\documentclass[a4paper,11pt]{article}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[french]{babel}
\usepackage[a4paper,top=3.2cm,bottom=2cm,left=2cm,right=2cm]{geometry}
\usepackage{graphicx}
\usepackage[export]{adjustbox}
\usepackage{fancyhdr}
\usepackage{qrcode}
\usepackage[table]{xcolor}
\usepackage{amsmath,amssymb}
\usepackage{enumitem}
% Puces nettes (babel-french remplace sinon la puce par un tiret « – » peu pro).
% Vaut pour TOUTES les fiches qui importent ce module. AtBeginDocument pour
% passer après la personnalisation des listes faite par babel-french.
\AtBeginDocument{%
  \setlist[itemize]{leftmargin=1.8em,itemsep=0.5ex,topsep=0.4ex,parsep=0pt}%
  \setlist[itemize,1]{label=\textcolor{red!75!black}{\small\textbullet}}%
  \setlist[itemize,2]{label=\textcolor{red!75!black}{\scriptsize\textbullet}}%
}
% Titres compacts (gain de place vertical, fiches plus denses) — vaut pour tous.
\usepackage{titlesec}
\titlespacing*{\section}{0pt}{1.4ex}{0.6ex}
\titlespacing*{\subsection}{0pt}{1.0ex}{0.3ex}
% multicol : listes denses (calcul mental) sur 2 colonnes — neutre tant qu'il
% n'est pas utilisé, dispo pour tous les générateurs.
\usepackage{multicol}
\setlength{\columnsep}{1.5em}
% needspace : empêche un titre d'exercice de rester orphelin en bas de page
% (réserve quelques lignes ; sinon le titre passe à la page suivante avec sa
% liste). Neutre tant qu'il n'est pas appelé — dispo pour tous les générateurs.
\usepackage{needspace}
\usepackage{tikz}
\usetikzlibrary{decorations.text}

\usepackage{tgheros}
\pagestyle{fancy}
\fancyhf{}
\setlength{\headheight}{75pt}
\setlength{\headsep}{10pt}

\newcommand{\logowouf}{%
\begin{tikzpicture}[baseline={(current bounding box.center)},scale=0.55,
                    line cap=round,line join=round]
    \path[use as bounding box] (-2.20,-1.45) rectangle (2.20,2.20);
    \path[
        postaction={decorate},
        decoration={
            text along path,
            text={|\fontfamily{qhv}\bfseries\fontsize{9}{9}\selectfont|site2wouf.fr},
            text align={center},
            raise=0.03ex
        }
    ] (-1.29,0.48) arc[start angle=160,end angle=20,radius=1.29];
    \begin{scope}[xshift=-0.06cm]
        \fill[black] (-0.66,0.14) ellipse (0.21 and 0.31);
        \fill[black] ( 0.66,0.14) ellipse (0.21 and 0.31);
        \fill[black] (-0.23,0.49) ellipse (0.22 and 0.35);
        \fill[black] ( 0.23,0.49) ellipse (0.22 and 0.35);
        \fill[black]
            (0,-0.10)
            .. controls (-0.36,-0.02) and (-0.63,-0.24) .. (-0.58,-0.58)
            .. controls (-0.52,-0.93) and (-0.16,-1.12) .. (0,-0.90)
            .. controls (0.16,-1.12) and (0.52,-0.93) .. (0.58,-0.58)
            .. controls (0.63,-0.24) and (0.36,-0.02) .. (0,-0.10)
            -- cycle;
    \end{scope}
\end{tikzpicture}%
}

\newcommand{\titre}{Trigonométrie}
\newcommand{\soustitre}{Cycle 4 --- Fiche n\textdegree 239}

\fancyhead[C]{%
\raisebox{4pt}{%
\begin{minipage}[c]{0.15\textwidth}
    \centering
    \logowouf
\end{minipage}
\begin{minipage}[c]{0.62\textwidth}
    \centering
    {\Large\bfseries \titre\par}
    {\small\color{gray} \soustitre}
\end{minipage}
\begin{minipage}[c]{0.21\textwidth}
    \centering
    \qrcode[height=1.5cm]{https://site2wouf.fr/exercices_trigonometrie.php?x=239}\\[2pt]
    {\small\ttfamily\bfseries\color{red} TRIG0239}
\end{minipage}%
}
}

\begin{document}

\sloppy

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

 Dans le triangle NZM rectangle en N, on sait que :
\begin{itemize}
\item ZM=5,1 cm
\item \(\widehat{ZMN}\) =  13\textdegree{}
\end{itemize}

