To solve the equation \( 9\sin(x) + 2 = 3 \) over the interval \( [0, 2\pi] \), follow these steps:

1. **Isolate the sine function:**
   \[
   9\sin(x) = 3 - 2
   \]
   \[
   9\sin(x) = 1
   \]
   \[
   \sin(x) = \frac{1}{9}
   \]

2. **Determine the general solution:**
   The sine function equals \(\frac{1}{9}\) at specific angles. Since \(\sin(x)\) is positive, we consider the angles in the first and second quadrants:
   \[
   x = \arcsin\left(\frac{1}{9}\right) \quad \text{or} \quad x = \pi - \arcsin\left(\frac{1}{9}\right)
   \]

3. **Find the specific solutions within the interval \( [0, 2\pi] \):**
   \[
   x_1 = \arcsin\left(\frac{1}{9}\right)
   \]
   \[
   x_2 = \pi - \arcsin\left(\frac{1}{9}\right)
   \]

These are the two solutions within the interval \( [0, 2\pi] \).

### Numerical Values
Let's calculate the numerical values for these solutions:

\[
x_1 \approx \arcsin\left(\frac{1}{9}\right) \approx 0.1116 \text{ radians}
\]
\[
x_2 \approx \pi - 0.1116 \approx 3.0300 \text{ radians}
\]

### Final Answer:
The solutions to the equation \( 9\sin(x) + 2 = 3 \) in the interval \( [0, 2\pi] \) are approximately:
\[
x \approx 0.1116 \text{ radians}, \quad x \approx 3.0300 \text{ radians}
\]

Would you like more details or have any other questions?

### Related Questions:
1. How do you solve a trigonometric equation involving cosine?
2. What are the properties of the sine function in different quadrants?
3. How do you solve a quadratic trigonometric equation?
4. How does the unit circle help in solving trigonometric equations?
5. What are the special angles for sine, cosine, and tangent?

### Tip:
Always remember the sine function is positive in the first and second quadrants, which helps determine the angles when solving equations involving sine.

Learn how to solve the trigonometric equation 9sin(x) + 2 = 3 over the interval [0, 2π] with detailed step-by-step instructions. This problem involves trigonometric functions and the arcsine function, focusing on finding solutions within a specified interval. Suitable for advanced high school students (Grades 10-12).

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