Is there a way to deduce minimum and maximum bounds for the maximum $r$ (given $\theta$) such that $re^{i\theta}$ lies within the Mandelbrot set?
I've attempted to find rough bounds for $r$ using the main cardioid, but even that proved tricky as the polar equation for such is shifted to the right by $\frac{1}{4}$.
$$\frac{1}{4}\bigg[\sqrt{1-\sin^\frac{2}{3}\theta}\pm\sqrt{\sin^\frac{2}{3}\theta-\frac{2\cos^\frac{2}{3}\theta}{\sqrt{1-\sin^\frac{2}{3}\theta}}+2}\bigg]\leq r\leq2$$
What can I do to improve these bounds without computational simulation?
Or rather, is there a way of finding $r(\theta)$ exactly?