5. Solve the system of equations $\left\{\begin{array}{l}4 \cos ^{2} 2 x \cdot \sin ^{2} \frac{x}{3}+4 \sin \frac{x}{3}-4 \sin ^{2} 2 x \cdot \sin \frac{x}{3}+1=0, \\ \sin \frac{x}{2}+\sqrt{\cos y}=0 .\end{array}\right.$

(10 points)
Solution: Let's solve the first equation of the system:

$4 \cos ^{2} 2 x \cdot \sin ^{2} \frac{x}{3}+4 \sin \frac{x}{3}-4 \sin ^{2} 2 x \cdot \sin \frac{x}{3}+1=0$

$4 \cos ^{2} 2 x \cdot \sin ^{2} \frac{x}{3}+4 \cos ^{2} 2 x \cdot \sin \frac{x}{3}+1=0 \quad \Leftrightarrow\left(2 \cos 2 x \sin \frac{x}{3}+\cos 2 x\right)^{2}+\sin ^{2} 2 x=0$

![](https://cdn.mathpix.com/cropped/2024_05_06_18575ac96ab43cb21cd9g-13.jpg?height=337&width=1624&top_left_y=1205&top_left_x=296)

$$
\left\{\begin{array} { l } 
{ x = - \frac { \pi } { 2 } + 6 \pi k , k \in Z } \\
{ x = - \frac { 5 \pi } { 2 } + 6 \pi k , k \in Z , } \\
{ x = \frac { \pi n } { 2 } , n \in Z }
\end{array} \Leftrightarrow \left[\begin{array}{l}
x=-\frac{\pi}{2}+6 \pi k, k \in Z \\
x=-\frac{5 \pi}{2}+6 \pi k, k \in Z
\end{array}\right.\right.
$$

Returning to the original system of equations, we get an equivalent system

$$
\left\{\begin{array} { l } 
{ x = - \frac { \pi } { 2 } + 6 \pi k , k \in Z , } \\
{ x = - \frac { 5 \pi } { 2 } + 6 \pi k , k \in Z , } \\
{ \operatorname { s i n } \frac { x } { 2 } + \sqrt { \operatorname { c o s } y } = 0 , }
\end{array} \Leftrightarrow \left[\begin{array} { l } 
{ \{ \begin{array} { l } 
{ x = - \frac { \pi } { 2 } + 1 2 \pi m , m \in Z , } \\
{ \operatorname { c o s } y = 1 / 2 }
\end{array} } \\
{ \{ \begin{array} { l } 
{ x = - \frac { 5 \pi } { 2 } + 6 \pi ( 2 m + 1 ) , m \in Z , } \\
{ \operatorname { c o s } y = 1 / 2 }
\end{array} }
\end{array} \Leftrightarrow \left[\begin{array}{l}
\left\{\begin{array}{l}
x=-\frac{\pi}{2}+12 \pi m, m \in Z \\
y= \pm \frac{\pi}{3}+2 \pi n, n \in Z
\end{array}\right. \\
\left\{\begin{array}{l}
x=\frac{7 \pi}{2}+12 \pi m, m \in Z \\
y= \pm \frac{\pi}{3}+2 \pi n, n \in Z
\end{array}\right.
\end{array}\right.\right.\right.
$$

Answer: $\left(-\frac{\pi}{2}+12 \pi m, \pm \frac{\pi}{3}+2 \pi n\right),\left(\frac{7 \pi}{2}+12 \pi m, \pm \frac{\pi}{3}+2 \pi n\right), \quad m, n \in Z$