How would you plot $|z|=\operatorname{arg}(z)$ on an argand diagram?

Question

Is there an easy way to see how to plot $|z|=\operatorname{arg}(z)$ on an argand diagram? Replacing the modulus and argument with the cartesian counterpart seems to be needlessly obtuse.

Answer 1

It was already often discussed here. See Solve $|z|=\arg z$

Answer 2

Could you not just plot $$z=\theta(\cos\theta+i\cdot\sin\theta)$$ The typical form is $z=r(\cos\theta+i\cdot\sin\theta)$ but given $|z|=r$ and $arg(z)=\theta$, for your case $r=\theta$