C H A P T E R 5 Analytic Trigonometry Section 5.1
Using Fundamental Identities
. . . . . . . . . . . . . . . 438
Section 5.2
Verifying Trigonometric Identities . . . . . . . . . . . . . 450
Section 5.3
Solving Trigonometric Equations
Section 5.4
Sum and Difference Formulas . . . . . . . . . . . . . . . 471
Section 5.5
Multiple-Angle and Product-to-Sum Formulas
. . . . . . . . . . . . . 458
. . . . . . 490
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Practice Test
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
C H A P T E R 5 Analytic Trigonometry Section 5.1
■
Using Fundamental Identities
You should know the fundamental trigonometric identities. (a) Reciprocal Identities sin u
1 csc u
csc u
1 sin u
cos u
1 sec u
sec u
1 cos u
tan u
1 sin u cot u cos u
cot u
1 cos u tan u sin u
(b) Pythagorean Identities sin2 u cos2 u 1 1 tan2 u sec2 u 1 cot2 u csc2 u (c) Cofunction Identities sin
2 u cos u
tan
2 u cot u
sec
2 u csc u
cos cot
2 u sin u
2 u tan u
csc
2 u sec u
(d) EvenOdd Identities sinx sin x
cscx csc x
cosx cos x
secx sec x
tanx tan x
cotx cot x
■
You should be able to use these fundamental identities to find function values.
■
You should be able to convert trigonometric expressions to equivalent forms by using the fundamental identities.
Vocabulary Check 1. tan u
2. cos u
3. cot u
4. csc u
5. cot2 u
6. sec2 u
7. cos u
8. csc u
9. cos u
10. tan u
438
Section 5.1
1. sin x tan x
3
2
, cos x
1 ⇒ x is in Quadrant II. 2
32 sin x 3 cos x 12
cot x
3 1 1 tan x 3 3
sec x
1 1 2 cos x 12
csc x
1 2 23 1 sin x 32 3 3
3. sec 2, sin cos
2
2
⇒ is in Quadrant IV.
2
1 1 tan
1 2 csc sin
5. tan x
5 13 , sec x ⇒ x is in 12 12
12 1 sec x 13
sin x 1 cos2 x
3
, cos x
3
2
x is in Quadrant III.
1 23
sin x
2
csc x
1 2 sin x
sec x
2 1 23 3 cos x 3
cot x
3 1 3 tan x 3
14 21
5 3 4. csc , tan 3 4
sin
1 3 csc 5
cos
sin 3 tan 5
sec
1 5 cos 4
cot
1 4 tan 3
4
4
35
10
6. cot 3, sin
10
is in Quadrant II.
Quadrant III. cos x
3
is in Quadrant I.
1 1 sec 2 2
sin 22 1 tan 22 cos cot
2. tan x
Using Fundamental Identities
cos cot sin 144 5 1 169 13
tan
1 1 cot 3
310 10
cot x
12 1 tan x 5
csc
1 10 sin
csc x
13 1 sin x 5
sec
10 1 10 cos 3 10 3
3 35 7. sec , csc ⇒ is in Quadrant IV. 2 5
8. cos
2 x 5, cos x 5, x is in Quadrant I. 3
4
sin
5 1 1 csc 355 3
sin x
1 45
cos
2 1 1 sec 32 3
tan x
sin x 3 cos x 5
tan
5 sin 53 cos 23 2
csc x
1 5 sin x 3
cot
1 1 25 2 tan 52 5 5
sec x
1 5 cos x 4
cot x
1 4 tan x 3
2
5
3 5 3
44
439
440
Chapter 5
9. sin x
Analytic Trigonometry
2 1 1 ⇒ sin x , tan x ⇒ x is 3 3 4
10. sec x 4, sin x > 0 x is in Quadrant I.
in Quadrant II. cos x 1 sin2 x
1
1 22 9 3
cos x
1 1 sec x 4
cot x
1 1 22 tan x 24
sin x
1 14
sec x
1 1 32 cos x 223 4
tan x
15 sin x cos x 4
csc x
1 1 3 sin x 13
csc x
1 4 415 sin x 15 15
cot x
15 1 1 tan x 15 15
11. tan 2, sin < 0 ⇒ is in Quadrant III. sec tan2 1 4 1 5 cos
5 1 1 5 sec 5
5 1 sin 2 1 1 cot tan 2
csc
13. sin 1, cot 0 ⇒
3 2
cos 1 sin2 0 sec is undefined.
