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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 = sin - 1 x x = sin y x = sin sin - 1 x

sin sin - 1 x = x

The composition sin 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 [ - 1,1 ] . 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 :

sin sin - 1 x = x ; x [ - 1,1 ]

cos cos - 1 x = x ; x [ - 1,1 ]

tan tan - 1 x = x ; x R

cot cot - 1 x = x ; x R

sec sec - 1 x = x ; x , 1 ] [ 1,

cosec cosec - 1 x = x ; x , 1 ] [ 1,

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 = sin - 1 x x = sin y y = sin - 1 sin y

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

sin - 1 sin x = x

The composition sin - 1 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. sin - 1 sin x evaluates to angle values lying in the interval [ - π / 2, π / 2 ] .

sin - 1 sin x = x ; x [ - π 2 , π 2 ]

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.

Sine function

Principle domain

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

Sine function

Additional domains for inversion

Our task now is to determine angles in any of these new intervals, say [ π / 2, 3 π / 2 ] , 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 :

Value diagrams

Value diagrams for positive and negative angles

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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
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what is the stm
Brian Reply
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How we are making nano material?
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What is STMs full form?
scanning tunneling microscope
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what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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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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