2)/y`

Explanation:

`(r,theta)` in polar coordinates is `(rcostheta,rsintheta)` in rectangular coordinates and

`(x,y)` in rectangular coordinates is `(sqrt(x^2+y^2),arctan(y/x))` in polar coordinates.

Note that `sintheta=y/r=y/(sqrt(x^2+y^2)`

Hence `r=3theta-csctheta` can be written as

`sqrt(x^2+y^2)=3arctan(y/x)-sqrt(x^2+y^2)/y`