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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
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\textsc{\textbf{Correction de  l'interrogation n$^{\circ}5$ }}
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\Exoi{\hfill \textit{Valeurs remarquables du sinus et du cosinus}-\textbf{6 points}\\
{\footnotesize A l'aide d'un cercle trigonométrique (ou de plusieurs pour plus de clarté) qui apparaitra sur votre copie }\\
En observant le cercle suivant on obtient les réponses à chacune des questions :
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\begin{enumerate}
\item Déterminer :
%\begin{multicols}{4}
\begin{enumerate}
\item $\cos \dfrac{3\pi}{4}=-\dfrac{\sqrt{2}}{2}$ ;
\item $ \sin \dfrac{5\pi}{4}=-\dfrac{\sqrt{2}}{2}$ ;
\item $\sin \left(-\dfrac{\pi}{6}\right)=-\dfrac{1}{2}$ ;
\item $ \cos\left(-\dfrac{11\pi}{3}\right)=\cos \dfrac{\pi}{3}=\dfrac{\sqrt{3}}{2}$
\end{enumerate}
%\end{multicols}
\item {\footnotesize Résoudre, dans l'intervalle $[0;2\pi]$, les deux équations suivantes :}
%\begin{multicols}{2}
\begin{enumerate}
\item $ \cos x=-\dfrac{1}{2}$ 
$$ \mathscr S=\left\{\dfrac{2\pi}{3};\dfrac{4\pi}{3}\right\}$$
\item $\sin x=\dfrac{\sqrt{2}}{2}$
$$ \mathscr S=\left\{\dfrac{\pi}{4};\dfrac{3\pi}{4}\right\}$$
\end{enumerate}
%\end{multicols}
\end{enumerate}
}

\Exoi{\hfill \textit{Angles de vecteurs}-\textbf{4 points}\\
{\footnotesize $ABC$ est un triangle équilatéral direct. $ABD$ est un triangle rectangle isocèle avec $\left(\overrightarrow{AB};\overrightarrow{AD}\right)=\dfrac{\pi}{2}$ rad. Enfin $BEC$ est un triangle rectangle isocèle avec $\left(\overrightarrow{BE};\overrightarrow{BC}\right)=\dfrac{\pi}{2}$ rad.}
%\begin{multicols}{2}
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\begin{enumerate}
\item  {\footnotesize Démontrer que $ \left(\overrightarrow{CD};\overrightarrow{CA}\right)=\dfrac{5\pi}{12}(2\pi)$.}\\
Dans le triangle isocèle $CAD$, on sait que $\left(\overrightarrow{AC};\overrightarrow{AD}\right)=\dfrac{\pi}{2}-\dfrac{\pi}{3}=\dfrac{\pi}{6}$, par conséquent comme la somme des angles dans un triangle vaut $\pi$ on obtient :
$$  \left(\overrightarrow{CD};\overrightarrow{CA}\right)=\dfrac{\pi-\frac{\pi}{6}}{2}=\dfrac{5\pi}{12}(2\pi)$$
\item {\footnotesize Déterminer une mesure, en radian, de $\left(\overrightarrow{CA}; \overrightarrow{CB}\right)$.}
\\
Un angle d'un triangle équilatéral vaut $\dfrac{\pi}{3}$ rad.
\item {\footnotesize Déterminer une mesure, en radian, de $\left(\overrightarrow{CB};\overrightarrow{CE}\right)$.}
\\$ECB$ est un triangle rectangle isocèle donc $ \left(\overrightarrow{CB};\overrightarrow{CE}\right)=\dfrac{\pi}{4}$
\item {\footnotesize En déduire que les points $E$, $C$ et $D$ sont alignés.}\\
$$ \left(\overrightarrow{CD};\overrightarrow{CE}\right)= \left(\overrightarrow{CD};\overrightarrow{CA}\right)+ \left(\overrightarrow{CA};\overrightarrow{CB}\right)+ \left(\overrightarrow{CB};\overrightarrow{CE}\right)=\dfrac{5\pi}{12}+\dfrac{\pi}{3}+\dfrac{\pi}{4}=\dfrac{5\pi+4\pi+3\pi}{12}=\pi$$
Les points $E$, $C$ et $D$ sont donc alignés.
\end{enumerate}
%\end{multicols}
}

\pagebreak
%\vfill

\setcounter{Exi}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
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\end{center}
\vspace{-3mm}
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}