4
1 15
1 1 csc 5
1 51
2
26 5
sin 1 cos 5
sec
1 5 56 cos 12 26
cot
1 12 26 tan 6
6 5 12 26
14. tan is undefined, sin > 0.
2 sin is undefined ⇒ cos 0 cos
sin 1 02 1
csc 1
15. sec x cos x sec x
4
tan
tan
tan is undefined.
15
is in Quadrant III.
cos
1 51 25 2 5 5
12. csc 5, cos < 0
sin
sin 1 cos2
2
1 1 sec x
The expression is matched with (d). 17. cot2 x csc2 x cot2 x 1 cot2 x 1 The expression is matched with (b).
csc
1 1 sin
sec
1 is undefined. cos
cot
cos 0 0 sin 1
16. tan x csc x
sin x cos x
1
1
sin x cos x sec x
Matches (a).
18. 1 cos2 xcsc x sin2 x Matches f .
1 sin x sin x
Section 5.1
19.
sin x sinx tan x cosx cos x
20.
sin2 x cos x cot x cos2 x sin x
21. sin x sec x sin x
22. cos2 xsec2 x 1 cos2 xtan2 x
1 tan x cos x
23. sec4 x tan4 x sec2 x tan2 xsec2 x tan2 x sec2 x tan2 x1 sec2 x tan2 x
cos x sin2 x 2
The expression is matched with f .
sin2 x Matches (c).
24. cot x sec x
cos x sin x
1
1
cos x sin x csc x
25.
Matches (a).
26.
sin2 x sec2 x 1 tan2 x 2 2 sin x sin x cos2 x
1
sin2 x sec2 x
The expression is matched with (e).
sin2 x sin x cos22 x sin x tan x sin x cos x cos x cos x
27. cot sec
cos sin
1
1
cos sin csc
Matches (d).
28. cos tan cos
sin sin cos
29. sin csc sin sin
1 sin2 sin
1 sin2 cos2
30. sec2 x1 sin2 x sec2 x sec2 x sin2 x sec2 x
1 cos2 x
sec2 x
sin2 x cos2 x
31.
cos xsin x cot x csc x 1sin x
sin2 x
cos x sin x
sin x 1
cos x
sec2 x tan2 x 1
32.
csc 1sin cos cot sec 1cos sin
33.
1 sin2 x cos2 x sin2 x cos2 x tan2 x cos2 x 2 2 2 csc x 1 cot x cos x sin2 x
34.
1 1 1 cos2 x tan2 x 1 sec2 x 1cos2 x
35. sec
sin 1 sin cot tan cos
36.
tan2 sin2 2 sec cos2
sin2 cos2
1
sec2 1
1cos2
441
The expression is matched with (b).
Matches (c).
The expression is matched with (e).
cos 2 x
Using Fundamental Identities
37. cos sin2 cos2 sin2 cos2
1 cos sin 1 cos sin
2 x sec x sin xsec x sin x
cos x cos x tan x 1
sin x
442
Chapter 5
Analytic Trigonometry
38. cot
2 x cos x tan x cos x cos x cos x sin x
sin x
39.
cos2 y 1 sin2 y 1 sin y 1 sin y
40. cos t1 tan2 t cos tsec2 t
41. sin tan cos sin
1 sin y1 sin y 1 sin y 1 sin y
1 cos t sec t cos2 t cos t
sin cos cos
42. csc tan sec
1 sin
sin
cos sec
sin2 cos2 cos cos
sin2 cos2 cos
2 sec
1 cos
1 sec cos
sec
43. cot u sin u tan u cos u
cos u sin u sin u cos u sin u cos u
44. sin sec cos csc
cos u sin u
sin cos cos sin
sin2 cos2 cos sin
1 cos sin
sec csc 45. tan2 x tan2 x sin2 x tan2 x1 sin2 x
46. sin2 x csc2 x sin2 x sin2 xcsc2 x 1 sin2 x cot2 x
tan2 x cos2 x
sin2 x cos2 x
sin2 x
cos2 x
cos2 x
sin2 x 47. sin2 x sec2 x sin2 x sin2 xsec2 x 1
cos2 x sin2 x
48. cos2 x cos2 x tan2 x cos2 x1 tan2 x cos2 xsec2 x
sin2 x tan2 x
cos2 x
cos1 x 2
1
49.
