61. sec 2 1+ cos 0 65. cos 64. tan = csc v - cot v sin (30) 67. cos (30) = 2 sin 0 cos 0 2.

Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter1: Trigonometry
Section1.3: Right Triangle Trigonometry
Problem 72E: A 20 meter line is used to tether a helium-filled balloon. The line makes an angle of approximately...
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exact area of a regular octagon whose side is 9 centimeters
(b) The area A of a regular octagon is also given by the formula
A = 2a? cot , where a is the length of a side. Find the
508 CHAPTER 7 Analytic Trigonometry
In Problems 51-72, establish each identity.
cot? 0 - 1
51. cos e - sin e = cos (20)
cot 0 - tan 0
52.
cot 0 + tan 0
53. cot (20) =
- cos (20)
2 cot 0
sec 0
sec? 0
54. cot (20)
56. csc (20) =
sec 0 csc e
(cot 0 - tan 0)
55. sec (20) =
2
cos (20)
1 + sin (20)
cot 0-1
A(8)
57. cos (2u) - sin (2u) = cos (4u)
58. (4 sin u cos u) (1 - 2 sin u) = sin (4u)
59.
cot 0 + 1
60. sin e cos 0 = - sin (20)
62. csc
2
2
61. sec
1 + cos e
1- cos 0
1- tan?
sec v + 1
63. cot?
65. cos 0 =
sec v - 1
64. tan
= csc v
cot v
1 + tan?
A(a) In ca
maxi
2
sin 0 + cos' 0
sin (30)
67.
cos (30)
66. 1 - sin (20) =
sin 0 + cos 0
sin 0
cos e
cos e + sin 0
68.
cos e - sin 0
cos 0 + sin 0
3 tan 0 - tan e
3 tan? 0
cos 0 - sin 6
= 2 tan (20)
69. tan (30) =
1
Sol
70. tan 0 + tan (0 + 120°) + tan (e + 240°) = 3 tan (36)
71. In sin e| = (In |1 – cos (260)| – In 2).
(b) Wh
72. In |cos 6| = (In |1 + cos (20)| – In 2)
(e) Gr
the
ar
ME Laser
or sca
In Problems 73-82, solve each equation on the interval 0 sO< 2m.
73. cos (20) + 6 sin? e = 4
74. cos (20) = 2 - 2 sin? 0
75. cos (20) = cos e
the
76. sin (20) = cos 0
77. sin (20) + sin (40) = 0
78. cos (20) + cos (40) = 0
by th
79. 3 - sin 0 = cos (20)
80. cos (20) + 5 cos 0 + 3 = 0
81. tan (20) + 2 sin 6 = 0
82. tan (20) + 2 cos 0 = 0
In Problems 83–94, find the exact value of each expression.
(a
83. sin 2 sin-1
84. sin 2 sin-1
85. cos( 2 sin
86. cos 2 cos
87. tan 2 cos
88. tan 2 tan
89. sin 2 cos
4
91. sir cos)
90. cos 2 tan
92. cos?
sin-
93. sec 2 tan
94. csc 2 sin
14
Applications and Extensions
In Problems 95–100, find the real zeros of each trigonometric
function on the interval 0 s e < 27.
101. Area of an Octagon
95. f(x) = sin (2x) - sin x
(a) The area A of a regular
octagon is given by the
formula A = 8r? tan
96. f(x) = cos (2x) + cos x
97. f(x) = cos ( 2x) + sin² x
where r is the apothem,
which is a line segment from
the center of the octagon
perpendicular to a side. See
the figure. Find the exact area of a regular octagon who
apothem is 12 inches.
d wod
8'
98. f(x) = 2 sin?x – sin (2x)
99. f(x) = sin (2x) + cos x
100. f(x) = cos ( 2x) – 5 cos x – 2
8'
Construce
fimctio
Transcribed Image Text:exact area of a regular octagon whose side is 9 centimeters (b) The area A of a regular octagon is also given by the formula A = 2a? cot , where a is the length of a side. Find the 508 CHAPTER 7 Analytic Trigonometry In Problems 51-72, establish each identity. cot? 0 - 1 51. cos e - sin e = cos (20) cot 0 - tan 0 52. cot 0 + tan 0 53. cot (20) = - cos (20) 2 cot 0 sec 0 sec? 0 54. cot (20) 56. csc (20) = sec 0 csc e (cot 0 - tan 0) 55. sec (20) = 2 cos (20) 1 + sin (20) cot 0-1 A(8) 57. cos (2u) - sin (2u) = cos (4u) 58. (4 sin u cos u) (1 - 2 sin u) = sin (4u) 59. cot 0 + 1 60. sin e cos 0 = - sin (20) 62. csc 2 2 61. sec 1 + cos e 1- cos 0 1- tan? sec v + 1 63. cot? 65. cos 0 = sec v - 1 64. tan = csc v cot v 1 + tan? A(a) In ca maxi 2 sin 0 + cos' 0 sin (30) 67. cos (30) 66. 1 - sin (20) = sin 0 + cos 0 sin 0 cos e cos e + sin 0 68. cos e - sin 0 cos 0 + sin 0 3 tan 0 - tan e 3 tan? 0 cos 0 - sin 6 = 2 tan (20) 69. tan (30) = 1 Sol 70. tan 0 + tan (0 + 120°) + tan (e + 240°) = 3 tan (36) 71. In sin e| = (In |1 – cos (260)| – In 2). (b) Wh 72. In |cos 6| = (In |1 + cos (20)| – In 2) (e) Gr the ar ME Laser or sca In Problems 73-82, solve each equation on the interval 0 sO< 2m. 73. cos (20) + 6 sin? e = 4 74. cos (20) = 2 - 2 sin? 0 75. cos (20) = cos e the 76. sin (20) = cos 0 77. sin (20) + sin (40) = 0 78. cos (20) + cos (40) = 0 by th 79. 3 - sin 0 = cos (20) 80. cos (20) + 5 cos 0 + 3 = 0 81. tan (20) + 2 sin 6 = 0 82. tan (20) + 2 cos 0 = 0 In Problems 83–94, find the exact value of each expression. (a 83. sin 2 sin-1 84. sin 2 sin-1 85. cos( 2 sin 86. cos 2 cos 87. tan 2 cos 88. tan 2 tan 89. sin 2 cos 4 91. sir cos) 90. cos 2 tan 92. cos? sin- 93. sec 2 tan 94. csc 2 sin 14 Applications and Extensions In Problems 95–100, find the real zeros of each trigonometric function on the interval 0 s e < 27. 101. Area of an Octagon 95. f(x) = sin (2x) - sin x (a) The area A of a regular octagon is given by the formula A = 8r? tan 96. f(x) = cos (2x) + cos x 97. f(x) = cos ( 2x) + sin² x where r is the apothem, which is a line segment from the center of the octagon perpendicular to a side. See the figure. Find the exact area of a regular octagon who apothem is 12 inches. d wod 8' 98. f(x) = 2 sin?x – sin (2x) 99. f(x) = sin (2x) + cos x 100. f(x) = cos ( 2x) – 5 cos x – 2 8' Construce fimctio
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