5.2 Periodic functions  (Page 5/5)

If “a” and “b” are non-zero real number and functions g(x) and h(x) are periodic functions having periods, “ $T_1$ ” and “ $T_2$ ”, then function $f\left(x\right)=ag\left(x\right)\pm bh\left(x\right)$ is also a periodic function. The period of f(x) is LCM of " $T_1$ " and " $T_2$ ".

Finding lcm

LCM of integral numbers is obtained easily. There is, however, difficulty in finding LCM when numbers are fractions (like 3/4, 1/3 etc.) or irrational numbers (like π, 2√2 etc.).

For rational fraction, we can find LCM using following formula :

$$\mathrm{LCM}=\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$$

Consider fractions 3/5 and 2/3. The LCM of numerators 3 and 2 is 6. The HCF of denominators is 1. Hence, LCM of two fractions is 6/1 i.e. 6.

This rule also works for irrational numbers of similar type like 2√2/3 , 3√2/5 etc or π/2, 3π/2 etc. However, we can not find LCM of irrational numbers of different kind like 2√2 and π. Similarly, there is no LCM for combination of rational and irrational numbers.

For example, LCM of π/3 and 3π/2 is :

$$\mathrm{LCM}=\frac{\text{LCM of numerators}}{\text{HCF of denominators}}=\frac{\text{LCM of (π,3π)}}{\text{HCF of (2,3)}}=\frac{\mathrm{3\pi }}{1}=\mathrm{3\pi }$$

Exception to lcm rule

LCM rule is not always true. There are exceptions to this rule. We do not apply this rule, when functions are co-functions of each other or when functions are even functions. Further, if individual periods are rational and irrational numbers respectively, then LCM is not defined. As such, this rule can not be applied in such situation as well.

Two functions f(x) and g(y) are cofunctions if x and y are complimentary angles. The functions sinx and consx are cofunctions as :

$$\sin x=\cos (\frac{\pi }{2}-x)$$

Similarly, |cosx| and |sinx| are cofunctions as :

$$\left|\cos x\right|=\left|\sin \right(\frac{\pi }{2}-x\left)\right|$$

In such cases where LCM rule is not applicable, we proceed to apply definition of periodic function to determine period.

Important results

The results obtained in earlier sections are summarized here for ready reference.

1: All trigonometric functions are periodic on “R”. The functions $\sin x,\cos x,\sec x\text{and}\mathrm{cosec}x$ have periodicity of " $2\pi$ ". On the other hand, periodicity of $\tan x\text{and}\cot x$ is $\pi$ .

2: Functions ${\sin }^{n}x,{\cos }^{n}x,{\mathrm{cosec}}^{n}x\text{and}{\sec }^{n}x$ are periodic on “R” with period “ $\pi$ ” when “n” is even and “ $2\pi$ ” when “n” is fraction or odd. On the other hand, Functions ${\tan }^{n}x\text{and}{\cot }^{n}x$ are periodic on “R” with period “ $\pi$ ” whether n is odd or even.

3: Functions $\left|\sin x\right|,\left|\cos x\right|,\left|\tan x\right|,\left|\mathrm{cosec}x\right|,\left|\sec x\right|\text{and}\left|\cot x\right|$ are periodic on “R” with period “ $\pi$ ”.

4: A constant function is a periodic function without any fundamental period. For example,

$$f\left(x\right)=c$$

$$f\left(x\right)={\sin }^{2}x+{\cos }^{2}x$$

5: If “T” is the period of f(x), then period of function of the form given below is “T/|b|” :

$$af(bx+c)+d;\mathrm{a,b,c,d}\in Z$$

6: If f(x) is a periodic function with period “T” and g(x) is one one function (bijection), then "gof" is also periodic with period “T”.

7: If f(x) is a periodic function with a period T and its domain is a proper subset of domain of g(x), then gof(x) is a periodic function with a period T.