Solve cos(x+frac{sqrt{y}}{6})=frac{sqrt{2}}{2}

$$\cos ( x + \frac { \sqrt { y } } { 6 } ) = \frac { \sqrt { 2 } } { 2 }$$

Solve for x (complex solution)

$$x=2\pi n_{1}-\frac{\sqrt{y}}{6}+\frac{\pi }{4}\text{, }n_{1}\in \mathrm{Z}$$

$$x=2\pi n_{1}-\frac{\sqrt{y}}{6}+\frac{7\pi }{4}\text{, }n_{1}\in \mathrm{Z}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}\\y=\frac{9\left(7\pi +8\pi n_{1}-4x\right)^{2}}{4}\text{, }n_{1}\in \mathrm{Z}\text{, }arg(12\pi n_{1}-6x+\frac{21\pi }{2})<\pi \text{; }y=\frac{9\left(\pi +8\pi n_{2}-4x\right)^{2}}{4}\text{, }n_{2}\in \mathrm{Z}\text{, }arg(12\pi n_{2}-6x+\frac{3\pi }{2})<\pi \text{, }&\text{unconditionally}\\y=0\text{, }&x=2\pi n_{2}+\frac{\pi }{4}\text{ or }x=2\pi n_{1}+\frac{7\pi }{4}\end{matrix}\right.$$

Solve for x

$$x=2\pi n_{1}-\frac{\sqrt{y}}{6}+\frac{7\pi }{4}\text{, }n_{1}\in \mathrm{Z}$$

$$x=2\pi n_{1}-\frac{\sqrt{y}}{6}+\frac{\pi }{4}\text{, }n_{1}\in \mathrm{Z}\text{, }y\geq 0$$

Solve for y

$$y=\frac{9\left(\pi +8\pi n_{1}-4x\right)^{2}}{4}\text{, }n_{1}\in \mathrm{Z}\text{, }12\pi n_{1}-6x+\frac{3\pi }{2}\geq 0$$

$$y=\frac{9\left(7\pi +8\pi n_{1}-4x\right)^{2}}{4}\text{, }n_{1}\in \mathrm{Z}\text{, }12\pi n_{1}-6x+\frac{21\pi }{2}\geq 0$$