Question

The value of integer $n$ for which the function $f(x) = \dfrac{{\sin nx}}{{\sin (x/n)}}$ has $4\pi$ as its period is
A: $2$
B: $3$
C: $4$
D: $5$

Answer 1

By using the definition of periodicity of a function which is given by the relation $f(x + T) = f(x)$ where $T$ is the period of the function. By substituting the function given in the equation mentioned above we can arrive at the solution.

Complete step-by-step answer:
Periodic function is nothing but a function which repeats its values at a regular interval of time.
We have an equation for periodicity of function given by,
$f(x + T) = f(x)$
Where $f(x)$ is the given function.
$T$ is the time period which is $4\pi$ given.
In the question they have given the function $f(x)$ as $f(x) = \dfrac{{\sin nx}}{{\sin (x/n)}}$ and also they have given the time period $T = 4\pi$. And they have asked us to find the integer $n$.
So, by substituting these values in the above equation, we get
$\dfrac{{\sin (n(x + 4\pi ))}}{{\sin \left( {\dfrac{{x + 4\pi }}{n}} \right)}} = \dfrac{{sinnx}}{{\sin \left( {\dfrac{x}{n}} \right)}}$
$\Rightarrow \dfrac{{\sin (nx + 4n\pi )}}{{\sin \left( {\dfrac{{x + 4\pi }}{n}} \right)}} = \dfrac{{sinnx}}{{\sin \left( {\dfrac{x}{n}} \right)}}$
From the definition of periodicity we can write the above equation as
$\Rightarrow \dfrac{{\sin nx}}{{\sin \left( {\dfrac{{x + 4\pi }}{n}} \right)}} = \dfrac{{sinnx}}{{\sin \left( {\dfrac{x}{n}} \right)}}$
The $\sin nx$ will get canceled on both the sides, then we get
$\Rightarrow \dfrac{1}{{\sin \left( {\dfrac{{x + 4\pi }}{n}} \right)}} = \dfrac{1}{{\sin \left( {\dfrac{x}{n}} \right)}}$
Cross multiply the above equation, we get
$\Rightarrow \sin \left( {\dfrac{x}{n}} \right) = \sin \left( {\dfrac{{x + 4\pi }}{n}} \right)$
On simplification,
$\dfrac{x}{n} + 2\pi = \dfrac{{x + 4\pi }}{n}$
Multiply $n$ throughout the equation and simplify, we get
$x + 2n\pi = x + 4\pi$
$\Rightarrow n = \dfrac{{x - x + 4\pi }}{{2\pi }}$
$\Rightarrow n = \dfrac{{4\pi }}{{2\pi }}$
$\Rightarrow n = 2$
Therefore, the value of integer $n$ for which the function $f(x) = \dfrac{{\sin nx}}{{\sin (x/n)}}$ has $4\pi$ as its period is $2$.
Hence option A is the correct answer.

Note: A function $f(x)$ is periodic only when there exists a positive real number $T$. If the value of $T$ is independent of $x$ then the function $f(x)$ is periodic, if $T$ is dependent on $x$ then the function $f(x)$ is non-periodic.