Solve the trigonometric equation if 3𝑠𝑖𝑛2𝜃−12𝜃+9=0 in the interval [0,4𝜋]

To solve the trigonometric equation \( 3\sin^2\theta - 12\sin\theta + 9 = 0 \) in the interval \([0, 4\pi]\), we can follow these steps:

### Step 1: Factorize the equation
From the previous problem, we already factorized the equation as:
\[
3(\sin\theta - 3)(\sin\theta - 1) = 0
\]
This gives us two possible equations to solve:
\[
\sin\theta - 3 = 0 \quad \text{or} \quad \sin\theta - 1 = 0
\]

### Step 2: Solve each equation

#### 1. Solve \( \sin\theta - 3 = 0 \):
\[
\sin\theta = 3
\]
Since the sine function has a range of \([-1, 1]\), \( \sin\theta = 3 \) has no solution, as 3 is outside this range.

#### 2. Solve \( \sin\theta - 1 = 0 \):
\[
\sin\theta = 1
\]
The sine function equals 1 at specific angles:
\[
\theta = \frac{\pi}{2} + 2k\pi \quad \text{for integer } k
\]

### Step 3: Find solutions in the interval \([0, 4\pi]\)
We need to find all angles \(\theta\) that satisfy \( \sin\theta = 1 \) within the interval \([0, 4\pi]\).

Starting with \( \theta = \frac{\pi}{2} + 2k\pi \), we find:

- For \( k = 0 \):
  \[
  \theta = \frac{\pi}{2}
  \]

- For \( k = 1 \):
  \[
  \theta = \frac{\pi}{2} + 2\pi = \frac{\pi}{2} + \frac{4\pi}{2} = \frac{5\pi}{2}
  \]

- For \( k = 2 \):
  \[
  \theta = \frac{\pi}{2} + 4\pi = \frac{\pi}{2} + \frac{8\pi}{2} = \frac{9\pi}{2}
  \]

The value \( \frac{9\pi}{2} \) exceeds \( 4\pi \), so it is not within the given interval.

### Step 4: List the solutions
The solutions within the interval \([0, 4\pi]\) are:
\[
\theta = \frac{\pi}{2}, \frac{5\pi}{2}
\]

### Final Answer
The solutions to the equation \( 3\sin^2\theta - 12\sin\theta + 9 = 0 \) in the interval \([0, 4\pi]\) are:
\[
\theta = \frac{\pi}{2}, \frac{5\pi}{2}
\]

Would you like any further clarification or have more questions on this?

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Here are 8 related questions you might find interesting:
1. How do you find all solutions to trigonometric equations within a specific interval?
2. What are the key properties of the sine function and how do they affect solving equations?
3. How do you solve trigonometric equations involving multiple angles?
4. How can we solve trigonometric equations using identities?
5. What happens when the sine function is set equal to a value outside its range?
6. How do you solve trigonometric equations with different trigonometric functions involved?
7. What is the general solution for \(\sin\theta = a\) for any real number \(a\)?
8. How do phase shifts affect the solutions of trigonometric equations?

**Tip:** When solving trigonometric equations, always consider the periodicity of the trigonometric functions to ensure you find all solutions within the given interval.

Learn how to solve the trigonometric equation 3sin^2(theta) - 12theta + 9 = 0 within the interval [0, 4pi]. This problem involves factoring trigonometric equations, understanding sine function properties, and finding solutions within specified intervals. Suitable for students in Grades 10-12.

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