Maximum value of `sin(cos x)` is-

Options

A. sin 1
B. 1
C. `sin(1/sqrt(2))`
D. `sin(sqrt(3)/2)`

Answer

To find the maximum value of \( \sin(\cos x) \), we can follow these steps:

### Step 1: Determine the range of \( \cos x \)
The cosine function, \( \cos x \), has a range of values from -1 to 1. This means that as \( x \) varies, \( \cos x \) will take all values in the interval:
\[ -1 \leq \cos x \leq 1 \]

### Step 2: Analyze the function \( \sin y \)
The sine function, \( \sin y \), has a maximum value of 1 when \( y = \frac{\pi}{2} + 2k\pi \) for any integer \( k \). However, since \( y \) in our case is \( \cos x \), we need to find the maximum value of \( \sin(\cos x) \) over the interval \( [-1, 1] \).

### Step 3: Evaluate \( \sin(\cos x) \) at the endpoints
To find the maximum value of \( \sin(\cos x) \), we can evaluate it at the endpoints of the interval:
- When \( \cos x = -1 \):
\[ \sin(-1) \approx -0.8415 \]
- When \( \cos x = 1 \):
\[ \sin(1) \approx 0.8415 \]

### Step 4: Find the maximum value in the interval
Since \( \cos x \) can take any value between -1 and 1, we can also check the value of \( \sin y \) at \( y = 0 \):
- When \( \cos x = 0 \):
\[ \sin(0) = 0 \]

Now, we compare the values:
- \( \sin(-1) \approx -0.8415 \)
- \( \sin(0) = 0 \)
- \( \sin(1) \approx 0.8415 \)

### Conclusion
The maximum value of \( \sin(\cos x) \) occurs at \( \cos x = 1 \), which gives us:
\[ \text{Maximum value of } \sin(\cos x) = \sin(1) \]

Thus, the answer is:
\[ \sin(1) \]