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\begin{document}

\sloppy

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

 Dans le triangle RCB rectangle en R, on sait que :
\begin{itemize}
\item RB=5,3 cm
\item CB=7,7 cm
\end{itemize}

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

 Dans le triangle RTN rectangle en R, on sait que :
\begin{itemize}
\item RN=1,7 cm
\item \(\widehat{RTN}\) = 71\textdegree{}
\end{itemize}

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

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

 Dans le triangle CDF rectangle en C, on sait que :
\begin{itemize}
\item CD=6,7 cm
\item \(\widehat{CDF}\) = 70\textdegree{}
\end{itemize}

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

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

 Dans le triangle TZA rectangle en T, on sait que :
\begin{itemize}
\item TZ=3,4 cm
\item \(\widehat{ZAT}\) =  45\textdegree{}
\end{itemize}

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

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

 Dans le triangle ZPK rectangle en Z, on sait que :
\begin{itemize}
\item ZK=5,6 cm
\item PK=8,1 cm
\end{itemize}

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

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

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

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

\par\addvspace{.5em}
sin(\(\widehat{RCB}\)) =  \(\dfrac{RB}{CB}\)
\par\addvspace{.5em}
 d'où sin(\(\widehat{RCB}\)) = \(\dfrac{5{,}3}{7{,}7}\)
\par\addvspace{.5em}
On a donc \(\widehat{RCB}\) =  ArcSin(\(\dfrac{5{,}3}{7{,}7}\)) \(\approx\) 43\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 2}

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

\par\addvspace{.5em}
tan(\(\widehat{RTN}\)) = \(\dfrac{RN}{RT}\)
\par\addvspace{.5em}
tan(71\textdegree{}) = \(\dfrac{1{,}7}{RT}\)
\par\addvspace{.5em}
 On a donc RT = \(\dfrac{1{,}7}{\tan(71^{\circ})}\) \(\approx\) 0,6 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 3}

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

\par\addvspace{.5em}
tan(\(\widehat{CDF}\)) = \(\dfrac{CF}{CD}\)
\par\addvspace{.5em}
d'où tan(70\textdegree{}) =\(\dfrac{CF}{6{,}7}\)
\par\addvspace{.5em}
On a donc CF = 6,7 \(\times\) tan(70\textdegree{}) \(\approx\) 18,4 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 4}

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\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
sin(\(\widehat{TAZ}\)) = \(\dfrac{TZ}{ZA}\)
\par\addvspace{.5em}
 D'où sin(45\textdegree{}) = \(\dfrac{3{,}4}{ZA}\)
\par\addvspace{.5em}
On a donc ZA =\(\dfrac{3{,}4}{\sin(45^{\circ})}\) \(\approx\) 4,8 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 5}

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  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (4.227,0.795) {?};
\end{tikzpicture}\end{center}


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

cos(\(\widehat{ZKP}\)) = \(\dfrac{ZK}{PK}\)
\par\addvspace{.5em}
  d'où    cos(\(\widehat{ZKP}\)) =  \(\dfrac{5{,}6}{8{,}1}\)
\par\addvspace{.5em}
On a donc \(\widehat{ZKP}\) = Arccos (\(\dfrac{5{,}6}{8{,}1}\)) \(\approx\) 46\textdegree{} \end{document}
