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\title{Nombres Complexes}
\author{Aubin SIONVILLE}
\date{MPI Clemenceau - 2021-2023}
\maketitle

\centering

\section*{Formules de trigonométrie}

\begin{multicols}{2}[Hypothèse : $(a,b) \in \mathbb{R}^2$]
    \[\boxed{\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)}\]

    \[\boxed{\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)}\]
\end{multicols}

\begin{multicols}{2}
    Hypothèse : $\displaystyle a + b \not\equiv \frac{\pi}{2}[\pi]$ :
    \[\boxed{\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}}\]
    
    Hypothèse : $\displaystyle a \not\equiv \frac{\pi}{2}[\pi]$ :
    \[\boxed{1+\tan^2(a) = \frac{1}{\cos^2(a)}}\]
\end{multicols}

\section*{Transformations géométriques}
\begin{align*}
    \text{Translation } : \quad &\boxed{M' = t_{\overrightarrow{\omega}}(M) \iff z_{M'} = z_M + z_{\overrightarrow{\omega}}} \\
    \text{Rotation } : \quad &\boxed{M' = \rho_{\Omega, \theta}(M) \iff z_{M'} = \omega + \mathrm{e}^{i\theta}(z_M-\omega)} \\
    \text{Homothétie } : \quad &\boxed{M' = h_{\Omega, \lambda}(M) \iff z_{M'} = \omega + \lambda(z_M-\omega)} \\
    \text{Similitude } : \quad &\boxed{M' = S_{\Omega, \lambda, \theta}(M) \iff z_{M'} = \omega + \lambda(\mathrm{e}^{i\theta}(z_M-\omega))} \\
    &S_{\Omega, \theta, \lambda} = \rho_{\Omega, \theta} \circ h_{\Omega, \lambda}
\end{align*}

\vspace{0.5cm}

\section*{Formules d'Euler}
\begin{multicols}{2}
    \[\boxed{\cos(x) = \frac{\mathrm{e}^{ix} + \mathrm{e}^{-ix}}{2}}\]
    
    \[\boxed{\sin(x) = \frac{\mathrm{e}^{ix} - \mathrm{e}^{-ix}}{2i}}\]    
\end{multicols}

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