6. Find the set of values of the function $\quad f(x)=\operatorname{arctg} \sqrt{6 \log _{0,25}^{-1}\left(\frac{\sin x}{\sin x+15}\right)}$.

points)

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# Solution:

Let

$t=\sin x, t \in[-1 ; 1] \Rightarrow u=\log _{0.25}(z), \quad z=\frac{t}{t+15}=1--$ (function $z(t)$ is increasing). For $t \in[-1 ; 1]$, we have $z \in\left[-\frac{1}{14} ; \frac{1}{16}\right]$, but by the properties of logarithms, $z>0$, hence, $z \in\left(0 ; \frac{1}{16}\right]$. The function $u=\log _{0.25} z$ is decreasing, and for $z \in\left(0 ; \frac{1}{16}\right]$, we have $u \in[2 ;+\infty)$. Further, we have $u^{-1} \in(0 ; 0.5]$, $\sqrt{6 u^{-1}} \in(0 ; \sqrt{3}]$, $\operatorname{arctg} \sqrt{6 u^{-1}} \in(0 ; \pi / 3]$. Thus, the range of the function $y=f(x)$ is the interval $(0 ; \pi / 3]$.

Answer: $E(y)=(0 ; \pi / 3]$.