The time interval between two waves is known as a Period whereas a function that repeats its values at regular intervals or periods is known as a Periodic Function. Or you can say, Periodic function is a function that repeats its values after every particular interval.

The period of the function is this particular interval mentioned above.

A function f will be periodic with period m, so if we have

**f (a + m) = f (a), For every m > 0.**

It Shows that the function f(a) possesses the same values after an interval of “m”. One can say that after every interval of “m” the function f repeats all its values.

**For example –** The sine function i.e. sin a has a period 2 Ï€ because 2 Ï€ is the smallest number for which sin (a + 2Ï€) = sin a, for all a.

We may also calculate the period using the formula derived from the basic sine and cosine equations. The period for function y = A sin( B a – c ) and y = A cos( B a – c ) is 2Ï€B radians.

The reciprocal of the period of a function = its frequency.

Frequency is defined as the number of cycles completed in one second. If the period of a function is denoted by P and f be its frequency, then –**f =**1/ P.

## Fundamental Period of a Function

The fundamental period of a function is the period of the function which are of the form,

**f(x+k)=f(x)**

f(x+k)=f(x), then k is called the period of the function and the function f is called a periodic function.

Now, Let us define the function h(t) on the interval [0,2] as follows:

If we extend the function h to all of R by the equation,

h(t+2)=h(t)

=> h is periodic with period 2.

The graph of the function is shown below.

## How to Find the Period of a Function?

- If a function repeats over at a constant period we say that is a periodic function.
- It is represented like f(x) = f(x + p), p is the real number and this is the period of the function.
- Period means the time interval between the two occurrences of the wave.
- To find the period of the periodic function we have to use the following formula, Where

**Period = 2pb, **where b is the coefficient of x

## Period of a Trigonometric Function

The distance between the repetition of any function is called the period of the function. For a trigonometric function, the length of one complete cycle is called a period. For any trigonometry graphÂ function, we can take x = 0 as the starting point.

In general, we have three basic trigonometric functions like sin, cos and tan functions, having 2Ï€, 2Ï€

and Ï€ period respectively.

Sine and cosine functions have the forms of a periodic wave:

**Period**: It is represented as “T”, Period is the distance among two repeating points on the graph function.**Amplitude**: It is represented as “A” It is the distance between the middle point to highest or lowest point on the graph function.

**sin(aÎ¸) = 2Ï€a and cos(aÎ¸) = 2Ï€a**

## Period of a Sine Function

If we have a function f(x) = sin (xs), where s > 0, then the graph of the function makes a complete cycles between 0 and 2Ï€ and each of the function have the period, p = 2Ï€s

Now, let’s discuss some examples based on sin function:

Let us discuss the graph of y = sin 2x

Period = Ï€ | Axis: y = 0 [x-axis ] | Amplitude: 1 | Maximum value = 1 |

Minimum value = -1 | Domain: { x : x âˆˆ R } | Range = [ -1, 1] | – |

## Period of a Tangent Function

If we have a function f(a) = tan (as), where s > 0, then the graph of the function makes complete cycles between âˆ’Ï€/2, 0 and Ï€/2 and each of the function have the period of p = Ï€s

### Periodic Functions Examples

Letâ€™s learn some of the examples of periodic functions.

**Example 1: **

Find the period of the given periodic function. Where f(x) = 9sin(6px7 + 5)

**Solution:**

Given periodic function is f(x) = 9sin(6px7+ 5)

period = 2pb, here period of the periodic function = 2p(6p7) = 146 = 73

**Example 2: **

What is the period of the following periodic function?

Where f(a) = 6 sin 5a

**Solution: **

The given periodic function is f(a) = 6 sin 5a. We have the formula for the period of the function.

period = 2Ï€/b, Where period of the periodic function = 2Ï€/5

**Example 3: **

Graph of y = 4 sin(a/2)

**Solution:**

- Period = 4Ï€
- Axis: y = 0 [x-axis ]
- Amplitude: 4
- Maximum value = 4
- Minimum value = -4
- Domain: { x : x âˆˆ R }
- Range = [ -4, 4]

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