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Inverse Functions
Inverse Sine Function
Inverse Cosine Function
Inverse Tangent Function
Remaining Inverse Circular Functions
Inverse Function Values
Inverse Functions
We first discussed inverse functions in Section 4.1. Review:
1. If a function f is one-to-one, then f has an inverse function f −1 .
2. In a one-to-one function, each x-value corresponds to only one y-value
and each y-value corresponds to only one x-value.
3. The domain of f is the range of f −1 , and the range of f is the domain of
f −1 .
4. The graphs of f and f −1 are reflections of each other about the line y = x.
5. To find f −1 (x) from f(x), follow these steps.
i. Replace f(x) with y and interchange x and y.
ii. Solve for y.
iii.
Replace y with f −1 (x).
Inverse Sine Function
From the graph of y = sin x, we notice that it fails the horizontal line test, and
hence, is not a one-to-one function. Therefore it does not have an inverse that is
 π π
a function unless we restrict the domain to a specified interval such as  − ,  .
 2 2
Inverse Sine Function
y = sin −1 x or y = arcsin x means that x = sin y, for -
π
2
≤ y≤
π
2
See picture and table on pp. 647.
Inverse Cosine Function
The function y = cos −1 x (or y = arccos x) is defined by restricting the domain of the
function y = cos x to the interval [ 0, π ]
Inverse Cosine Function
y = cos −1 x or y = arccos x means that x = cos y, for 0 ≤ y ≤ π
See graph on pp. 648.
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Inverse Tangent Function
 π π
Restricting the domain of the function y = tan x to the open interval  − , 
 2 2
yields a one-to-one function. By interchanging the roles of x and y, we obtain the
inverse tangent function given by y = tan −1 x or y = arctan x .
Inverse Tangent Function
y = yan −1 x or y = arctan x means that x = tan y for −
π
2
< y<
π
2
See graph on pp. 649.
Remaining Inverse Circular Functions. The remaining three inverse
trigonometric functions are defined similarly; their graphs are shown in figure 18
on pp. 649. All six inverse trigonometric functions with their domains and ranges
are given in the table below.
Inverse
Domain
Range
Range
Function
Interval
Quadrants of the Unit Circle
I and IV
y = sin −1 x
 π π
[ −1,1]
 − 2 , 2 
I and II
y = cos −1 x
[ −1,1]
[0, π ]
y = tan −1 x
( −∞, ∞ )
y = cot −1 x
( −∞, ∞ )
y = sec −1 x
(−∞, −1] ∪ [1, ∞)
−1
y = csc x
(−∞, −1] ∪ [1, ∞)
 π π
− , 
 2 2
( 0, π )
[0, π ] , y ≠
 π π
π*
I and IV
I and II
I and II
2
*
 − 2 2  , y ≠ 0
I and IV
*The inverse secant and inverse cosecant functions are sometimes defined
with different ranges. We use intervals that match their reciprocal functions
(except for one missing point)
Homework exercises 1 – 6. Complete each statement
2. The domain of y = arcsin x equals the ______ of y = sin x
π 
4. The point  ,1 lies on the graph of y = tan x. Therefore, the point ____
4 
lies on the graph of ______.
6. How can the graph of f −1 be sketched if the graph of f is known?
Homework exercises 7 – 12. Write short answers.
8. Consider the inverse cosine function, defined by y = cos −1 x or y = arccos x
(a) What is its domain?
(b) What is its range?
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(c) Is this function increasing or decreasing?
4π
 1  2π
 1
(d) Arc cos  −  =
. Why is arccos  −  not equal to −
?
3
 2 3
 2
10. Give the domain and range of the three other inverse trigonometric
functions, as defined in this section.
(a) inverse cosecant function
(b) inverse secant function
(c) inverse cotangent function
Homework exercises 13 - 32. Find the exact value of each real number y.
do not use a calculator.
Homework exercises 33 – 40. Give the degree measure of θ . Do not use a
calculator.
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Homework exercises 41 – 46. Use a calculator to give each value in decimal
degrees.
Homework exercises 47 – 52. Use a calculator to give each real number value.
(Be sure the calculator is in radian mode.)
Homework exercises 53 – 57. Graph each inverse function as defined in the
text.
Homework exercises 63 – 78. Give the exact value of each expression
without using a calculator. See Examples 5 and 6.
1
1 



 1 
64. sin  arccos  66. sec  sin −1 (− )  68. cos  2sin −1   
4
5 
 4 




 1 
70. tan  2 cos −1   
 4 

72. cos ( 2 tan −1 (−2) )
74. csc(csc −1 2)

3
5
3

 3 
− sin −1  −  
76. cos  sin −1 + cos −1  78. tan  cos −1
5
13 
2
 5 


Homework exercises 79 – 82. Use a calculator to find each value. Give
answers as real numbers.
80. sin(cos −1 .25)
82. cot ( arccos(.58236841) )
Homework exercises 83 – 92. Write each expression as an algebraic
(nontrigonometric) expression in u, u > 0. See example 7.
3

84. tan(arccos u ) 86. cot(arcsin u )
88. cos  tan −1 
u

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
u 
90. sec  cos −1

u2 + 5 

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