Which of the following is correct?

Options

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

Answer

To solve the problem of comparing \(\sin(1^\circ)\) and \(\sin(1 \text{ radian})\), we can follow these steps:

### Step 1: Convert radians to degrees
We know that \( \pi \) radians is equal to \( 180^\circ \). Therefore, to convert 1 radian to degrees, we use the conversion factor:
\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]
Calculating this gives:
\[ 1 \text{ radian} \approx 57.32^\circ \]

### Step 2: Understand the sine function in the first quadrant
The sine function is an increasing function in the first quadrant, which means that as the angle increases from \(0^\circ\) to \(90^\circ\), the value of sine also increases. Specifically:
\[ \sin(0^\circ) = 0 \quad \text{and} \quad \sin(90^\circ) = 1 \]

### Step 3: Compare the angles
Since \(1^\circ\) is less than \(57.32^\circ\), we can conclude that:
\[ 1^\circ < 57.32^\circ \]

### Step 4: Apply the property of the sine function
Using the property of the sine function being increasing in the first quadrant, we can write:
\[ \sin(1^\circ) < \sin(57.32^\circ) \]

### Step 5: Substitute back to radians
Since we know that \(\sin(57.32^\circ) = \sin(1 \text{ radian})\), we can replace it in our inequality:
\[ \sin(1^\circ) < \sin(1 \text{ radian}) \]

### Conclusion
Thus, the correct relation is:
\[ \sin(1^\circ) < \sin(1 \text{ radian}) \]

### Final Answer
The correct option is:
**Option B: \(\sin(1^\circ) < \sin(1 \text{ radian})\)**

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