\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. $\forall x \in \R, \cos^2 x + \sin^2 x = 1$ \subsection{cos} %TODO Complete subsection Formule d'Euler $\forall \theta \in \R, cos(\theta) = \frac{e^{i\theta} + e^{-i \theta}}{2}$ $\cos 0 = 1$ $\cos \frac{\pi}{2} = 0$ $\cos \pi = -1$ $\cos(-\frac{\pi}{2}) = 0$ $\cos(\frac{\pi}{2} + t) = -\sin(t)$ $\cos(\pi + t) = -\cos(t)$ $\cos(\pi - t) = -\cos(t)$ $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$ $\cos\frac{\pi}{3} = \frac{1}{2}$ $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ $\forall (a,b) \in \R$ $\cos(a + b) = \cos a \cos b + \sin a \sin b$ $\cos(a - b) = \cos a \cos b - \sin a \sin b$ $\cos a + \cos b = 2 \cos(\frac{a + b}{2}) \cos(\frac{a - b}{2} )$ $\cos a - \cos b = -2 \sin(\frac{a + b}{2}) \sin(\frac{a - b}{2} )$ $\cos a \cos b = \frac{\cos(a + b) + \cos(a - b)}{2}$ $\forall t \in \R, \cos(\frac{\pi}{2} - t) = \sin(t)$ $\lim\limits_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$ $\frac{d}{dx} \cos x = -\sin x$ $\forall x \in [-1, 1], \cos(\arcsin(x)) = \sqrt{1-x^{2}}$ $\forall x \in [-1, 1], \cos(\arccos(x)) = x$ $\forall x\in\R, \cos^2 x = \frac{1 + \cos(2x)}{2}$ \subsection{sin} %TODO Complete subsection Formule d'Euler $\forall \theta \in \R, sin(\theta) = \frac{e^{i\theta} - e^{-i \theta}}{2i}$ $\sin 0 = 0$ $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$ $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ $\sin \frac{\pi}{6} = \frac{1}{2}$ $\sin \frac{\pi}{2} = 1$ $\sin(\frac{\pi}{2} + t) = -\cos(t)$ $\sin(\pi - t) = \sin(t)$ $\sin(\pi + t) = -\sin(t)$ $\sin(\frac{\pi}{2} - t) = \cos(t)$ $\forall (a,b) \in \R$ $\sin(a + b) = \sin a \cos b + \sin b \cos a$ $\sin(a - b) = \sin a \cos b - \sin b \cos a$ $\sin a - \sin b = 2 \cos (\frac{a+b}{2}) \sin (\frac{a-b}{2})$ $\sin a + \sin b = 2 \sin (\frac{a+b}{2}) \cos (\frac{a-b}{2})$ $\sin a\sin b = \frac{\cos(a - b) - \cos(a + b)}{2}$ $\sin a \cos b = \frac{\sin(a - b) - \sin(a + b)}{2}$ $\lim\limits_{t \to 0} \frac{\sin t}{t} = 1$ $\frac{d}{dx} \sin x = \cos x$ $\forall x \in [-1, 1], \sin(\arcsin(x)) = x$ $\forall x \in [-1, 1], \sin(\arccos(x)) = \sqrt{1-x^{2}}$ $\forall x\in\R, \sin^2 x = \frac{1 - \cos(2x)}{2}$ \subsection{tan} %TODO Complete subsection $\tan 0 = 0$ $\tan \frac{\pi}{3} = \sqrt{3}$ $\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$ $\tan \frac{\pi}{4} = 1$ $\tan(\frac{\pi}{2} - x) = \frac{1}{\tan x}$ $\tan(\frac{\pi}{2} + x) = -\frac{1}{\tan x}$ $\tan(a + b) = \frac{\tan(a) + \tan(b)}{1- \tan(a)\tan(b)}$ $\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}$ \subsection{Combinaisons} %TODO Complete subsection $\forall (a,b) \in \R$ $\sin a \cos b = \frac{\sin(a + b) + \sin(a - b)}{2}$ \langsection{Fonctions hyperboliques}{Hyperbolic functions} \subsection{cosh} $cosh\ x = \frac{e^x + e^{-x}}{2} = \frac{e^{2x} + 1}{2e^x} = \frac{1 + e^{-2x}}{2e^{-x}}$ \subsection{sinh} $sinh\ x = \frac{e^x - e^{-x}}{2} = \frac{e^{2x} - 1}{2e^x} = \frac{1 - e^{-2x}}{2e^{-x}}$ \subsection{tanh} $tanh\ x = \frac{sinh\ x}{cosh\ x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^{2x} - 1}{e^{2x} + 1}$