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\begin{document}

\begin{flushright}
\itshape\bfseries\Huge\color{LightSeaGreen}Trigonométrie
\end{flushright}

\titre{Angles associés}

\begin{minipage}{.33\linewidth}
\begin{displaymath}
\left\lbrace
\begin{array}{l}
\cos\left(\frac{\pi}{2}+x\right)=-\sin x\\
\sin\left(\frac{\pi}{2}+x\right)=\cos x
\end{array}
\right.
\end{displaymath}

\begin{displaymath}
\left\lbrace
\begin{array}{l}
\cos\left(\pi-x\right)=-\cos x\\
\sin\left(\pi-x\right)=\sin x
\end{array}
\right.
\end{displaymath}

\begin{displaymath}
\left\lbrace
\begin{array}{l}
\cos\left(\pi+x\right)=-\cos x\\
\sin\left(\pi+x\right)=-\sin x
\end{array}
\right.
\end{displaymath}

\end{minipage}
\begin{minipage}{.34\linewidth}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\draw (-2,0) -- (2,0) node[right] {$\cos$};
\draw (0,-2) -- (0,2) node[above] {$\sin$};
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\end{tikzpicture}
\end{minipage}
\begin{minipage}{.33\linewidth}
\begin{displaymath}
\left\lbrace
\begin{array}{l}
\cos\left(\frac{\pi}{2}-x\right)=\sin x\\
\sin\left(\frac{\pi}{2}-x\right)=\cos x
\end{array}
\right.
\end{displaymath}

\begin{displaymath}
\left\lbrace
\begin{array}{l}
\cos\left(-x\right)=\cos x\\
\sin\left(-x\right)=-\sin x
\end{array}
\right.
\end{displaymath}
\end{minipage}

\bigskip

\titre{Egalités remarquables}

\[ \cos^2x+\sin^2x = 1 \qquad ; \qquad 1+\tan^2x=\frac{1}{\cos^2x}\]

\titre{Formules d'addition}

\begin{minipage}{.5\textwidth}
\[ \cos(a-b)=\cos a\cos b + \sin a \sin b \]
\[ \cos(a+b)=\cos a\cos b - \sin a \sin b \]
\[ \tan(a-b) = \frac{\tan a-\tan b}{1+\tan(a)\tan(b)}\]
\end{minipage}
\begin{minipage}{.5\textwidth}
\[ \sin(a-b)=\sin a\cos b - \cos a \sin b \]
\[ \sin(a+b)=\sin a\cos b + \cos a \sin b \]
\[ \tan(a+b) = \frac{\tan a+\tan b}{1-\tan(a)\tan(b)}\]
\end{minipage}

\bigskip

\titre{Formules de duplication}

\begin{minipage}[t][][t]{.33\textwidth}
\[ 
\begin{array}[t]{ll}
\cos (2a) & =\cos^2a-\sin^2a\\
 & =2\cos^2a-1\\
 & =1-2\sin^2a
\end{array}
\]
\end{minipage}
\begin{minipage}[t][][t]{.33\textwidth}
\[ \sin (2a)=2\sin(a)\cos(a)\]
\end{minipage}
\begin{minipage}[t][][t]{.33\textwidth}
\[ \tan (2a)=\frac{2\tan a}{1-\tan^2a}\]
\end{minipage}

\[ \cos(a)\cos(b)=\frac{1}{2}\left[\cos(a-b)+\cos(a+b) \right]\]
\[ \cos(a)\sin(b)=\frac{1}{2}\left[\sin(a+b)-\sin(a-b) \right]\]
\[ \sin(a)\sin(b)=\frac{1}{2}\left[\cos(a-b)-\cos(a+b) \right]\]

\titre{Formules diverses}

\begin{minipage}{.5\textwidth}
\[ \cos p + \cos q = 2\cos\left(\frac{p+q}{2}\right)\cos\left(\frac{p-q}{2}\right)\]
\[ \sin p + \sin q = 2\sin\left(\frac{p+q}{2}\right)\cos\left(\frac{p-q}{2}\right)\]
\end{minipage}
\begin{minipage}{.5\textwidth}
\[ \cos p - \cos q = -2\sin\left(\frac{p+q}{2}\right)\sin\left(\frac{p-q}{2}\right)\]
\[ \sin p - \sin q = 2\cos\left(\frac{p+q}{2}\right)\sin\left(\frac{p-q}{2}\right)\]
\end{minipage}

\bigskip

\titre{Equations trigonométriques}

\[ \cos a = \cos b \Longleftrightarrow \left(\begin{array}{l} a = b + 2k\pi,\ k \in \mathbb{Z} \\ ou \\ a = -b + 2k\pi,\ k \in \mathbb{Z}\end{array} \right.\]

\[ \sin a = \sin b \Longleftrightarrow \left(\begin{array}{l} a = b + 2k\pi,\ k \in \mathbb{Z} \\ ou \\ a = \pi-b + 2k\pi,\ k \in \mathbb{Z}\end{array} \right.\]

\[ \tan a = \tan b \Longleftrightarrow a=b+ k\pi,\ k \in \mathbb{Z}\]

\titre{Représentations paramétriques}

En posant $t=\tan\left(\frac{a}{2}\right)$, on a :

\[ \cos a = \frac{1-t^2}{1+t^2} \qquad ; \qquad \sin a = \frac{2t}{1+t^2} \qquad ; \qquad \tan a = \frac{2t}{1-t^2}\]
\end{document}