Give the exponential and trigonometric forms of the complex numbers $1 + \mathrm{i}$ and $1 - \mathrm{i}$.
For every natural number $n$, we define $$S_{n} = (1 + \mathrm{i})^{n} + (1 - \mathrm{i})^{n}.$$ a. Determine the trigonometric form of $S_{n}$. b. For each of the two following statements, say whether it is true or false by justifying your answer. An unjustified answer will not be taken into account and the absence of an answer is not penalised. Statement A: For every natural number $n$, the complex number $S_{n}$ is a real number. Statement B: There exist infinitely many natural numbers $n$ such that $S_{n} = 0$.
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 $\_\_\_\_$.