The number of values of x in the interval `[0, 3pi]` satisfying the equation `3sin^(2)x-7sinx+2=0` is

Options

A. 0
B. 5
C. 4
D. 10

Answer

To solve the equation \(3\sin^2 x - 7\sin x + 2 = 0\) and find the number of values of \(x\) in the interval \([0, 3\pi]\), we can follow these steps:

### Step 1: Substitute \(y = \sin x\)
We rewrite the equation in terms of \(y\):
\[ 3y^2 - 7y + 2 = 0 \]

### Step 2: Factor the quadratic equation
Next, we factor the quadratic equation:
\[ 3y^2 - 6y - y + 2 = 0 \]
Grouping the terms gives:
\[ 3y(y - 2) - 1(y - 2) = 0 \]
Factoring out \((y - 2)\):
\[ (3y - 1)(y - 2) = 0 \]

### Step 3: Solve for \(y\)
Setting each factor to zero gives us:
1. \(3y - 1 = 0 \implies y = \frac{1}{3}\)
2. \(y - 2 = 0 \implies y = 2\)

### Step 4: Analyze the solutions
Since \(\sin x\) can only take values in the range \([-1, 1]\), the solution \(y = 2\) is not valid. Therefore, we only consider:
\[ \sin x = \frac{1}{3} \]

### Step 5: Find the values of \(x\)
The sine function is positive in the first and second quadrants. Thus, we can find the angles corresponding to \(\sin x = \frac{1}{3}\):
1. In the first quadrant:
\[ x_1 = \arcsin\left(\frac{1}{3}\right) \]
2. In the second quadrant:
\[ x_2 = \pi - \arcsin\left(\frac{1}{3}\right) \]

### Step 6: Consider the interval \([0, 3\pi]\)
Since we are looking for solutions in the interval \([0, 3\pi]\), we need to find all possible angles:
- For \(x_1\) and \(x_2\) in the first cycle \([0, 2\pi]\):
- \(x_1 = \arcsin\left(\frac{1}{3}\right)\)
- \(x_2 = \pi - \arcsin\left(\frac{1}{3}\right)\)

- For the second cycle \([2\pi, 3\pi]\):
- \(x_3 = 2\pi + \arcsin\left(\frac{1}{3}\right)\)
- \(x_4 = 2\pi - \arcsin\left(\frac{1}{3}\right)\)

### Step 7: Count the solutions
Thus, we have four solutions in the interval \([0, 3\pi]\):
1. \(x_1 = \arcsin\left(\frac{1}{3}\right)\)
2. \(x_2 = \pi - \arcsin\left(\frac{1}{3}\right)\)
3. \(x_3 = 2\pi + \arcsin\left(\frac{1}{3}\right)\)
4. \(x_4 = 2\pi - \arcsin\left(\frac{1}{3}\right)\)

### Final Answer
The number of values of \(x\) in the interval \([0, 3\pi]\) satisfying the equation is **4**.