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

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

 Dans le triangle RDC rectangle en R, on sait que :
\begin{itemize}
\item RC=5 cm
\item DC=7,2 cm
\end{itemize}

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

 Dans le triangle MWG rectangle en M, on sait que :
\begin{itemize}
\item MW=9,4 cm
\item \(\widehat{WGM}\) =  39\textdegree{}
\end{itemize}

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

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

 Dans le triangle PJS rectangle en P, on sait que :
\begin{itemize}
\item PS=5,7 cm
\item \(\widehat{PJS}\) = 49\textdegree{}
\end{itemize}

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

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

 Dans le triangle RMV rectangle en R, on sait que :
\begin{itemize}
\item RM=2,7 cm
\item MV=7,3 cm
\end{itemize}

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

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

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

\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[black] (1.050,2.566) -- (1.047,2.421) -- (1.038,2.281) -- (1.043,2.140) -- (1.050,1.997) -- (1.061,1.856) -- (1.048,1.718) -- (1.058,1.578) -- (1.050,1.434);
  \draw[black] (1.050,1.434) -- (1.397,1.439) -- (1.751,1.432) -- (2.102,1.442) -- (2.453,1.426) -- (2.800,1.428) -- (3.151,1.440) -- (3.498,1.423) -- (3.852,1.425) -- (4.199,1.435) -- (4.550,1.440) -- (4.900,1.427) -- (5.247,1.446) -- (5.598,1.435) -- (5.950,1.434);
  \draw[black] (5.950,1.434) -- (5.621,1.510) -- (5.296,1.578) -- (4.965,1.650) -- (4.640,1.729) -- (4.322,1.820) -- (3.990,1.884) -- (3.662,1.966) -- (3.333,2.034) -- (3.016,2.125) -- (2.686,2.201) -- (2.360,2.269) -- (2.034,2.347) -- (1.705,2.409) -- (1.378,2.491) -- (1.050,2.566);
  \draw[black] (1.050,1.547) -- (1.044,1.511) -- (1.050,1.471) -- (1.050,1.434);
  \draw[black] (1.050,1.434) -- (1.087,1.432) -- (1.127,1.435) -- (1.163,1.434);
  \draw[black] (1.050,1.547) -- (1.088,1.546) -- (1.126,1.549) -- (1.163,1.547);
  \draw[black] (1.163,1.547) -- (1.168,1.511) -- (1.160,1.472) -- (1.163,1.434);
  \draw[black] (5.560,1.524) -- (5.557,1.517) -- (5.555,1.510) -- (5.559,1.502) -- (5.547,1.496) -- (5.555,1.487) -- (5.547,1.480) -- (5.553,1.472) -- (5.554,1.464) -- (5.550,1.457) -- (5.542,1.450) -- (5.551,1.442) -- (5.555,1.434);
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (0.783,1.301) {N};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (0.783,2.632) {Z};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (6.050,1.301) {M};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (0.550,1.933) {?};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.667,2.000) {5,1 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (5.288,1.443) {13°};
\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
sin(\(\widehat{NMZ}\)) = \(\dfrac{NZ}{ZM}\)
\par\addvspace{.5em}
 D'où sin(13\textdegree{}) = \(\dfrac{NZ}{5{,}1}\)

On a donc NZ = 5,1 \(\times\) sin(13\textdegree{}) \(\approx\) 1,1 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 2}

