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\begin{document}
\pagestyle{fancy}
\chapHead{La trigonométrie en \\Terminale S}

\section{Formules élémentaires}

\begin{itemize}
\item Pour tout réel $x$, $\cos^2x+\sin^2x=1$.
\item Pour tout réel $x\neq\dfrac{\pi}{2}+k\pi$ ($k\in\Z$), $\tan x=\dfrac{\sin x}{\cos x}$.
\item Pour tout réel $x$, $-1\leqslant\cos x\leqslant1$ et $-1\leqslant\sin x\leqslant1$.
\end{itemize}

\section{Formules de rotation}
\vspace*{-1em}
\begin{multicols}{3}
\begin{itemize}
\item $\cos(x\pm\pi)=-\cos x$
\item $\sin(x\pm\pi)=-\sin x$
\end{itemize}
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\columnbreak

\begin{itemize}
\item $\cos\left(x+\dfrac{\pi}{2}\right)=-\sin x$
\item $\sin\left(x+\dfrac{\pi}{2}\right)=\cos x$
\end{itemize}

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\columnbreak

\begin{itemize}
\item $\cos\left(\dfrac{\pi}{2}-x\right)=\sin x$
\item $\sin\left(\dfrac{\pi}{2}-x\right)=\cos x$
\end{itemize}

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\end{multicols}

\section{Formules de duplication}
\vspace*{-1em}
\setlength{\columnseprule}{0pt}
\begin{multicols}{2}
\begin{itemize}
\item $\begin{aligned}[t]
\cos(2x) & = \cos^2x-\sin^2x\\
& = 2\cos^2x-1\\
& = 1-2\sin^2x
\end{aligned}$

\item $\begin{aligned}[t]
\sin(2x) & = 2\sin x\cos x
\end{aligned}$
\end{itemize}
\end{multicols}

\section{Formules d'addition}
\vspace*{-1em}
\begin{multicols}{2}
\begin{itemize}
\item $\cos(a+b)=\cos a\cos b-\sin a\sin b$
\item $\cos(a-b)=\cos a\cos b+\sin a\sin b$
\item $\sin(a+b)=\sin a\cos b+\sin b\cos a$
\item $\sin(a-b)=\sin a\cos b-\sin b\cos a$
\end{itemize}
\end{multicols}

\newpage

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\section{Valeurs remarquables}

\begin{center}
\arrayrulecolor{purple}
\begin{tabular}{|>{\columncolor{purple!10}}Sc|*5{Sc|}}
\hline\rowcolor{purple!10}
$x$ & 0 & $\dfrac{\pi}{6}$ & $\dfrac{\pi}{4}$ & $\dfrac{\pi}{3}$ & $\dfrac{\pi}{2}$\\
\hline
$\cos x$ & 1 & $\dfrac{\sqrt3}{2}$ & $\dfrac{\sqrt2}{2}$ & $\dfrac{1}{2}$ & 0\\
\hline
$\sin x$ & 0 & $\dfrac{\sqrt3}{2}$ & $\dfrac{\sqrt2}{2}$ & $\dfrac{1}{2}$ & 1\\
\hline
\end{tabular}
\end{center}

\section{\'Equations trigonométriques}

\begin{itemize}
\item $\cos x=\cos a \iff x=a+2k\pi$ ou $x=-a+2k\pi$, $k\in\Z$.
\item $\sin x=\sin a \iff x=a+2k\pi$ ou $x=\pi-a+2k\pi$, $k\in\Z$.
\item $\cos x=\sin a \iff \cos x=\cos\left(\dfrac{\pi}{2}-a\right) \iff x=\dfrac{\pi}{2}-a+2k\pi$ ou $x=a-\dfrac{\pi}{2}+2k\pi$, $k\in\Z$.
\end{itemize}

\section{Fonctions paires, impaires, périodiques}
Soit $f$ une fonction définie sur un ensemble $\cal D$ centré en 0, de courbe représentative ${\cal C}$.
\begin{itemize}
\item $f$ est paire si, pour tout $x\in{\cal D}$ : $f(-x)=f(x)$.

\hfill{\color{purple}\textit{Conséquence : ${\cal C}$ est symétrique par rapport à l'axe des ordonnées.}}

\item $f$ est impaire si, pour tout $x\in{\cal D}$ : $f(-x)=-f(x)$.

\hfill{\color{purple}\textit{Conséquence : ${\cal C}$ est symétrique par rapport à l'origine du repère.}}

\item $f$ est $T$-périodique si, pour tout $x\in{\cal D}$ : $f(x+T)=f(x)$.

\hfill{\color{purple}\textit{Conséquence : le motif de ${\cal C}$ sur un intervalle d'amplitude $T$ va se répéter à l'infini.}}
\end{itemize}

\section{Dérivation}
\vspace*{-1em}
\begin{multicols}{2}
\begin{center}
\begin{tabular}{|Sc@{\tikz\draw[->,>=latex] (0,0) -- (1.5,0) node[midway,above,scale=0.75] {dérivée};}Sc|}
\hline
$\sin x$ & $\cos x$\\
$\sin(ax+b)$ & $a\cos(ax+b)$\\
$\sin\big[u(x)\big]$ & $u'(x)\cos\big[u(x)\big]$\\
\hline
\end{tabular}

\begin{tabular}{|Sc@{\tikz\draw[->,>=latex] (0,0) -- (1.5,0) node[midway,above,scale=0.75] {dérivée};}Sc|}
\hline
$\cos x$ & $-\sin x$\\
$\cos(ax+b)$& $-a\sin(ax+b)$\\
$\cos\big[u(x)\big]$ & $-u'(x)\sin\big[u(x)\big]$\\
\hline
\end{tabular}
\end{center}
\end{multicols}

\section{Limites}
\vspace*{-1em}
\begin{multicols}{2}
\begin{itemize}
\item $\limite{x}{0}{\dfrac{\sin x}{x}}=1$
\item $\limite{x}{0}{\dfrac{\cos x-1}{x}}=0$
\end{itemize}
\end{multicols}

\section{Courbes}

\begin{center}
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\end{center}
\end{document}