\Exoi{\hfill \textit{Valeurs remarquables du sinus et du cosinus}-\textbf{6 points}\\
{\footnotesize A l'aide d'un cercle trigonométrique (ou de plusieurs pour plus de clarté) qui apparaitra sur votre copie }\\
En observant le cercle suivant on obtient les réponses à chacune des questions :
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\endpspicture
\end{center}
\begin{enumerate}
\item Déterminer :
\begin{multicols}{2}
\begin{enumerate}
\item $\sin \dfrac{3\pi}{4}=\dfrac{\sqrt{2}}{2}$ ;
\item $ \cos \dfrac{5\pi}{3}=\cos \dfrac{\pi}{3}=\dfrac{1}{2}$ ;
\item $\cos \left(-\dfrac{\pi}{3}\right)=\dfrac{1}{2}$ ;
\item $ \sin\left(-\dfrac{11\pi}{6}\right)=\sin \dfrac{\pi}{6}=\dfrac{1}{2}$
\end{enumerate}
\end{multicols}
\item Résoudre, dans l'intervalle $[0;2\pi]$, les deux équations suivantes :
\begin{enumerate}
\item $ \sin x=-\dfrac{1}{2}$ 
$$\mathscr S=\left\{\dfrac{7\pi}{6};\dfrac{11\pi}{6}\right\} $$
\item $\cos x=\dfrac{\sqrt{2}}{2}$
$$ \mathscr S=\left\{\dfrac{\pi}{4};\dfrac{7\pi}{4}\right\}$$
\end{enumerate}
\end{enumerate}
}

\Exoi{\hfill \textit{Angles de vecteurs}-\textbf{4 points}\\
\begin{footnotesize}
La figure ci-dessous est formée de deux carrés de même centre $O$ avec $\left(\overrightarrow{OA};\overrightarrow{OE}\right)=-\dfrac{\pi}{3}$ rad.
\end{footnotesize}
%\begin{multicols}{2}
\begin{center}
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\begin{scriptsize}
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\end{center} 
\begin{enumerate}
\item {\footnotesize En utilisant la donnée de l'énoncé et la relation de Chasles, déterminer une mesure de $\left(\overrightarrow{OA};\overrightarrow{OF}\right)$.}
\\
$$ \left(\overrightarrow{OA};\overrightarrow{OF}\right)=\left(\overrightarrow{OA};\overrightarrow{OE}\right)+\left(\overrightarrow{OE};\overrightarrow{OF}\right)=-\dfrac{\pi}{3}+\dfrac{\pi}{2}=\dfrac{-2\pi+3\pi}{6}=\dfrac{\pi}{6}$$
\item {\footnotesize Déduire de la question précédente une mesure de $\left(\overrightarrow{OA};\overrightarrow{OG}\right)$.}\\
$$ \left(\overrightarrow{OA};\overrightarrow{OG}\right)=\left(\overrightarrow{OA};\overrightarrow{OF}\right)+\left(\overrightarrow{OF};\overrightarrow{OG}\right)=\dfrac{\pi}{6}+\dfrac{\pi}{2}=\dfrac{\pi+3\pi}{6}=\dfrac{2\pi}{3}$$
\item \begin{footnotesize}
Soit $I$ et $J$ les milieux respectifs des segments $[AB]$ et $[FG]$.
\\ Déterminer une mesure des angles $\left(\overrightarrow{OI};\overrightarrow{OA}\right)$ et $\left(\overrightarrow{OF};\overrightarrow{OJ}\right)$
\end{footnotesize}\\
On a $\left(\overrightarrow{OI};\overrightarrow{OA}\right)=-\dfrac{\pi}{4}$ et $\left(\overrightarrow{OF};\overrightarrow{OJ}\right)=\dfrac{\pi}{4}$
\item {\footnotesize En déduire la mesure principale de $\left(\overrightarrow{OI};\overrightarrow{OJ}\right)$}\\
$$ \left(\overrightarrow{OI};\overrightarrow{OJ}\right)=\left(\overrightarrow{OI};\overrightarrow{OA}\right)+\left(\overrightarrow{OA};\overrightarrow{OF}\right)+\left(\overrightarrow{OF};\overrightarrow{OJ}\right)=-\dfrac{\pi}{4}+\dfrac{\pi}{6}+\dfrac{\pi}{4}=\dfrac{\pi}{6}$$
\end{enumerate}
%\end{multicols}
}

\end{document}
