let's take a look on Archimedean spiral. the polar equation is $r = \theta$. click here to look.
but $\tan (\theta) = y/x$ and $r = \sqrt{x^2+y^2}$,
so $r = \theta \rightarrow \tan(\sqrt{x^2+y^2}) = y/x$. click here to look.
but wolfram alpha shows that the equations are different.
where is my mistake? or this is error on wolfram alpha?
thank you.
If you look carefully, it actually does plot the same spiral. The second plot will of course also plot the spiral's 180 degree rotation (since $(-x,-y)$ solves the same equation).
It also looks to be plotting "numerically unstable" concentric circles of radii $\approx\pi(n+\frac 1 2)$. Presumably this is because of tan's singularity there and a numerical loss of precision that causes unexpected equality. That looks wrong, maybe worth a bug report, but shouldn't phase you.