Polar equation of the curve y = sinx

Question

I am looking for the polar equation of the following curve given in Cartesian Coordinates.

y = sinx

Any kind of hint or help is appreciated.

Answer 1

With $x = r \cos(\theta)$ and $y = r \sin(\theta)$, this becomes $$ r \sin(\theta) = \sin(r \cos(\theta))$$

That's as simple as it gets: in particular you're not going to get an explicit closed form for $r$ as a function of $\theta$, or $\theta$ as a function of $r$.

Answer 2

As $$\tan(\theta)=\frac yx=\frac{\sin(x)}x=\text{sinc}(x),$$ where $\text{sinc}$ denotes the cardinal sine function, you have

$$\color{green}{r=\sqrt{(\text{sinc}^{-1}(\tan(\theta)))^2+\sin^2(\text{sinc}^{-1}(\tan(\theta)))}}.$$

Unfortunately, the inverse $\text{sinc}$ function is a wild animal.