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Complex numbers

Results​

Proposition : Moivre formula

For θ∈R\theta \in \mathbb R and n∈Zn \in \mathbb Z, we have (eiθ)n=einθ(e^{i\theta})^n = e^{in\theta} or even, by definition of eiθe^{i\theta}

cos⁡θ+isin⁡(θ)n=cos⁡(nθ)+isin⁡(nθ).\cos\theta + i\sin(\theta)^n = \cos(n\theta) + i\sin(n\theta).
Proposition : Euler formula

For θ∈R\theta \in \mathbb R, we have cos⁡θ=eiθ+e−iθ2andsin⁡θ=eiθ−e−iθ2i\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}.