<< Chapter < Page Chapter >> Page >

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

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Contemporary math applications' conversation and receive update notifications?