# 6. Solution:

Let's find the range of the function $z=g(x)=30+14 \cos x-7 \cos 2 x$. The function $g(x)$ is defined for all real numbers. We will make a substitution. Let $t=\cos x$. Then $z=30+14 t-7\left(2 t^{2}-1\right)=37-14\left(t^{2}-t\right)=40.5-14(t-0.5)^{2}$ for $t \in[-1 ; 1]$, and $E_{g}=[9 ; 40.5]$. The function $u=\frac{\pi}{54} z$ takes all values from the interval $\left[\frac{\pi}{6} ; \frac{3 \pi}{4}\right]$. The range of the function $f(x)$ coincides with the range of the function $y=\sin u$, where $u \in\left[\frac{\pi}{6} ; \frac{3 \pi}{4}\right]$. Therefore, $E_{f}=[0.5 ; 1]$
Answer: $E_{f}=[0,5 ; 1]$.