sec2 x 1 sec x 1sec x 1 sec x 1 sec x 1
50.
cos2 x 4 cos x 2cos x 2 cos x 2 cos x 2
sec x 1
cos x 2
51. tan4 x 2 tan2 x 1 tan2 x 12
52. 1 2 cos2 x cos4 x 1 cos2 x2
sec2 x2 sec4 x
sin2 x2 sin4 x
Section 5.1 53. sin4 x cos4 x sin2 x cos2 xsin2 x cos2 x
Using Fundamental Identities
54. sec4 x tan4 x sec2 x tan2 xsec2 x tan2 x
1sin2 x cos2 x
sec2 x tan2 x1
sin2 x cos2 x
sec2 x tan2 x
55. csc3 x csc2 x csc x 1 csc2 xcsc x 1 1csc x 1 csc2 x 1csc x 1 cot2 xcsc x 1 56. sec3 x sec2 x sec x 1 sec2 xsec x 1 sec x 1 sec2 x 1sec x 1 tan2 xsec x 1 57. sin x cos x2 sin2 x 2 sin x cos x cos2 x
58. cot x csc xcot x csc x cot2 x csc2 x 1
sin2 x cos2 x 2 sin x cos x 1 2 sin x cos x 59. 2 csc x 22 csc x 2 4 csc2 x 4
61.
60. 3 3 sin x3 3 sin x 9 9 sin2 x
4csc2 x 1
91 sin2 x
4 cot2 x
9 cos2 x
1 1 1 cos x 1 cos x 1 cos x 1 cos x 1 cos x1 cos x
62.
1 1 sec x 1 sec x 1 sec x 1 sec x 1 sec x 1sec x 1
2 1 cos2 x
sec x 1 sec x 1 sec2 x 1
2 sin2 x
2 tan2 x
2 csc2 x
2
tan x 1
2
2 cot2 x
63.
1 sin x cos2 x 1 sin x2 cos2 x 1 2 sin x sin2 x cos x 1 sin x cos x cos x1 sin x cos x1 sin x
2 2 sin x cos x1 sin x
21 sin x cos x1 sin x
2 cos x
2 sec x
64. tan x
sec2 x tan2 x sec2 x tan x tan x
1 cot x tan x
65.
1 cos2 y sin2 y 1 cos y 1 cos y
1 cos y1 cos y 1 cos y 1 cos y
443
444
66.
Chapter 5 5 tan x sec x
Analytic Trigonometry
tan x sec x
5tan x sec x tan2 x sec2 x
5tan x sec x 1
tan x sec x
67.
3 sec x tan x
sec x tan x
3sec x tan x sec2 x tan2 x
3sec x tan x 1
sec x tan x
3sec x tan x
5sec x tan x
68.
csc x 1
tan2 x csc x 1
69. y1 cos
csc x 1
2 x, y
2
tan2 xcsc x 1 tan2 xcsc x 1 tan2 xcsc x 1 tan2 x tan4 xcsc x 1 2 csc x 1 cot2 x
sin x
1
x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
y1
0.1987
0.3894
0.5646
0.7174
0.8415
0.9320
0.9854
0
y2
0.1987
0.3894
0.5646
0.7174
0.8415
0.9320
0.9854
Conclusion: y1 y2
2
0
70. y1 sec x cos x, y2 sin x tan x
6
x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
y1
0.0403
0.1646
0.3863
0.7386
1.3105
2.3973
5.7135
2
0 0
y2
0.0403
71. y1
0.1646
0.3863
0.7386
1.3105
2.3973
5.7135
It appears that y1 y2.
cos x 1 sin x , y 1 sin x 2 cos x
12
x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
y1
1.2230
1.5085
1.8958
2.4650
3.4082
5.3319
11.6814
y2
1.2230
1.5085
1.8958
2.4650
3.4082
5.3319
11.6814
2
0
72. y1 sec4 x sec2 x, y2 tan2 x tan4 x
Conclusion: y1 y2
1200
x
0.2
0.4
0.6
0.8
1.0
1.2
1.4
y1
0.0428
0.2107
0.6871
2.1841
8.3087
50.3869
1163.6143
y2
1
2
0
0.0428
0.2107
0.6871
2.1841
8.3087
50.3869
0
1163.6143
It appears that y1 y2.
