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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.
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 FunctionExample: 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 T2.
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| Trigonometry Values | Secant Function | Sin Cos Formulas |
| Law of Tangents | Law of Sines | |
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:
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.
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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| Heights and Distances | Value of Sin 180 | Sin 30 Degrees |
| Cosine Rule | Value of Cos 180 | Angle Between Two Lines |
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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