Polar Form Addition without Rectangular Form?

Question

As I've learned, and also described this answer that

In rectangular form, complex numbers are easy to add; just add their components.

In polar form, complex numbers are easy to multiply; just multiply their magnitudes and add their arguments.

Thou the polar form presents an easier way to multiply two values, would it even be possible to add two polar form values without transforming them to the rectangular form? If so, how?

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The answers in this question (How to add real number and complex number in polar form) are still somewhat particular. I am looking of a generic addition in the phasor form, such as,

$$(A_1 \angle \theta_1 ) + (A_2 \angle \theta_2 ) = ?$$

Answer 1

You can extract the rectangular parts from the polar expression, add, and then re-express to do this, but it ain't pretty.

Recall that if $z=a+bi$ where $a,b \in \mathbb{R}$, then the modulus is $|z|=\sqrt{a^2+b^2}$ and the argument is $\mathrm{arg}(z)$ set of angles associated with $z$'s polar form.

$z=|z|e^{\mathrm{arg}(z)i}=|z|\cos(\mathrm{arg}(z))+i|z|\sin(\mathrm{arg}(z))$

$w+z=(|z|\cos(\mathrm{arg}(z))+|w|\cos(\mathrm{arg}(w)))+i(|z|\sin(\mathrm{arg}(z))+|w|\sin(\mathrm{arg}(w)))$

Then the modulus of $w+z$ is $\sqrt{(|z|\cos(\mathrm{arg}(z))+|w|\cos(\mathrm{arg}(w)))^2+(|z|\sin(\mathrm{arg}(z))+|w|\sin(\mathrm{arg}(w)))^2}$.

The argument is trickier to extract. Say a representative of the argument lies strictly between $-\pi/2$ and $\pi/2$ (in particular, the imaginary part isn't zero), you could use arctangent to get at a representative angle: $\mathrm{arctan}\left(\frac{|z|\sin(\mathrm{arg}(z))+|w|\sin(\mathrm{arg}(w))}{|z|\cos(\mathrm{arg}(z))+|w|\cos(\mathrm{arg}(w))}\right)$.

In the end, it's nasty.