The number of values of x in the interval `[0,5pi]` satisfying the equation.
`3sin^(2)x -7sinx + 2=0` is-

Options

A. 0
B. 5
C. 6
D. 10

Explanation

To solve the equation \(3\sin^2 x - 7\sin x + 2 = 0\) and find the number of values of \(x\) in the interval \([0, 5\pi]\), we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ 3\sin^2 x - 7\sin x + 2 = 0 \]

### Step 2: Factor the quadratic equation
We can factor the quadratic equation:
\[ 3\sin^2 x - 6\sin x - \sin x + 2 = 0 \]
Grouping the terms, we get:
\[ (3\sin x - 1)(\sin x - 2) = 0 \]

### Step 3: Set each factor to zero
Now we set each factor to zero:
1. \(3\sin x - 1 = 0\)
2. \(\sin x - 2 = 0\)

From the first equation:
\[ 3\sin x = 1 \implies \sin x = \frac{1}{3} \]

From the second equation:
\[ \sin x = 2 \]
However, we know that the sine function has a maximum value of 1. Therefore, \(\sin x = 2\) is not possible.

### Step 4: Solve for \(x\) when \(\sin x = \frac{1}{3}\)
Now we need to find the values of \(x\) for \(\sin x = \frac{1}{3}\).

### Step 5: Determine the general solutions
The sine function is positive in the first and second quadrants. Therefore, the general solutions for \(\sin x = \frac{1}{3}\) are:
\[ x = \arcsin\left(\frac{1}{3}\right) + 2k\pi \quad \text{(for the first quadrant)} \]
\[ x = \pi - \arcsin\left(\frac{1}{3}\right) + 2k\pi \quad \text{(for the second quadrant)} \]
where \(k\) is any integer.

### Step 6: Find solutions in the interval \([0, 5\pi]\)
Now we need to find how many values of \(x\) satisfy these equations in the interval \([0, 5\pi]\).

1. For \(k = 0\):
- \(x_1 = \arcsin\left(\frac{1}{3}\right)\)
- \(x_2 = \pi - \arcsin\left(\frac{1}{3}\right)\)

2. For \(k = 1\):
- \(x_3 = \arcsin\left(\frac{1}{3}\right) + 2\pi\)
- \(x_4 = \pi - \arcsin\left(\frac{1}{3}\right) + 2\pi\)

3. For \(k = 2\):
- \(x_5 = \arcsin\left(\frac{1}{3}\right) + 4\pi\)
- \(x_6 = \pi - \arcsin\left(\frac{1}{3}\right) + 4\pi\)

### Step 7: Count the solutions
Now we count the solutions:
- For \(k = 0\): 2 solutions
- For \(k = 1\): 2 solutions
- For \(k = 2\): 2 solutions

Thus, the total number of solutions in the interval \([0, 5\pi]\) is:
\[ 2 + 2 + 2 = 6 \]

### Final Answer
The number of values of \(x\) in the interval \([0, 5\pi]\) satisfying the equation is **6**.

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