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5.6 Inverse trigonometric functions  (Page 2/15)

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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 .
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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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