# 3.14 Inverse trigonometric functions  (Page 3/4)

 Page 3 / 4

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

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

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

$\text{Domain of secant}=\left[0,\pi /2\right)\cup \left(\pi /2,\pi \right]=\left[0,\pi \right]-\left\{\pi /2\right\}$

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

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 :

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

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

Therefore, we define arcsecant function as :

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

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

## 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 $\pi$ 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.

$\text{Domain of cotangent}=\left(0,\pi \right)$

$\text{Range of cotangent}=R$

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 :

$\text{Domain of arccotangent}=R$

$\text{Range of arccotangent}=\left(0,\pi \right)$

Therefore, we define arccotangent function as :

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

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

## Example

Problem : Find y when :

$y=\mathrm{tan}{}^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

Solution : There are multiple angles for which :

$⇒\mathrm{tan}y=x=-\frac{1}{\sqrt{3}}$

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 :

$⇒y=2\pi -\frac{\pi }{6}=\frac{11\pi }{6}$

Corresponding negative angle is :

$⇒y=\frac{11\pi }{6}-2\pi =-\frac{\pi }{6}$

Problem : Find domain of the function given by :

$f\left(x\right)=\frac{{\mathrm{cos}}^{-1}\left(x\right)}{\left[x\right]}$

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

$f\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}$

The domain of quotient is given by :

$D={D}_{1}\cap {D}_{2}-\left\{x:x\phantom{\rule{1em}{0ex}}\text{when h(x)}=0\right\}$

Here, $g\left(x\right)={\mathrm{cos}}^{-1}\left(x\right)$ . The domain of arccosine is [-1,1]. Hence,

${D}_{1}=\text{Domain of “g”}=\left[-1,1\right]$

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

${D}_{2}=\text{Domain of “h”}=R$

The intersection of two domains is :

$⇒{D}_{1}\cap {D}_{2}=\left[-1,1\right]\cap R=\left[-1,1\right]$

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

$D={D}_{1}\cap {D}_{2}-\left[0,1\right)$

$D=\left[-1,1\right]-\left[0,1\right)=-1\le x<0\cup \left\{1\right\}$

how can chip be made from sand
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
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
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
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x

#### Get Jobilize Job Search Mobile App in your pocket Now! By Lakeima Roberts By Sandhills MLT By Alec Moffit By Madison Christian By Cath Yu By David Bourgeois By Eddie Unverzagt By Nick Swain By OpenStax By OpenStax