the denominato SECTION 7.4 Trigonometric Identities 485 Skill Building In Problems 11-20, simplify each trigonometric expression by following the indicated direction. \I1. Rewrite in terms of sine and cosine functions: he numerator and or by 1 +sin 12. Rewrite in terms of sine and cosine functions: tan 0. csc 0. cot 0 sec 0. cos 0 1- sin 0 1 + sin e -by 1 + sin 0 \ 13 Multiply sin 6 1 + cos 0 16. Rewrite as a single quotient: 1-cos 0 by 1 cos 0 14. Multiply 15. Rewrite as a single quotient: sin 0 + cos 6 cos e - sin 0 Cos e sin 0 1. 1. = cos e (sin e + cos 0) (sin 0 + cos 0) - 1 cos v 1 + cos v 17. Multiply and simplify: (tan 0 + 1) (tan 0 + 1) - sec? 0 sin e cos 0 18. Multiply and simplify: 3 sin e + 4 sin e + 1 tan 0 19. Factor and simplify. sin0 + 2 sin e + 1 cos 0 - 1 20. Factor and simplify: cos 0 - cos 0 In Problems 21-100, establish each identity. \21. csc 0. cos 0 = cot 6 22. sec 0. sin 0 = tan 0 establish ideni 23. 1 + tan? (-0) = sec? e 24. 1 + cot? (-0) = csc? 0 25. cos 0 (tan 0 + cot 0) = csc 0 26. sin 0 (cot 0 + tan 0) = sec 0 27. tan u cot u - cos u = sin? u 28. sin u csc u - cos u = sin' u \29. (sec 0 – 1) (sec 0 + 1) = tan 0 30. (csc 0 - 1) (csc 0 + 1) = cot? e 31. (sec 0 + tan 0) (sec e - tan 0) = 1 32. (csc 0 + cot 0) (csc 0 - cot 0) = 1 33. cos? e (1 + tan? 0) = 1 34. (1 - cos e) (1 + cot? 0) = 1 35. (sin 0 + cos 0)2 + (sin 0 - cos 0)2 = 2 containing the m 36. tan 0 cos? e + cot2 e sin? e = 1 37. sec* e - sec? 0 = tan 0 + tan? e 38. csc* 0 - csc2 0 = cot 0 + cot? e cotient. cos u sin u 39. sec u - tan u 40. csc u - cot u = 41. 3 sin? e + 4 cos 0 = 3 + cos e 1 + sin u 1 + cos u nd cosine funcio cos? 0 sin? 0 1- cos e 42. 9 sec? 0 - 5 tan? 0 = 5 + 4 sec? 0 43. 1 = sin 0 44. 1 - = - cos 0 1 + sin 0 e of the expresin le. 1 + tan v 45. 1 sec e 47. cot v + 1 csc v - 1 46. 1- sin v sin 6 1+sin v = 2 tan 0 cos 0 tan v cot v - 1 csc v +1 csc e csc 0 + 1 csc e - 1 48. 1 + sin 0 49. 1 cos 0 + 1 50. cos 0 - 1 cot e 1 + sec 0 %3D %3D cot 0 csc e + 1 sin 6 csc 0 -1 1 - sec 0 cos V 1 + sin v 1 - sin v 51. cos v sin 0 1 = 2 sec v 52. = 2 sec v 53. sin 0 - cos 6 %3D cos v 1- sin v sin v cCs V 1- cot 0 1 - cos 0 56. 1 + cos 0 listed in rol sin? 0 sin e 54. 1 = cos 0 = (sec 0 - tan 0)2 (csc 0 - cot 0)2 1 + cos e 1 + sin 9 Os 8- sin. p cot 0 tan 0 cos 0 57. 1 - tan 0 sin 0 sin 0 + cos 58. 