9-1. Find all positive $a$ for which the equation

$$
\frac{2 \pi a + \arcsin (\sin x) + 2 \arccos (\cos x) - a x}{\tan^2 x + 1} = 0
$$

has exactly three distinct solutions in the set $(-\infty; 7\pi]$. In your answer, provide the sum of all such $a$ (if no such $a$ exists, indicate 0; if the number $a$ is not an integer, round it to the nearest hundredth).
Solution. The domain of permissible values - $x \in \mathbb{R} \backslash\{\pi / 2+\pi n\}_{n=1}^{+\infty}$. On the domain of permissible values, we solve the equation

$$
\arcsin (\sin x)+2 \arccos (\cos x)=a x-2 \pi a
$$

The function

$$
f(x)=\arcsin (\sin x)+2 \arccos (\cos x)
$$

is periodic with a period of $2 \pi$, and it is linear on each of the sets $[0 ; \pi / 2],[\pi / 2 ; \pi]$, $[\pi ; 3 \pi / 2],[3 \pi / 2 ; 2 \pi]$. Since

$$
f(0)=f(2 \pi)=0, \quad f(\pi / 2)=3 \pi / 2, \quad f(\pi)=\pi, \quad f(3 \pi / 2)=\pi / 2
$$

we can now construct the graph of the function on the entire number line.

![](https://cdn.mathpix.com/cropped/2024_05_06_997c53c06135e2de9c16g-09.jpg?height=574&width=1216&top_left_y=1164&top_left_x=474)

Fig. 1:

The graph of the function $y=a(x-2 \pi)$ represents a family of lines passing through the point $(2 \pi ; 0)$.

Next, we select the lines that provide three solutions from the specified set. These values are $a=1 / 3, a=2 / 3, a=3 / 5$; their sum is 1.6.

Answer: 1.6.