diff --git a/contents/trigonometry.tex b/contents/trigonometry.tex index 237b125..f7989f9 100644 --- a/contents/trigonometry.tex +++ b/contents/trigonometry.tex @@ -23,7 +23,7 @@ $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$ $\cos\frac{\pi}{3} = \frac{1}{2}$ -$\forall (a,b) \in \R^2$ +$\forall (a,b) \in \R$ $\cos(a + b) = \cos a \cos b + \sin a \sin b$ @@ -46,7 +46,7 @@ $\sin \frac{\pi}{2} = 1$ %$\sin(\frac{\pi}{2} + t) = -\cos(t)$ -$\forall (a,b) \in \R^2$ +$\forall (a,b) \in \R$ $\sin(a + b) = \sin a \cos b + \sin b \cos a$ @@ -76,8 +76,20 @@ $\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}$ \subsection{Combinaisons} %TODO Complete subsection -$\forall (a,b) \in \R^2$ +$\forall (a,b) \in \R$ $\sin a \cos b = \frac{\sin(a + b) + \sin(a - b)}{2}$ +\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} + +$tanh\ x = \frac{sinh\ x}{cosh\ x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^{2x} - 1}{e^{2x} + 1}$