1 = 1 + tan 0 + cot 0 1 - cot 0 tan 0 cot 0 tan 0 + sec 0- 1 61. tan 0 - sec 0 + 1 cos 0 sin 0 cos 0 tan 0 = tan 0 + sec 0 59. tan 0 + = sec 0 60. cos e - sin? 0 1- tan? 0 1 + sin 0 sin 0 sin 0 - cos 0 +1 62, tan 0 - cot 0 63. tan 0 + cot 0 sec 0 - cos 0 64. sec 0 + cos 0 sin 0 + 1 = sin? 0 – cos² 0 quations i 1 + cos? 0 sin 0 + cos 0 - 1 cos 0 tan u - cot u 66. tan u + cot u sec 0 + tan 0 67. cot 0 + cos 0 tan u - cot u + 2 cos? u = 1 = tan 0 sec 0 65. tan u + cot u + 1 = 2 sin? u niste 1 - tan? 0 69. 1 + tan 6 1- cot? 0 70. sind 1 - cos 0 + 2 cos? 0 = 1 seç 0 + 1 = 2 cos? 0 68. 1+ sec 0 1 + cot? 0 %3D sin 0 sin? e - tan 0 72. cos? 0 - cot 0 sec e - csc 0 71. tan 0 73. sec 0 - cos 0 = sin 0 tan 0 %3D = sin 0 - cos 0 sec 0 csc 0 1 + sin 0 76. 1 1 - sin 0 74. tan A + cot 0 75. 2 sec? 0 = 4 tan 0 sec 0 sin A 1. + sin A denominato
the denominato SECTION 7.4 Trigonometric Identities 485 Skill Building In Problems 11-20, simplify each trigonometric expression by following the indicated direction. \I1. Rewrite in terms of sine and cosine functions: he numerator and or by 1 +sin 12. Rewrite in terms of sine and cosine functions: tan 0. csc 0. cot 0 sec 0. cos 0 1- sin 0 1 + sin e -by 1 + sin 0 \ 13 Multiply sin 6 1 + cos 0 16. Rewrite as a single quotient: 1-cos 0 by 1 cos 0 14. Multiply 15. Rewrite as a single quotient: sin 0 + cos 6 cos e - sin 0 Cos e sin 0 1. 1. = cos e (sin e + cos 0) (sin 0 + cos 0) - 1 cos v 1 + cos v 17. Multiply and simplify: (tan 0 + 1) (tan 0 + 1) - sec? 0 sin e cos 0 18. Multiply and simplify: 3 sin e + 4 sin e + 1 tan 0 19. Factor and simplify. sin0 + 2 sin e + 1 cos 0 - 1 20. Factor and simplify: cos 0 - cos 0 In Problems 21-100, establish each identity. \21. csc 0. cos 0 = cot 6 22. sec 0. sin 0 = tan 0 establish ideni 23. 1 + tan? (-0) = sec? e 24. 1 + cot? (-0) = csc? 0 25. cos 0 (tan 0 + cot 0) = csc 0 26. sin 0 (cot 0 + tan 0) = sec 0 27. tan u cot u - cos u = sin? u 28. sin u csc u - cos u = sin' u \29. (sec 0 – 1) (sec 0 + 1) = tan 0 30. (csc 0 - 1) (csc 0 + 1) = cot? e 31. (sec 0 + tan 0) (sec e - tan 0) = 1 32. (csc 0 + cot 0) (csc 0 - cot 0) = 1 33. cos? e (1 + tan? 0) = 1 34. (1 - cos e) (1 + cot? 0) = 1 35. (sin 0 + cos 0)2 + (sin 0 - cos 0)2 = 2 containing the m 36. tan 0 cos? e + cot2 e sin? e = 1 37. sec* e - sec? 0 = tan 0 + tan? e 38. csc* 0 - csc2 0 = cot 0 + cot? e cotient. cos u sin u 39. sec u - tan u 40. csc u - cot u = 41. 3 sin? e + 4 cos 0 = 3 + cos e 1 + sin u 1 + cos u nd cosine funcio cos? 0 sin? 0 1- cos e 42. 9 sec? 0 - 5 tan? 0 = 5 + 4 sec? 0 43. 1 = sin 0 44. 1 - = - cos 0 1 + sin 0 e of the expresin le. 1 + tan v 45. 