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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

what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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