# 3.15 Composition of trigonometric function and its inverse

 Page 1 / 3

Trigonometric and inverse trigonometric functions are inverse to each other. We can use them to compose new functions. In such composition, trigonometric function represents value of trigonometric ratio, whereas inverse trigonometric function represents angle. The composite function either evaluates to value or angle, depending on particular composition.

## Composition representing value of trigonometric function

Sine inverse trigonometric function is given by :

$y={\mathrm{sin}}^{-1}x\phantom{\rule{1em}{0ex}}⇒x=\mathrm{sin}y\phantom{\rule{1em}{0ex}}⇒x=\mathrm{sin}{\mathrm{sin}}^{-1}x$

$⇒\mathrm{sin}{\mathrm{sin}}^{-1}x=x$

The composition $\mathrm{sin}{\mathrm{sin}}^{-1}x$ evaluates to a value. Clearly, x is a value of sine trigonometric function which falls within the range of sine function i.e $x\in \left[-1,1\right]$ . It is important to note that domain of inverse function is same as range of the corresponding trigonometric function. We write six compositions denoting value of trigonometric functions as :

$\mathrm{sin}{\mathrm{sin}}^{-1}x=x;\phantom{\rule{1em}{0ex}}x\in \left[-1,1\right]$

$\mathrm{cos}{\mathrm{cos}}^{-1}x=x;\phantom{\rule{1em}{0ex}}x\in \left[-1,1\right]$

$\mathrm{tan}{\mathrm{tan}}^{-1}x=x;\phantom{\rule{1em}{0ex}}x\in R$

$\mathrm{cot}{\mathrm{cot}}^{-1}x=x;\phantom{\rule{1em}{0ex}}x\in R$

$\mathrm{sec}{\mathrm{sec}}^{-1}x=x;\phantom{\rule{1em}{0ex}}x\in \left(-\infty ,-1\right]\cup \left[1,\infty \right)$

$\mathrm{cosec}{\mathrm{cosec}}^{-1}x=x;\phantom{\rule{1em}{0ex}}x\in \left(-\infty ,-1\right]\cup \left[1,\infty \right)$

## Composition representing angle

We shall discuss this composition with respect to individual inverse trigonometric ratio.

## Composition with arcsine

Sine inverse trigonometric function is given by :

$y={\mathrm{sin}}^{-1}x\phantom{\rule{1em}{0ex}}⇒x=\mathrm{sin}y\phantom{\rule{1em}{0ex}}⇒y={\mathrm{sin}}^{-1}\mathrm{sin}y$

In order to maintain generality, we replace y by x as :

$⇒{\mathrm{sin}}^{-1}\mathrm{sin}x=x$

The composition ${\mathrm{sin}}^{-1}\mathrm{sin}x$ evaluates to an angle. Clearly, x is angle value – not the value of trigonometric ratio. However, we know that we use a truncated domain of trigonometric function for defining range of inverse function. The values in the interval are selected such that all unique values of sine trigonometric function are represented. It means that expression on LHS of the equation i.e. ${\mathrm{sin}}^{-1}\mathrm{sin}x$ evaluates to angle values lying in the interval $\left[-\pi /2,\pi /2\right]$ .

${\mathrm{sin}}^{-1}\mathrm{sin}x=x;\phantom{\rule{1em}{0ex}}x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]$

However, x as argument of sine function can assume angle values belonging to real number set. It means angles represented by LHS and RHS can be different if we consider angle values beyond principal set selected to render corresponding trigonometric function invertible.

Let us consider adjacent intervals such that all sine values are included once. Such intervals are $\left[\pi /2,3\pi /2\right],\left[3\pi /2,5\pi /2\right]$ etc on the right side and $\left[-3\pi /2,-\pi /2\right],\left[-5\pi /2,-3\pi /2\right]$ etc on the left side of the principal interval.

Our task now is to determine angles in any of these new intervals, say $\left[\pi /2,3\pi /2\right]$ , corresponding to angles in the principal interval. We make use of value diagram which allows to determine angles having same trigonometric values. Let us consider a positive acute angle “θ” in the principal interval. This lies in the first quadrant. The new interval represents second and third quadrants. However, sine is positive in second quadrant and negative in third quadrant. Let the angle corresponding to positive acute angle in principal interval be x. Clearly, x corresponding to positive acute angle θ lies in second quadrant and is given by :

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
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

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

 By OpenStax By Janet Forrester By Madison Christian By OpenStax By George Turner By Samuel Madden By OpenStax By Saylor Foundation By Edgar Delgado By Monty Hartfield