Question

For what integral value of $n$ is $3\pi$ the period of the function $\cos \left(nx\right)\sin \left(\frac{5x}{n}\right)?$

Answer 1

$\begin{array}{c}\cos \left(n\pi \right)\sin \left(\frac{5\pi }{n}\right)\\ iftheperiodiss\pi \\ \cos \left(n\pi +3\pi \right)\sin \left(\frac{5\pi }{n}+3\pi \right)=\cos \left(n\pi \right)\sin \left(\frac{5\pi }{n}\right)\\ \Rightarrow \cos \left(2\pi +\pi +n\pi \right)\sin \left(3\pi +\frac{5\pi }{n}\right)=\cos \left(n\pi \right)\sin \left(\frac{5\pi }{3}\right)\\ \Rightarrow \cos \left(\pi +n\pi \right)\sin \left(3\pi +\frac{5\pi }{n}\right)=\cos \left(n\pi \right)\sin \left(\frac{5\pi }{3}\right)\\ \Rightarrow -\cos \pi \sin \left(3\pi +\frac{5\pi }{n}\right)=\cos \left(n\pi \right)\sin \left(\frac{5\pi }{3}\right)\\ onsolving\\ n=\pm 1,\pm 3,\pm 5\pm 15\end{array}$