# Thread: Determining the Period of a function

1. ## Determining the Period of a function

Hi

The question is whether the function [cos(pi*x)]^n is periodic with period 1?

I cannot remember how to determine the period of a function. However, I do remember the definition:

f(x+h)=f(x), the h is the period.

I just thought that the period of, for instanse, cosinus was 2Pi. So how can the question be if the period is 1 or not?

Really appreciate the help

2. Originally Posted by Ase
Hi

The question is whether the function [cos(pi*x)]^n is periodic with period 1?

I cannot remember how to determine the period of a function. However, I do remember the definition:

f(x+h)=f(x), the h is the period.

I just thought that the period of, for instanse, cosinus was 2Pi. So how can the question be if the period is 1 or not?

Really appreciate the help

The answer ( the period ) depends on whether n is even or odd , do you know why ?

3. Hi

I am not sure. Could you elaborate?

4. we know that $\displaystyle \cos[ \pi (x + 1 )] = \cos(\pi x + \pi ) = - \cos(x)$

However , if n is even , we have $\displaystyle \cos^n[ \pi (x + 1 )]$

$\displaystyle = [- \cos(x)]^n = (-1)^n \cos^n(\pi x) = \cos^n(\pi x )$

so it has a period 1 for n is even , but if n is odd , it only has a period 2 .

5. Hi

Thanks. That explained a lot. I am however a bit confused about something. You write cos(pi*(x+1)). I thought it was cos(pi*x+1).