Question

How do you convert `r = 2 / (1+sin(theta))` into rectangular form?

Answer 1

The rectangular equation is `y=-x^2/4+1`.

Explanation:

`r=2/(1+sintheta)`

`r=2/(1+sintheta)color(red)(*(1-sintheta)/(1-sintheta))`

`r=(2-2sintheta)/(1-sin^2theta)`

`r=(2-2sintheta)/cos^2theta`

`r^2=(2r-2rsintheta)/cos^2theta`

`r^2cos^2theta=2r-2rsintheta`

`(rcostheta)^2=2r-2rsintheta`

Using the substitutions `y=rsintheta` and `x=rcostheta` and `r=sqrt(x^2+y^2)`:

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

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

`x^4/4+yx^2+y^2=x^2+y^2`

`x^4/4+yx^2=x^2`

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

`x^2/4+y=1`

`y=-x^2/4+1`

That's the equation; it's a parabola. Here's what it looks like:

graph{-x^2/4+1 [-10, 10, -5, 5]}