1 sec e 47. cot v + 1 csc v - 1 46. 1- sin v sin 6 1+sin v = 2 tan 0 cos 0 tan v cot v - 1 csc v +1 csc e csc 0 + 1 csc e - 1 48. 1 + sin 0 49. 1 cos 0 + 1 50. cos 0 - 1 cot e 1 + sec 0 %3D %3D cot 0 csc e + 1 sin 6 csc 0 -1 1 - sec 0 cos V 1 + sin v 1 - sin v 51. cos v sin 0 1 = 2 sec v 52. = 2 sec v 53. sin 0 - cos 6 %3D cos v 1- sin v sin v cCs V 1- cot 0 1 - cos 0 56. 1 + cos 0 listed in rol sin? 0 sin e 54. 1 = cos 0 = (sec 0 - tan 0)2 (csc 0 - cot 0)2 1 + cos e 1 + sin 9 Os 8- sin. p cot 0 tan 0 cos 0 57. 1 - tan 0 sin 0 sin 0 + cos 58. 1 = 1 + tan 0 + cot 0 1 - cot 0 tan 0 cot 0 tan 0 + sec 0- 1 61. tan 0 - sec 0 + 1 cos 0 sin 0 cos 0 tan 0 = tan 0 + sec 0 59. tan 0 + = sec 0 60. cos e - sin? 0 1- tan? 0 1 + sin 0 sin 0 sin 0 - cos 0 +1 62, tan 0 - cot 0 63. tan 0 + cot 0 sec 0 - cos 0 64. sec 0 + cos 0 sin 0 + 1 = sin? 0 – cos² 0 quations i 1 + cos? 0 sin 0 + cos 0 - 1 cos 0 tan u - cot u 66. tan u + cot u sec 0 + tan 0 67. cot 0 + cos 0 tan u - cot u + 2 cos? u = 1 = tan 0 sec 0 65. tan u + cot u + 1 = 2 sin? u niste 1 - tan? 0 69. 1 + tan 6 1- cot? 0 70. sind 1 - cos 0 + 2 cos? 0 = 1 seç 0 + 1 = 2 cos? 0 68. 1+ sec 0 1 + cot? 0 %3D sin 0 sin? e - tan 0 72. cos? 0 - cot 0 sec e - csc 0 71. tan 0 73. sec 0 - cos 0 = sin 0 tan 0 %3D = sin 0 - cos 0 sec 0 csc 0 1 + sin 0 76. 1 1 - sin 0 74. tan A + cot 0 75. 2 sec? 0 = 4 tan 0 sec 0 sin A 1. + sin A denominato
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter8: Complex Numbers And Polarcoordinates
Section8.4: Roots Of A Complex Number
Problem 42PS
Related questions
Question
67,73,85
Transcribed Image Text:\ II. Rewrite in terms of sine and cosine functions:
the denominato
In Problems 11-20, simplify each trigonometric expression by following the indicated direction.
19. Factor and simplify: sin 0 + 2 sin e + 1
SECTION 7.4 Trigonometric Identities 485
Skill Building
he numerator and
or by 1 + sing
12. Rewrite in terms of sine and cosine functions:
tan e. csc 0.
cot 0 sec 0.
cos 0
1- sin 0
1 + sin e
-by
1 + sin e
\ 1L. Multiply
sin 6
1 - cos 0
by
1- cos 0"
14. Multiply
15. Rewrite as a single quotient:
sin 0 + cos 6
1 + cos 0
cos e - sin 0
16. Rewrite as a single quotient:
Cos e
sin 0
= cos 6
(sin e + cos 0) (sin e + cos 0) - 1
cos v
1 + cos v
17. Multiply and simplify:
(tan 0 + 1) (tan 0 + 1) - sec? 0
sin e cos 0
18. Multiply and simplify:
3 sin e + 4 sin e + 1
tan 0
20. Factor and simplify:
cos 0 - 1
cos 0 - cos 0
In Problems 21-100, establish each identity.
