The value of `n in N` for which the function
`f(x) = (sin(nx))/(sin (x/n))` has a period of `4pi`, is

Options

A. 1
B. 2
C. 4
D. 8

Answer

To find the value of \( n \in \mathbb{N} \) for which the function

\[ f(x) = \frac{\sin(nx)}{\sin\left(\frac{x}{n}\right)} \]

has a period of \( 4\pi \), we can follow these steps:

### Step 1: Determine the period of the numerator
The function in the numerator is \( \sin(nx) \). The period of \( \sin(kx) \) is given by

\[ \frac{2\pi}{k} \]

Thus, the period of \( \sin(nx) \) is

\[ \frac{2\pi}{n} \]

### Step 2: Determine the period of the denominator
The function in the denominator is \( \sin\left(\frac{x}{n}\right) \). The period of \( \sin(kx) \) is again

\[ \frac{2\pi}{k} \]

In this case, \( k = \frac{1}{n} \), so the period of \( \sin\left(\frac{x}{n}\right) \) is

\[ 2\pi n \]

### Step 3: Find the combined period of the function
When we have a function defined as the ratio of two functions, the period can be found using the formula:

\[ \text{Period} = \frac{\text{lcm}(\text{Period of numerator}, \text{Period of denominator})}{\text{hcf}(\text{Period of numerator}, \text{Period of denominator})} \]

Here, the periods we have are:
- Period of numerator: \( \frac{2\pi}{n} \)
- Period of denominator: \( 2\pi n \)

### Step 4: Calculate the lcm and hcf
1. **Calculate lcm**:
- The least common multiple of \( \frac{2\pi}{n} \) and \( 2\pi n \) is \( 2\pi n \) since \( n \) is a natural number.

2. **Calculate hcf**:
- The highest common factor of \( \frac{2\pi}{n} \) and \( 2\pi n \) is \( 2\pi \).

### Step 5: Combine to find the period of \( f(x) \)
Now, substituting into the formula for the period:

\[ \text{Period of } f(x) = \frac{\text{lcm}(2\pi/n, 2\pi n)}{\text{hcf}(2\pi/n, 2\pi n)} = \frac{2\pi n}{2\pi} = n \]

### Step 6: Set the period equal to \( 4\pi \)
We want this period to equal \( 4\pi \):

\[ n = 4\pi \]

### Step 7: Solve for \( n \)
Since \( n \) must be a natural number, we can express \( n \) in terms of \( \pi \):

\[ n = 4 \quad (\text{since } \pi \text{ is not a natural number}) \]

Thus, the value of \( n \) that satisfies the condition is

\[ \boxed{2} \]