Question

The value of integer $n$ for which the function $f\left(x\right)=\frac{\sin nx}{\sin \left(x/n\right)}$ has $4\pi$ as its period is

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is A $2$

$f(x+T)=f\left(x\right)$
$\therefore \frac{sin\left(n\right(x+4\pi \left)\right)}{sin\left(\frac{x+4\pi }{n}\right)}=\frac{sinnx}{sin\left(\frac{x}{n}\right)}$
$sin\frac{x}{n}=sin\left(\frac{x+4\pi }{n}\right)[\because sin(nx+4n\pi )=sinnx]$
$\frac{x}{n}+2\pi =\frac{x+4\pi }{n}$
$n=2$