Question

The period of $f\left(x\right)=\sin \left(\frac{\pi x}{n-1}\right)+\cos \left(\frac{\pi x}{n}\right)$, $n\in z,n>2$ is

• A. $2\pi n(n-1)$
• B. $4n(n-1)$
• C. $2n(n-1)$
• D. None of these

Answer 1

The correct option is C $2n(n-1)$

Given, $f\left(x\right)=\sin \left(\frac{\pi x}{n-1}\right)+\cos \left(\frac{\pi x}{n}\right)$
Period of $\sin \left(\frac{\pi x}{n-1}\right)=\frac{2\pi }{\frac{\pi }{(n-1)}}=2(n-1)$
and period of $\cos \left(\frac{\pi x}{n}\right)=\frac{2\pi }{\frac{\pi }{n}}=2n$
Hence, the period of $f\left(x\right)$ is LCM of $2n$ and $2(n-1)=2n(n-1)$.