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\begin{document}

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\title{Trigonométrie}

\section*{Cercle trigonométrique}

\begin{center}
\begin{tikzpicture}[scale=4]
\node[below left] at (0,0) {O};
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% 30°
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\end{center}

\subsection*{Placer un angle sur le cercle trigonométrique (méthode)}

\begin{itemize}
\item On souhaite placer l'angle $\dfrac{37\pi}{3}$. Pour cela, on doit écrire l'angle sous la forme $k\pi+\alpha$. Le multiple de 3 (dénominateur) le plus proche de 37 est 36 donc:
\[
\frac{37\pi}{3} = \dfrac{36\pi}{3}+\dfrac{1\pi}{3}=12\pi+\dfrac{\pi}{3} \to \frac{\pi}{3}\text{ car }12\pi\text{ est au même endroit que 0 sur le cercle.}
\]

\item On souhaite placer l'angle $\dfrac{38\pi}{3}$. Pour cela, on doit écrire l'angle sous la forme $k\pi+\alpha$. Le multiple de 3 (dénominateur) le plus proche de 38 est 39 donc:
\[
\frac{38\pi}{3} = \dfrac{39\pi}{3}-\dfrac{1\pi}{3}=13\pi+\dfrac{\pi}{3} \to \pi-\frac{\pi}{3}=\frac{2\pi}{3}\text{ car }13\pi\text{ est au même endroit que }\pi\text{ sur le cercle.}
\]
\end{itemize}

\section*{Conversions}

\begin{minipage}{0.5\linewidth}
\begin{center}
\begin{minipage}{0.5\linewidth}
\begin{center}
\subsection*{De Degrés à Radians}
\end{center}
\end{minipage}
\end{center}
\[
 x^\circ = x\times\dfrac{\pi}{180}\text{ rad}
\]
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\begin{minipage}{0.5\linewidth}
\begin{center}
\subsection*{De Radians à Degrés}
\end{center}
\end{minipage}
\end{center}
\[
 x \text{ rad} = x\times\dfrac{180}{\pi}^\circ
\]
\end{minipage}

\bigskip

\section*{Égalité fondamentale}
\[
\forall x\in\R,\qquad \cos^2(x) + \sin^2(x) = 1.
\]

\section*{Fonctions sinus et cosinus}

\subsection*{Parité}

\begin{itemize}
\item La fonction $x \mapsto \sin(x)$ est impaire : $\sin(-x) = -\sin(x)$.
\item La fonction $x \mapsto \cos(x)$ est paire : $\cos(-x) = \cos(x)$.
\end{itemize}

\subsection*{Périodicité}

Les fonctions $\sin$ et $\cos$ sont $2\pi$-périodiques:
\[
\sin(x+2\pi)=\sin(x)\qquad\text{et}\qquad\cos(x+2\pi)=\cos(x).
\]

\subsection*{Courbes représentatives}

\begin{center}
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\node[below right] at ({2*3.14159},0) {$2\pi$};
%
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\end{document}
