5.2 Periodic functions  (Page 4/5)

 Page 4 / 5

In the nutshell, if “T” is the period of f(x), then period of function of the form given below id “T/|b|” :

$af\left(bx+c\right)+d;\phantom{\rule{1em}{0ex}}\mathrm{a,b,c,d}\in Z$

Problem : What is the period of function :

$f\left(x\right)=3+2\mathrm{sin}\left\{\frac{\left(\pi x+2\right)}{3}\right\}$

Solution : Rearranging, we have :

$f\left(x\right)=3+2\mathrm{sin}\left(\frac{\pi }{3}x+\frac{2}{3}\right)$

The period of sine function is “ $2\pi$ ”. Comparing with function form " $af\left(bx+c\right)+d$ ", magnitude of b i.e. |b| is π/3. Hence, period of the given function is :

$⇒T\prime =\frac{T}{|b|}=\frac{2\pi }{\frac{\pi }{3}}=6$

Modulus of trigonometric functions and periods

The graphs of modulus of a function are helpful to determine periods of modulus of trigonometric functions like |sinx|, |cosx|, |tanx| etc. We know that modulus operation on function converts negative function values to positive function values with equal magnitude. As such, we draw graph of modulus function by taking mirror image of the corresponding core graph in x-axis. The graphs of |sinx| and |cotx| are shown here :

From the graphs, we observe that periods of |sinx| and |cotx| are π. Similarly, we find that periods of modulus of all six trigonometric functions are π.

Integral exponentiation of trigonometric function and periods

The periods of trigonometric functions which are raised to integral powers, depend on the nature of exponents. The periods of trigonometric exponentiations are different for even and odd powers. Following results with respect these exponentiated trigonometric functions are useful :

Functions ${\mathrm{sin}}^{n}x,{\mathrm{cos}}^{n}x,{\mathrm{cosec}}^{n}x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathrm{sec}}^{n}x$ 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 ${\mathrm{tan}}^{n}x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathrm{cot}}^{n}x$ are periodic on “R” with period “ $\pi$ ” whether n is odd or even.

Problem : Find period of ${\mathrm{sin}}^{2}x$ .

Solution : Using trigonometric identity,

$⇒{\mathrm{sin}}^{2}x=\frac{1+\mathrm{cos}\mathrm{2x}}{2}$

$⇒{\mathrm{sin}}^{2}x=\frac{1}{2}+\frac{\mathrm{cos}\mathrm{2x}}{2}$

Comparing with $af\left(bx+c\right)+d$ , the magnitude of “b” i.e. |b| is 2. The period of cosine is 2π. Hence, period of ${\mathrm{sin}}^{2}x$ is :

$⇒T=\frac{\mathrm{2\pi }}{2}=\pi$

Problem : Find period of function :

$f\left(x\right)={\mathrm{sin}}^{3}x$

Writing identity for " ${\mathrm{sin}}^{3}x$ ", we have :

$⇒f\left(x\right)={\mathrm{sin}}^{3}x=\frac{3\mathrm{sin}x-\mathrm{sin}3x}{4}=\frac{3}{4}\mathrm{sin}x-\frac{3}{4}\mathrm{sin}3x$

We know that period of “ag(x)” is same as that of “g(x)”. The period of first term of “f(x)”, therefore, is equal to the period of “sinx”. Now, period of “sinx” is “2π”. Hence,

$⇒{T}_{1}=2\pi$

We also know that period of g(ax+b) is equal to the period of g(x), divided by “|a|”. The period of second term of “f(x)”, therefore, is equal to the period of “sinx”, divided by “3”. Now, period of “sinx” is “2π”. Hence,

$⇒{T}_{2}=\frac{2\pi }{3}$

Applying LCM rule,

$⇒T=\frac{\text{LCM of}\phantom{\rule{1em}{0ex}}2\pi \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}2\pi }{\text{HCF of}\phantom{\rule{1em}{0ex}}1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}3}=\frac{\mathrm{2\pi }}{1}=\pi$

Lcm rule for periodicity

When two periodic functions are added or subtracted, the resulting function is also a periodic function. The resulting function is periodic when two individual periodic functions being added or subtracted repeat simultaneously. Consider a function,

$\mathrm{f\left(x\right)}=\mathrm{sinx}+\mathrm{sin}\frac{x}{2}$

The period of sinx is 2π, whereas period of sinx/2 is 4π. The function f(x), therefore, repeats after 4π, which is equal to LCM of (least common multiplier) of the two periods. It is evident from the graph also.

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