Let $z \in \mathbb{C}$. We set $z = a + ib$, where $a, b \in \mathbb{R}$.
Let $n \in \mathbb{N}^*$. Determine the modulus and an argument of $\left(1 + \frac{z}{n}\right)^n$ as a function of $a$, $b$ and $n$.

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 $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Justify that $\theta$ and $R$ are well defined.

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 $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ When $z$ takes successively the values $z_1 = 4$, $z_2 = 2\mathrm{i}$ and $z_3 = 1 - \mathrm{i}\sqrt{3}$, calculate $R(z)$, $\theta(z)$ and $(R(z))^2$.

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 $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Verify that $\theta(z) \in ]-\pi, \pi[$ and that $R(z) \in \mathcal{P} = \{Z \in \mathbb{C},\, \operatorname{Re}(Z) > 0\}$.

For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument of $z$, with $- \pi < \arg ( z ) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg ( \omega ) < \pi$. Let
$$\alpha = \arg \left( \sum _ { n = 1 } ^ { 2025 } ( - \omega ) ^ { n } \right) .$$
Then the value of $\frac { 3 \alpha } { \pi }$ is $\_\_\_\_$.

If $z$ and $\omega$ are two complex numbers such that $| z \omega | = 1$ and $\arg ( z ) - \arg ( \omega ) = \frac { 3 \pi } { 2 }$, then $\arg \left( \frac { 1 - 2 \bar { z } \omega } { 1 + 3 \bar { z } \omega } \right)$ is: (Here $\arg ( z )$ denotes the principal argument of complex number $z$)
(1) $\frac { \pi } { 4 }$
(2) $- \frac { 3 \pi } { 4 }$
(3) $- \frac { \pi } { 4 }$
(4) $\frac { 3 \pi } { 4 }$

The value of $\left( \frac { 1 + \sin \frac { 2 \pi } { 9 } + i \cos \frac { 2 \pi } { 9 } } { 1 + \sin \frac { 2 \pi } { 9 } - i \cos \frac { 2 \pi } { 9 } } \right) ^ { 3 }$ is
(1) $\frac { - 1 } { 2 } ( 1 - i \sqrt { 3 } )$
(2) $\frac { 1 } { 2 } ( 1 - i \sqrt { 3 } )$
(3) $\frac { - 1 } { 2 } ( \sqrt { 3 } - i )$
(4) $\frac { 1 } { 2 } ( \sqrt { 3 } + i )$

The complex number $z = \frac{i-1}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}$ is equal to:
(1) $\sqrt{2}i\left(\cos\frac{5\pi}{12} - i\sin\frac{5\pi}{12}\right)$
(2) $\cos\frac{\pi}{12} - i\sin\frac{\pi}{12}$
(3) $\sqrt{2}\left(\cos\frac{\pi}{12} + i\sin\frac{\pi}{12}\right)$
(4) $\sqrt{2}\left(\cos\frac{5\pi}{12} + i\sin\frac{5\pi}{12}\right)$