# Fiche de mathématiques

### I. Relations de base

$\cos^2 \theta + \sin^2 \theta = 1$

 $\cos (-\theta) = \cos \theta$ $\cos (\pi - \theta) = - \cos \theta$ $\cos (\pi + \theta) = - \cos \theta$ $\cos \left(\dfrac{\pi}{2} - \theta\right) = \sin \theta$ $\cos \left(\dfrac{\pi}{2} + \theta\right) = - \sin \theta$ $\sin (-\theta) = - \sin \theta$ $\sin (\pi - \theta) = \sin \theta$ $\sin (\pi + \theta) = - \sin \theta$ $\sin \left(\dfrac{\pi}{2} - \theta\right) = \cos \theta$ $\sin \left(\dfrac{\pi}{2} + \theta\right) = \cos \theta$ $\tan (-\theta) = - \tan \theta$ $\mathrm{cotan} (-\theta) = - \mathrm{cotan} \theta$

### II. Formules en t

$t = \tan \dfrac{\theta}{2}$
$1 + t^2 = \dfrac{1}{\cos^2\frac{\theta}{2}}$

$\cos \theta = \dfrac{1-t^2}{1+t^2}$
$\sin \theta = \dfrac{2t}{1+t^2}$
$\tan \theta = \dfrac{2t}{1-t^2}$

 $\cos (a - b) = \cos a \cos b + \sin a \sin b$ $\sin (a - b) = \sin a \cos b - \cos a \sin b$ $\cos (a + b) = \cos a \cos b - \sin a \sin b$ $\sin (a + b) = \sin a \cos b + \cos a \sin b$ $\tan (a + b) = \dfrac{\tan a + \tan b}{1 - \tan a \tan b}$ $\tan (a - b) = \dfrac{\tan a - \tan b}{1 + \tan a \tan b}$ $\cos 2a = \cos^2a - \sin^2a$           $= 2 \cos^2a - 1$           $= 1 - 2 \sin^2a$ $\cos^2 a = \dfrac{1 + \cos 2a}{2}$ $\sin^2 a = \dfrac{1 - \cos 2a}{2}$ $\sin 2a = 2 \sin a \cos a$ $\tan 2a = \dfrac{2 \tan a}{1 - \tan^2 a}$

### IV. Arcs remarquables

 $\theta$ (°) 0 30 45 60 90 $\theta$ (rad) 0 $\dfrac{\pi}{6}$ $\dfrac{\pi}{4}$ $\dfrac{\pi}{3}$ $\dfrac{\pi}{2}$ $\sin \theta$ 0 $\dfrac{1}{2}$ $\dfrac{\sqrt{2}}{2}$ $\dfrac{\sqrt{3}}{2}$ 1 $\cos \theta$ 1 $\dfrac{\sqrt{3}}{2}$ $\dfrac{\sqrt{2}}{2}$ $\dfrac{1}{2}$ 0 $\tan \theta$ 0 $\dfrac{\sqrt{3}}{3}$ 1 $\sqrt{3}$ $\infty$

### V. Formules de transformation de produit en somme

$\cos a \cos b = \dfrac{1}{2} (\cos(a - b) + \cos(a + b))$
$\sin a \sin b = \dfrac{1}{2} (\cos(a - b) - \cos (a + b))$
$\sin a \cos b = \dfrac{1}{2} (\sin(a + b) + \sin (a - b))$
$\cos a \sin b = \dfrac{1}{2} (\sin(a + b) - \sin (a - b))$

Retrouvez cette page sur l'île des mathématiques