# 5.2 Periodic functions  (Page 2/5)

 Page 2 / 5

## Basic periodic functions

Not many of the functions that we encounter are periodic. There are few functions, which are periodic by their very definition. We are, so far, familiar with following periodic functions in this course :

• Constant function, (c)
• Trigonometric functions, (sinx, cosx, tanx etc.)
• Fraction part function, {x}

Six trigonometric functions are most commonly used periodic functions. They are used in various combination to generate other periodic functions. In general, we might not determine periodicity of each function by definition. It is more convenient to know periods of standard functions like that of six trigonometric functions, their integral exponents and certain other standard forms/ functions. Once, we know periods of standard functions, we use different rules, properties and results of periodic functions to determine periods of other functions, which are formed as composition or combination of standard periodic functions.

## Constant function

For constant function to be periodic function,

$f\left(x+T\right)=\mathrm{f\left(x\right)}$

By definition of constant function,

$f\left(x+T\right)=\mathrm{f\left(x\right)}=c$

Clearly, constant function meets the requirement of a periodic function, but there is no definite, fixed or least period. The relation of periodicity, here, holds for any change in x. We, therefore, conclude that constant function is a periodic function without period.

## Trigonometric functions

Graphs of trigonometric functions (as described in the module titled trigonometric function) clearly show that periods of sinx, cosx, cosecx and secx are “2π” and that of tanx and cotx are “π”. Here, we shall mathematically determine periods of few of these trigonometric functions, using definition of period.

## Sine function

For sinx to be periodic function,

$\mathrm{sin}\left(x+T\right)=\left(x\right)$

$x+T=n\pi +{\left(-1\right)}^{n}x;\phantom{\rule{1em}{0ex}}n\in Z$

The term ${\left(-1\right)}^{n}$ evaluates to 1 if n is an even integer. In that case,

$x+T=n\pi +x$

Clearly, T = nπ, where n is an even integer. The least positive value of “T” i.e. period of the function is :

$T=2\pi$

## Cosine function

For cosx to be periodic function,

$\mathrm{cos}\left(x+T\right)=\mathrm{cos}x$

$⇒x+T=2n\pi ±x;\phantom{\rule{1em}{0ex}}n\in Z$

Either,

$⇒x+T=2n\pi +x$

$⇒T=2n\pi$

or,

$⇒x+T=2n\pi -x$

$⇒T=2n\pi -2x$

First set of values is independent of “x”. Hence,

$T=2n\pi ;\phantom{\rule{1em}{0ex}}n\in Z$

The least positive value of “T” i.e. period of the function is :

$T=2\pi$

## Tangent function

For tanx to be periodic function,

$\mathrm{tan}\left(x+T\right)=\mathrm{tan}x$ $x+T=n\pi +x;\phantom{\rule{1em}{0ex}}n\in Z$

Clearly, T = nπ; n∈Z. The least positive value of “T” i.e. period of the function is :

$T=\pi$

## Fraction part function (fpf)

Fraction part function (FPF) is related to real number "x" and greatest integer function (GIF) as $\left\{x\right\}=x-\left[x\right]$ . We have seen that greatest integer function returns the integer which is either equal to “x” or less than “x”. For understanding the nature of function, let us compute few function values as here :

--------------------------------- x [x]x – [x] ---------------------------------1 1 0 1.25 1 0.251.5 1 0.5 1.75 1 0.752 2 0 2.25 2 0.252.5 2 0.5 2.75 2 0.753 3 0 3.25 3 0.253.5 3 0.5 3.75 3 0.754 4 0 ---------------------------------

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x By Qqq Qqq       By By By