Question

How do you convert `r = -sin3θ` into rectangular form?

Answer 1

We know the relations

`x=rcostheta and y =rsintheta`

where r and `theta` are the polar coordinate of a point having rectangular coordinate `(x,y)`
So `r^2=x^2+y^2`

Now the given relation is

`r=-sin3theta=4sin^3theta-3sintheta`

`=>r^4=4r^3sintheta-3r^3sintheta`

`=>r^4=4(rsintheta)^3-3r^2*rsintheta`

`=>(x^2+y^2)^2=4y^3-3(x^2+y^2)y`

This is the rectangular form of the given polar equation.