diff --git a/1G_math/03_Radians/3B_sin_cos.pdf b/1G_math/03_Radians/3B_sin_cos.pdf index 9b428dc..e95bab7 100644 Binary files a/1G_math/03_Radians/3B_sin_cos.pdf and b/1G_math/03_Radians/3B_sin_cos.pdf differ diff --git a/1G_math/03_Radians/3B_sin_cos.tex b/1G_math/03_Radians/3B_sin_cos.tex index 831e136..f0cc27b 100644 --- a/1G_math/03_Radians/3B_sin_cos.tex +++ b/1G_math/03_Radians/3B_sin_cos.tex @@ -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}