\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[black] (2.149,3.400) -- (2.156,3.052) -- (2.160,2.697) -- (2.142,2.352) -- (2.161,2.000) -- (2.150,1.648) -- (2.145,1.300) -- (2.139,0.953) -- (2.149,0.600);
  \draw[black] (2.149,0.600) -- (2.488,0.602) -- (2.822,0.589) -- (3.161,0.606) -- (3.498,0.589) -- (3.837,0.606) -- (4.179,0.613) -- (4.516,0.602) -- (4.851,0.600);
  \draw[black] (4.851,0.600) -- (4.594,0.848) -- (4.352,1.102) -- (4.114,1.362) -- (3.859,1.613) -- (3.619,1.869) -- (3.384,2.137) -- (3.138,2.388) -- (2.893,2.646) -- (2.641,2.893) -- (2.387,3.138) -- (2.149,3.400);
  \draw[black] (2.149,0.870) -- (2.145,0.781) -- (2.144,0.691) -- (2.149,0.600);
  \draw[black] (2.149,0.600) -- (2.238,0.596) -- (2.331,0.594) -- (2.419,0.600);
  \draw[black] (2.149,0.870) -- (2.240,0.875) -- (2.331,0.876) -- (2.419,0.870);
  \draw[black] (2.419,0.870) -- (2.415,0.782) -- (2.418,0.689) -- (2.419,0.600);
  \draw[black] (2.430,3.109) -- (2.405,3.098) -- (2.383,3.085) -- (2.366,3.066) -- (2.343,3.054) -- (2.323,3.038) -- (2.302,3.021) -- (2.274,3.023) -- (2.252,3.004) -- (2.226,3.002) -- (2.200,2.999) -- (2.174,3.010) -- (2.149,2.998);
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (1.882,0.467) {R};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (1.882,3.467) {D};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (4.951,0.467) {C};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.500,0.267) {5 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.667,2.000) {7,2 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (2.398,2.715) {?};
\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
sin(\(\widehat{RDC}\)) =  \(\dfrac{RC}{DC}\)
\par\addvspace{.5em}
 d'où sin(\(\widehat{RDC}\)) = \(\dfrac{5}{7{,}2}\)
\par\addvspace{.5em}
On a donc \(\widehat{RDC}\) =  ArcSin(\(\dfrac{5}{7{,}2}\)) \(\approx\) 44\textdegree{}
\needspace{4\baselineskip}
\subsection*{
Exercice 3}

\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[black] (1.771,3.400) -- (1.774,3.049) -- (1.782,2.703) -- (1.780,2.350) -- (1.777,1.999) -- (1.770,1.652) -- (1.760,1.302) -- (1.769,0.950) -- (1.771,0.600);
  \draw[black] (1.771,0.600) -- (2.118,0.601) -- (2.461,0.612) -- (2.808,0.592) -- (3.155,0.606) -- (3.502,0.588) -- (3.849,0.605) -- (4.192,0.587) -- (4.539,0.597) -- (4.885,0.598) -- (5.229,0.600);
  \draw[black] (5.229,0.600) -- (4.963,0.819) -- (4.701,1.035) -- (4.429,1.247) -- (4.170,1.465) -- (3.903,1.683) -- (3.634,1.892) -- (3.363,2.101) -- (3.109,2.331) -- (2.833,2.534) -- (2.577,2.759) -- (2.298,2.962) -- (2.033,3.174) -- (1.771,3.400);
  \draw[black] (1.771,0.880) -- (1.766,0.787) -- (1.769,0.692) -- (1.771,0.600);
  \draw[black] (1.771,0.600) -- (1.866,0.605) -- (1.959,0.599) -- (2.051,0.600);
  \draw[black] (1.771,0.880) -- (1.865,0.876) -- (1.959,0.878) -- (2.051,0.880);
  \draw[black] (2.051,0.880) -- (2.055,0.785) -- (2.048,0.694) -- (2.051,0.600);
  \draw[black] (4.912,0.856) -- (4.912,0.828) -- (4.884,0.820) -- (4.877,0.797) -- (4.877,0.771) -- (4.869,0.751) -- (4.845,0.736) -- (4.853,0.710) -- (4.834,0.691) -- (4.829,0.669) -- (4.826,0.646) -- (4.835,0.622) -- (4.830,0.600);
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (1.504,0.467) {M};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (1.504,3.467) {W};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (5.329,0.467) {G};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (1.271,1.933) {9,4 cm};
  \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 (4.600,0.756) {39°};
\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
tan(\(\widehat{MGW}\)) = \(\dfrac{MW}{MG}\)
\par\addvspace{.5em}
d'où  tan(39\textdegree{}) =  \(\dfrac{9{,}4}{MG}\)
\par\addvspace{.5em}
On a donc MG = \(\dfrac{9{,}4}{\tan(39^{\circ})}\) \(\approx\) 11,6 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 4}

