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x = π θ

θ = π x

Hence,

sin - 1 sin x = π x ; x [ π 2 , 3 π 2 ]

In order to find expression corresponding to negative angle interval [ - 3 π / 2, - π / 2 ] , we need to construct negative value diagram. We know that equivalent negative angle is obtained by deducting “-2π” to the positive angle. Thus, corresponding to expression for positive angles in four quadrants, the expression in terms of negative angles are “-θ”,“-π+θ”,“-π-θ” and “-2π+θ” in four quadrants counted in clockwise direction in the value diagram. Now, we estimate from the sine plot that an angle, corresponding to a positive acute angle, θ, in the principal interval, lies in third negative quadrant. Therefore,

x = - π θ

θ = - π x

Hence,

sin - 1 sin x = - π x ; x [ - 3 π 2 , π 2 ]

Combining three results,

|-π-x; x∈ [-3π/2, -π/2] sin⁻¹ sinx = | x; x∈ [-π/2, π/2]| π- x; x∈ [π/2, 3π/2]

We can similarly find expressions for more such intervals.

Graph of sin⁻¹sinx

Using three expressions obtained above, we can draw plot of the composition function. We extend the plot, using the fact that composition is a periodic function with a period of 2π. The equation of plot, which is equivalent to plot y=x shifted by 2π towards right, is :

y = x 2 π

The equation of plot, which is equivalent to plot y=x shifted by 2π towards left, is :

y = x + 2 π

Sine inverse of sine

The function is periodic with period 2π.

We see that graph of composition is continuous. Its domain is R. Its range is [ - π / 2, π / 2 ] . The function is periodic with period 2π.

Composition with arccosine

The composition cos - 1 cos x evaluates to angle values lying in the interval [0, π].

cos - 1 cos x = x ; x [ 0, π ]

Let us consider adjacent intervals such that all cosine values are included once. Such intervals are [π, 2π], [2π, 3π]etc on the right side and [-π, 0], [-2π, -π]etc on the left side of the principal interval.

Cosine function

Additional domains for inversion.

The new interval [ π , 2 π ] represents third and fourth quadrants. The angle x, corresponding to positive acute angle θ, lies in fourth quadrant. Then,

Value diagrams

Value diagrams for positive and negative angles

x = 2 π θ

θ = 2 π x

Hence,

cos - 1 cos x = 2 π x ; x [ π , 2 π ]

In order to find expression corresponding to negative angle interval [ - π , 0 ] , we estimate from the cosine plot that an angle corresponding to a positive acute angle, θ, in the principal interval lies in first negative quadrant. Therefore,

x = - θ

θ = x

Hence,

cos - 1 cos x = - x ; x [ π , 0 ]

Combining three results,

|-x; x∈ [-π, 0] cos⁻¹ cosx = | x; x∈[0, π]|2π- x; x∈ [π, 2π]

We can similarly find expressions for other intervals.

Graph of cos⁻¹cosx

Using three expressions obtained above, we can draw plot of the composition function. We have extended the plot, using the fact that composition is a periodic function with a period of 2π. The equation of plot, which is equivalent to plot y=x shifted by 2π towards right, is :

y = x 2 π

The equation of plot, which is equivalent to plot y=x shifted by 2π towards left, is :

y = x + 2 π

Cosine inverse of cosine

The function is periodic with period 2π.

We see that graph of composition is continuous. Its domain is R. Its range is [ 0, π ] . The function is periodic with period 2π.

Questions & Answers

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
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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

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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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