Résoudre y=3cos(pix-4)

$$y = 3 \cos ( \pi x - 4 )$$

Calculer x (solution complexe)

$$x=\left(2n_{1}\pi +4+\left(-i\right)\ln(\frac{1}{3}y+\left(-\frac{1}{3}\right)\left(y^{2}-9\right)^{\frac{1}{2}})\right)\pi ^{-1}\text{, }n_{1}\in \mathrm{Z}$$

$$x=\left(-i\right)\left(2i\pi n_{2}+4i+\ln(\frac{1}{3}y+\frac{1}{3}\left(y^{2}-9\right)^{\frac{1}{2}})\right)\pi ^{-1}\text{, }n_{2}\in \mathrm{Z}$$

Calculer y (solution complexe)

$$y=\frac{3\left(e^{\pi ix-4i}+e^{-\pi ix+4i}\right)}{2}$$

Calculer x

$$x=\frac{\arccos(\frac{y}{3})+4}{\pi }+2n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$x=\frac{-\arccos(\frac{y}{3})+4}{\pi }+2n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }|y|\leq 3$$

Calculer y

$$y=3\cos(\pi x-4)$$