Resolver y=2+2sin(2x+pi)

$$y = 2 + 2 \sin ( 2 x + \pi )$$

Resolva para x (complex solution)

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

$$x=-\frac{i\ln(-\frac{i\sqrt{y^{2}-4y}}{2}+\frac{iy}{2}-i)}{2}+\pi n_{2}-\frac{\pi }{2}\text{, }n_{2}\in \mathrm{Z}$$

Resolva para y (complex solution)

$$y=2+ie^{-2ix-\pi i}-ie^{2ix-\pi i}$$

Resolva para x

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

$$x=\frac{-\arcsin(\frac{2-y}{2})+2\pi n_{2}+\pi }{2}\text{, }n_{2}\in \mathrm{Z}\text{, }y\geq 0\text{ and }y\leq 4$$

Resolva para y

$$y=2\left(-\sin(2x)+1\right)$$