So, I was trying to find the polar equation for a sine curve and here's what I did :
Now, some assumptions that I've made here are : $\theta\in\left(0,\dfrac\pi2\right)$ and $C_x<\pi$, where $C_x$ is the abscissa of $C$.
So, $\sin\theta=\dfrac{BC}{r}\implies BC=r\cdot\sin\theta$ and $\cos\theta=\dfrac{AB}{r}\implies AB=r\cdot\cos\theta$. Also, if the length of $AB$ is $k$ units, then
$$\sin(k~\mathrm{rads})=BC\quad\implies\quad\color{red}{\sin(r\cdot\cos\theta)=r\cdot\sin\theta}$$
which is the trigonometric equation that we need to solve.
My knowledge of trigonometry is only limited to 11th grade and so far, I haven't been able to think of any way to approach this equation.
Now, I don't have any formal education regarding polar coordinates and I am just trying to do this for fun, which means that my belief that solving for $r$ in terms of $\theta$ will give the equation of the sine curve when plotted in the polar plane might be wrong but I'm still fairly confident that at least the part of the sine wave lying in the interval $(0,\pi)$ will be plotted.
I would greatly appreciate any help and hints, thank you!