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Periodic Formulas with Examples

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  • Last Updated : 24 Jul, 2022
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A period is defined as the time interval between two points in time, and a periodic function is defined as a function that repeats itself at regular intervals or periods in time. In other words, a periodic function is a function whose values recur after a specific time interval. A periodic function is represented as f(x + p) = f(x), where “p” is the period of the function. Sine wave, triangular wave, square wave, and sawtooth wave are some examples of periodic functions. Below are graphs of some periodic functions, and we can observe that each periodic function’s graph has translational symmetry.

Examples of Periodic Function

Fundamental period of a function

A periodic function’s domain encompasses all real number values while its range is specified for a fixed interval. A periodic function is one in which there exists a positive real number P such that f (x + p) = f (x), for all x being real numbers. The fundamental period of a function is the least value of the positive real number P or the period during which a function repeats itself.

f(x + P) = f(x)

where,

P is the period of the function and f is the periodic function.

How to determine the Period of a Function?

  1. A periodic function is defined as a function that repeats itself at regular intervals or periods.
  2. It is represented as f(x + p) = f(x), where “p” is the period of the function, p ∈ R.
  3. Period means the time interval between the two occurrences of the wave.

Periods of Trigonometric Functions

Trigonometric Functions are periodic functions and the period of Trigonometric Functions are as follows

  • The period of Sin x and Cos x is .

i.e. sin(x + 2π) = sin x and cos(x + 2π) = cos x

  • The period of Tan x and Cot x is π.

i.e. tan(x + π) = tan x and cot(x + π) = cot x

  • The period of Sec x and Cosec x is 2π.

i.e. sec(x + 2π) = sec x and cosec(x + 2π) = cosec x

The period of the function is referred to as the distance between the repetitions of any function. The period of a trigonometric function is the length of one complete cycle. Amplitude is defined as the maximum displacement of a particle in a wave from equilibrium. In simple words, it is the distance between the highest or lowest point and the middle point on the graph of a function. In trigonometry, there are three fundamental functions, namely, sin, cos, and tan, whose periods are 2π, 2π, and π periods, respectively. The starting point of the graph of any trigonometric function is taken as x = 0.

For example, if we observe the cosine graph given below, we can see that the distance between two occurrences is 2π, i.e., the period of the cosine function is 2π. Its amplitude is 1.

Cosine function Graph

Cosine graph

Periodic Formulae

  • If “p” is the period of the periodic function f (x), then 1/f (x) is also a periodic function and will have the same fundamental period of p as f(x).

If f (x + p) = f (x),

F (x) = 1/f (x), then F (x + p) = F (x).

  • If “p” is the period of the periodic function f(x), then f (ax + b), a>0 is also a periodic function with a period of p/|a|.
  • The period of Sin (ax + b) and Cos (ax + b) is 2π/|a|.
  • The period of Tan (ax + b) and Cot (ax + b) is π/|a|.
  • The period of Sec (ax + b) and Cosec (ax + b) is 2π/|a|.
  • If “p” is the period of the periodic function f(x), then af(x) + b, a>0 is also a periodic function with a period of p.
  • The period of [a Sin x + b] and [a Cos x + b] is 2π.
  • The period of [a Tan x + b] and [a Cot x + b] is π.
  • The period of [a Sec x + b] and [a Cosec x + b] is 2π.

Practice Problems based on Periodic function

Problem 1: Determine the period of the periodic function cos(5x + 4).

Solution:

Given function: cos (5x + 4)

The coefficient of x = a = 5.

We know that,

The period of cos x is 2π.

So, the period of cos(5x + 4) is 2π/ |a| = 2π/5.

Hence, the period of cos(5x + 4) is 2π/5.

Problem 2: Find the period of f(x) = cot 4x + sin 3x/2.

Solution:

Given periodic function: f(x) = cot 4x + sin 3x/2

We know that,

The period of cot x is π and the period of sin x is 2π.

So, the period of cot 4x is π/4.

So, the period of sin 3x/2 is 2π/(3/2) = 4π/3.

Now, the calculation of the period of the function f(x) = cot 4x + sin 3x/2 is,

Period of f(x) = (LCM of π and 4π)/(HCF of 3 and 4) = 4π/1 = 4π.

Therefore, the period of cot 4x + sin 3x/2 is 4π.

Problem 3: Sketch the graph of y = 3 sin 3x+ 5.

Solution:

Given that y = 3 sin 3x + 5

The given wave is in the form of y = a sin bx + c

From the above graph, we can write the following:

  1. Period = 2π/|b| = 2π/3
  2. Axis: y = 0 [x-axis ]
  3. Amplitude: 3
  4. Maximum value = (3 × 1) + 5 = 8
  5. Minimum value = (3 × -1) + 5 = 2
  6. Domain: { x : x ∈ R }
  7. Range = [ 8, 2]

Problem 4: Determine the period of the given periodic function 5 sin(2x + 3).

Solution:

Given function: 5 sin(2x + 3)

The coefficient of x = a = 2.

We know that,

The period of cos x is 2π.

So, the period of 5 sin(2x + 3) is 2π/ |a| = 2π/2 = π.

Hence, the period of 5 sin(2x + 3) is π.

Problem 5: Find the period of f (x) = tan 3x + cos 5x.

Solution:

Given periodic function: f(x) =tan 3x + cos 6x.

We know that,

The period of tan x is π and the period of cos x is 2π.

So, the period of tan 3x is π/3.

So, the period of cos 6x is 2π/5.

Now, the calculation of the period of the function f(x) = tan 3x + cos 6x is,

Period of f(x) = (LCM of π and 2π)/(HCF of 3 and 5) = 2π/1 = 2π.

Therefore, the period of f (x) = tan 3x + cos 5x is 2π.

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