Question

How to convert `r^2 = sintheta` from polar to rectangular form?

Through a bit of questionable math, I got `x^6+x^4y^2+y^4x^2+y^6-x^2`, but that's definitely not right, and I'm not sure what to do now. Any and all help is greatly appreciated.

Answer 1

I'm not sure what I can possibly do with my equation either

Explanation:

`r^2= sintheta`

Multiply both sides by `r`:
`r^2*r= r*sintheta`

Since `rsintheta= y` and `r^2= x^2+y^2`:

`(x^2+y^2)*r= y`

And `r= +-sqrt(x^2+y^2)`

First let's move the `r` over to the left side by dividing by `r` on both sides:

`(x^2+y^2)= y/r`

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

Square both sides to see where this goes:

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

`x^4+2x^2y^2+y^4= y^2/(x^2+y^2)`

`y^2=(x^4+2x^2y^2+y^4)(x^2+y^2)`

`y^2= x^6+x^4y^2+2x^4y^2+2y^4x^2+y^4x^2+y^6`

`y^2= x^6+x^4y^2+2x^4y^2+2y^4x^2+y^4x^2+y^6`

Gave me this graph
graph{y^2= x^6+x^4y^2+2x^4y^2+2y^4x^2+y^4x^2+y^6 [-10, 10, -5, 5]}