The period of `f(x)=sin(sin(x/5))`, is

Options

A. `2pi`
B. `(2pi)/(5)`
C. `10 pi`
D. `pi`

Explanation

To find the period of the function \( f(x) = \sin(\sin(x/5)) \), we can follow these steps:

### Step 1: Identify the inner function
The inner function is \( g(x) = \sin(x/5) \). We need to determine its period first.

### Step 2: Determine the period of \( g(x) = \sin(x/5) \)
The standard period of the sine function \( \sin(x) \) is \( 2\pi \). When the argument of the sine function is modified to \( kx \), the period changes to \( \frac{2\pi}{k} \).

In our case, \( k = \frac{1}{5} \). Therefore, the period of \( g(x) = \sin(x/5) \) is:
\[ \text{Period of } g(x) = \frac{2\pi}{\frac{1}{5}} = 2\pi \cdot 5 = 10\pi \]

### Step 3: Determine the period of \( f(x) = \sin(g(x)) \)
Now, we need to find the period of the outer function \( f(x) = \sin(g(x)) = \sin(\sin(x/5)) \). The sine function is periodic with a period of \( 2\pi \).

### Step 4: Analyze the effect of the inner function's period
Since \( g(x) \) has a period of \( 10\pi \), we need to check if \( f(x) \) will repeat its values after \( 10\pi \):
\[ f(x + 10\pi) = \sin(g(x + 10\pi)) = \sin(\sin((x + 10\pi)/5)) = \sin(\sin(x/5 + 2\pi)) = \sin(\sin(x/5)) = f(x) \]
This shows that \( f(x) \) also has a period of \( 10\pi \).

### Conclusion
Thus, the period of the function \( f(x) = \sin(\sin(x/5)) \) is \( 10\pi \).

### Final Answer
The period of \( f(x) = \sin(\sin(x/5)) \) is \( 10\pi \).

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