I try to find a manner to convert cartesian rectangular equation like $y = f(x)$ to it's polar equation $r = r(\theta)$.
Given the function $f(x) = y = 5x^4$, lets try to get the polar equation:
$$x = r\cdot \cos(\theta), \quad y = r\cdot \sin(\theta)$$
So:
$$r\cdot \sin(\theta) = 5\cdot r^4 \cdot \cos^4(\theta)$$ $$ \Rightarrow r^3= \frac{\sin(\theta)}{5\cdot \cos^4(\theta)} $$ $$ \Rightarrow r(\theta) = \sqrt[3] {\frac {\sin(\theta)}{5\cdot \cos^4(\theta)}}$$
My question is, That's really the correct answer? if so, how we can convert it back to rectangular equation? I tried to convert the polar equation we received to a rectangular equation using the following equations
$$\theta = \arctan\Bigr(\frac yx\Bigr), \quad r = \sqrt{x^2+y^2}$$
The result I got was everything except $y = 5x^4$.
Can someone point me the right direction to get the correct solution?
Thanks for Help!!