Period of A Function

Periodic Function

The time interval between two waves is called a Period whereas a function that repeats its values at regular intervals or periods can be defined as a Periodic Function. Or we can say that a periodic function is a function that repeats its values after every particular interval. This is the periodic function definition.

Suppose we have a function f that would be periodic with period m, so if we can write 

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

This shows that the given function f(a) possesses the same values after the given interval value of “m”. One can also say that after every interval of “m” the given function f repeats all its values.

Periodic Functions Examples – The sine function, sin a has a period 2 π because 2 π is the smallest number for which the value of sin (a + 2π) = sin a, for all values of a.

We can always 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 equal to 2πB radians.

The reciprocal of the period of a function is equal to its frequency.

Frequency can be defined as the number of cycles completed per second (in a period of one second). If we denote the period of a function by P and let f be its frequency, then the formula for frequency can be written as –f =1/ P.

Fundamental Period of a Function

According to periodic function definition the fundamental period of a function can be defined  as the period of the function which are of the form,

f(x+k)= f(x)

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

Periodic functions examples :-

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

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


Then, h is periodic with period 2.

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Period of Trigonometric Functions -

When you plot the trigonometric functions on a graph, regularly-repeating wave shapes are produced by the trigonometric functions. Like in any wave, the shapes have recognizable features such as peaks (which are also known as high points) and troughs (which are also known as low points). The word period tells you the angular “distance” of one full cycle of the wave that is usually measured between two adjacent peaks or the troughs. For this reason, in Mathematics, you have to measure a function’s period in angle units. This is known as the period of trigonometric function

Here’s for example, is suppose we start at an angle of zero, the trigonometric function produces a smooth curve that rises to a maximum of a value of 1 at π / 2 radians (which equals to 90 degrees), crosses zero at π radians (which is equal to180 degrees), decreases to a minimum of −1 at 3π / 2 radians (which is equal to 270 degrees) and reaches zero again at 2π radians (that is 360 degrees). And after this point, the cycle repeats indefinitely, producing the same features and the same values as the angle increases in the positive x direction.

Different Trigonometric Functions:

For Sine and Cosine Functions:

The sine and cosine functions are trigonometric functions and both have a period of 2π radians. The cosine function is very similar to the sine function, except that it is “ahead” of the sine by a value of π / 2 radians. The sine function takes the value of zero at zero degrees, where as the cosine equals to1 at the same point

For the Tangent Function:

You basically get the tangent function by dividing sine function by cosine function. Its period lies from π radians or 180 degrees. The graph of tangent (x) is zero at angle zero, curves upward and reaches 1 at π / 4 radians (that is equal to 45 degrees), then curves upward again where the tangent function reaches a divide-by-zero point at π / 2 radians. The tangent function then becomes negative infinity and it traces out a mirror image below the y axis, reaching −1 at 3π / 4 radians, and crosses the y axis at π radians that is 180 degrees. Although it has x values at which it becomes undefined, the tangent function does have a definable period.

For Secant, Cosecant and Cotangent Functions:

The three other trigonometric functions, cosecant, secant and cotangent, are the reciprocals of sine, cosine and tangent functions, respectively. In other words, cosecant (x) can be written as 1 / sin(x), secant(x) can be written as 1 / cos(x) and cot(x) can be written as 1 / tan(x). Although their graphs have undefined points, for each of the period of trigonometric functions are the same as for sine, cosine and tangent functions.


How to Find the Period of a Function?

If a function repeats over at a constant period we can call it a periodic function.

According to periodic function definition the period of a function is represented like f(x) = f(x + p), p is equal to the real number and this is the period of the given function f(x).

Period can be defined as the time interval between the two occurrences of the wave.

So, how to find the period of a function actually?

To find the period of the periodic function we can use the following formula, where

Period is equal to 2pb, where b is equal to the coefficient of x.

Periodic functions examples and Questions to be solved :

Question 1) How to find the period of a function for the given periodic function, where f(x) = 9sin(6px7 + 5)

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

Given period = 2pb, here the period of the periodic function = 2p(6p7) = 146  which is equal to 73.

FAQ (Frequently Asked Questions)

1. What is the formula of period and what is the difference between period and frequency?

The formula for time can be written as: T (period) = 1 / f (frequency), λ = c / f = wave speed c (m/s) / frequency f ( in Hz). The unit hertz (Hz) can also be known as cps = cycles per second.

Periods can be defined as the amount of time taken by a wave to complete one full cycle of oscillation or vibration. Frequency, on the contrary, can be defined as the number of complete cycles or oscillations that occurs per second. Period is a quantity that is related to time, whereas frequency is related to the rate.