3.11 Trigonometric functions  (Page 2/5)

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$\text{Range}=\left[-A,A\right]$

We now consider yet another form of sine function which is given as :

$f\left(x\right)=A\mathrm{sin}\left(kx\right)$

Multiplying argument x of sine function by a constant k does not change the nature of plot. However, it changes the periodicity of the function. Recall that if T is the period of function f(x), then period of function $af\left(kx±b\right)$ is $\frac{T}{|k|}$ Clearly, the period of sin(kx) is $\frac{T}{|k|}$ . If |k| is less than 1, then period is more than 2π and if |k| is greater than 1, then period is less than 2π.

Problem : Find domain and range of function :

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

Solution : We know that domain of sinx is real number set R and range is [-1,1]. The given function is real for all real values of x. Hence, its domain remains R. On the other hand, minimum and maximum values of function changes from that corresponding to sinx function :

${y}_{\text{min}}=-1+2=1$ ${Y}_{\mathrm{max}}=1+2=3$

Hence, range of given function is [1,3]. It is evident that graph of function is that of graph of sinx shifted up by 2 units.

Cosine function

For each real number “x”, there is a cosine function defined as :

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

The plot of cos(x) .vs. x is shown here.

The plot, here, is continuous and period is "2π". Think period of the function in term of minimum segment which can be used to extend the plot on either side. Further as cos(-x) = cosx, cosine function is an even function. This fact is also substantiated by the fact that plot is symmetric about y-axis.

Since function holds for all values of “x”, its domain is “R”. On the other hand, the values of cosine function is bounded between “-1” and “1”, inclusive of end points. Hence, domain and range of sine function are :

$\text{Domain}=R$

$\text{Range}=\left[-1,1\right]$

When cosine function is given as f(x) = Acosx, maximum and minimum values of function becomes -A and A. The range is modified as :

$\text{Range}=\left[-A,A\right]$

When cosine function is given as f(x) = Acos(kx), the period of cosine function is given by $\frac{T}{|k|}$ .

Problem : Find domain range of the function :

$f\left(x\right)=12\mathrm{sin}x+5\mathrm{cos}x$

Solution : The given function comprises of sine and cosine functions. Here, we reduce given function in terms of one trigonometric function and then find range of the function. This reduction is required as otherwise it would be difficult to estimate when two trigonometric functions together evaluates to minimum and maximum values. Let us put,

$a\mathrm{cos}\alpha =12$ $a\mathrm{sin}\alpha =5$

Clearly, $a=\sqrt{{12}^{2}+{5}^{2}}=13$ . Putting these values/ expression in function,

$f\left(x\right)=13\left(\mathrm{cos}\alpha \mathrm{sin}x+\mathrm{sin}\alpha \mathrm{cos}x\right)=13\mathrm{sin}\left(x+\alpha \right)$

We know that range of sine function is [-1,1]. Hence, range of given function is :

$\text{Range}\left[-13,13\right]$

Tangent function

For a real number “x”, there is a tangent function defined as :

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

Note that defining statement defines the function for a real number “x” – not for "each" real “x” as in the case of sine and cosine functions. It is so because, tangent function is not defined for all real values of “x”. Let us recall that :

$⇒\mathrm{tan}x=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$

This is a rational polynomial form, which is defined for $\mathrm{cos}\left(x\right)\ne 0$ . Now, cos(x) evaluates to zero for certain values of “x”, which appears at a certain interval given by the condition,

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
hey
Giriraj
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x