Periodic function repeats its values in regular interval. The regular interval is referred to as the period. A function is said to be periodic if it gives same value after a same period. And functions which are not periodic are called aperiodic.

Trigonometric functions are the most important examples of the periodic functions. There is repetition of function values with respect to independent variable.

Not only in the field of mathematics but these functions are used throughout science to describe waves, periodicity and oscillations etc. In this section we will learn about periodic function.

## Definition

Suppose f is a function from set A to set B. It is a non constant real valued function of a real variable.
If there is a non zero real number p such that:

i) x belongs to A implies that x + p also belongs to A and

ii)
f(x) = f(x + p), for all x belonging to A.
Then f is called a periodic function and p is the period of f. If p is the least positive period of f, the p is called the principal period (or just the period) of f.

## Formula

The formula for a periodic function usually varies from one function to another. For example, we know that the trigonometric point function,
$f(\theta)$ = $f(\theta + 2n\pi )$
For all $\theta \epsilon R$ and for all $n \epsilon Z$. Thus, the periods of this function are:
...,-8$\pi$, -6$\pi$, -4$\pi$,-2$\pi$, 2$\pi$, 4$\pi$, 6$\pi$, 8$\pi$, ....
If f has a principal period p, then

$f(\theta)$ = $f(\theta + p).$ $\forall \theta \epsilon R$ and if 0 < q < p, then $f(\theta)$ $\neq f(\theta + q)$ for some $\theta \epsilon R$.
Similarly various other trigonometric functions have different periods. We shall look at some of them in a while.

## How to Find Periodic Function

The simplest way to find the period of a function is to graph it. The distance between two consecutive points on the function that have the same y value but different x values would be the period for that function.

Let us try to understand this better with the help of an example:

Example 1: Find the period of the function y = 3 sin (2x).

Solution: We plot the graph of this function first using a graphing calculator or manually using various x and y values.

The graph of this function would look as follows:

Now in this function look at any two points that have the same y value as shown in the following figure.

From the above figure we see that the distance between those two points at same level is $\pi$. Therefore the period of this function is $\pi$ (Pie).

There are other methods as well to find the period of trigonometric functions. We shall have a look at them later once we have understood about the periods of parent trigonometric functions.

## Period of Sine and Cosine Function

We know that the trigonometric point function f assigns to every real number a unique point P on the unit circle. Based on that now let us define the sine function. The sine function is actually the y co ordinate of the point on the unit circle.

Thus of the co ordinates of the point P on a unit circle are (x, y) and the angle made by the radius OP with the positive x axis is $\theta$, then we say that $Sin(\theta)$ = y. This can be understood better with the help of the figure below:

As we can see from the above picture the same y value for any other point on the same circle with the same angle theta, will appear again only after one full rotation. Thus the period of a sine function is one rotation or 2$\pi$.

Similarly the cosine function is defined as the x value of that same point P. Thus we say that $Cos(\theta)$ = x. As in case of the sine function the period of cosine function would also be 2$\pi$.
Period of tangent function:

The tangent function is also a periodic function. It is defined as the ratio of sine to cosine. Thus,
$\tan(\theta)$ = $\frac{\sin \theta}{\cos \theta}$
From the above figure we also note that the point Pâ€™ if located in exactly opposite quadrant of point P would have both sine and cosine values as negative of the sine and cosine of the angle $\theta$. However their ratio would still be positive as the ratio of two negative numbers is positive. Thus the period of a tangent function is $\pi$.

## Examples

Example 1: Find the solution for sin x = $\frac{1}{2}$.

Solution: Consider the periodic function equation: sin x = $\frac{1}{2}$.
This equation can be solved using the unit circle and special angle values so that we get,

x = $\frac{\pi}{6}$
However since the period of the sine function is 2$\pi$, the solution to the above equation would be:

x = $\frac{\pi}{6}$ $\pm 2n\pi$

Now there is another value of x for which sin(x) would be $\frac{1}{2}$. That is because the y co ordinates of a unit circle would be positive in two quadrants the I and the II quadrants. Therefore,
$sin$ $\frac{5\pi}{6}$ = $\frac{1}{2}$ as well

Thus another solution to the same equation would be:

x=$\frac{5\pi}{6}$ $\pm 2n\pi$

Example 2: What is the period for the function Sin($\frac{\theta}{3}$).

Solution: We know that sin$\theta$ is a periodic function with period 2$\pi$

Also the formula for finding period of any trigonometric function i.e. P = $\frac{2\pi}{K}$; k = Multiple of $\theta$

Given Sin($\frac{\theta}{3}$), here the value of k = $\frac{1}{3}$

Now P = $\frac{2\pi}{\frac{1}{3}}$ = 6 $\pi$.

Period for the given function is 6 $\pi$.