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The argument to the sine and cosine functions

If you think of x as being a measurement of time in seconds, and 1/n being a measurement of frequency in cycles/second, the arguments to the sine and cosine functions can be viewed as:

2*pi(radians/cycle)*(x sec)*(1/n)( cycle/sec)

If you cancel out like terms, this reduces to:

2*pi(radians)*(x )*(1/n )

Thus, with a fixed value of n , for each value of x , the argument represents an angle in radians, which is what is required for use withthe functions of the Java Math library.

Where does sin(arg) equal zero ?

The value of the sine of an angle goes through zero at every integer multiple of pi radians. This explanation will probably make more sense if you refer back to Figure 3 .

The curve in Figure 3 was calculated and plotted for n equal to 50. The sine curve has a zero crossing for every value of x such that x is a multiple of n/2, or 25.

Where are the peaks in the cosine function ?

Similarly, the peaks in the cosine curve in Figure 3 occur for every value of x such that x is a multiple of n/2, or 25.

Composition and decomposition

In theory, it is possible to decompose any time series into a number (quite possibly a very large number) of sine and cosine functions each having its own amplitude and frequency. (In a future module, we will learn how this is possible using a Fourier series or a Fourier transform.)

Conversely, it is theoretically possible to create any time series by adding together just the right combination of sine and cosine functions, each havingits own amplitude and frequency.

An approximate square waveform

As an example of composition, suppose that I need to create a time series that approximates a square waveform, as shown at the bottom of Figure 6 .

Figure 5. An approximate square waveform.
missing image

I can create such a waveform by adding together the correct combination of sinusoids, each having its own frequency and amplitude.

Figure 6. An improved approximate square waveform.
missing image

Successive approximations

The ten curves plotted in Figure 5 and Figure 6 show successive approximations to the creation of the desired square waveform. The bottom curvein Figure 6 is a plot of the following sinusoidal expression containing the algebraic sum of ten sinusoidal terms.

cos(2*pi*x/50) - cos(2*pi*x*3/50)/3+ cos(2*pi*x*5/50)/5 - cos(2*pi*x*7/50)/7+ cos(2*pi*x*9/50)/9 - cos(2*pi*x*11/50)/11+ cos(2*pi*x*13/50)/13 - cos(2*pi*x*15/50)/15+ cos(2*pi*x*17/50)/17 - cos(2*pi*x*19/50)/19

Each curve contains more sinusoidal terms

The top curve in Figure 5 is a plot of only the first sinusoidal term shown above. It is a pure cosine curve.

Each successive plot, moving down the page in Figure 5 and Figure 6 adds another term to the expression being plotted, until all ten terms are includedin the bottom curve in Figure 6 .

Reasonably good approximation

As you can see, the bottom curve in Figure 6 is a reasonably good approximation to a square wave, but it is not perfect.

(A perfect square wave would have square corners, a flat top, no ripple, and perfectly vertical sides.)

Each term improves the approximation

If you start at the top of Figure 5 and examine the successive curves, you will see that the approximation to a square wave improves as each new sinusoidalterm is added.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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