Find the maimum and minimum values of the function f(x)= sin (sin x)

Answer

To find the maximum and minimum values of the function \( f(x) = \sin(\sin x) \), we can follow these steps:

### Step 1: Determine the range of \( \sin x \)

The sine function, \( \sin x \), oscillates between -1 and 1 for all real numbers \( x \). Therefore, we have:
\[ -1 \leq \sin x \leq 1 \]

### Step 2: Find the range of \( f(x) = \sin(\sin x) \)

Next, we need to evaluate \( f(x) = \sin(\sin x) \) based on the range of \( \sin x \). Since \( \sin x \) takes values between -1 and 1, we can substitute these limits into the sine function:
\[ \sin(-1) \leq \sin(\sin x) \leq \sin(1) \]

### Step 3: Calculate \( \sin(-1) \) and \( \sin(1) \)

Using a calculator or trigonometric tables, we find:
- \( \sin(1) \) is approximately \( 0.8415 \)
- \( \sin(-1) = -\sin(1) \) is approximately \( -0.8415 \)

### Step 4: Establish the maximum and minimum values

From the calculations above, we can conclude:
- The maximum value of \( f(x) = \sin(\sin x) \) is \( \sin(1) \approx 0.8415 \)
- The minimum value of \( f(x) = \sin(\sin x) \) is \( \sin(-1) \approx -0.8415 \)

### Final Answer

Thus, the maximum and minimum values of the function \( f(x) = \sin(\sin x) \) are:
- Maximum value: \( \sin(1) \approx 0.8415 \)
- Minimum value: \( \sin(-1) \approx -0.8415 \)

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