Vyriešiť y=log(cos(x))

$$y= \log ( \cos ( x ) )$$

Riešenie pre x (complex solution)

$$x=-i\ln(10^{y}+\sqrt{100^{y}-1})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$x=-i\ln(10^{y}-\sqrt{100^{y}-1})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }Im(\ln(10^{y}))-\ln(10)Im(y)=0$$

Riešenie pre y (complex solution)

$$y=\log(\cos(x))$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}$$

Riešenie pre x

$$x=-\arccos(10^{y})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$x=\arccos(10^{y})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }y\leq 0\text{ and }10^{y}\leq 1$$

Riešenie pre y

$$y=\log(\cos(x))$$

$$\exists n_{1}\in \mathrm{Z}\text{ : }\left(x>2\pi n_{1}-\frac{\pi }{2}\text{ and }x<2\pi n_{1}+\frac{\pi }{2}\right)$$