# 1.4: The Period of a Periodic Function

• • Contributed by Marcia Levitus

A function $$f(x)$$ is said to be periodic with period $$P$$ if $$f(x) = f(x + P)$$. In plain English, the function repeats itself in regular intervals of length $$P$$. The period of the function of Figure $$\PageIndex{1}$$ is $$2 \pi$$. Figure $$\PageIndex{4}$$: A periodic function with period $$P = 2 \pi$$ (CC BY-NC-SA; Marcia Levitus)

We know that the period of $$\sin (x)$$ is $$2 \pi$$, but what is the period of the function $$\sin (nx)$$?

The period of $$\sin (x)$$ is $$2 \pi$$, so:

$\sin (nx) = \sin (nx + 2 \pi) \nonumber$

By definition, for a periodic function of period $$P$$, the function repeats itself if we add $$P$$ to $$x$$:

$sin (nx) = \sin (n(x + P)) = \sin (nx + nP)) \nonumber$

Comparing the two equations: $$2 \pi = nP$$, and therefore $$\textcolor{red}{P = 2π/n}$$.

For example, the period of $$\sin (2x)$$ is $$\pi$$, and the period of $$\sin (3x)$$ is $$2 \pi/3$$ (see Figure $$\PageIndex{2}$$). Figure $$\PageIndex{2}$$: Some examples of the family of functions $$\sin (nx)$$. From left to right: $$\sin (x)$$, $$\sin (2x)$$, $$\sin (3x)$$ and $$\sin (10x)$$ (CC BY-NC-SA; Marcia Levitus)

You can follow the same logic to prove that the period of $$\cos (nx)$$ is $$2 \pi/n$$. These are important results that we will use later in the semester, so keep them in mind!

Test yourself with this short quiz! http://tinyurl.com/k4wop6l