1.4: The Period of a Periodic Function

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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 \(2 \pi\).

Screen Shot 2019-10-18 at 2.41.42 PM.png — 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 \(\PageIndex{2}\)).

Screen Shot 2019-10-18 at 2.42.41 PM.png — 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