The equation `|"sin" x| = "sin" x + 3"has in" [0, 2pi]`

Options

A. no root
B. only one root
C. two roots
D. more than two roots

Explanation

To solve the equation \( |\sin x| = \sin x + 3 \) in the interval \([0, 2\pi]\), we need to analyze the equation based on the definition of the absolute value function.

### Step 1: Analyze the Absolute Value
The absolute value function \( |\sin x| \) can be expressed in two cases:
1. **Case 1:** When \( \sin x \geq 0 \), then \( |\sin x| = \sin x \).
2. **Case 2:** When \( \sin x < 0 \), then \( |\sin x| = -\sin x \).

### Step 2: Case 1 - \( \sin x \geq 0 \)
In this case, we substitute \( |\sin x| \) with \( \sin x \):
\[ \sin x = \sin x + 3 \]
Subtracting \( \sin x \) from both sides gives:
\[ 0 = 3 \]
This is a contradiction, indicating that there are no solutions in this case.

### Step 3: Case 2 - \( \sin x < 0 \)
In this case, we substitute \( |\sin x| \) with \( -\sin x \):
\[ -\sin x = \sin x + 3 \]
Adding \( \sin x \) to both sides results in:
\[ 0 = 2\sin x + 3 \]
Rearranging gives:
\[ 2\sin x = -3 \]
Dividing both sides by 2, we find:
\[ \sin x = -\frac{3}{2} \]
### Step 4: Analyze the Result
The value \( -\frac{3}{2} \) is outside the range of the sine function, which is \([-1, 1]\). Therefore, there are no solutions in this case either.

### Conclusion
Since both cases lead to contradictions or values outside the range of the sine function, we conclude that the equation \( |\sin x| = \sin x + 3 \) has no solutions in the interval \([0, 2\pi]\).

### Final Answer
The correct option is: **no roots.**

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