Periodic Function


These are the function, whose value repeats after a fixed constant interval called period, and which makes a class of a widely used function.

        A function f of x, such that:

        f(T + x) = f(x) ∀ x ε domain of f.

The least positive real value of T for, which above relation is true, is called the fundamental period or just the period of the function.

        e.g. for f(x) = sin x ∀ x ε R.

We know that sin (2∏ + x) = sin x, ∀ x ε R

so f(x) = sin x is a periodic function with a period of 2∏ radians.

Rules for finding the period of the Periodic Functions

(i) If f(x) is periodic with period p, then a f(x) + b, where a, b ε R (a≠0) is also a periodic function with period p.

(ii) If f(x) is periodic with period, then f(ax + b), where a ε R -{0} and b ε R, is also periodic with period p/|a|.

(iii) let us suppose that f(x) is periodic with period p and g(x) is periodic with period q. Let r be the L.C.M. of p and q, if it exists.

(a) If f(x) and g(x) cannot be interchanged by adding a least positive number k, then r is the period of f(x) + g(x).

(b) If f(x) and g(x) can be interchanged by adding a least positive number k and if k < r, then k is the period of f(x) + g(x). Otherwise r is the period.

Illustration:  Find the period of the following functions

(i) f(x) = sinx + {x}   

(ii) f(x) = tan(x/3) + sin 2x.

(iii) f(x) = |sinx| + |cosx|      

(iv) f(x) = ((1+sin x)(1+sec x))/((1+cos x)(1+cosec x))


(i) Here f(x) = sinx + {x}

Period of sinx is 2p and that of {x} is 1. But the L.C.M. of 2p and 1 does not exist. Hence sinx + {x} is not periodic.

(ii) Here f(x) = tanx/3 + sin2x. Here tan(x/3). Here tan(x/3) is periodic with period 3p and sin2x is periodic with period p. 

Hence f(x) will be periodic with period 3p.

(iii) Here f(x) = |sinx| + |cosx|

Now, |sinx| = √sin2x = √((1+cos2x)/2),              which is periodic with period ∏.

Similarly, |cosx| is periodic with period ∏.

Hence, according to rule of LCM, period of f(x) must be ∏.

But |sin((∏/2)+x)| = |cos x| and |cos((∏/2)+x)| = |sin x|.

Since ∏/2 < ∏, period of f(x) is ∏/2.

(iv)   f(x) =((1+sin x)(1+sec x))/((1+cos x)(1+cosec x))  =

               ((1+sin x)(1+cos x)sin x)/((1+cos x)(1+sin x)cos x) = tan x

Hence f(x) has period ∏. 

Note:        For f(x) = |sin x| + |cos x|

                The period of both |sin x| and |cos x| is ∏

                But they are related with phase difference ∏/2  i.e.

                |sin x| = |cos (x + ∏/2)|

                |cos x| = |sin (x + ∏/2)|

So the period of the function f(x) is ∏/2.

Illustration:       Prove that the period of y = sin x is 2∏


        Let T be the period of f(x) = sin x

i.e.    f(T + x) = f(x)

        =>  sin (T + x) = sin x

        => T + x = n∏ + (-1)nx, n ε 1                        .........(i)

Let    n = 0

        T + x = x

        => T = 0

But we want a positive real value for T.

Let  n = 1,

        T + x = ∏ - x

        => T - ∏ = 2x              (Is it possible? Think)

No it is not possible because LHS is constant and RHS is a continuous variable.

Now, Let n = 2 in equation (i)

        T + x = 2∏ + x

       => T = 2∏

Therefore period of y = sin x is 2∏

Note: If we cannot find T independent of x, then y = f(x) is not periodic.


Find the period of y = cos √x and y = x sin x if possible

Ans. These are non-periodic function 

You can register at the website to have a free demo lecture and have a feel of the live online classroom programmes offered by askIITians for IIT JEE, AIEEE and other engineering examinations. A comprehensive study material for IIT JEE, AIEEE and other engineering examinations is available online free of cost at

To read more, Buy study materials of Set Relations and Functions comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.

Course Features

  • Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution