Whch of the following is correct?

Options

A. `cos 1^(@) gt cos1`
B. `cos 1^(@) lt cos1`
C. `cos1^(@)=sin 1^(@)`
D. `sin1^(@) = 1/(180^(@)) sin1^(@)`

Answer

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

### Step 1: Understand the relationship between degrees and radians
We know that:
\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]
This means that:
\[ 1^\circ = \frac{\pi}{180} \text{ radians} \]

### Step 2: Compare the values of 1 degree and 1 radian
To compare \( 1 \text{ radian} \) and \( 1^\circ \), we need to find out if \( 1 \text{ radian} \) is greater than or less than \( 1^\circ \):
\[ 1 \text{ radian} \approx 3.14 \text{ degrees} \]
Since \( 3.14 \text{ degrees} > 1 \text{ degree} \), we have:
\[ 1 \text{ radian} > 1^\circ \]

### Step 3: Analyze the behavior of the cosine function
The cosine function is decreasing in the first quadrant (from \( 0 \) to \( \frac{\pi}{2} \) radians or \( 0^\circ \) to \( 90^\circ \)). Therefore, if \( x < y \), then:
\[ \cos(x) > \cos(y) \]
In our case, since \( 1 \text{ radian} > 1^\circ \), we can conclude:
\[ \cos(1 \text{ radian}) < \cos(1^\circ) \]

### Step 4: State the final conclusion
Thus, we can conclude that:
\[ \cos(1^\circ) > \cos(1 \text{ radian}) \]

### Step 5: Identify the correct option
Based on the conclusion, the correct option is:
\[ \text{Option A: } \cos(1^\circ) > \cos(1 \text{ radian}) \]