Question

How do you convert `r = 1 + 2sin(t)` to rectangular form?

Answer 1

`x^4 -x^2 -4x^2y +2x^2y^2+3y^2-4y^3+y^4=0`

Explanation:

Use the substitutions:

`r^2 = x^2 + y^2`
`y = rsintheta`
`x = rcostheta`

Now make substitutions when possible.

`r = 1 + 2sintheta`

`r^2 = r + 2rsintheta`

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

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

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

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

`x^4-4x^2y+2x^2y^2+4y^2-4y^3+y^4 = x^2 + y^2`

`x^4 -x^2 -4x^2y +2x^2y^2+3y^2-4y^3+y^4=0`

You could expand and simplify this further but this is a good stopping point.

Final Answer