Question

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

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

Answer 1

Answer: (C)

$2n(n - 1)$

Answer 2

$f(x) = \sin\left( \frac{\pi x}{n - 1} \right) + \cos\left( \frac{\pi x}{n} \right)$

Period of $\sin\left( \frac{\pi x}{n - 1} \right) = \frac{2\pi}{\left( \frac{\pi}{n - 1} \right)} = 2(n - 1)$ and period of $\cos\left( \frac{\pi x}{n} \right) = \frac{2\pi}{\left( \frac{\pi}{n} \right)} = 2n$

Hence period of $f(x)$ is LCM of $2n$ and $2(n - 1) \Rightarrow 2n(n - 1)$.