\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[black] (1.889,3.400) -- (1.902,3.051) -- (1.876,2.703) -- (1.882,2.350) -- (1.900,2.001) -- (1.879,1.652) -- (1.896,1.299) -- (1.880,0.952) -- (1.889,0.600);
  \draw[black] (1.889,0.600) -- (2.251,0.613) -- (2.602,0.603) -- (2.960,0.588) -- (3.323,0.593) -- (3.679,0.602) -- (4.036,0.608) -- (4.396,0.606) -- (4.755,0.594) -- (5.111,0.600);
  \draw[black] (5.111,0.600) -- (4.834,0.825) -- (4.578,1.076) -- (4.299,1.291) -- (4.037,1.533) -- (3.779,1.774) -- (3.504,2.003) -- (3.228,2.229) -- (2.955,2.459) -- (2.703,2.708) -- (2.429,2.934) -- (2.154,3.162) -- (1.889,3.400);
  \draw[black] (1.889,0.880) -- (1.893,0.788) -- (1.890,0.694) -- (1.889,0.600);
  \draw[black] (1.889,0.600) -- (1.983,0.594) -- (2.078,0.594) -- (2.169,0.600);
  \draw[black] (1.889,0.880) -- (1.982,0.876) -- (2.076,0.876) -- (2.169,0.880);
  \draw[black] (2.169,0.880) -- (2.175,0.785) -- (2.170,0.695) -- (2.169,0.600);
  \draw[black] (2.191,3.138) -- (2.179,3.110) -- (2.157,3.091) -- (2.133,3.074) -- (2.111,3.055) -- (2.077,3.057) -- (2.053,3.041) -- (2.027,3.031) -- (2.002,3.016) -- (1.973,3.017) -- (1.946,3.006) -- (1.919,2.993) -- (1.889,3.004);
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (1.623,0.467) {P};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (1.623,3.467) {J};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (5.211,0.467) {S};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.500,0.267) {5,7 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.667,2.000) {?};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (2.166,2.727) {49°};
\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
sin(\(\widehat{PJS}\)) =  \(\dfrac{PS}{JS}\)
\par\addvspace{.5em}
 d'où sin(49\textdegree{}) = \(\dfrac{5{,}7}{JS}\)
\par\addvspace{.5em}
 On a donc JS = \(\dfrac{5{,}7}{\sin(49^{\circ})}\) \(\approx\) 7,6 cm
\needspace{4\baselineskip}
\subsection*{
Exercice 5}

\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[black] (1.050,2.975) -- (1.040,2.732) -- (1.051,2.486) -- (1.049,2.246) -- (1.055,1.999) -- (1.046,1.754) -- (1.058,1.509) -- (1.060,1.265) -- (1.050,1.025);
  \draw[black] (1.050,1.025) -- (1.398,1.028) -- (1.749,1.019) -- (2.102,1.026) -- (2.447,1.038) -- (2.798,1.023) -- (3.147,1.021) -- (3.501,1.012) -- (3.849,1.016) -- (4.202,1.022) -- (4.549,1.022) -- (4.898,1.026) -- (5.247,1.035) -- (5.597,1.028) -- (5.950,1.025);
  \draw[black] (5.950,1.025) -- (5.626,1.168) -- (5.295,1.286) -- (4.973,1.421) -- (4.646,1.547) -- (4.324,1.685) -- (3.986,1.799) -- (3.662,1.931) -- (3.334,2.067) -- (3.009,2.197) -- (2.677,2.318) -- (2.350,2.447) -- (2.029,2.585) -- (1.704,2.719) -- (1.383,2.856) -- (1.050,2.975);
  \draw[black] (1.050,1.220) -- (1.048,1.155) -- (1.046,1.091) -- (1.050,1.025);
  \draw[black] (1.050,1.025) -- (1.116,1.029) -- (1.181,1.021) -- (1.245,1.025);
  \draw[black] (1.050,1.220) -- (1.117,1.217) -- (1.181,1.222) -- (1.245,1.220);
  \draw[black] (1.245,1.220) -- (1.245,1.155) -- (1.241,1.091) -- (1.245,1.025);
  \draw[black] (5.572,1.175) -- (5.578,1.159) -- (5.577,1.146) -- (5.562,1.138) -- (5.566,1.124) -- (5.566,1.111) -- (5.558,1.100) -- (5.556,1.087) -- (5.552,1.075) -- (5.545,1.063) -- (5.553,1.050) -- (5.552,1.037) -- (5.544,1.025);
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (0.783,0.891) {R};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (0.783,3.042) {M};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (6.050,0.891) {V};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (0.550,1.933) {2,7 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (3.667,2.000) {7,3 cm};
  \node[black, anchor=base, font=\fontsize{22.8}{27.3}\selectfont] at (5.295,1.084) {?};
\end{tikzpicture}\end{center}


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

\par\addvspace{.5em}
sin(\(\widehat{RVM}\)) = \(\dfrac{RM}{MV}\)
\par\addvspace{.5em}
d'où sin(\(\widehat{RVM}\)) =\(\dfrac{2{,}7}{7{,}3}\)
\par\addvspace{.5em}
 On a donc \(\widehat{RVM}\) = ArcSin(\(\dfrac{2{,}7}{7{,}3}\)) \(\approx\) 22\textdegree{}\end{document}
