Formulaire de trigonométrie

Formules d'addition

  • \(\cos(\alpha+\beta)=\cos \alpha.\cos \beta - \sin \alpha.\sin \beta\)
  • \(\cos(\alpha-\beta)=\cos \alpha.\cos \beta + \sin \alpha.\sin \beta\)
  • \(\sin(\alpha+\beta)=\sin \alpha.\cos \beta + \sin \beta.\cos \alpha \)
  • \(\sin(\alpha-\beta)=\sin \alpha.\cos \beta - \sin \beta.\cos \alpha \)
  • \(\tan (\alpha+\beta)=\dfrac{\tan \alpha + \tan \beta}{1-\tan \alpha.\tan \beta} \)
  • \(\tan (\alpha-\beta)=\dfrac{\tan \alpha - \tan \beta}{1+\tan \alpha.\tan \beta} \)

Formules de duplication

  • $\cos \left(2\alpha\right) = \cos^2 \alpha-\sin^2 \alpha$
    • $\cos \left(2\alpha\right) = 2.\cos^2 \alpha-1$
    • $\cos \left(2\alpha\right) = 1-2.\sin^2 \alpha$
  • $\sin \left(2\alpha\right) = 2.\sin \alpha.\cos \alpha$
  • $\tan \left(2\alpha\right) = \dfrac{2\tan \alpha}{1-\tan^2 \alpha}$

Formules de Carnot

  • $\cos^2 \alpha = \dfrac{1 + \cos 2 \alpha}{2}$
  • $\sin^2 \alpha = \dfrac{1 - \cos 2 \alpha}{2}$
  • $\tan^2 \alpha = \dfrac{1 - \cos 2\alpha}{1 + \cos 2\alpha}$

Formules de Simpson (somme → produit)

  • \(\cos \alpha + \cos \beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\)
  • \(\sin \alpha + \sin \beta = 2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\)
  • \(\cos \alpha - \cos \beta = -2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)\)
  • \(\sin \alpha - \sin \beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)\)
  • \(\tan \alpha + \tan \beta = \dfrac{\sin\left(\alpha+\beta\right)}{\cos \alpha \cos \beta}\)
  • \(\tan \alpha - \tan \beta = \dfrac{\sin\left(\alpha-\beta\right)}{\cos \alpha \cos \beta}\)

Formules de linéarisation (produit → somme)

  • $\cos \alpha\cdot\cos \beta=\dfrac{{\cos{\left(\alpha+\beta\right)}+\cos{\left(\alpha-\beta\right)}}}{2}$
  • $\sin \alpha\cdot\cos \beta=\dfrac{{\sin{\left(\alpha+\beta\right)}+\sin{\left(\alpha-\beta\right)}}}{2}$
  • $\sin \alpha\cdot\sin \beta=\dfrac{{\cos{\left(\alpha-\beta\right)}-\cos{\left(\alpha+\beta\right)}}}{2}$
  • $\cos \alpha\cdot\sin \beta=\dfrac{{\sin{\left(\alpha+\beta\right)}-\sin{\left(\alpha-\beta\right)}}}{2}$

Formules de demi-angle

  • $\sin \alpha = \dfrac{2\tan\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}$
  • $\cos \alpha = \dfrac{1-\tan^2\frac{\alpha}{2}}{1+\tan^2\frac{\alpha}{2}}$
  • $\tan \alpha = \dfrac{2\tan\frac{\alpha}{2}}{1-\tan^2\frac{\alpha}{2}}$