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\begin{document}

\sloppy

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

 Dans le triangle ZPT rectangle en Z, on sait que :
\begin{itemize}
\item ZT=4,7 cm
\item PT=8,5 cm
\end{itemize}

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

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

 Dans le triangle JFR rectangle en J, on sait que :
\begin{itemize}
\item FR=1,1 cm
\item \(\widehat{FRJ}\) =  13\textdegree{}
\end{itemize}

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

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

 Dans le triangle JFR rectangle en J, on sait que :
\begin{itemize}
\item JF=7,5 cm
\item \(\widehat{FRJ}\) =  32\textdegree{}
\end{itemize}

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

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

 Dans le triangle ZSP rectangle en Z, on sait que :
\begin{itemize}
\item ZS=1,4 cm
\item \(\widehat{SPZ}\) =  23\textdegree{}
\end{itemize}

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

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

 Dans le triangle TNR rectangle en T, on sait que :
\begin{itemize}
\item TN=3 cm
\item TR=4,9 cm
\end{itemize}

Après avoir fait un schéma, calcule l'arrondi au degré près de la mesure de l'angle \(\widehat{TNR}\). \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 (3.842,0.849) {?};
\end{tikzpicture}\end{center}


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

cos(\(\widehat{ZTP}\)) = \(\dfrac{ZT}{PT}\)
\par\addvspace{.5em}
  d'où    cos(\(\widehat{ZTP}\)) =  \(\dfrac{4{,}7}{8{,}5}\)
\par\addvspace{.5em}
On a donc \(\widehat{ZTP}\) = Arccos (\(\dfrac{4{,}7}{8{,}5}\)) \(\approx\) 56\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 2}

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

\par\addvspace{.5em}
sin(\(\widehat{JRF}\)) = \(\dfrac{JF}{FR}\)
\par\addvspace{.5em}
 D'où sin(13\textdegree{}) = \(\dfrac{JF}{1{,}1}\)

On a donc JF = 1,1 \(\times\) sin(13\textdegree{}) \(\approx\) 0,2 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 3}

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

\par\addvspace{.5em}
sin(\(\widehat{JRF}\)) = \(\dfrac{JF}{FR}\)
\par\addvspace{.5em}
 D'où sin(32\textdegree{}) = \(\dfrac{7{,}5}{FR}\)
\par\addvspace{.5em}
On a donc FR =\(\dfrac{7{,}5}{\sin(32^{\circ})}\) \(\approx\) 14,2 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 4}

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

\par\addvspace{.5em}
tan(\(\widehat{ZPS}\)) = \(\dfrac{ZS}{ZP}\)
\par\addvspace{.5em}
d'où  tan(23\textdegree{}) =  \(\dfrac{1{,}4}{ZP}\)
\par\addvspace{.5em}
On a donc ZP = \(\dfrac{1{,}4}{\tan(23^{\circ})}\) \(\approx\) 3,3 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 5}

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  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.500,0.267) {4,9 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (1.539,2.752) {?};
\end{tikzpicture}\end{center}


 Dans le triangle TNR rectangle en T, on cherche une relation entre l'angle aigu   \(\widehat{TNR}\) son coté adjacent et son coté opposé.

\par\addvspace{.5em}
tan(\(\widehat{TNR}\)) = \(\dfrac{TR}{TN}\)
\par\addvspace{.5em}
d'où tan(\(\widehat{TNR}\)) = \(\dfrac{4{,}9}{3}\)
\par\addvspace{.5em}
 On a donc TNR = ArcTan(\(\dfrac{4{,}9}{3}\)) \(\approx\) 59\textdegree{}\end{document}
