Period of a Function: Formulas & Examples

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Periodic functions refer to the functions that are periodic and repeat their values at regular intervals. Graphically, the period of a function is represented as the time interval between two waves.

Waves are represented to show the oscillating or periodic movement of the values of the periodic functions.
The period of a function is always a positive number.
Aperiodic or non-periodic functions are functions that do not repeat or oscillate their values.
For a periodic function of the form, f(x+T)=f(x), the period of the function is T.
It indicates that the function f(x) possesses the same values after an interval of x.
The formula for finding the period of a function is given as:

\(\frac{T}{\mid a \mid}\) or \(\frac{2 \prod}{\mid a \mid}\),

where is the coefficient of the variable, x.

Table of Content

Fundamental Period of a Function
How to find Period of a Function?
Periods of a Trigonometric Functions
Things to Remember
Sample Questions

Key Terms: Periodic Functions, Period of a Function, Functions, Interval, Waves, Period, Oscillate, Algebraic Functions, Trigonometric Functions, Real Numbers, Domain

Fundamental Period of a Function

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Periodic functions are functions that have oscillating or periodic values; that is, the values are repetitive. While plotting a periodic function graphically, the period of a function is represented using the waves.

The period is the time interval between two waves.
It is also known as a cyclic function or periodic waveform.
The repeated part of the function is called the cycle.
It is used in the field of science to describe the oscillation of waveform.

In general, a function is considered periodic if a positive real number T exists such that f (x+T) = f (x) for all x in the domain of f. T is the representation for the period of the function.

Example of Fundamental Period of a Function

Example: Given a periodic function, sin(x+2\(\prod\)) = sinx. The period of the given function, T = 2\(\prod\).

How to find Period of a Function?

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Steps to calculate period of a function are as follows:

A function is known as periodic function if it represent over a constant period of time.
If the period of a function f(x+T)=f(x) is given by T, then the period of the function of the form, f(ax+b)+c is given as \(\frac{T}{\mid a \mid}\) or \(\frac{2 \prod}{\mid a \mid}\).
Period indicates the time interval between occurences of two wave.
If f(x) has a period of T₁ and g(x) has a period of T₂, the period of both the functions f(x) and g(x) are given by the LCM of T₁ and T_2.

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Periods of a Trigonometric Functions

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The periodic function can be represented graphically. When a trigonometric function is plotted on the graph, then it will form a regularly-repeating wave that resembles the periodic function.

The length of one complete cycle on a trigonometric function is called a period.
The distance when two functions start repetition is known as the period of the function.
Sin, Cos and tan are three basic trigonometric functions.
These functions have a period of -2π, 2π and π respectively.

Period of a Sine Function

Consider a function f(x) = sin (xb), where b > 0 then the graph of the function forms complete cycle from 0 to 2π. The period of a sine function is given as 2π/b.

It can be tabulated as follows:

Category	Data
Period	π
Axis	0 [x-axis ]
Minimum value	-1
Domain	{ x : x ∈ R }
Range	[ -1, 1]
Amplitude	1
Maximum value	1

Period of a Tan Function

Consider a function f(x) = tan (xb), where b > 0 then the graph of the function forms complete cycle from -π/2, 0 and π/2. The period of a tan function is given as π/b.

Period of a Function-01.jpg

Periods of Some Standard Periodic Functions

Things to Remember

Period of a function refers to the distance between repetitive function.
It is divided into two categories namely periodic function and aperiodic function.
Functions that have repeating values are called periodic functions.
The functions that do not have repeating values are called aperiodic functions.
Trigonometric functions such as sinx, cosx, cosecx, tanx, etc are examples of periodic functions.
Algebraic functions such as x,x²,, x³, x+2 are aperiodic functions.
For the standard periodic function of the form, f(x+T)=f(x), the period of the function is given by T.

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Sample Questions

Ques. What is the period of the function sin4x + tan2x (2 Marks)

Ans. We have the periodic function, sin4x + tan2x

The period of sin4x = \(\frac{\prod}{2}\)

Also, the period of tan2x = \(\frac{\prod}{2}\)

Hence, we find that the period of the function sin4x + tan2x is \(\frac{\prod}{2}\)

Ques. Find the period of the function f(x) = 6 cos5x (2 Marks)

Ans. We have the periodic function, f(x) = 6 cos5x

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a=5

Hence, the period of the function f(x) = 6 cos5x is \(\frac{2 \prod}{5}\)

Ques. What is the period of the function 9 sin (6x+5) (2 Marks)

Ans. We have the periodic function, 9 sin (6x+5)

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a = 6

Hence, the period of the function 9 sin (6x+5) is \(\frac{2 \prod}{6}\) or \(\frac{\prod}{3}\)

Ques. What is the period of the function y= 2 sin (3x+1) (2 Marks)

Ans. We have the periodic function, y= 2 sin (3x+1)

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a=3

Hence, the period of the function y = 2 sin(3x+1) is \(\frac{2 \prod}{3}\)

Ques. What is the period of the function y = sin (-2x+1) (2 Marks)

Ans. We have the periodic function, y = sin (-2x+1)

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

In the given periodic function, the coefficient of x or a= -2

We know that the period is always a positive number.

So, the period of the function y = sin(-2x+1) is \(??\frac{2 \prod}{-2}\) = 2\(\prod\)

Hence, the period of the function y = sin(-2x+1) is 2\(\prod\)

Ques. Is it possible to find the period of the function y = log (2x) (2 Marks)

Ans. We have the given function, y = log (2x)

Here, y = log (2x) is a logarithmic function.

