How to prove the formula for the argument of a complex number?

Question

$\arg(x + iy) = 2 \cdot \arctan(\dfrac{y}{x + r})$ there is always the mark, this is derived from the 'Half-angle formula'

How can I come from $\tan(\phi) = \tan(\phi + k \pi) = \dfrac{y}{x}$ to $\phi = \arg(x + iy)$?

Answer 1

Look at the next figure

$\hspace4cm$

Clearly ${\rm Arg}(z)=2\theta=2\arctan\dfrac{y}{x+r}$. Note that this gives the principal determination of the argument of $z\in\mathbb{C} \setminus((-\infty,0]\times\mathbb{R})~$.