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On these restricted domains, we can define the inverse trigonometric functions .

  • The inverse sine function     y = sin 1 x means x = sin y . The inverse sine function is sometimes called the arcsine    function, and notated arcsin x .
    y = sin 1 x has domain [ −1 , 1 ] and range [ π 2 , π 2 ]
  • The inverse cosine function     y = cos 1 x means x = cos y . The inverse cosine function is sometimes called the arccosine    function, and notated arccos x .
    y = cos 1 x has domain [ −1 , 1 ] and range [ 0 , π ]
  • The inverse tangent function     y = tan 1 x means x = tan y . The inverse tangent function is sometimes called the arctangent    function, and notated arctan x .
    y = tan 1 x has domain ( −∞ , ) and range ( π 2 , π 2 )

The graphs of the inverse functions are shown in [link] , [link] , and [link] . Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that sin 1 x has domain [ −1 , 1 ] and range [ π 2 , π 2 ] , cos 1 x has domain [ −1 ,1 ] and range [ 0 , π ] , and tan 1 x has domain of all real numbers and range ( π 2 , π 2 ) . To find the domain and range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line y = x .

A graph of the functions of sine of x and arc sine of x. There is a dotted line y=x between the two graphs, to show inverse nature of the two functions
The sine function and inverse sine (or arcsine) function
A graph of the functions of cosine of x and arc cosine of x. There is a dotted line at y=x to show the inverse nature of the two functions.
The cosine function and inverse cosine (or arccosine) function
A graph of the functions of tangent of x and arc tangent of x. There is a dotted line at y=x to show the inverse nature of the two functions.
The tangent function and inverse tangent (or arctangent) function

Relations for inverse sine, cosine, and tangent functions

For angles in the interval [ π 2 , π 2 ] , if sin y = x , then sin 1 x = y .

For angles in the interval [ 0 , π ] , if cos y = x , then cos 1 x = y .

For angles in the interval ( π 2 , π 2 ) , if tan y = x , then tan 1 x = y .

Writing a relation for an inverse function

Given sin ( 5 π 12 ) 0.96593 , write a relation involving the inverse sine.

Use the relation for the inverse sine. If sin y = x , then sin 1 x = y .

In this problem, x = 0.96593 , and y = 5 π 12 .

sin 1 ( 0.96593 ) 5 π 12

Given cos ( 0.5 ) 0.8776, write a relation involving the inverse cosine.

arccos ( 0.8776 ) 0.5

Finding the exact value of expressions involving the inverse sine, cosine, and tangent functions

Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically π 6 (30°), π 4 (45°), and π 3 (60°), and their reflections into other quadrants.

Given a “special” input value, evaluate an inverse trigonometric function.

  1. Find angle x for which the original trigonometric function has an output equal to the given input for the inverse trigonometric function.
  2. If x is not in the defined range of the inverse, find another angle y that is in the defined range and has the same sine, cosine, or tangent as x , depending on which corresponds to the given inverse function.

Evaluating inverse trigonometric functions for special input values

Evaluate each of the following.

  1. sin 1 ( 1 2 )
  2. sin 1 ( 2 2 )
  3. cos 1 ( 3 2 )
  4. tan 1 ( 1 )
  1. Evaluating sin 1 ( 1 2 ) is the same as determining the angle that would have a sine value of 1 2 . In other words, what angle x would satisfy sin ( x ) = 1 2 ? There are multiple values that would satisfy this relationship, such as π 6 and 5 π 6 , but we know we need the angle in the interval [ π 2 , π 2 ] , so the answer will be sin 1 ( 1 2 ) = π 6 . Remember that the inverse is a function, so for each input, we will get exactly one output.
  2. To evaluate sin 1 ( 2 2 ) , we know that 5 π 4 and 7 π 4 both have a sine value of 2 2 , but neither is in the interval [ π 2 , π 2 ] . For that, we need the negative angle coterminal with 7 π 4 : sin 1 ( 2 2 ) = π 4 .
  3. To evaluate cos 1 ( 3 2 ) , we are looking for an angle in the interval [ 0 , π ] with a cosine value of 3 2 . The angle that satisfies this is cos 1 ( 3 2 ) = 5 π 6 .
  4. Evaluating tan 1 ( 1 ) , we are looking for an angle in the interval ( π 2 , π 2 ) with a tangent value of 1. The correct angle is tan 1 ( 1 ) = π 4 .

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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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
what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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