What are polar coordinates of the origin and why is $\arg(0) =$ undefined, an option? I have come here from reading this and but it didn't really answered my question.
Wikipedia says the cartesian co-ordinates are converted using $$x=r \cos \theta$$ $$y = r \sin \theta$$
where $r≥0$ but that doesn't make sense because those conversions are found using the definitions of $\sin$ and $\cos$ which are $$\sin \theta = \frac{y}{r},$$ $$\cos \theta =\frac{x}{r}$$ where $r>0$ and thus $r≠0$.
So for $(0,0)$ those conversions are not valid.
Also from this answer, we may let $\theta = \arg (0,0)$ be undefined or $\mathbb{R}$.
My question is why is "$\arg (0) =$ undefined" even an option when all real numbers are perfect candidates for it? Why is the "convention" of leaving it undefined even a thing? What are we afraid of? Multiple values? Because surely, $\arg(z)$ has multiple values even when $z≠0$. So why "undefined" for $z=0$?

The origin has polar coordinates $ (r,\theta)$ defined as
$$ ( 0, t )$$
for any polar angle $t$.
Let me say with some caution... in rectangular coordinates since the origin is
$$ (0,0)$$
we tend to continue the habit or custom and assume it as some sort of convention.
In cases where our curve is going through the origin it presents a minor difficulty. For example the circle with polar equation $r= \sin (\theta-\dfrac{\pi}{4} )$ can still  be thought of as having polar coordinates $(0,0)$ at the origin. However we like to take the point representation as
$$(0,  \dfrac{\pi}{4} )$$
rather than $(0,0) $ in order to avoid discontinuity in the polar angle at $\theta=\dfrac{\pi}{4}$, derivatives takes taken would be smooth continuous.
