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Sine function

Redefined domain of function

Domain of sine = [ - π 2 , π 2 ]

Range of sine = [ - 1, 1 ]

This redefinition renders sine function invertible. Clearly, the domain and range are exchanged for the inverse function. Hence, domain and range of the inverse function are :

Domain of arcsine = [ - 1, 1 ]

Range of arcsine = [ - π 2, π 2 ]

Therefore, we define arcsine function as :

f : [ - 1,1 ] [ - π 2 , π 2 ] by f(x) = arcsin(x)

The arcsin(x) .vs. x graph is shown here.

Arcsine function

The arcsine function .vs. real value

Arccosine function

The arccosine function is inverse function of trigonometric cosine function. From the plot of cosine function, it is clear that an interval between 0 and π includes all possible values of cosine function only once. Note that end points are included. The redefinition of domain of trigonometric function, however, does not change the range.

Cosine function

Redefined domain of function

Domain of cosine = [ 0, π ]

Range of cosine = [ - 1, 1 ]

This redefinition renders cosine function invertible. Clearly, the domain and range are exchanged for the inverse function. Hence, domain and range of the inverse function are :

Domain of arccosine = [ - 1,1 ]

Range of arccosine = [ 0, π ]

Therefore, we define arccosine function as :

f : [ - 1,1 ] [ 0, π ] by f(x) = arccos(x)

The arccos (x) .vs. x graph is shown here.

Arccosine function

The arccosine function .vs. real value

Arctangent function

The arctangent function is inverse function of trigonometric tangent function. From the plot of tangent function, it is clear that an interval between - π / 2 and π / 2 includes all possible values of tangent function only once. Note that end points are excluded. The redefinition of domain of trigonometric function, however, does not change the range.

Tangent function

Redefined domain of function

Domain of tangent = - π 2, π 2

Range of tangent = R

This redefinition renders tangent function invertible. Clearly, the domain and range are exchanged for the inverse function. Hence, domain and range of the inverse function are :

Domain of arctangent = R

Range of arctangent = - π / 2, π / 2

Therefore, we define arctangent function as :

f : R - π 2 , π 2 by f(x) = arctan (x)

The arctan(x) .vs. x graph is shown here.

Arctangent function

The arctangent function .vs. real value

Arccosecant function

The arccosecant function is inverse function of trigonometric cosecant function. From the plot of cosecant function, it is clear that union of two disjointed intervals between “ - π / 2 and 0” and “0 and π / 2 ” includes all possible values of cosecant function only once. Note that zero is excluded, but “ - π / 2 “ and “ π / 2 ” are included . The redefinition of domain of trigonometric function, however, does not change the range.

Cosecant function

Redefined domain of function

Domain of cosecant = [ - π / 2, π / 2 ] { 0 }

Range of cosecant = - , - 1 ] [ 1, = R - 1, 1

This redefinition renders cosecant function invertible. Clearly, the domain and range are exchanged for the inverse function. Hence, domain and range of the inverse function are :

Domain of arccosecant = R - 1, 1

Range of arccosecant = [ - π / 2, π / 2 ] { 0 }

Therefore, we define arccosecant function as :

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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