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\begin{document}

\sloppy

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

 Dans le triangle PFR rectangle en P, on sait que :
\begin{itemize}
\item PF=6,1 cm
\item \(\widehat{FRP}\) =  38\textdegree{}
\end{itemize}

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

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

 Dans le triangle ZST rectangle en Z, on sait que :
\begin{itemize}
\item ZS=1,5 cm
\item ZT=4,6 cm
\end{itemize}

Après avoir fait un schéma, calcule l'arrondi au degré près de la mesure de l'angle \(\widehat{ZST}\).
\needspace{4\baselineskip}
\section*{
Exercice 3}

 Dans le triangle ZVJ rectangle en Z, on sait que :
\begin{itemize}
\item ZV=1,6 cm
\item ZJ=4,1 cm
\end{itemize}

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

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

 Dans le triangle MKT rectangle en M, on sait que :
\begin{itemize}
\item KT=7,3 cm
\item \(\widehat{KTM}\) =  15\textdegree{}
\end{itemize}

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

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

 Dans le triangle ZGV rectangle en Z, on sait que :
\begin{itemize}
\item GV=1,8 cm
\item \(\widehat{GVZ}\) =  32\textdegree{}
\end{itemize}

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

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

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

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\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
sin(\(\widehat{PRF}\)) = \(\dfrac{PF}{FR}\)
\par\addvspace{.5em}
 D'où sin(38\textdegree{}) = \(\dfrac{6{,}1}{FR}\)
\par\addvspace{.5em}
On a donc FR =\(\dfrac{6{,}1}{\sin(38^{\circ})}\) \(\approx\) 9,9 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 2}

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

\par\addvspace{.5em}
tan(\(\widehat{ZST}\)) = \(\dfrac{ZT}{ZS}\)
\par\addvspace{.5em}
d'où tan(\(\widehat{ZST}\)) = \(\dfrac{4{,}6}{1{,}5}\)
\par\addvspace{.5em}
 On a donc ZST = ArcTan(\(\dfrac{4{,}6}{1{,}5}\)) \(\approx\) 72\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 3}

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  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (0.550,1.933) {1,6 cm};
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\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
tan(\(\widehat{ZJV}\)) = \(\dfrac{ZV}{ZJ}\)
\par\addvspace{.5em}
d'où  tan(\(\widehat{ZJV}\)) =  \(\dfrac{1{,}6}{4{,}1}\)
\par\addvspace{.5em}
 On a donc \(\widehat{ZJV}\) =  ArcTan(\(\dfrac{1{,}6}{4{,}1}\)) \(\approx\) 21\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 4}

\begin{center}\begin{tikzpicture}[line width=0.6pt]
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 Dans le triangle MKT rectangle en M, on cherche une relation entre l'angle aigu   \(\widehat{MTK}\) son coté opposé et l'hypoténuse du triangle.

\par\addvspace{.5em}
sin(\(\widehat{MTK}\)) = \(\dfrac{MK}{KT}\)
\par\addvspace{.5em}
 D'où sin(15\textdegree{}) = \(\dfrac{MK}{7{,}3}\)

On a donc MK = 7,3 \(\times\) sin(15\textdegree{}) \(\approx\) 1,9 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) {?};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.667,2.000) {1,8 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (5.100,0.717) {32°};
\end{tikzpicture}\end{center}


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

cos(\(\widehat{ZVG}\)) = \(\dfrac{ZV}{GV}\)
\par\addvspace{.5em}
  d'où    cos(32\textdegree{}) =  \(\dfrac{ZV}{1{,}8}\)

On a donc ZV = 1,8 \(\times\) cos(32\textdegree{}) \(\approx\) 1{,}5 cm\end{document}
