The number of solutions of the equation `sin^(2)x=1//4i`

Options

A. 2
B. 3
C. 4
D. none of these

Answer

To solve the equation \( \sin^2 x = \frac{1}{4i} \), we will analyze the properties of the sine function and the nature of the right-hand side of the equation.

### Step-by-Step Solution:

1. **Understanding the Range of \( \sin^2 x \)**:
The sine function, \( \sin x \), takes values in the range \([-1, 1]\). Therefore, when we square it, \( \sin^2 x \) will take values in the range \([0, 1]\). This means that \( \sin^2 x \) is always a non-negative real number.

2. **Analyzing the Right-Hand Side**:
The right-hand side of the equation is \( \frac{1}{4i} \). Here, \( i \) is the imaginary unit, which means that \( \frac{1}{4i} \) is an imaginary number.

3. **Comparing Both Sides**:
Since \( \sin^2 x \) is always a non-negative real number (between 0 and 1), and \( \frac{1}{4i} \) is an imaginary number, there cannot be any \( x \) for which \( \sin^2 x \) equals \( \frac{1}{4i} \).

4. **Conclusion**:
Since there are no values of \( x \) that satisfy the equation \( \sin^2 x = \frac{1}{4i} \), we conclude that there are **no solutions** to the equation.

### Final Answer:
The number of solutions of the equation \( \sin^2 x = \frac{1}{4i} \) is **none of these** (Option 4).

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