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feat(1G_math): ajoute dernier bialn

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Bertrand Benjamin
2025-09-26 12:31:15 +02:00
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1G_math/03_Radians/3B_sin_cos.pdf
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1G_math/03_Radians/3B_sin_cos.tex
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@@ -14,4 +14,150 @@
\setcounter{section}{3}
\section{Sinus et cosinus d'un angle}
\begin{definition}[Sinus et Cosinus]
\begin{minipage}{0.6\textwidth}
Soit $M$ un point sur le cercle trigonométrique.
Pour tout réel $x$ d'image $M$ sur le cercle trigonométrique:
\begin{itemize}
\item le \textbf{cosinus} de $x$, noté $\cos{x}$ est \textbf{l'abscisse} de $M$.
\item le \textbf{sinus} de $x$, noté $\sin{x}$ est \textbf{l'ordonnée} de $M$.
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
\begin{tikzpicture}[scale=2]
\cercleTrigo
\draw[->, thick] (120:1.2) arc (120:150:1.2);
\node[] at (-1,1) {$+$};
% Illustration du dépliement
\draw[red,very thick] (0:1) arc (0:45:1) node[midway, right] {$x$};
\draw[->, very thick] (0:0.3) arc (0:45:0.3);
\draw[] (0; 0) -- (45:1) node[above right] {\large$M$};
\draw[red, very thick] (0.7,0) node {\bullet} node[below] {$\cos{x}$} -- (45:1) ;
\draw[red, very thick] (0,0.7) node {\bullet} node[left] {$\sin{x}$} -- (45:1) ;
\end{tikzpicture}
\end{minipage}
\end{definition}
\begin{propriete}[Inégalités]
Pour tout $x \in \R$, on a
\begin{itemize}
\item $-1 \leq \cos{x} \leq 1$
\item $-1 \leq \sin{x} \leq 1$
\item $\cos^2{x} + \sin^2{x} = 1$
\end{itemize}
\end{propriete}
\paragraph{Démonstration}
\enclasse{laissez un dizaine de lignes}
\begin{propriete}[Valeurs particulières]
\begin{center}
\renewcommand{\arraystretch}{2}
\begin{tabular}[c]{|l|*{6}{p{1cm}|}}
\hline
$x$ & $0$ & $\dfrac{\pi}{6}$ & $\dfrac{\pi}{4}$ & $\dfrac{\pi}{3}$ & $\dfrac{\pi}{2}$ & $\pi$ \\
\hline
$\cos{x}$ & & & & & & \\
\hline
$\sin{x}$ & & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tikzpicture}[scale=3]
\cercleTrigo
% Points sur le cercle - tous les quadrants
\draw[fill] (0:1) circle (0.02) node[above right] {$0$};
\draw[fill] (30:1) circle (0.02) node[above right] {$\frac{\pi}{6}$};
\draw[fill] (45:1) circle (0.02) node[above right] {$\frac{\pi}{4}$};
\draw[fill] (60:1) circle (0.02) node[above right] {$\frac{\pi}{3}$};
\draw[fill] (90:1) circle (0.02) node[above right] {$\frac{\pi}{2}$};
\draw[fill] (120:1) circle (0.02) node[above left] {$\frac{2\pi}{3}$};
\draw[fill] (135:1) circle (0.02) node[above left] {$\frac{3\pi}{4}$};
\draw[fill] (150:1) circle (0.02) node[above left] {$\frac{5\pi}{6}$};
\draw[fill] (180:1) circle (0.02) node[above left] {$\pi$};
\draw[fill] (210:1) circle (0.02) node[below left] {$\frac{7\pi}{6}$};
