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\begin{document}

\sloppy

\needspace{4\baselineskip}
\section*{
Exercice 1}

 Dans le triangle GTD rectangle en G, on sait que :
\begin{itemize}
\item GD=6,5 cm
\item TD=9 cm
\end{itemize}

Après avoir fait un schéma, calcule l'arrondi au degré près de la mesure de l'angle \(\widehat{GDT}\).

\needspace{4\baselineskip}
\section*{
Exercice 2}

 Dans le triangle GFS rectangle en G, on sait que :
\begin{itemize}
\item GF=7,3 cm
\item \(\widehat{GFS}\) = 57\textdegree{}
\end{itemize}

Après avoir fait un schéma, calcule la longueur du segment [GS]. (Arrondir au dixième)

\needspace{4\baselineskip}
\section*{
Exercice 3}

 Dans le triangle MVZ rectangle en M, on sait que :
\begin{itemize}
\item MZ=8,3 cm
\item \(\widehat{VZM}\) =  42\textdegree{}
\end{itemize}

Après avoir fait un schéma, calcule la longueur du segment [ZV]. (Arrondir au dixième)

\needspace{4\baselineskip}
\section*{
Exercice 4}

 Dans le triangle NKA rectangle en N, on sait que :
\begin{itemize}
\item NA=4,7 cm
\item KA=9,8 cm
\end{itemize}

Après avoir fait un schéma, calcule l'arrondi au degré près de la mesure de l'angle \(\widehat{NKA}\).
\needspace{4\baselineskip}
\section*{
Exercice 5}

 Dans le triangle ZFL rectangle en Z, on sait que :
\begin{itemize}
\item ZL=9,4 cm
\item \(\widehat{ZFL}\) = 47\textdegree{}
\end{itemize}

Après avoir fait un schéma, calcule la longueur du segment [ZF]. (Arrondir au dixième)

\newpage
\section*{\textcolor{red}{Correction}}

\needspace{4\baselineskip}
\subsection*{
Exercice 1}

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  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (4.343,0.782) {?};
\end{tikzpicture}\end{center}


 Dans le triangle GTD rectangle en G, on cherche une relation entre l'angle aigu   \(\widehat{GDT}\) son coté adjacent et l'hypoténuse du triangle.

cos(\(\widehat{GDT}\)) = \(\dfrac{GD}{TD}\)
\par\addvspace{.5em}
  d'où    cos(\(\widehat{GDT}\)) =  \(\dfrac{6{,}5}{9}\)
\par\addvspace{.5em}
On a donc \(\widehat{GDT}\) = Arccos (\(\dfrac{6{,}5}{9}\)) \(\approx\) 44\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 2}

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 Dans le triangle GFS rectangle en G, on cherche une relation entre l'angle aigu   \(\widehat{GFS}\) son coté adjacent et son coté opposé.

\par\addvspace{.5em}
tan(\(\widehat{GFS}\)) = \(\dfrac{GS}{GF}\)
\par\addvspace{.5em}
d'où tan(57\textdegree{}) =\(\dfrac{GS}{7{,}3}\)
\par\addvspace{.5em}
On a donc GS = 7,3 \(\times\) tan(57\textdegree{}) \(\approx\) 11,2 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 3}

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\end{tikzpicture}\end{center}


 Dans le triangle MVZ rectangle en M, on cherche une relation entre l'angle aigu   \(\widehat{MZV}\) son coté adjacent et l'hypoténuse du triangle.

cos(\(\widehat{MZV}\)) = \(\dfrac{MZ}{VZ}\)
\par\addvspace{.5em}
  d'où    cos(42\textdegree{}) =  \(\dfrac{8{,}3}{VZ}\)
\par\addvspace{.5em}
 On a donc VZ =  \(\dfrac{8{,}3}{\cos(42^{\circ})}\) \(\approx\) 11,2 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 4}

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\end{tikzpicture}\end{center}


 Dans le triangle NKA rectangle en N, on cherche une relation entre l'angle aigu   \(\widehat{NKA}\) son coté opposé et l'hypoténuse du triangle.

\par\addvspace{.5em}
sin(\(\widehat{NKA}\)) =  \(\dfrac{NA}{KA}\)
\par\addvspace{.5em}
 d'où sin(\(\widehat{NKA}\)) = \(\dfrac{4{,}7}{9{,}8}\)
\par\addvspace{.5em}
On a donc \(\widehat{NKA}\) =  ArcSin(\(\dfrac{4{,}7}{9{,}8}\)) \(\approx\) 29\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 5}

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  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (2.265,2.722) {47°};
\end{tikzpicture}\end{center}


 Dans le triangle ZFL rectangle en Z, on cherche une relation entre l'angle aigu   \(\widehat{ZFL}\) son coté adjacent et son coté opposé.

\par\addvspace{.5em}
tan(\(\widehat{ZFL}\)) = \(\dfrac{ZL}{ZF}\)
\par\addvspace{.5em}
tan(47\textdegree{}) = \(\dfrac{9{,}4}{ZF}\)
\par\addvspace{.5em}
 On a donc ZF = \(\dfrac{9{,}4}{\tan(47^{\circ})}\) \(\approx\) 8,8 cm\end{document}
