If the period of ` (sin nx )/( sin (x//n))` be `4 pi` then the value of ` n, n in Z` ( the set of integers ) I

Options

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

Answer

To solve the problem, we need to find the value of \( n \) such that the period of the function \( \frac{\sin(nx)}{\sin\left(\frac{x}{n}\right)} \) is \( 4\pi \).

### Step-by-Step Solution:

1. **Identify the period of \( \sin(nx) \)**:
The period of \( \sin(nx) \) is given by the formula:
\[ \text{Period of } \sin(nx) = \frac{2\pi}{n} \]

2. **Identify the period of \( \sin\left(\frac{x}{n}\right) \)**:
The period of \( \sin\left(\frac{x}{n}\right) \) is:
\[ \text{Period of } \sin\left(\frac{x}{n}\right) = 2\pi n \]

3. **Determine the period of the function \( \frac{\sin(nx)}{\sin\left(\frac{x}{n}\right)} \)**:
The period of a function that is a ratio of two periodic functions can be determined using the least common multiple (LCM) and the highest common factor (HCF) of their periods. The formula is:
\[ \text{Period} = \text{LCM}(\text{Period of } \sin(nx), \text{Period of } \sin\left(\frac{x}{n}\right)) / \text{HCF}(\text{Period of } \sin(nx), \text{Period of } \sin\left(\frac{x}{n}\right)) \]

4. **Calculate the LCM and HCF**:
- The periods we have are \( \frac{2\pi}{n} \) and \( 2\pi n \).
- The LCM of \( \frac{2\pi}{n} \) and \( 2\pi n \) is:
\[ \text{LCM}\left(\frac{2\pi}{n}, 2\pi n\right) = 2\pi n \]
- The HCF of \( \frac{2\pi}{n} \) and \( 2\pi n \) is:
\[ \text{HCF}\left(\frac{2\pi}{n}, 2\pi n\right) = \frac{2\pi}{n} \]

5. **Substituting into the period formula**:
Now substituting these into the period formula:
\[ \text{Period} = \frac{2\pi n}{\frac{2\pi}{n}} = n^2 \]

6. **Set the period equal to \( 4\pi \)**:
We know from the problem statement that this period equals \( 4\pi \):
\[ n^2 = 4\pi \]

7. **Solve for \( n \)**:
Taking the square root of both sides gives:
\[ n = \sqrt{4\pi} = 2\sqrt{\pi} \]
Since \( n \) must be an integer, we consider integer values of \( n \) that satisfy this equation. The only integer solution that satisfies \( n^2 = 4\pi \) is \( n = 2 \) (as \( \pi \) is approximately 3.14, \( 2\sqrt{\pi} \) is not an integer).

8. **Conclusion**:
Thus, the value of \( n \) is:
\[ n = 2 \]