Formulaire de Trigonométrie

Formules d'Addition

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

Formules de Duplication

En posant $a = b$ :

$$\cos(2a) = \cos^2(a) - \sin^2(a) = 2\cos^2(a) - 1 = 1 - 2\sin^2(a)$$
$$\sin(2a) = 2\sin(a)\cos(a)$$

Exemple : calcul de $\cos\left(\frac{5\pi}{12}\right)$

$$\frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6}$$

$$\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}$$