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+\subsection*{Définition} + +\begin{minipage}{0.6\textwidth} +Un nombre complexe peut être décrit de façon \textbf{trigonométrique}, pour cela il est décrit par deux grandeurs +\begin{itemize} + \item \textbf{Le module}, $r$, c'est sa distance avec l'origine. + \item \textbf{L'argument}, $\theta$, c'est l'angle orienté qu'il fait avec l'axe des abscisses. +\end{itemize} +On écrira alors +\[ + z = r(\cos(\theta) + i\sin(\theta)) +\] +\end{minipage} +\hfill +\begin{minipage}{0.3\textwidth} + \begin{tikzpicture}[yscale=.8, xscale=.8] + \repereNoGrid{-1}{5}{-1}{5} + \draw (0,0) -- (3,3) node [above, midway, sloped] {$r$} node [above right] {$M(a+ib)$}; + \draw [->] (2,0) arc (0:45:2) node [midway, right] {$\theta$}; + \draw [dashed] (3,0) node [below] {$a$} -- (3,3); + \draw [dashed] (0,3) node [left] {$b$} -- (3,3); + \end{tikzpicture} +\end{minipage} + +\subsection*{Trigonométrique vers algébrique} + +On a un nombre complexe sous forme trigonométrique $z = r(\cos(\theta) + i\sin(\theta))$. Sa forme algébrique est alors +\[ + a = r\cos(\theta) \mbox{ et } b = r\sin(\theta) +\] + +\paragraph{Exemple:} Forme algébrique de $z = 2(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}))$ + +\afaire{à convertir} + + + +\subsection*{Algébrique vers trigonométrique} + +On a un nombre complexe sous forme algébrique $z = a + ib$. On peut calculer son module et son argument ainsi +\[ + r = \sqrt{a^2+b^2} \qquad \mbox{ et } \theta \mbox{ se détermine avec } \qquad \cos(\theta) = \frac{a}{r} \qquad \sin(\theta) = \frac{b}{r} +\] + +\paragraph{Exemple:} Retrouver le module et l'argument de $z = \sqrt{2} + i\sqrt{2}$ + +\afaire{à convertir} + +\end{document} diff --git a/TST_sti2d/03_Complexes/exercises.tex b/TST_sti2d/03_Complexes/exercises.tex index 8843c5c..997a42e 100644 --- a/TST_sti2d/03_Complexes/exercises.tex +++ b/TST_sti2d/03_Complexes/exercises.tex @@ -62,6 +62,28 @@ \end{multicols} \end{exercise} +\begin{exercise}[subtitle={Algébrique vers trigonométrique}, step={1}, origin={Création}, topics={Complexes}, tags={Complexes, Trigonométrie}] + Placer les points suivant sur le plan complexe puis déterminer leur module et argument. + \begin{minipage}{0.3\textwidth} + \begin{itemize} + \item $z_A = 2i + 4$ + \item $z_B = -2i + 1$ + \item $z_C = i$ + \item $z_D = -3i - 3$ + \item $z_E = 2i + 2\sqrt{3}$ + \item $z_F = -3i + 3$ + \item $z_G = $ + \item $z_H = $ + \end{itemize} + \end{minipage} + \begin{minipage}{0.7\textwidth} + \begin{tikzpicture}[yscale=.6, xscale=.6] + \repere{-5}{5}{-5}{5} + \draw (-4,-1) node {$\times$} node[below left] {$G$}; + \draw (-4,4) node {$\times$} node[below left] {$H$}; + \end{tikzpicture} + \end{minipage} +\end{exercise} \collectexercisesstop{banque} diff --git a/TST_sti2d/03_Complexes/index.rst b/TST_sti2d/03_Complexes/index.rst index 5bde846..c805005 100644 --- a/TST_sti2d/03_Complexes/index.rst +++ b/TST_sti2d/03_Complexes/index.rst @@ -2,7 +2,7 @@ Complexes ######### :date: 2020-09-29 -:modified: 2020-10-01 +:modified: 2020-10-08 :authors: Benjamin Bertrand :tags: Complexes, Trigonométrie :category: TST_sti2d @@ -30,6 +30,10 @@ On pourra ajouter une exercice en lien avec la physique. Cours: Définition de la notation trigonométrique. Passage de la forme algébrique à la forme trigonométrique. +.. image:: ./2B_module_argument.pdf + :height: 200px + :alt: Forme trigonométrique d'un nombre complexe. + Exercices techniques pour le passage d'une forme à l'autre avec toujours le lien avec le plan complexe. Étape 3: Transformation géométriques