.....

r  =      1  / [3 + 3 sin θ]

3r  =  1 / [ 1 + sinθ]

3r  =  1 / [ 1  +  y /r]

3r  =  r /  [ r + y]

r + y  =  1/3

r  =  (1/3)  - y

√[x^2 + y^2]  =  (1/3) - y       square both sides

x^2  + y^2   =  1/9  - (2/3)y + y^2

x^2  =  1/9 - (2/3)y

9x^2 =  1  + 6y

6y  =   -9x^2 + 1

y  = (-3/2)x^2  + 1/6

This is a parabola  ...  see  here  :  https://www.desmos.com/calculator/m0v1m8zsrd

Nice work Chris!

Quilly, if this is what you intended, and I expect it is, then you need to learn to use BRACKETS

I read it as you had written it which is     y=     1/3     +    3sin(theta)

And so there is no way I could have given this answer!

You are lucky that CPhill has been practicing his forensic mathematics!