What Is A Periodic Function?

A periodic function is a function that repeats its values on regular intervals or "periods." Think of it like a heartbeat or the underlying rhythm in a song: It repeats the same activity on a steady beat. The graph of a periodic function looks like a single pattern is being repeated over and over again.

The most famous periodic functions are trigonometric functions: sine, cosine, tangent, cotangent, secant, cosecant, etc. Other examples of periodic functions in nature include light waves, sound waves and phases of the moon. Each of these, when graphed on the coordinate plane, makes a repeating pattern on the same interval, making it easy to predict.

The period of a periodic function is the interval between two "matching" points on the graph. In other words, it's the distance along the ​x​-axis that the function has to travel before it starts to repeat its pattern. The basic sine and cosine functions have a period of 2π, while tangent has a period of π.

Another way to understand period and repetition for trig functions is to think about them in terms of the unit circle. On the unit circle, values go around and around the circle when they increase in size. That repetitive motion is the same idea that's reflected in the steady pattern of a periodic function. And for sine and cosine, you have to make a full path around the circle (2π) before the values start to repeat.

A periodic function can also be defined as an equation with this form:

\(f(x + nP) = f(x)\)

Where ​P​ is the period (a nonzero constant) and ​n​ is a positive integer.

For example, you can write the sine function in this way:

\(\sin(x + 2π) = \sin(x)\)

​n​ = 1 in this case, and the period, ​P​, for a sine function is 2π.

Test it by trying out a couple of values for ​x​, or look at the graph: Pick any ​x​-value, then move 2π in either direction along the ​x​-axis; the ​y​-value should stay the same.

Now try it when ​n​ = 2:

\(\sin(x + (2×2π)) = \sin(x) \
\sin(x + 4π) = \sin(x)\)

Calculate for different values of ​x​: ​x​ = 0, ​x​ = π, ​x​ = π/2, or check it on the graph.

The cotangent function follows the same rules, but its period is π radians instead of 2π radians, so its graph and its equation look like this:

\(\cot(x + nπ) = \cot(x)\)

Notice that tangent and cotangent functions are periodic, but they are not continuous: There are "breaks" in their graphs.