Download pg 349 #1-16 all, 47, 48, 49

333202_0407.qxd
12/7/05
11:10 AM
Page 349
Section 4.7
4.7
349
Inverse Trigonometric Functions
Exercises
VOCABULARY CHECK: Fill in the blanks.
Function
Alternative Notation
Domain
Range
≤ y ≤
2
2
1. y arcsin x
__________
__________
2. __________
y cos1 x
1 ≤ x ≤ 1
__________
3. y arctan x
__________
__________
__________
PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.
In Exercises 1–16, evaluate the expression without using a
calculator.
2. arcsin 0
3. arccos 12
4. arccos 0
5. arctan
2 3
9. arctan 3
2
11. arccos 13. sin1
π
2
π
4
6. arctan1
3
7. cos1 1
3
2
15. tan1 0
8. sin1 2 12. arcsin
17. f x sin x,
2
14. tan1 3
3
4
)
−2
(
π
,6
1
2
)
x
−1
x
θ
θ
4
4
39.
40.
x+2
x+1
θ
20. arcsin 0.45
21. arcsin0.75
22. arccos0.7
23. arctan3
24. arctan 15
cos1
)π
x
5
19. arccos 0.28
0.26
27. arccos0.41
28. arcsin0.125
29. arctan 0.92
30. arctan 2.8
3
31. arcsin 4
1
32. arccos 3 7
33. tan1 2
(
y = arccos x
In Exercises 37–42, use an inverse trigonometric function
to write as a function of x.
37.
38.
gx arcsin x
26.
−1,
2
x
3
π
,−6
16. cos1 1
In Exercises 19–34, use a calculator to evaluate the expression. Round your result to two decimal places.
0.31
(
)
π
(−1, )
2
18. f x tan x, gx arctan x
25.
π
,4
(
1 2
(− 3, )
10. arctan 3
y
36.
y = arctan x
−3 −2
2
In Exercises 17 and 18, use a graphing utility to graph f, g,
and y x in the same viewing window to verify geometrically that g is the inverse function of f. (Be sure to restrict
the domain of f properly.)
sin1
y
35.
1. arcsin 12
3
In Exercises 35 and 36, determine the missing coordinates
of the points on the graph of the function.
95
34. tan1 7 θ
10
41.
42.
2x
x−1
θ
θ
x+3
x2 − 1
In Exercises 43–48, use the properties of inverse trigonometric functions to evaluate the expression.
43. sinarcsin 0.3
44. tanarctan 25
45. cosarccos0.1
46. sinarcsin0.2
47. arcsinsin 3
48. arccos cos
7
2
333202_0407.qxd
12/7/05
350
11:10 AM
Chapter 4
Page 350
Trigonometry
In Exercises 49–58, find the exact value of the expression.
(Hint: Sketch a right triangle.)
49. sinarctan 34 51. costan
1
50. secarcsin 45 2
1
52. sin cos
5
5
5
53. cosarcsin 13 5
54. csc arctan 12 2
57. sin arccos 3 5
58. cot arctan 8 3
55. sec arctan 5 3
56. tanarcsin 4 60. sinarctan x
61. cosarcsin 2x
62. secarctan 3x
63. sinarccos x
64. secarcsinx 1
66. cot arctan
1
x
xh
r
85. f x arctan2x 3
cos1
2
9
arcsin,
x
s
5 ft
θ
In Exercises 75 and 76, sketch a graph of the function and
compare the graph of g with the graph of f x arcsin x.
75. g x arcsinx 1
Use a graphing utility to graph both forms of the function.
What does the graph imply?
0 ≤ x ≤ 6
3
arcsin
73. arccos
x 2 2x 10
x2
arctan, x 2 ≤ 2
74. arccos
2
76. gx arcsin
A
.
B
91. Docking a Boat A boat is pulled in by means of a winch
located on a dock 5 feet above the deck of the boat (see
figure). Let be the angle of elevation from the boat to the
winch and let s be the length of the rope from the winch to
the boat.
x0
arccos,
A cos t B sin t A2 B2 sin t arctan
90. f t 4 cos t 3 sin t
In Exercises 71–74, fill in the blank.
36 x 2
23
1 89. f t 3 cos 2t 3 sin 2t
2x
1 4x2
4 x 2
x
70. f x tan arccos , gx 2
x
6
84. f x arcsin4x
In Exercises 89 and 90, write the function in terms of the
sine function by using the identity
69. f x sinarctan 2x, gx 72. arcsin
x
4
In Exercises 83– 88, use a graphing utility to graph the
function.
88. f x In Exercises 69 and 70, use a graphing utility to graph f and
g in the same viewing window to verify that the two
functions are equal. Explain why they are equal. Identify
any asymptotes of the graphs.
71. arctan
arctan x
2
87. f x sin1
2
68. cos arcsin
80. f x 86. f x 3 arctan x
67. csc arctan
79. f x) arctan 2x
83. f x 2 arccos2x
x
78. gt arccost 2
82. f x arccos
59. cotarctan x
77. y 2 arccos x
81. hv tanarccos v
In Exercises 59–68, write an algebraic expression that is
equivalent to the expression. (Hint: Sketch a right triangle,
as demonstrated in Example 7.)
x
65. tan arccos
3
In Exercises 77–82, sketch a graph of the function.
x
2
(a) Write as a function of s.
(b) Find when s 40 feet and s 20 feet.