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Properties of Periodicity

  • Constant function is a periodic function without any period. This happens because of the non-existence of the least positive real number which is due to the continuity of real number system.
  • If f(x) has it’s period T then f(ax + b) has its period .
  • If f(x) has its period T1 and g(x) has its period                T2 then   (af(x) + bg(x))  has  its  period                         £ L.C. M.(T1, T2).  Moreover if f(x) and g(x) are basic trigonometric functions then period of                        [af(x) + bg(x)]  =  L.C.M. (T1, T2)

Examine whether sin x is a periodic function or not. If so, find its period.

Given f(x) = sin x. Let’s assume sin x to be periodic. So, it must have some positive value independent of x say T such that f(x + T) = f(x)

  • sin (x + T) = sin x
  • x + T = n p + (–1)n x where n = 0, ± 1, ± 2 ……

The positive values of T independent of x are given by n p where n = 2, 4, 6……..

Further according to definition for periodic number, it should be least. So, here we have T = 2p.

Thus, it is proved that sin x is periodic function having periodicity 2p.

Prove that f(x) = sin√x  is not a periodic function.

Proof : Let the positive real number T be such that f (x + T) = f(x)

mat2.1

This above relation does not give any positive value of T independent of x because it holds only when T = 0.

f(x) is non-periodic function.

Let f(x) = x – [x] where [x] is the greatest integer less than or equal to x. Find out the periodicity of f(x). Assume f(x + T) = f(x)

  • (x + T) – [x + T] = x- [x]
  • T = [x + T] – [x] = an integer.

Hence least positive value of T independent of x is 1.

Thus f(x) is a periodic function of period 1.

This can be explained through graphs. As in the case of algebraic function, we can have same idea about the nature of a trigonometric function by its graph.

The variations in the values of the trigonometric ratios generated the concept of graph in trigonometric functions.

mat2.2

From the graph, we observe that:

  • The value of sin x repeats itself after an interval of 2p. So sin x is a periodic function with period of 2p. Actually a revolution of 2p is the complete revolution.
  • sin x takes value from –1 to 1.

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