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\newcommand{\ie}{\leqslant}  % inferieur ou egal
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keywordstyle =\color{red}\usefont{OT1}{cmtt}{b}{n},basicstyle=\ttfamily\footnotesize\color{red},commentstyle=\normalfont\scriptsize\slshape,breaklines=true,backgroundcolor=\color{0.9white}}


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\newcommand\exot[2][\thenum]{\refstepcounter{num}
\vspace{0.5cm}
{\bfseries\large \textsf{Exercice #1\ \ #2}}\\ 
\par\nobreak 
\normalsize
\vspace{0.25cm}
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\begin{document}


\setlength{\parindent}{0mm}






%\begin{multicols}{2}

\fbox{%
\begin{minipage}{\textwidth}
\smallskip
{\bfseries \sffamily \begin{center} CORRIGÉ du Devoir Surveillé n°1 \hfill Jeudi 2 octobre 2008 \\
                                  BTS domotique 1\hfill 1 heure \end{center}}
\smallskip
\end{minipage}
}
\bigskip




\exot{}

\begin{enumerate}
\item $\displaystyle z_1z_2=3\sqrt{2}+\sqrt{6}+{\rm i}
  \left(3\sqrt{2}-\sqrt{6}\right)\qquad \qquad\displaystyle \frac{z_1}{z_2}=\frac{\sqrt{2}\left(\sqrt{3}-{\rm i}\right)\times
  \sqrt{3}\left(1-{\rm                   i}\right)}{3                  +
  3}=\frac{\sqrt{6}}{6}\left(\sqrt{3}-1+{\rm i}\left(-1-\sqrt{3}\right)\right)$

\item $\displaystyle z_1=2\sqrt{2}\left(\frac{\sqrt{3}}{2}-\frac{1}{2}{\rm i}\right)=2
  \sqrt{2}{\rm e}^{-{\rm i}\frac{\pi}{6}} \qquad\qquad \displaystyle z_2=\sqrt{6}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}{\rm
      i}\right)=\sqrt{6}{\rm e}^{{\rm i}\frac{\pi}{4}}$

$\displaystyle z_1z_2=2     \sqrt{2}\sqrt{6}{\rm     e}^{{\rm     i}\frac{\pi}{4}-{\rm
    i}\frac{\pi}{6}}=4 \sqrt{3}{\rm e}^{{\rm i}\frac{\pi}{12}} \qquad \qquad\displaystyle \frac{z_1}{z_2}=\frac{2 \sqrt{2}}{\sqrt{6}}{\rm e}^{-{\rm i}\frac{\pi}{4}-{\rm
    i}\frac{\pi}{6}}=\frac{2 \sqrt{3}}{3}{\rm e}^{{-\rm i}\frac{5\pi}{12}}$
\end{enumerate}

\exot{}

On utilise les formules d'Euler~:

%@Commande-1
\begin{lstlisting}
trig2exp(cos(x));trig2exp(sin(x))
\end{lstlisting}
{\MarqueLaTeXGiac{\[ \frac{(e^{i x}+\frac{1}{e^{i x}})}{2},\frac{(e^{i x}-\frac{1}{e^{i x}})}{2*i} \]}}

et les formules de développement~:

%@Commande-2
\begin{lstlisting}
expand((a+b)^3); expand((a-b)^2)
\end{lstlisting}
{\MarqueLaTeXGiac{\[ 3 a^{2} b+3 a b^{2}+b^{3}+a^{3},-\left(2 a b\right)+\left(-b\right)^{2}+a^{2} \]}}

et on obtient

%@Commande-3
\begin{lstlisting}
tcollect((cos(x))^3)
\end{lstlisting}
{\MarqueLaTeXGiac{\[ \frac{3}{4} \cdot \cos\left(x\right)+\frac{1}{4} \cdot \cos\left(3 x\right) \]}}

%@Commande-4
\begin{lstlisting}
tcollect((sin(2*x))^2)
\end{lstlisting}
{\MarqueLaTeXGiac{\[ \frac{1}{2}+\frac{1}{-2} \cdot \cos\left(4 x\right) \]}}

\exot{}

On résout l'équation~:

%@Commande-5
\begin{lstlisting}
S:=csolve(z^2-2*sqrt(2)*z+4=0,z)
\end{lstlisting}
{\MarqueLaTeXGiac{\[ [\sqrt{2}\cdot i+\sqrt{2},-\left(\sqrt{2}\cdot i\right)+\sqrt{2}] \]}}

On cherche module et argument de la première solution~:

%@Commande-6
\begin{lstlisting}
abs(S[0]);arg(S[0])
\end{lstlisting}
{\MarqueLaTeXGiac{\[ 2,\frac{\pi }{4} \]}}

puis de la deuxième~:

%@Commande-7
\begin{lstlisting}
simplifier(abs(S[1]));simplifier(arg(S[1]))
\end{lstlisting}
{\MarqueLaTeXGiac{\[ 2,\frac{(-\pi )}{4} \]}}


\exot{}

%@Commande-8
\begin{lstlisting}
z1:=2*exp(i*pi/4)
\end{lstlisting}
{\MarqueLaTeXGiac{\[ 2 (\frac{\sqrt{2}}{2}+\frac{i \sqrt{2}}{2}) \]}}

%@Commande-9
\begin{lstlisting}
z2:=3*exp(-i*pi/6)
\end{lstlisting}
{\MarqueLaTeXGiac{\[ 3 (\frac{\sqrt{3}}{2}+i (-\frac{1}{2})) \]}}

%@Commande-10
\begin{lstlisting}
z1+z2
\end{lstlisting}
{\MarqueLaTeXGiac{\[ 2 (\frac{\sqrt{2}}{2}+\frac{i \sqrt{2}}{2})+3 (\frac{\sqrt{3}}{2}+\frac{-i}{2}) \]}}

%@Commande-11
\begin{lstlisting}
approx(abs(z1+z2))
\end{lstlisting}
{\MarqueLaTeXGiac{\[      4.013207 \]}}

%@Commande-12
\begin{lstlisting}
approx(arg(z1+z2))
\end{lstlisting}
{\MarqueLaTeXGiac{\[Psr 2, Mod 5.74456, Heu 1.6, Min1.6
GCD dim 1, n=10 maxpqdeg0 4(Vector [4])
// Using Heu gcd 
// mrref line 0\]}}
{\MarqueLaTeXGiac{\[     -0.021378 \]}}


%\end{multicols}

\end{document}


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