Which of the following is correct ? [Hint: 1 radian = 180°/π = 57°30’ approx.]

Options

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

Answer

To determine which of the given options is correct regarding the comparison between sin 1 degree and sin 1 radian, we can follow these steps:

### Step 1: Convert Radians to Degrees
We know that:
1 radian = 180°/π ≈ 57.3°

### Step 2: Identify the Values
Since 1 radian is approximately 57.3 degrees, we can see that:
- 1 degree < 1 radian (approximately 57.3 degrees)

### Step 3: Understand the Sine Function in the First Quadrant
The sine function is increasing in the first quadrant (0° to 90°). This means that as the angle increases, the value of sin(θ) also increases.

### Step 4: Compare sin(1°) and sin(1 radian)
- Since 1 degree is much smaller than 57.3 degrees (1 radian), we can conclude that:
- sin(1°) corresponds to a very small angle.
- sin(1 radian) corresponds to an angle that is approximately 57.3 degrees.

### Step 5: Conclusion
Since sin(θ) is increasing in the first quadrant:
- sin(1°) < sin(1 radian)

Thus, we conclude that:
- **sin(1 degree) is less than sin(1 radian)**.

### Final Answer
The correct option is the second one: **sin 1 degree is less than sin 1 radian**.

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