\21. csc 0- cos 0 = cot 6
22. sec e. sin 0 = tan 0
23. 1 + tan? (-0) = sec? e
establish iden
24. 1 + cot? (-0) = csc² 0
25. cos e (tan 0 + cot 0) = csc 0
26. sin 0 (cot 0 + tan 0) = sec 0
27. tan u cot u - cos u = sin? u
28. sin u csc u - cos u = sin? u
\ 29. (sec 0 - 1) (sec 0 + 1) = tan? 0
30. (csc 0 - 1) (csc 0 + 1) = cot? 0
31. (sec 0 + tan 6) (sec 0 - tan 0) = 1 32. (csc 0 + cot 0) (csc 0 - cot 0) = 1
33. cos? 0 (1 + tan? 0) = 1
34. (1 - cos 0) (1 + cot? 0) = 1
35. (sin 0 + cos 0)2 + (sin 0 - cos 0)2 = 2
containing the m
36. tan? 0 cos? 0 + cot? 0 sin? e = 1
37. sec* e - sec? 0 = tan 0 + tan? e
38. csc* 0 - csc² 0 = cot 0 + cot? e
cotient.
cos u
sin u
39. sec u - tan u
40. csc u - cot u =
41. 3 sin? 0 + 4 cos? 0 = 3 + cos 0
1 + sin u
1 + cos u
nd cosine funcion
cos? 0
sin? 0
1 - cos 0
42. 9 sec? 0 - 5 tan? 0 = 5 + 4 sec? 0
43. 1
= sin 0
44. 1 -
= - cos 0
1 + sin 0
e of the expresin
le.
1 + tan v
45.
1
cot v + 1
1 - sin v
csc v - 1
46.
csc v +1
sec e
47.
csc 0
sin 6
= 2 tan 0
cos 0
tan v
cot v - 1
1+ sin v
csc e - 1
48.
cot 0
1 + sin 0
49.
1
cos 0 + 1
50.
cos 0 - 1
cot e
csc 0 + 1
1 + seç 0
%3D
csc e + 1
sin 6
csc 0
·1
1 - sec 0
1 - sin v
51.
CoS V
1 + sin v
sin 0
cos v
1
= 2 sec v
52.
= 2 sec v
53.
sin 0 - cos 6
%3D
cos v
1- sin v
sin v
cos V
1- cot 0
listed in rad
1 - cos 0
56.
1 + cos 0
sin? 0
sin o
54. 1
= cos 0
= (sec 0 - tan 0)2
(csc 0 – cot 0)2
1 + cos e
1+sin 9
Os 0- sine p
cot 0
tan 0
cos 0
57.
1- tan 0
sin 0
sin 0 + cos
58.
1
= 1 + tan 0 + cot 0
cot 0
1 - cot 0
tan 0
1
tan 0 + sec 0 - 1
61.
tan 0 - sec 0 + 1
cos e
sin 0 cos 0
tan 0
= tan 0 + sec 0
59. tan 0 +
1 + sin 0
= sec 0
60.
cos 0 – sin? 0
1- tan? 0
sin 0
sin 0 - cos 0 + 1
62,
sin 0 + cos 0 - 1
tan 0 - cot 0
63.
tan 0 + cot 0
sec 0 - cos 0
64.
sec 0 + cos 0
sin 0 + 1
= sin? 0 – cos² e
quations i
1 + cos? 0
cos 0
tan u - cot u
66.
tan u + cot u
sec 0 + tan 0
67.
cot 0 + cos 0
tan u - cot u
+ 2 cos? u = 1
= tan 0 sec 0
65.
tan u + cot u
+ 1 = 2 sin? u
1 – tan² 0
69.
1 + tan? 6
1 - cot? 0
70.
1 + cot? 0
sind
1 - cos 0
+ 2 cos? 0 = 1
seç 0
+ 1 = 2 cos? 0
68.
1+ sec 0
sin? 0
sin? 0 – tan 0
72.
cos? 0 - cot 0
71 sec 0 – csc 0
tan? 0
73. sec 0 - cos 0 = sin 0 tan 0
71.
sec 0 csc 0
= sin 0 - cos 0
1 + sin 0
76.
