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.