147 lines
4.2 KiB
TeX
147 lines
4.2 KiB
TeX
\documentclass[11pt,xcolor=table]{classPres}
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\setlength\columnsep{0pt}
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\title{Formules trigonométriques}
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\date{Octobre 2019}
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\begin{document}
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\begin{frame}{Angle opposé}
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\begin{minipage}{0.5\textwidth}
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\begin{tikzpicture}[scale=2.3]
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\cercleTrigo
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\draw (0,0) -- (40:1);
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\draw[->, very thick, red] (0.8,0) arc (0:40:0.8) node [midway, left] {$a$};
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\pause
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\draw (0,0) -- (-40:1);
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\draw[->, very thick, red] (0.8,0) arc (0:-40:0.8) node [midway, left] {$-a$};
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\end{tikzpicture}
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\end{minipage}
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\begin{minipage}{0.4\textwidth}
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\begin{block}{Propriété}
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Soit $a$ un angle alors
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\[
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\cos(-a) =
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\]
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\[
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\sin(-a) =
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\]
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\end{block}
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\end{minipage}
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\end{frame}
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\begin{frame}{Additions d'angles}
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\begin{block}{Propriété}
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Soit $a$ et $b$ deux angles alors
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\[
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\cos(a+b) = \ldots
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\]
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\end{block}
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\begin{minipage}{0.4\textwidth}
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\pause
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\begin{tikzpicture}[scale=2]
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\cercleTrigo
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\draw (0,0) -- (40:1) node [above right] {$A$};
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\draw[->, very thick, red] (0.8,0) arc (0:40:0.8) node [midway, left] {$a$};
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\draw (0,0) -- (-20:1) node [below right] {$B$};
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\draw[->, very thick, blue] (0.8,0) arc (0:-20:0.8) node [midway, left] {$b$};
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\draw[->, very thick, blue] (0.8,0) arc (0:-20:0.8) node [midway, left] {$b$};
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\pause
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\end{tikzpicture}
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\end{minipage}
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\begin{minipage}{0.55\textwidth}
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\begin{enumerate}
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\item Exprimer les coordonnées de $A$ et $B$.
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\item Calculer $\vec{OA}\cdot \vec{OB}$ avec les deux formules.
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\item En déduire une formule pour calculer le cosinus et le sinus d'une somme de 2 angles.
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\end{enumerate}
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\end{minipage}
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\end{frame}
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\begin{frame}{Additions d'angles}
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\begin{block}{Propriété}
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Soit $a$ et $b$ deux angles alors
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\[
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\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)
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\]
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\[
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\sin(a+b) = \cos(a)\sin(b) + \sin(a)\cos(b)
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\]
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\end{block}
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\begin{block}{Exemple}
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On note que $\dfrac{7\pi}{12} = \dfrac{3\pi}{4} - \dfrac{\pi}{6}$.
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Calculer
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$\cos(\dfrac{7\pi}{12}) = $
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\end{block}
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\begin{block}{Exercices}
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\begin{enumerate}
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\item $\dfrac{\pi}{12} = \dfrac{\pi}{3} - \dfrac{\pi}{4}$.
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Calculer $ \cos(\frac{\pi}{12})$ et $\sin(\frac{\pi}{12})$
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\item
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$\cos(2x+\dfrac{\pi}{6}) = \ldots \qquad \sin(\dfrac{x}{3} - \dfrac{\pi}{4}) = \ldots$
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\end{enumerate}
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\end{block}
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\end{frame}
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\begin{frame}{Formules de duplications}
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\begin{block}{Propriété}
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Soit $a$ et $b$ deux angles alors
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\[
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\cos(2a) = \ldots
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\qquad
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\qquad
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\qquad
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\qquad
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\sin(2a) = \ldots
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\]
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\end{block}
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\pause
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\begin{block}{Propriété}
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Soit $a$ un angle
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\[
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\cos^2(a) + \sin^2(a) = 1
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\]
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\end{block}
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\pause
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\begin{block}{Propriété}
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Soit $a$ et $b$ deux angles alors
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\[
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\cos(2a) = \ldots
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\qquad
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\qquad
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\qquad
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\qquad
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\sin(2a) = \ldots
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\]
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\end{block}
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\end{frame}
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\begin{frame}{Formules de duplications}
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\begin{block}{Propriété}
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Soit $a$ et $b$ deux angles alors
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\[
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\cos^2(a) = \ldots
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\]
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\[
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\sin^2(a) = \ldots
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\]
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\end{block}
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\begin{block}{Exercices}
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Linéariser les quantités suivantes
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\begin{enumerate}
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\item $\cos^2(2t+\dfrac{\pi}{6})$
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\item $\sin^2(3t+\dfrac{\pi}{8})$
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\end{enumerate}
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\end{block}
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\end{frame}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "master"
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%%% End:
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