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. `"sin"1^(@)=pi/(180)"sin"1`

Answer

To determine which of the following statements is correct regarding the sine function for 1 degree and 1 radian, we will analyze the options step by step:

### Step 1: Understand the conversion between degrees and radians
1 radian is equal to \( \frac{180}{\pi} \) degrees. To find out how many degrees are in 1 radian, we can calculate:

\[ 1 \text{ radian} = \frac{180}{\pi} \approx \frac{180}{3.14} \approx 57.3 \text{ degrees} \]

### Step 2: Compare 1 degree and 1 radian
Since \( 1 \text{ degree} < 1 \text{ radian} \) (because \( 1 \text{ radian} \approx 57.3 \text{ degrees} \)), we can conclude that:

\[ 1 \text{ degree} < 57.3 \text{ degrees} \]

### Step 3: Apply the sine function
The sine function is increasing in the interval \( [0, \frac{\pi}{2}] \). Therefore, if \( x < y \), then \( \sin(x) < \sin(y) \) for \( x, y \) in this interval.

Since we have established that:

\[ 1 \text{ degree} < 1 \text{ radian} \]

we can apply the sine function:

\[ \sin(1 \text{ degree}) < \sin(1 \text{ radian}) \]

### Conclusion
Thus, the correct statement is:

\[ \sin(1 \text{ degree}) < \sin(1 \text{ radian}) \]

This means option B is correct: **sine 1 degree is less than sine 1 radian**.

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