# 3.14 Inverse trigonometric functions  (Page 2/4)

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$\text{Domain of sine}=\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$

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

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 :

$\text{Domain of arcsine}=\left[-1,1\right]$

$\text{Range of arcsine}=\left[-\frac{\pi }{2,}\frac{\pi }{2}\right]$

Therefore, we define arcsine function as :

$f:\left[-1,1\right]\to \left[-\frac{\pi }{2},\frac{\pi }{2}\right]\phantom{\rule{1em}{0ex}}\text{by f(x)}=\text{arcsin(x)}$

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

## 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 $\pi$ 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.

$\text{Domain of cosine}=\left[0,\pi \right]$

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

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 :

$\text{Domain of arccosine}=\left[-1,1\right]$

$\text{Range of arccosine}=\left[0,\pi \right]$

Therefore, we define arccosine function as :

$f:\left[-1,1\right]\to \left[0,\pi \right]\phantom{\rule{1em}{0ex}}\text{by f(x)}=\text{arccos(x)}$

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

## 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 $-\pi /2$ and $\pi /2$ 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.

$\text{Domain of tangent}=\left(-\frac{\pi }{2,}\frac{\pi }{2}\right)$

$\text{Range of tangent}=R$

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 :

$\text{Domain of arctangent}=R$

$\text{Range of arctangent}=\left(-\pi /2,\pi /2\right)$

Therefore, we define arctangent function as :

$f:R\to \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\phantom{\rule{1em}{0ex}}\text{by f(x)}=\text{arctan (x)}$

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

## 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 “ $-\pi /2$ and 0” and “0 and $\pi /2$ ” includes all possible values of cosecant function only once. Note that zero is excluded, but “ $-\pi /2$ “ and “ $\pi /2$ ” are included . The redefinition of domain of trigonometric function, however, does not change the range.

$\text{Domain of cosecant}=\left[-\pi /2,\pi /2\right]-\left\{0\right\}$

$\text{Range of cosecant}=\left(-\infty ,-1\right]\cup \left[1,\infty \right)=R-\left(-1,1\right)$

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 :

$\text{Domain of arccosecant}=R-\left(-1,1\right)$

$\text{Range of arccosecant}=\left[-\pi /2,\pi /2\right]-\left\{0\right\}$

Therefore, we define arccosecant function as :

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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.
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
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
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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