# 5.2 Periodic functions  (Page 4/5)

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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.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x