We know that a logarithmic function is never periodic as they cannot repeat themselves. The logarithmic functions either increase or decrease.

Hence, the logarithmic function,y = log (2x) is not a periodic function and it is not possible to find its period.

Ques. Find the period of the function sinx + sin2x (2 Marks)

Ans. We have the periodic function, sinx + sin2x

The period of sinx = 2\(\prod\)

Also, the period of sin2x =\(\prod\)

The LCM of and 2 is 2\(\prod\)

Hence, we find that the period of the function sinx + sin2x is 2\(\prod\)

Ques. What is the period of the function y = 2sinx (2 Marks)

Ans. We have the periodic function, y= 2 sinx

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a=1

Hence, the period of the function y= 2 sinx is 2\(\prod\)

Ques. What is the period of the function f(x) = cos (7x-5) (2 Marks)

Ans. We have the periodic function, f(x) = cos (7x-5)

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a=7

Hence, the period of the function f(x) = cos (7x-5) is \(\frac{2\prod}{7}\)

Ques. Find the period of the function y = sinx + cos3x (2 Marks)

Ans. We have the periodic function, y = sinx + cos3x

The period of sinx = 2\(\prod\)

Also, the period of cos3x = \(\frac{2\prod}{3}\)

The LCM of 2 and \(\frac{2\prod}{3}\) is 2\(\prod\)

Hence, the period of the function y = sinx + cos3x is 2\(\prod\)

Ques. What is the period of the function y = 5 cos3x (2 Marks)

Ans. We have the periodic function, y = 5 cos3x

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a=3

Hence, the period of the functiony = 5 cos3x is \(\frac{2\prod}{3}\)

Ques. Find the period of the function f(x) = cot3x – cos(4x+3) (2 Marks)

Ans. We have the periodic function, f(x) = cot3x – cos(4x+3)

The period of cot3x = \(\frac{\prod}{3}\)

Also, the period of cos(4x+3) = \(\frac{\prod}{2}\)

The LCM of \(\frac{\prod}{3}\) and \(\frac{\prod}{2}\) is \(\prod\)

Hence, the period of the function f(x) = cot3x – cos(4x+3) is \(\prod\)

Ques. Find the period of the function y = 3 cos(-6x+2) (2 Marks)

Ans. We have the periodic function, y = 3 cos(-6x+2)

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

In the given periodic function, the coefficient of x or a = -6

We know that the period is always a positive number.

So, the period of the function y = 3 cos(-6x+2) is \(\frac{2\prod}{-6}\) = \(\frac{\prod}{3}\)

Hence, the period of the function y = 3 cos(-6x+2) is \(\frac{\prod}{3}\)

Ques. Find the period of the function y = sin\(\frac{x}{2}\) – cos\(\frac{x}{3}\) (2 Marks)

Ans. We have the periodic function, y = sin\(\frac{x}{2}\) - cos\(\frac{x}{3}\)

The period of sin\(\frac{x}{2}\) = 4\(\prod\)

Also, the period of cos\(\frac{x}{3}\) = 6\(\prod\)

The LCM of 4\(\prod\) and 6\(\prod\) is 12\(\prod\)

Hence, the period of the function y = sin\(\frac{x}{2}\) – cos\(\frac{x}{3}\) is 12\(\prod\)

Ques. Find the period of the function f(x) = 5cos4x (2 Marks)

Ans. We have the periodic function, f(x) = 5 cos4x

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a=4

Hence, the period of the function f(x) = 5 cos4x is \(\frac{2 \prod}{4}\)

Ques. What is the period of the function 10 sin (5x+8) (2 Marks)

Ans. We have the periodic function, 10 sin (5x+8)

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a = 5

Hence, the period of the function 10 sin (5x + 8) is \(\frac{2 \prod}{5}\)

Ques. What is the period of the function y= 12 sin (3x+10) (2 Marks)

Ans. We have the periodic function, y= 12 sin (3x+10)

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a=3

Hence, the period of the function y = 12 sin(3x+10) is \(\frac{2 \prod}{3}\)

Ques. Is it possible to find the period of the function y = log (3x) (2 Marks)

Ans. We have the given function, y = log 32x)

Here, y = log (3x) is a logarithmic function.

We know that a logarithmic function is never periodic as they cannot repeat themselves. The logarithmic functions either increase or decrease.

Hence, the logarithmic function,y = log (3x) is not a periodic function and it is not possible to find its period.

Ques. What is the period of the function y= sin (13x+9) (2 Marks)

Ans. We have the periodic function, y= sin (13x+9)

The formula for finding periodic function is given as, \(\frac{2 \prod}{\mid a \mid}\)

Here, in the given periodic function, the coefficient of x or a=13

Hence, the period of the function y = sin(13x+9) is \(\frac{2 \prod}{13}\)

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Period of a Function: Formulas & Examples

Fundamental Period of a Function

Example of Fundamental Period of a Function

How to find Period of a Function?

Periods of a Trigonometric Functions

Period of a Sine Function

Period of a Tan Function

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

Comments

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Period of a Function: Formulas & Examples

Fundamental Period of a Function

Example of Fundamental Period of a Function

How to find Period of a Function?

Periods of a Trigonometric Functions

Period of a Sine Function

Period of a Tan Function

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2.Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6.A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

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1.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.