Complex numbers

Results

Proposition : Moivre formula

For $\theta \in \mathbb R$ and $n \in \mathbb Z$, we have $(e^{i\theta})^n = e^{in\theta}$ or even, by definition of $e^{i\theta}$

$$\cos\theta + i\sin(\theta)^n = \cos(n\theta) + i\sin(n\theta).$$

Proposition : Euler formula

For $\theta \in \mathbb R$, we have $\quad \cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \quad \text{and} \quad \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$.