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\begin{document}

\sloppy

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

 Dans le triangle ZAR rectangle en Z, on sait que :
\begin{itemize}
\item ZA=2,3 cm
\item AR=7,5 cm
\end{itemize}

Après avoir fait un schéma, calcule l'arrondi au degré près de la mesure de l'angle \(\widehat{ZAR}\).
\needspace{4\baselineskip}
\section*{
Exercice 2}

 Dans le triangle JMD rectangle en J, on sait que :
\begin{itemize}
\item MD=6,3 cm
\item \(\widehat{JMD}\) = 48\textdegree{}
\end{itemize}

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

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

 Dans le triangle STD rectangle en S, on sait que :
\begin{itemize}
\item ST=4 cm
\item \(\widehat{STD}\) = 50\textdegree{}
\end{itemize}

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

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

 Dans le triangle PJF rectangle en P, on sait que :
\begin{itemize}
\item JF=0,7 cm
\item \(\widehat{JFP}\) =  27\textdegree{}
\end{itemize}

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

\needspace{4\baselineskip}
\section*{
Exercice 5}

 Dans le triangle VCZ rectangle en V, on sait que :
\begin{itemize}
\item VC=1,9 cm
\item VZ=4,9 cm
\end{itemize}

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

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

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

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\end{tikzpicture}\end{center}


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

cos(\(\widehat{ZAR}\)) = \(\dfrac{ZA}{AR}\)
\par\addvspace{.5em}
 d'où cos(\(\widehat{ZAR}\)) = \(\dfrac{2{,}3}{7{,}5}\)
\par\addvspace{.5em}
On a donc \(\widehat{ZAR}\) =ArcCos(\(\dfrac{2{,}3}{7{,}5}\)) \(\approx\)72\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 2}

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

\par\addvspace{.5em}
sin(\(\widehat{JMD}\)) =  \(\dfrac{JD}{MD}\)
\par\addvspace{.5em}
 d'où sin(48\textdegree{}) = \(\dfrac{JD}{6{,}3}\)

On a donc JD = 6,3 \(\times\) sin(48\textdegree{}) \(\approx\) 4,7 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 3}

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\end{tikzpicture}\end{center}


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

cos(\(\widehat{STD}\)) = \(\dfrac{ST}{TD}\)
\par\addvspace{.5em}
 d'où cos(50\textdegree{}) = \(\dfrac{4}{TD}\)
\par\addvspace{.5em}
On a donc TD = \(\dfrac{4}{\cos(50^{\circ})}\)\(\approx\) 6,2 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 4}

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

\par\addvspace{.5em}
sin(\(\widehat{PFJ}\)) = \(\dfrac{PJ}{JF}\)
\par\addvspace{.5em}
 D'où sin(27\textdegree{}) = \(\dfrac{PJ}{0{,}7}\)

On a donc PJ = 0,7 \(\times\) sin(27\textdegree{}) \(\approx\) 0,3 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 5}

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  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.500,0.717) {4,9 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (5.295,1.106) {?};
\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
tan(\(\widehat{VZC}\)) = \(\dfrac{VC}{VZ}\)
\par\addvspace{.5em}
d'où  tan(\(\widehat{VZC}\)) =  \(\dfrac{1{,}9}{4{,}9}\)
\par\addvspace{.5em}
 On a donc \(\widehat{VZC}\) =  ArcTan(\(\dfrac{1{,}9}{4{,}9}\)) \(\approx\) 21\textdegree{}\end{document}
