The number of solutions of equation `sin.(5x)/(2)-sin.(x)/(2)=2` in `[0,2pi]` is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the equation \( \frac{\sin(5x)}{2} - \frac{\sin(x)}{2} = 2 \) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Simplify the Equation
We start with the given equation:
\[ \frac{\sin(5x)}{2} - \frac{\sin(x)}{2} = 2 \]
Multiplying through by 2 to eliminate the fractions gives:
\[ \sin(5x) - \sin(x) = 4 \]

### Step 2: Analyze the Range of the Sine Function
The sine function, \( \sin(x) \), has a range of \([-1, 1]\). Therefore, both \( \sin(5x) \) and \( \sin(x) \) also have ranges of \([-1, 1]\).

### Step 3: Determine Feasibility of the Equation
Since the maximum value of \( \sin(5x) \) is 1 and the minimum value of \( \sin(x) \) is -1, the maximum value of \( \sin(5x) - \sin(x) \) is:
\[ 1 - (-1) = 1 + 1 = 2 \]
This means the left-hand side can achieve a maximum value of 2, which matches the right-hand side of our equation. However, we need to check if this condition can be satisfied.

### Step 4: Set Conditions for Equality
For the equation \( \sin(5x) - \sin(x) = 4 \) to hold, we would require:
\[ \sin(5x) = 1 \quad \text{and} \quad \sin(x) = -1 \]
This is because \( 1 - (-1) = 2 \).

### Step 5: Solve for \( x \)
1. **For \( \sin(5x) = 1 \)**:
\[ 5x = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}) \]
Thus,
\[ x = \frac{\pi}{10} + \frac{2n\pi}{5} \]

2. **For \( \sin(x) = -1 \)**:
\[ x = \frac{3\pi}{2} + 2m\pi \quad (m \in \mathbb{Z}) \]

### Step 6: Check for Common Solutions
Now we need to find values of \( n \) and \( m \) such that:
\[ \frac{\pi}{10} + \frac{2n\pi}{5} = \frac{3\pi}{2} + 2m\pi \]
This is a complex equation to solve, but we can analyze the feasibility.

### Step 7: Analyze Parity
Notice that:
- The left-hand side \( \frac{\pi}{10} + \frac{2n\pi}{5} \) is an even function of \( n \).
- The right-hand side \( \frac{3\pi}{2} + 2m\pi \) is also an even function of \( m \).

However, since the left-hand side can only equal 2 at specific points and the right-hand side can only equal 2 at specific points, we need to check if there are any values of \( n \) and \( m \) that satisfy both conditions.

### Conclusion
Since there are no integer values of \( n \) and \( m \) that satisfy both conditions simultaneously, we conclude that there are **no solutions** to the original equation in the interval \([0, 2\pi]\).

Thus, the number of solutions is:
\[ \boxed{0} \]