Finding all integer $n$ so that period of $\cos( n x)\sin(5x/n)$ is $3\pi$

Question

I want to find all integer $n$ so that the function has period $3\pi$. I am unable to take a start, as there is no general rule for periid of product of periodic functions. Please give a starting.

Answer 1

Let $f(x)=\cos(nx)\sin(\frac{5x}{n})$.

Note that $f(0)=0$ and $f(3\pi)=(-1)^{n}\sin(\frac{15\pi}{n})$. So if $f$ is $3\pi$-periodic, we must have $\sin(\frac{15\pi}{n})=0$, and $n$ must divide $15$.

Conversely, suppose that $n$ divides $15$. Then $d=\frac{15}{n}$ is an odd integer. We deduce that $f(x)=\cos(nx)\sin(\frac{dx}{3})$. From $\cos(n(x+3\pi))=-\cos(nx)$ and $\sin(\frac{d(x+3\pi)}{3})=-\sin(\frac{dx}{3})$ it follows that $f$ is $3\pi$-periodic. Done.