73. y1 cos x cot x sin x csc x cos x cot x sin x cos x
x sin x cos sin x
cos2 x sin2 x sin x sin x
cos2 x sin2 x 1 csc x sin x sin x
4
−2
2
−4
Section 5.1
Using Fundamental Identities
74. y1 sec x csc x tan x cot x sec x csc x tan x
75. y1
1 cos x
1
6
sin x
sin x cos x
1 sin2 x cos x sin x cos x sin x
1 sin2 x cos x sin x
cos2 x cos x cot x cos x sin x sin x
− 2
−6
1 1 cos x tan x sin x cos x
5
1 1 1 cos x cos x sin x cos x sin x cos x sin x
76. y1
− 2
sin2 x sin x 1 cos2 x tan x sin x cos x sin x cos x cos x
1 1 sin cos 2 cos 1 sin
4
1 1 2 sin sin2 cos2 2 cos 1 sin
1 1 2 sin 1 2 cos 1 sin
1 2 2 sin 2 cos 1 sin
1 1 sin sec cos 1 sin cos
2
−5
cos 1 1 sin 1 sin 1 1 sin cos cos 2 cos 1 sin 2 cos 1 sin cos 1 sin
2
− 2
2
−4
77. Let x 3 cos , then 9 x2 9 3 cos 2 9 9 cos2 91 cos2
9 sin2 3 sin . 78. Let x 2 cos . 64 16x2 64 162 cos 2
79. Let x 3 sec , then x2 9 3 sec 2 9
641 cos2
9 sec2 9
64 sin2
9sec2 1
8 sin
9 tan2 3 tan .
445
446
Chapter 5
Analytic Trigonometry
80. Let x 2 sec .
81. Let x 5 tan , then
x 4 2 sec 4
x2 25 5 tan 2 25
4sec2 1
25 tan2 25
4 tan2
25tan2 1
2 tan
25 sec2
2
2
5 sec . 82. Let x 10 tan .
83. Let x 3 sin , then 9 x2 3 becomes
x2 100 10 tan 2 100
9 3 sin 2 3
100tan2 1
9 9 sin2 3
100 sec2
91 sin2 3
10 sec
9 cos2 3
3 cos 3 cos 1 sin 1 cos2 1 12 0. 84. x 6 sin
85. Let x 2 cos , then 16 4x2 22 becomes 16 42 cos 2 22
3 36 x2 36 6 sin 2
16 16 cos2 22
361 sin2
161 cos2 22
36 cos2
16 sin2 22
6 cos cos
4 sin 22
3 1 6 2
sin ± 1
sin cos2
1 12 3 ± 4 ±
±
86.
2
2
cos 1 sin2
1 21 1 2
2
3
2
x 10 cos
2
2
.
87. sin 1 cos2
53 100 x2
Let y1 sin x and y2 1 cos2 x, 0 ≤ x ≤ 2.
100 10 cos 2
y1 y2 for 0 ≤ x ≤ , so we have
1001 cos2
sin 1 cos2 for 0 ≤ ≤ .
