\langchapter{Trigonométrie}{Trigonometry}
%TODO Complete chapter

\langsection{Cercle unitaire}{Unit circle}
%TODO Complete section

Le cercle unitaire est un cercle de centre $(0,0)$ et de rayon 1.

\subsection{cos}
%TODO Complete subsection

$\cos 0 = 1$

$\cos \frac{\pi}{2} = 0$

$\cos \pi = -1$

$\cos(-\frac{\pi}{2}) = 0$

$\cos(\pi + t) = -\cos(t)$

$\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$

$\cos\frac{\pi}{3} = \frac{1}{2}$

$\forall (a,b) \in \R^2$

$\cos(a + b) = \cos a \cos b + \sin a \sin b$

$\cos(a - b) = \cos a \cos b - \sin a \sin b$

$\cos a + \cos b = 2 \cos(\frac{a + b}{2}) \cos(\frac{a - b}{2} )$

\subsection{sin}
%TODO Complete subsection

$\sin 0 = 0$

$\sin(\pi - t) = \sin(t)$

$\sin(\frac{\pi}{2} - t) = \cos(t)$

$\sin \frac{\pi}{6} = \frac{1}{2}$

$\sin \frac{\pi}{2} = 1$

%$\sin(\frac{\pi}{2} + t) = -\cos(t)$

$\forall (a,b) \in \R^2$

$\sin(a + b) = \sin a \cos b + \sin b \cos a$

$\sin(a - b) = \sin a \cos b - \sin b \cos a$

$\sin a - \sin b = 2 \cos (\frac{a+b}{2}) \sin (\frac{a-b}{2})$

$\sin a\sin b = \frac{\cos(a - b) - \cos(a + b)}{2}$

\subsection{tan}
%TODO Complete subsection

$\tan 0 = 0$

$\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$

$\tan \frac{\pi}{4} = 1$

$\tan(\frac{\pi}{2} - x) = \frac{1}{\tan x}$

$\tan(\frac{\pi}{2} + x) = -\frac{1}{\tan x}$

$\tan(a + b) = \frac{\tan(a) + \tan(b)}{1- \tan(a)\tan(b)}$

$\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}$

\subsection{Combinaisons}
%TODO Complete subsection

$\forall (a,b) \in \R^2$

$\sin a \cos b = \frac{\sin(a + b) + \sin(a - b)}{2}$