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Trigonométrie et fonctions circulaires

Les basiques

    \[ \cos(a+b)=\cos(a)\cos(b) - \sin(a)\sin(b) \qquad \sin(a+b)=\sin(a)\cos(b) + \sin(b)\sin(a) \qquad \tan(a+b) = \dfrac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)} \]

    \[ \cos(2a)=\cos^2(a)-\sin^2(a) = 2\cos^2(a)-1 = 1 - 2\sin^2(a) \qquad \qquad \cos^2(a) = \dfrac{1+\cos(2a)}{2} \qquad \qquad \sin^2(a) = \dfrac{1-\cos(2a)}{2} \]

    \[ \sin(2a)=2\sin(a)\cos(a) \qquad \qquad \tan(2a) = \dfrac{2\tan(a)}{1-\tan^2(a)} \]

Les produits

    \[ \cos(a)\cos(b) = \dfrac{1}{2}(\cos(a+b)+\cos(a-b)) \qquad \qquad \sin(a)\sin(b) = -\dfrac{1}{2}(\cos(a+b)-\cos(a-b)) \]

    \[ \sin⁡(a)\cos⁡(b)=\dfrac{1}{2}(\sin⁡(a+b)+\sin⁡(a-b)) \]

Les sommes

    \[ \cos⁡(p) + \cos⁡(q) = 2 \cos(\dfrac{p+q}{2}) \cos(\dfrac{p-q}{2}) \qquad \qquad \cos⁡(p) - \cos⁡(q) = -2 \sin(\dfrac{p+q}{2}) \sin(\dfrac{p-q}{2}) \]

    \[ \sin⁡(p) + \sin⁡(q) = 2 \sin(\dfrac{p+q}{2}) \cos(\dfrac{p-q}{2}) \qquad \qquad \sin⁡(p) - \sin⁡(q) = 2 \sin(\dfrac{p-q}{2}) \cos(\dfrac{p+q}{2}) \]

Changement de variable

    \[ \text{si on pose } t = \tan(\dfrac{\theta}{2}) \qquad \text{alors} \qquad \cos\theta = \dfrac{1 - t^2}{1 + t^2}} \qquad \qquad \sin\theta = \dfrac{2t}{1 + t^2} \qquad \qquad \tan\theta = \dfrac{2t}{1 - t^2} \]

Fonctions réciproques

    \[ \forall x \in [-1,1] \qquad \cos⁡(\arccos⁡(x)) = x \qquad \sin⁡(\arcsin⁡(x)) = x \qquad \qquad \qquad \forall x \in \mathbb{R} \qquad \tan⁡(\arctan⁡(x)) = x \]

    \[ \forall x \in [0,\pi] \qquad \arccos⁡(\cos⁡(x)) = x \qquad \qquad \qquad \forall x \in [-\dfrac{\pi}{2},\dfrac{\pi}{2}] \qquad \arcsin⁡(\sin⁡(x)) = x \qquad \arctan⁡(\tan⁡(x))=x \]

    \[ \forall x \in [-1,1] \qquad \arccos⁡(x) + \arcsin⁡(x) = \dfrac{\pi}{2} \qquad \qquad \qquad \forall x \in \mathbb{R}^{*+} \qquad \arctan⁡(x) + \arctan(\dfrac{1}{x}) = \dfrac{\pi}{2} \]