The number of values of x in `[0,2pi]` satisfying the equation `3cos2x-10cosx+7=0` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the equation \(3 \cos 2x - 10 \cos x + 7 = 0\) for the number of values of \(x\) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Use the double angle formula for cosine
We know that:
\[ \cos 2x = 2 \cos^2 x - 1 \]
Substituting this into the equation gives:
\[ 3(2 \cos^2 x - 1) - 10 \cos x + 7 = 0 \]

### Step 2: Simplify the equation
Expanding and simplifying:
\[ 6 \cos^2 x - 3 - 10 \cos x + 7 = 0 \]
This simplifies to:
\[ 6 \cos^2 x - 10 \cos x + 4 = 0 \]

### Step 3: Rearrange into standard quadratic form
The equation can be rearranged as:
\[ 6 \cos^2 x - 10 \cos x + 4 = 0 \]

### Step 4: Factor the quadratic equation
We will factor the quadratic equation. We need two numbers that multiply to \(6 \times 4 = 24\) and add to \(-10\). The numbers are \(-6\) and \(-4\):
\[ 6 \cos^2 x - 6 \cos x - 4 \cos x + 4 = 0 \]
Factoring by grouping:
\[ 6 \cos x (\cos x - 1) - 4 (\cos x - 1) = 0 \]
Factoring out \((\cos x - 1)\):
\[ (6 \cos x - 4)(\cos x - 1) = 0 \]

### Step 5: Solve for \(\cos x\)
Setting each factor to zero gives us:
1. \(6 \cos x - 4 = 0\)
\[ \Rightarrow \cos x = \frac{4}{6} = \frac{2}{3} \]
2. \(\cos x - 1 = 0\)
\[ \Rightarrow \cos x = 1 \]

### Step 6: Find the values of \(x\)
Now we need to find the values of \(x\) for each case in the interval \([0, 2\pi]\).

1. **For \(\cos x = \frac{2}{3}\)**:
- The solutions for \(\cos x = \frac{2}{3}\) occur in the first and fourth quadrants.
- Let \(x_1 = \cos^{-1}\left(\frac{2}{3}\right)\) (first quadrant)
- Let \(x_2 = 2\pi - \cos^{-1}\left(\frac{2}{3}\right)\) (fourth quadrant)

2. **For \(\cos x = 1\)**:
- The solution for \(\cos x = 1\) is:
- \(x = 0\) (only one solution)

### Step 7: Count the total number of solutions
From the above, we have:
- 2 solutions from \(\cos x = \frac{2}{3}\)
- 1 solution from \(\cos x = 1\)

Thus, the total number of solutions in the interval \([0, 2\pi]\) is:
\[ 2 + 1 = 3 \]

### Final Answer
The number of values of \(x\) in \([0, 2\pi]\) satisfying the equation \(3 \cos 2x - 10 \cos x + 7 = 0\) is **3**.