# 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 :

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
Sahil
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Santosh
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Rafiq
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Mahi
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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 is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x