1 - sin 0
1 - sin 0
1 + sin 0
1
= 2 seç? 0
= 4 tan 0 sec 0
74. tan 0 + cot 0 = sec 0 csc 0
75.
sin 0
1 + sin 0
denominatoR
Transcribed Image Text:113. Why do you think it is usually preferable to start with t
486 CHAPTER 7 Analytic Trigonometry
(sec v - tan v)2 + 1
= 2 tan y
79.
csc v ( sec v- tan v)
1 + sin 6
cos' e
1 + sin 0
78.
sec 0
= (sec 0 + tan 0)2
77
1- sin
sin 0
sin 0 - cos 0
= sec 0 csc 0
sin 0 + cos 0
81.
cos 0
sin 0
sec' v
80.
tan v + tan v
= sin v + cos v
sin 0 + cos 6
83.
sec v
= 1 - sin 0 cos 0
sin 0 + cos 0
82.
cos 0 - sin 0
= sec 0 csc 0
sin 0 + cos 0
cos 0 + sin 0 - sin? e
86.
sin 0
cos 0
cos? 0 - sin? 0
85.
1- tan? 0
= cot 0 + cos
sin' e + cos' 0
84.
sec 0 - sin 6
= cos? 0
sin 6
1- 2 cos e
tan 0 - 1
1 + sin 0 + cos 0
89.
1 + sin 0 - cos 0
1 + cos 0
(2 cos 0 – 1)?
87.
cos 0 - sin' e
1- 2 cos? 0
88.
sin 0 cos 0
= 1- 2 sin? e
= tan 0
cot 0
sin 0
91. (a sin 0 + b cos 0)2 + (a cos 0 – b sin 0)² = q² + 12
1 + cos 0 + sin 0
90.
1 + cos 0
= sec 0 + tan 0
- sin e
tan a + tan B
93.
cot a + cot B
= tan a tan B
92. (2a sin 0 cos 0)2 + a? (cos? 0 – sin? 0)² = a²
94. (tan a + tan B) (1 - cot a cot B) + (cot a + cot B) (1 - tan a tan B) = 0
95. (sin a + cos B)2 + (cos B + sin a) (cos B - sin a) = 2 cos B(sin a + cos B)
96. (sin a – cos B)2 + (cos B + sin a) (cos B - sin a) = -2 cos B(sin a – cos B)
97. In |sec e = -In |cos 0|
98. In tan 0| = In |sin 0| – In |cos 0
99. In 1 + cos 0 + In |1 – cos e = 2 In |sin 0
100. In sec 0 + tan 0 + In sec 0 – tan 0 = 0
In Problems 101–104, show that the functions f and g are identically equal.
101. f(x) = sin x• tan x
g(x) = secx - cos x
102. f(x) = cos x• cot x
g(x) = csc x – sin x
1 - sin 0
cos 0
1 + sin 0
103. f(0) =
8(0) = 0
104. f(0) = tan 0 + sec 0
g(0)
cos e
cos e
1 - sin 0
A 105. Show that V16 + 6 tan² 0 = 4 sec 0 if -< 0 <.
A 106. Show that V9 sec? - 9 = 3 tan 0 if T so<
Applications and Extensions
107. Searchlights A searchlight at the grand opening of a new car
dealership casts a spot of light on a wall located 75 meters
from the searchlight. The acceleration i of the spot of light is
found to be i = 1200 sec 0 (2 sec² 0 – 1). Show that this is
108. Optical Measurement Optical methods of measurement
often rely on the interference of two light waves. If two light
waves, identical except for a phase lag, are mixed together,
the resulting intensity, or irradiance, is given by
sin o
cos 0
Source: Adapted from Hibbeler, Engineering Mechanics:
1+
equivalent to i = 1200
(csc e - 1) (sec 0 + tan 0)
I = 4A
csc 0 sec 0
Show that this is equivalent to I, = (2A cos 0)“.
Source: Experimental Techniques, July/August 2002
Dynamics, 13th ed., Pearson 2013.
109. Challenge Problem Prove: sin (-x) =
- sin 'x
110. Challenge Problem Prove: cot¬lx = tan
Explaining Concepts: Discussion and Writing
111. Write a few paragraphs outlining your strategy for establishing
identities.
112. Write down the three Pythagorean Identities.
side containing the more complicated expression w
establishing an identity?
114. Make up an identity that is not a basic identity.
- Retain Your Knowledge
Problems 115-124 are based on material learned earlier in the course. The purpose of these probla
your mind so that you are better prepared for the final exam
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