100 sin2
2
10 sin
y2
3 53 sin 10 2
cos 1 sin 2
2
0
y1
1 2 3
2
−2
1 2
Section 5.1
Using Fundamental Identities
89. sec 1 tan2
88. cos 1 sin2 2
Let y1
1 and y2 1 tan2 x, 0 ≤ x ≤ 2. cos x
2
0
447
y1 y2 for 0 ≤ x <
3 and < x ≤ 2, so we have 2 2
−2
sec 1 tan2 for 0 ≤ <
3 ≤ ≤ 2 2
3 and < < 2. 2 2
4
y2 2
0
y1 −4
90. csc 1 cot2
91. ln cos x ln sin x ln
2
0 < <
cos x ln cot x sin x
2
0
−2
92. ln sec x ln sin x ln sec x sin x
1 ln cos x
93. ln cot t ln1 tan2 t ln cot t 1 tan2 t
sin x
ln tan x
94. lncos2 t ln1 tan2 t lncos2 t1 tan2 t lncos2 t sec2 t
ln cos2 t
1 cos2 t
96. tan2 1 sec2
(b) csc2
97. cos
346
ln
1 ln csc t sec t sin t cos t
1 sec 346 cos 346
(b)
2
2 sin 80
cos90 80 sin 80
1.0622
tan 3.12 1 1.00173 sec 3.12
0.9848 0.9848
0.8
(b)
3.1
cos 3.1 1
cos 2
1.00173
cos2 t
2 2 cot2
1.6360 0.6360 1 7 7
(a)
tan 346 1 1.0622
1
cos t sin t
95. (a) csc2 132 cot2 132 1.8107 0.8107 1
2
2
ln
ln1 0
(a)
ln cot t sec2 t
2 0.8 sin 0.8 0.7174 0.7174
448
Chapter 5
Analytic Trigonometry
98. sin sin
99. W cos W sin
250
(a)
sin250 0.9397
W sin tan W cos
sin 250 0.9397
(b)
1 2
2 0.4794
sin
sin
1
1
0.4794 2
100. csc x cot x cos x
1 cos x cos x sin x sin x
cos x cos x sin2 x
cos x sin2 x cos x sin2 x
cos x1 sin2 x sin2 x
cos x cos2 x cos x cot2 x sin2 x
102. False. A cofunction identity can be used to transform a tangent function so that it can be represented by a cotangent function.
104. As x → 0, cos x → 1 and sec x
106. As x → , sin x → 0 and csc x
1 → 1. cos x
1 → . sin x
101. True. For example, sinx sin x means that the graph of sin x is symmetric about the origin.
103. As x →
, sin x → 1 and csc x → 1. 2
105. As x →
, tan x → and cot x → 0. 2
107. cos 1 sin2 is not an identity. cos2 sin2 1 ⇒ cos ± 1 sin2
108. The equation is not an identity.
109.
cot ± csc2 1
sin k tan k cos k 111. sin csc 1 is an identity.
110. The equation is not an identity.
1 1 1 1 sec 5 sec 5 cos 5 cos 5 112. The equation is not an identity. The angles are not the same. sin csc sin
sin k tan is not an identity. cos k
1
sin
sin sin 1, in general
sin
1
sin 1, provided sin 0.
Section 5.1 113. Let x, y be any point on the terminal side of .
yr xr 2
449
114. Divide both sides of sin2 cos2 1 by cos2 :
Then, r x2 y2 and sin2 cos2
Using Fundamental Identities
sin 2 cos 2 1 2 cos cos 2 cos 2
2
tan2 1 sec2 Divide both sides of sin2 cos2 1 by sin2 :
y2 x2 r2
sin 2 cos 2 1 sin 2 sin 2 sin 2
r2 r2
1 cot2 csc2
1.
Discussion for remembering identities will vary, but one key is first to learn the identities that concern the sine and cosine functions thoroughly, and then to use these as a basis to establish the other identities when necessary.
115. x 5x 5 x 52 x 25
116. 2z 32 2z 22z 3 32
2
2
4z 12z 9
117.
1 x x 8 xx 5 x5 x8 x 5x 8
119.
