# 6. Solution:

Let's find the range of the function $z=g(x)=14 \cos 2 x+28 \sin x+15$. The function $g(x)$ is defined for all real numbers. We will make a substitution. Let $t=\sin x$. Then $z=14\left(1-2 t^{2}\right)+28 t+15=29-28\left(t^{2}-t\right)=36-28(t-0.5)^{2}$ for $t \in[-1 ; 1]$, and $E_{g}=[-27 ; 36]$. The function $u=\frac{\pi}{108} z$ takes all values from the interval $\left[-\frac{\pi}{4} ; \frac{\pi}{3}\right]$. The range of the function $f(x)$ coincides with the range of the function $y=\cos u$, where $u \in\left[-\frac{\pi}{4} ; \frac{\pi}{3}\right]$. Therefore, $E_{f}=[0.5 ; 1]$.
Answer: $E_{f}=[0,5 ; 1]$.