The value of integer n for which the function `f(x)=(sinx)/(sin(x / n)` has `4pi` its period is

Options

A. 2
B. 3
C. 5
D. 4

Answer

To find the value of integer \( n \) for which the function \( f(x) = \frac{\sin x}{\sin\left(\frac{x}{n}\right)} \) has a period of \( 4\pi \), we can follow these steps:

### Step 1: Understand the periodicity condition
A function \( f(x) \) is periodic with period \( T \) if:
\[ f(x) = f(x + T) \]
In our case, we need to find \( n \) such that:
\[ f(x) = f(x + 4\pi) \]

### Step 2: Write the equation for periodicity
Substituting \( x + 4\pi \) into the function:
\[ f(x + 4\pi) = \frac{\sin(x + 4\pi)}{\sin\left(\frac{x + 4\pi}{n}\right)} \]

Using the property of sine:
\[ \sin(x + 4\pi) = \sin x \]
Thus, we have:
\[ f(x + 4\pi) = \frac{\sin x}{\sin\left(\frac{x + 4\pi}{n}\right)} \]

### Step 3: Set the two expressions equal
For \( f(x) \) to be periodic with period \( 4\pi \), we need:
\[ \frac{\sin x}{\sin\left(\frac{x}{n}\right)} = \frac{\sin x}{\sin\left(\frac{x + 4\pi}{n}\right)} \]

### Step 4: Cancel out \( \sin x \)
Assuming \( \sin x \neq 0 \), we can cancel \( \sin x \) from both sides:
\[ \sin\left(\frac{x}{n}\right) = \sin\left(\frac{x + 4\pi}{n}\right) \]

### Step 5: Use the sine periodicity property
The sine function has the property that:
\[ \sin A = \sin B \Rightarrow A = B + 2k\pi \text{ or } A = \pi - B + 2k\pi \text{ for } k \in \mathbb{Z} \]
Thus, we can write:
\[ \frac{x}{n} = \frac{x + 4\pi}{n} + 2k\pi \]
or
\[ \frac{x}{n} = \pi - \frac{x + 4\pi}{n} + 2k\pi \]

### Step 6: Solve the first equation
From the first equation:
\[ \frac{x}{n} - \frac{x + 4\pi}{n} = 2k\pi \]
This simplifies to:
\[ -\frac{4\pi}{n} = 2k\pi \]
Dividing both sides by \( \pi \):
\[ -\frac{4}{n} = 2k \]
Thus:
\[ n = -\frac{4}{2k} = -\frac{2}{k} \]

### Step 7: Solve the second equation
From the second equation:
\[ \frac{x}{n} + \frac{x + 4\pi}{n} = \pi + 2k\pi \]
This simplifies to:
\[ \frac{2x + 4\pi}{n} = \pi + 2k\pi \]
Multiplying both sides by \( n \):
\[ 2x + 4\pi = n(\pi + 2k\pi) \]

### Step 8: Find integer values for \( n \)
To find integer values of \( n \), we can set \( k = -1 \) to find a positive integer:
\[ n = -\frac{2}{-1} = 2 \]

### Conclusion
Thus, the value of integer \( n \) for which the function \( f(x) = \frac{\sin x}{\sin\left(\frac{x}{n}\right)} \) has a period of \( 4\pi \) is:
\[ \boxed{2} \]