Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Draw on a figure the circle $\mathcal{C}$ with center $O$ and radius $|z|$ and the points $M$ with affixe $z$ and $B$ with affixe $-|z|$. By considering well-chosen angles, show that $$\theta(z) = \operatorname{Arg}(z) = 2\operatorname{Arg}(z + |z|)$$ where $\operatorname{Arg}(z)$ denotes the principal determination of the argument of the complex number $z$.
Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote
$$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$
Draw on a figure the circle $\mathcal{C}$ with center $O$ and radius $|z|$ and the points $M$ with affixe $z$ and $B$ with affixe $-|z|$.\\
By considering well-chosen angles, show that
$$\theta(z) = \operatorname{Arg}(z) = 2\operatorname{Arg}(z + |z|)$$
where $\operatorname{Arg}(z)$ denotes the principal determination of the argument of the complex number $z$.