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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

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
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Sherica
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Sherica
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Tamia
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Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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ramon Reply
Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Porter
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AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
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Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
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Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
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Uday
I'm interested in Nanotube
Uday
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Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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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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