\draw[fill] (225:1) circle (0.02) node[below left] {$\frac{5\pi}{4}$};
\draw[fill] (240:1) circle (0.02) node[below left] {$\frac{4\pi}{3}$};
\draw[fill] (270:1) circle (0.02) node[below right] {$\frac{3\pi}{2}$};
\draw[fill] (300:1) circle (0.02) node[below right] {$\frac{5\pi}{3}$};
\draw[fill] (315:1) circle (0.02) node[below right] {$\frac{7\pi}{4}$};
\draw[fill] (330:1) circle (0.02) node[below right] {$\frac{11\pi}{6}$};
% Traits verticaux et horizontaux pour tous les angles
\draw[red, dashed] (30:1) -- ({cos(30)},0);
\draw[blue, dashed] (30:1) -- (0,{sin(30)});
\draw[red, dashed] (45:1) -- ({cos(45)},0);
\draw[blue, dashed] (45:1) -- (0,{sin(45)});
\draw[red, dashed] (60:1) -- ({cos(60)},0);
\draw[blue, dashed] (60:1) -- (0,{sin(60)});
\draw[red, dashed] (120:1) -- ({cos(120)},0);
\draw[blue, dashed] (120:1) -- (0,{sin(120)});
\draw[red, dashed] (135:1) -- ({cos(135)},0);
\draw[blue, dashed] (135:1) -- (0,{sin(135)});
\draw[red, dashed] (150:1) -- ({cos(150)},0);
\draw[blue, dashed] (150:1) -- (0,{sin(150)});
\draw[red, dashed] (210:1) -- ({cos(210)},0);
\draw[blue, dashed] (210:1) -- (0,{sin(210)});
\draw[red, dashed] (225:1) -- ({cos(225)},0);
\draw[blue, dashed] (225:1) -- (0,{sin(225)});
\draw[red, dashed] (240:1) -- ({cos(240)},0);
\draw[blue, dashed] (240:1) -- (0,{sin(240)});
\draw[red, dashed] (300:1) -- ({cos(300)},0);
\draw[blue, dashed] (300:1) -- (0,{sin(300)});
\draw[red, dashed] (315:1) -- ({cos(315)},0);
\draw[blue, dashed] (315:1) -- (0,{sin(315)});
\draw[red, dashed] (330:1) -- ({cos(330)},0);
\draw[blue, dashed] (330:1) -- (0,{sin(330)});
% Points sur les axes pour marquer les projections
\draw[red, fill] ({cos(30)},0) circle (0.015);
\draw[red, fill] ({cos(45)},0) circle (0.015);
\draw[red, fill] ({cos(60)},0) circle (0.015);
\draw[red, fill] ({cos(120)},0) circle (0.015);
\draw[red, fill] ({cos(135)},0) circle (0.015);
\draw[red, fill] ({cos(150)},0) circle (0.015);
\draw[red, fill] ({cos(210)},0) circle (0.015);
\draw[red, fill] ({cos(225)},0) circle (0.015);
\draw[red, fill] ({cos(240)},0) circle (0.015);
\draw[red, fill] ({cos(300)},0) circle (0.015);
\draw[red, fill] ({cos(315)},0) circle (0.015);
\draw[red, fill] ({cos(330)},0) circle (0.015);
\draw[blue, fill] (0,{sin(30)}) circle (0.015);
\draw[blue, fill] (0,{sin(45)}) circle (0.015);
\draw[blue, fill] (0,{sin(60)}) circle (0.015);
\draw[blue, fill] (0,{sin(120)}) circle (0.015);
\draw[blue, fill] (0,{sin(135)}) circle (0.015);
\draw[blue, fill] (0,{sin(150)}) circle (0.015);
\draw[blue, fill] (0,{sin(210)}) circle (0.015);
\draw[blue, fill] (0,{sin(225)}) circle (0.015);
\draw[blue, fill] (0,{sin(240)}) circle (0.015);
\draw[blue, fill] (0,{sin(300)}) circle (0.015);
\draw[blue, fill] (0,{sin(315)}) circle (0.015);
\draw[blue, fill] (0,{sin(330)}) circle (0.015);
\end{tikzpicture}
\end{center}
\end{propriete}
\end{document}