Which of the following is correct ?

Options

A. `sin 1^(@) gt sin 1`
B. `sin 1^(@) lt sin 1`
C. `sin 1^(@) = sin 1`
D. `sin1^(@) = ( pi )/( 180) sin 1`

Answer

To determine which of the given options is correct regarding the sine function, we will analyze the statements one by one.

### Step-by-Step Solution

1. **Understanding the Sine Function**:
The sine function is defined for angles in both degrees and radians. We know that as the angle increases in the first quadrant (from 0° to 90°), the value of sine increases.

2. **Evaluating the Options**:
- **Option 1**: \( \sin(1^\circ) > \sin(1) \)
- **Option 2**: \( \sin(1^\circ) < \sin(1) \)
- **Option 3**: \( \sin(1^\circ) = \sin(1) \)
- **Option 4**: \( \sin(1^\circ) = \frac{\pi}{180} \cdot \sin(1) \)

3. **Analyzing Option 1**:
- Since \(1^\circ\) is a small angle, we can say that \( \sin(1^\circ) \) is a value corresponding to a very small angle.
- As the angle increases, the sine value increases, so \( \sin(1^\circ) \) cannot be greater than \( \sin(1) \) (where \(1\) is in radians).

4. **Analyzing Option 2**:
- Given that \( \sin(1^\circ) \) is a sine value of a small angle and \( \sin(1) \) (where \(1\) is in radians) corresponds to a larger angle, it follows that \( \sin(1^\circ) < \sin(1) \).
- This option seems plausible.

5. **Analyzing Option 3**:
- The sine values of different angles cannot be equal unless the angles are the same. Since \(1^\circ\) and \(1\) radian are not equal, this option is incorrect.

6. **Analyzing Option 4**:
- The statement \( \sin(1^\circ) = \frac{\pi}{180} \cdot \sin(1) \) is incorrect because \( \sin(1^\circ) \) should not be equal to a multiple of \( \sin(1) \) in this manner.

7. **Conclusion**:
- After analyzing all options, we conclude that **Option 2**: \( \sin(1^\circ) < \sin(1) \) is the correct statement.

### Final Answer
The correct option is: **Option 2: \( \sin(1^\circ) < \sin(1) \)**.