I'm getting pretty frustrate with polar coordinates and I can't find any videos online that are very helpful. So I have this and a few other questions I'm posting.

The first problem, I thought, would be simple but my graphs dont match:

Convert$\displaystyle r=sin(2 \theta)$ into cartesian

1. I changed it to $\displaystyle r=2sin(\theta) cos(\theta)$

2. Then multiplied by r twice $\displaystyle r^3=2rsin(\theta) rcos(\theta)$

3. $\displaystyle x=rcos(\theta)$ and $\displaystyle y=rsin(\theta)$

4. so: $\displaystyle r^3=2xy$

5. and since $\displaystyle r^2=x^2+y^2$

6. finally I get: $\displaystyle (x^2+y^2)(\sqrt(x^2+y^2))=2xy$

The graphs do not match so I'm getting desperate to find my mistake here.