Graph Polar equation: $r = \sin^2(\theta)$ in polar plane

Question

That's the answer, but I don't understand how to get there. Especially how to draw it. I tried looking it up but I am unable to find resources on graphing on polar plane.

Also, how am I supposed to convert $r = \sin^2(\theta)$ to cartesian?

Answer 1

$$x=r\cos\theta=\sin^2\theta\cos\theta\\y=r\sin\theta=\sin^3\theta$$ From the second equation $$\sin\theta=\sqrt[3]y$$ You can then plug it into the first equation, and use $\cos\theta=\pm\sqrt{1-\sin^2\theta}$: $$x=\pm y^\frac23\sqrt{1-y^\frac23}$$ Notice that you plot $x(y)$.