Find the range of `f(x)=sin(cosx).`

Explanation

To find the range of the function \( f(x) = \sin(\cos x) \), we will follow these steps:

### Step 1: Determine the range of \( \cos x \)
The cosine function oscillates between -1 and 1 for all real values of \( x \). Therefore, we can state that:
\[ -1 \leq \cos x \leq 1 \]

### Step 2: Apply the sine function to the range of \( \cos x \)
Next, we need to find the range of \( f(x) = \sin(\cos x) \). Since the input to the sine function (which is \( \cos x \)) ranges from -1 to 1, we will evaluate \( \sin(y) \) where \( y \) is in the interval \([-1, 1]\).

### Step 3: Find the values of \( \sin(y) \) for \( y \in [-1, 1] \)
We know that:
- The sine function is continuous and increasing in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
- The maximum value of \( \sin(y) \) occurs at \( y = 1 \) and the minimum value occurs at \( y = -1 \).

Calculating these values:
\[ \sin(-1) \quad \text{and} \quad \sin(1) \]
Using a calculator or sine values:
- \( \sin(-1) \) is approximately \(-0.8415\)
- \( \sin(1) \) is approximately \(0.8415\)

### Step 4: Combine the results to find the range of \( f(x) \)
Thus, the range of \( f(x) = \sin(\cos x) \) can be expressed as:
\[ -0.8415 \leq f(x) \leq 0.8415 \]

### Conclusion
The range of the function \( f(x) = \sin(\cos x) \) is:
\[ [-\sin(1), \sin(1)] \]