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f : R - 1, 1 [ - π 2 , π 2 ] { 0 } by f(x) = arccosec (x)

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

Arccosecant function

The arccosecant function .vs. real value

Arcsecant function

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

Secant function

Redefined domain of function

Domain of secant = [ 0, π / 2 ) ( π / 2, π ] = [ 0, π ] { π / 2 }

Range of secant = ( - , - 1 ] [ 1, ) = R - 1,1

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

Domain of arcsecant = R - 1, 1

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

Therefore, we define arcsecant function as :

f : R - 1, 1 [ 0, π ] { π / 2 } by f(x) = arcsec (x)

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

Arcsecant function

The arcsecant function .vs. real value

Arccotangent function

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

Cotangent function

Redefined domain of function

Domain of cotangent = 0, π

Range of cotangent = R

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

Domain of arccotangent = R

Range of arccotangent = 0, π

Therefore, we define arccotangent function as :

f : R 0, π by f(x) = arccot (x)

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

Arccotangent function

The arccotangent function .vs. real value

Example

Problem : Find y when :

y = tan - 1 - 1 3

Solution : There are multiple angles for which :

tan y = x = - 1 3

However, range of sine function is [-π/2, π/2]. We need to find angle, which falls in this range. Now, acute angle corresponding to the value of 1/√3 is π/6. In accordance with sign diagram, tangent is negative in second and fourth quarters. But range is [-π/2, π/2]. Hence, we need to find angle in fourth quadrant. The angle in the fourth quadrant whose tangent has magnitude of 1/√3 is given by :

y = 2 π - π 6 = 11 π 6

Corresponding negative angle is :

y = 11 π 6 - 2 π = - π 6

Problem : Find domain of the function given by :

f x = cos - 1 x [ x ]

Solution : The given function is quotient of two functions having rational form :

f x = g x h x

The domain of quotient is given by :

D = D 1 D 2 { x : x when h(x) = 0 }

Here, g x = cos - 1 x . The domain of arccosine is [-1,1]. Hence,

D 1 = Domain of “g” = [ - 1,1 ]

The denominator function h(x) is greatest integer function. Its domain is “R”.

D 2 = Domain of “h” = R

The intersection of two domains is :

D 1 D 2 = [ - 1,1 ] R = [ - 1,1 ]

Intersection of domains

The intersection of domains result in common interval.

Now, greatest integer function becomes zero for values of “x” in the interval [0,1). Hence, domain of given function is :

Domain of function

The domain of function is obtained by subtracting interval, which is not permitted.

D = D 1 D 2 [ 0,1 )

D = [ - 1,1 ] - [ 0,1 ) = - 1 x < 0 { 1 }

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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