Problem 6. Find the set of values of the function $f(x)=\operatorname{arctg} \sqrt{\log _{0.5}^{-1}\left(\frac{\sin x}{\sin x+7}\right)}$. (10 points)

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# Solution:

Let $\quad t=\sin x, t \in[-1 ; 1] \Rightarrow u=\log _{0.5}(z), \quad z=\frac{t}{t+7}=1-\frac{7}{t+7} \quad$ (the function is increasing). For $\mathrm{t} \in[-1 ; 1]$ we have $\mathrm{z} \in\left[-\frac{1}{6} ; \frac{1}{8}\right]$, but by the properties of logarithms $\mathrm{z}>0$, hence, $\mathrm{z} \in\left(0 ; \frac{1}{8}\right]$. The function $u=\log _{0.5} z$ is decreasing, and for $\mathrm{z} \in\left(0 ; \frac{1}{8}\right]$ we have $u \in[3 ;+\infty) . \quad$ Further, we have $u^{-1} \in(0 ; 1 / 3], \quad \sqrt{u^{-1}} \in(0 ; 1 / \sqrt{3}], \quad \operatorname{arctg} \sqrt{u^{-1}} \in(0 ; \pi / 6]$. Thus, the range of the function $y=f(x)$ is the interval $(0 ; \pi / 6]$. Answer: $E(y)=(0 ; \pi / 6]$.