118.
x2 6x 8 x 5x 8
2x 7 2xx 4 7x2 4 x2 4 x 4 x2 4x 4
121. f x
120.
x2
6x 3 x4
32x 1 x4
x x2x 5 x2 x 25 x 5 x 5x 5 x 5x 5
2x2 8x 7x2 28 x2 4x 4
x x3 5x2 x 5x 5
5x2 8x 28 x2 4x 4
x1 x2 5x x 5x 5
xx2 5x 1 x2 25
1 sinx 2
122. f x 2 tan
y
2x
y 3
2
Amplitude: 2
1 Amplitude: 2
1
2 Period: 2
x 1 −1
3
Period:
2 2
1 1 3 1 0, 0, , , 1, 0, , , 2, 0 2 2 2 2
−3
Two consecutive vertical asymptotes: x 1, x 1
−2
Key points:
6x 3 6x 3 x4 4x x4 x4
Key points:
21, 2, 0, 0, 12, 2
−1
x 1 −1 −2 −3
3
450
Chapter 5
123. f x
Analytic Trigonometry
1 sec x 2 4
y 4 3
1 cos x first. 2 4
Sketch the graph of y
2 1
π 2 −2
1 2
Amplitude:
3π 2π 2
x
−3
Period: 2
−4
One cycle: x 0 ⇒ x 4 4 x
7 2 ⇒ x 4 4
The x-intercepts of y x
1 1 cos x correspond to the vertical asymptotes of f x sec x . 2 4 2 4
5 ,x , . . . 4 4
124. f x
3 cosx 3 2
Using y a cos bx, a b 1 so the period is
y 5 4
3 3 so the amplitude is . 2 2
3
1
2 2. 1
−π
π
2π
x
−2
x shifts the graph right by and 3 shifts the graph upward by 3.
Section 5.2
−1 −3
Verifying Trigonometric Identities
■
You should know the difference between an expression, a conditional equation, and an identity.
■
You should be able to solve trigonometric identities, using the following techniques. (a) Work with one side at a time. Do not “cross” the equal sign. (b) Use algebraic techniques such as combining fractions, factoring expressions, rationalizing denominators, and squaring binomials. (c) Use the fundamental identities. (d) Convert all the terms into sines and cosines.
Vocabulary Check 1. identity
2. conditional equation
3. tan u
4. cot u
5. cos2 u
6. sin u
7. csc u
8. sec u
1. sin t csc t sin t
sin t 1 1
3. 1 sin 1 sin 1 sin2 cos2
2. sec y cos y
cos1 y cos y 1
4. cot2 ysec2 y 1 cot2 y tan2 y 1
Section 5.2 5. cos2 sin2 1 sin2 sin2
Verifying Trigonometric Identities
6. cos2 sin2 cos2 1 cos2
1 2 sin2
2 cos2 1
7. sin2 sin4 sin2 1 sin2
8. cos x sin x tan x cos x sin x
1 cos2 cos2 cos2 cos4
cos x
cos2 x sin2 x cos x
1 cos x
sec x
9.
csc2 1 csc2 cot cot
10.
cot3 t cot t cot2 t csc t csc t
csc2 tan
sin sin1 cos
sin1 cos1
2
cot tcsc2 t 1 csc t
cos t csc2 t 1 sin t 1 sin t
csc sec
cos t sin t csc2 t 1 sin t
cos tcsc2 t 1
11.
cot2 t cos2 t csc t sin2 t
sin t
12.
cos2 t sin t
1 sin2 t 1 sin2 t sin t sin t sin t
1 1 tan2 tan tan tan
sec2 tan
csc t sin t 13. sin12 x cos x sin52 x cos x sin12 x cos x1 sin2 x sin12 x cos x
cos2 x cos3 xsin x
14. sec6 xsec x tan x sec4 xsec x tan x sec4 xsec x tan xsec2 x 1 sec4 xsec x tan x tan2 x sec5 x tan3 x
15.
1 cos x cos x cot x cos x sec x tan x sin x
cos2 x sin x
1 sin2 x sin x
1 sin x sin x
csc x sin x
16.
sec 1 sec 1 1 cos 1 1sec
sec
sec
sec sec 1 sec 1
sec
sin x
451
452
Chapter 5
17. csc x sin x
Analytic Trigonometry 1 sin2 x sin x sin x
18. sec x cos x
1 sin2 x sin x
1 cos2 x cos x
cos2 x sin x
sin2 x cos x
cos x 1
cos x sin x
sin x
cos x cot x
19.
20.
1 1 csc x sin x sin x csc x sin x csc x
cot x tan x 1
tan x cot x
csc x sin x 1
csc x sin x
cos cot cos cot 1 sin 1 1 sin 1 sin cos
cos 1 sin sin 1 sin
22.
