Answer:
To find the four smallest positive solutions to the equation \( \csc(3x) - 9 = 0 \), we need to follow these steps:
1. Solve for \(\csc(3x)\):
\[
\csc(3x) - 9 = 0 \implies \csc(3x) = 9
\]
2. Recall that \(\csc(3x) = \frac{1}{\sin(3x)}\). Therefore, we have:
\[
\frac{1}{\sin(3x)} = 9 \implies \sin(3x) = \frac{1}{9}
\]
3. Find the general solutions for \(3x\) when \(\sin(3x) = \frac{1}{9}\):
\[
3x = \arcsin\left(\frac{1}{9}\right) + 2k\pi \quad \text{or} \quad 3x = \pi - \arcsin\left(\frac{1}{9}\right) + 2k\pi \quad \text{for} \; k \in \mathbb{Z}
\]
4. Solve for \(x\):
\[
x = \frac{1}{3} \arcsin\left(\frac{1}{9}\right) + \frac{2k\pi}{3} \quad \text{or} \quad x = \frac{\pi}{3} - \frac{1}{3} \arcsin\left(\frac{1}{9}\right) + \frac{2k\pi}{3}
\]
Now, we will calculate the four smallest positive solutions for \(x\).
### Calculation Steps
Let's denote \(\theta = \arcsin\left(\frac{1}{9}\right)\).
1. First solution (\(k=0\)):
\[
x_1 = \frac{\theta}{3}
\]
2. Second solution (\(k=0\)):
\[
x_2 = \frac{\pi}{3} - \frac{\theta}{3}
\]
3. Third solution (\(k=1\)):
\[
x_3 = \frac{\theta}{3} + \frac{2\pi}{3}
\]
4. Fourth solution (\(k=1\)):
\[
x_4 = \frac{\pi}{3} - \frac{\theta}{3} + \frac{2\pi}{3} = \frac{\pi}{3} - \frac{\theta}{3} + \frac{2\pi}{3} = \pi - \frac{\theta}{3}
\]
### Numerical Values
1. Calculate \(\theta\):
\[
\theta = \arcsin\left(\frac{1}{9}\right)
\]
Let's calculate the numerical values of these solutions.
\[
\theta \approx \arcsin\left(\frac{1}{9}\right) \approx 0.111
\]
Using this approximation:
1. \(x_1 \approx \frac{0.111}{3} \approx 0.037\)
2. \(x_2 \approx \frac{\pi}{3} - \frac{0.111}{3} \approx 1.047 - 0.037 \approx 1.010\)
3. \(x_3 \approx \frac{0.037 + 2\pi/3} \approx 0.037 + 2.094 \approx 2.131\)
4. \(x_4 \approx \pi - 0.037 \approx 3.105\)
Thus, the four smallest positive solutions for \(x\) are approximately:
\[
0.037, 1.010, 2.131, 3.105
\]