In books we have seen that $\arg{zw} = \arg{z} + \arg{w}$ (z and w are complex numbers), is here the arg referred to the general argument not concerning the principal one only ? So for principal argument it would be $\def\Arg{\operatorname{Arg}} \Arg{z} + \Arg{w} \pm 2\pi = \Arg{zw}$ ? Likewise for divison we would have $\arg\frac{z}{w} = \arg{z} - \arg{w}$ and $\Arg\frac{z}{w} = \Arg{z} - \Arg{w} \pm 2\pi$ ?
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The complex argument is an equivalence, and is generally written as $\theta \mod 2\pi$, although other notations are also used, such as $\mathbb{R}/2\pi\mathbb{Z}$ at Wikipedia.
JMP
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The formula $\textrm{arg}zw=\textrm{arg}z+\textrm{arg}w$ is referred to the general argument, and the reason is the following: let $z=\vert z\vert e^{i\textrm{arg}z}$ and $w=\vert w\vert e^{i\textrm{arg}w}$. Then: $$zw=\vert z\vert \vert w\vert e^{i\textrm{arg}z+i\textrm{arg}w}=zw=\vert z\vert \vert w\vert e^{i(\textrm{arg}z+\textrm{arg}w)}$$ On the other hand, the principal argument is defined in $(-\pi,\pi]$ (or in $[0,2\pi$), it depends on the author). So if you want to calculate $\textrm{Arg}zw$, by the first formula you obtain an argument in the interval $(-2\pi,2\pi]$, and usually you will need to add or to substract $2\pi$ to return to the interval $(-\pi,\pi]$.
Earnur
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\text{by}r_2e^{i\theta_2} = r_2[\cos(\theta_2 + i \sin(\theta_2)].$$ – user2661923 Apr 24 '22 at 07:40