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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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What is power set
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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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