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\newcommand{\titre}{Trigonométrie}
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\begin{document}

\sloppy

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

 Dans le triangle CAL rectangle en C, on sait que :
\begin{itemize}
\item CL=6,6 cm
\item AL=8 cm
\end{itemize}

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

 Dans le triangle GJB rectangle en G, on sait que :
\begin{itemize}
\item GJ=9,8 cm
\item \(\widehat{GJB}\) = 53\textdegree{}
\end{itemize}

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

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

 Dans le triangle DRG rectangle en D, on sait que :
\begin{itemize}
\item DG=9,9 cm
\item \(\widehat{RGD}\) =  35\textdegree{}
\end{itemize}

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

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

 Dans le triangle NCT rectangle en N, on sait que :
\begin{itemize}
\item NT=5,3 cm
\item \(\widehat{CTN}\) =  23\textdegree{}
\end{itemize}

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

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

 Dans le triangle ARL rectangle en A, on sait que :
\begin{itemize}
\item AR=2,4 cm
\item AL=5,3 cm
\end{itemize}

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

\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 (1.767,2.744) {?};
\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
sin(\(\widehat{CAL}\)) =  \(\dfrac{CL}{AL}\)
\par\addvspace{.5em}
 d'où sin(\(\widehat{CAL}\)) = \(\dfrac{6{,}6}{8}\)
\par\addvspace{.5em}
On a donc \(\widehat{CAL}\) =  ArcSin(\(\dfrac{6{,}6}{8}\)) \(\approx\) 56\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 2}

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

\par\addvspace{.5em}
tan(\(\widehat{GJB}\)) = \(\dfrac{GB}{GJ}\)
\par\addvspace{.5em}
d'où tan(53\textdegree{}) =\(\dfrac{GB}{9{,}8}\)
\par\addvspace{.5em}
On a donc GB = 9,8 \(\times\) tan(53\textdegree{}) \(\approx\) 13 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 3}

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\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
tan(\(\widehat{DGR}\)) = \(\dfrac{DR}{DG}\)
\par\addvspace{.5em}
d'où  tan(35\textdegree{}) =  \(\dfrac{DR}{9{,}9}\)
\par\addvspace{.5em}
On a donc DR =  9,9 \(\times\) tan(35\textdegree{})  \(\approx\) 6,9 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 4}

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  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.667,2.000) {?};
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 Dans le triangle NCT rectangle en N, on cherche une relation entre l'angle aigu   \(\widehat{NTC}\) son coté adjacent et l'hypoténuse du triangle.

cos(\(\widehat{NTC}\)) = \(\dfrac{NT}{CT}\)
\par\addvspace{.5em}
  d'où    cos(23\textdegree{}) =  \(\dfrac{5{,}3}{CT}\)
\par\addvspace{.5em}
 On a donc CT =  \(\dfrac{5{,}3}{\cos(23^{\circ})}\) \(\approx\) 5,8 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 5}

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  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.500,0.557) {5,3 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (5.298,0.965) {?};
\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
tan(\(\widehat{ALR}\)) = \(\dfrac{AR}{AL}\)
\par\addvspace{.5em}
d'où  tan(\(\widehat{ALR}\)) =  \(\dfrac{2{,}4}{5{,}3}\)
\par\addvspace{.5em}
 On a donc \(\widehat{ALR}\) =  ArcTan(\(\dfrac{2{,}4}{5{,}3}\)) \(\approx\) 24\textdegree{}\end{document}
