3.15 Composition of trigonometric function and its inverse  (Page 2/3)

 Page 2 / 3

$x=\pi -\theta$

$⇒\theta =\pi -x$

Hence,

$⇒{\mathrm{sin}}^{-1}\mathrm{sin}x=\pi -x;\phantom{\rule{1em}{0ex}}x\in \left[\frac{\pi }{2},\frac{3\pi }{2}\right]$

In order to find expression corresponding to negative angle interval $\left[-3\pi /2,-\pi /2\right]$ , 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,

$x=-\pi -\theta$

$⇒\theta =-\pi -x$

Hence,

$⇒{\mathrm{sin}}^{-1}\mathrm{sin}x=-\pi -x;\phantom{\rule{1em}{0ex}}x\in \left[-\frac{3\pi }{2},-\frac{\pi }{2}\right]$

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 :

$y=x-2\pi$

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

$y=x+2\pi$

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

Composition with arccosine

The composition ${\mathrm{cos}}^{-1}\mathrm{cos}x$ evaluates to angle values lying in the interval [0, π].

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

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.

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

$x=2\pi -\theta$

$⇒\theta =2\pi -x$

Hence,

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

In order to find expression corresponding to negative angle interval $\left[-\pi ,0\right]$ , 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,

$x=-\theta$

$⇒\theta =-x$

Hence,

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

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 :

$y=x-2\pi$

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

$y=x+2\pi$

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

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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x