# Conversion of polar equations

How would you say have some equation in the polar coordinate system as: $$r=3\sin3\theta$$

I know how to find the are of one petal of this shape using polar coordinate integration, but say if I convert this to its Cartesian form this exact polar equation using $$x = r \cos \theta$$ , $$y= r \sin\theta$$, then how would I find the area in when it is in the Cartesian form?

Could anyone give me some clear steps, this will really help me understand better. Thanks in advance.

As you know we have $$\sin 3\theta = 3 \sin \theta -4 \sin ^3 \theta$$
Thus your equation in Cartesian coordinates gets a little bit complicated. $$r = 3 \sin (3\theta ) \implies r^2=3r (3\sin \theta -4\sin ^3 \theta )$$ $$x^2+y^2=9y -12y \frac {y^2}{x^2+y^2}$$
• You have to cut it into parts where $y$ is a function of $x$ and use symmetry for other parts. Good luck – Mohammad Riazi-Kermani Oct 21 '18 at 16:33