Question

How do you convert `r=1-2cos2t` to rectangular form?

Answer 1

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

Explanation:

The relation between polar coordinates `(r,t)` and rectangular coordinates `(x,y)` are given by `x=rcost` and `y=rsint` and hence `r=sqrt(x^2+y^2)`

Thus `r=1-2cos2t` can be written as

`r=1-2(2cos^2t-1)=3-4cos^2t` or

`r=3-4cos^2t=3-4x^2/r^2` or

`r^3=3r^2-4x^2` or

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

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

graph{(x^2+y^2)^(3/2)=3y^2-x^2 [-10, 10, -5, 5]}