1 sin cos 1 sin 2 cos2 cos 1 sin cos 1 sin
sin
sin
cos2 sin sin2 sin 1 sin 1 sin sin 1 sin
sin x cos x
sin x tan x
1 1 cot x tan x tan x cot x tan x cot x
21.
1 cos x cos x
1 sin
1 2 sin sin2 cos2 cos 1 sin
2 2 sin cos 1 sin
21 sin cos 1 sin
2 cos
2 sec
csc
23.
1 1 csc x 1 sin x 1 sin x 1 csc x 1 sin x 1csc x 1
24. cos x
cos x1 tan x cos x cos x 1 tan x 1 tan x
sin x csc x 2 sin x csc x sin x csc x 1
cos x tan x 1 tan x
sin x csc x 2 1 sin x csc x 1
cos xsin xcos x 1 sin xcos x
sin x csc x 2 sin x csc x 2
sin x cos x cos x sin x
sin x cos x sin x cos x
1
25. tan
2 tan cot tan
tan1 tan
1
26.
cos2 x sin x tan x sin2 x cos x
27.
cos x
cos x
cscx 1sinx secx 1cosx
cosx sinx
cos x sin x
cot x
Section 5.2
28. 1 sin y1 siny 1 sin y1 sin y
29.
Verifying Trigonometric Identities
tan x cot x 1 sec x cos x cos x
1 sin2 y cos2 y
tan x tan y 30. 1 tan x tan y
32.
1 1 cot x cot y 1 1 1 cot x cot y
cot x cot y cot x cot y
1 1 cot x tan y tan x cot y 31. tan x cot y 1 1 cot x tan y
cot y cot x cot x cot y 1
cot x tan y
cot x tan y
tan y cot x
sin x sin y cos x cos y cos x cos ycos x cos y sin x sin ysin x sin y sin x sin y cos x cos y sin x sin ycos x cos y
cos2 x cos2 y sin2 x sin2 y sin x sin ycos x cos y
cos2 x sin2 x cos2 y sin2 y sin x sin ycos x cos y
0
33.
1 sin 1 sin 11 sin sin 1 sin 1 sin
35. cos2 cos2
37. sin t csc
34.
1 cos 1 cos 11 cos cos 1 cos 1 cos
11 sin sin
11 cos cos
1 cossin
1 sincos
1 sin cos
1 cos sin
2
2
2
2
2 cos
2
sin2 1
2 t sin t sec t sin t cos1 t
−5
38. sec2
(b) 5
−5
2 2sin x2 sin x2 cos x2 2 cos x cos x2 and y2 1. Let y1
Identity —CONTINUED—
2
2
2 y sec
2
2 x 1 csc
sin t tan t cos t
5
39. (a)
36. sec2 y cot2
2
2
Identity
2
y tan2 y 1
x 1 cot2 x
453
454
Chapter 5
Analytic Trigonometry
39. —CONTINUED— (c) 2 sec2 x 2 sec2 x sin2 x sin2 x cos2 x 2 sec2 x1 sin2 x sin2 x cos2 x 2 sec2 xcos2 x 1 2
1 cos2 x
cos2 x 1
21 1 40. (a)
(b)
3
−2
2 −1
Identity
Identity (c) csc xcsc x sin x
sin x cos x cos x cot x csc2 x csc x sin x 1 cot x sin x sin x csc2 x 1 1 cot x cot x csc2 x
41. (a)
(b)
5
y2 y1
−2
2
−1
Not an identity (c) 2 cos2 x 3 cos4 x 1 cos2 x2 3 cos2 x
Let y1 2 cos x2 3cos x4 and
sin2 x2 3 cos2 x
y2 sin x23 2cos x2.
sin2 x3 2 cos2 x
Not an identity 42. (a)
(b)
5
−
−5
Not an identity
Not an identity (c) tan4 x tan2 x 3
sin4 x sin2 x 3 4 cos x cos2 x
sin4 x 1 sin2 x 3 2 cos x cos2 x
1 sin4 x sin2 x cos2 x 3 cos2 x cos2 x
1 sin2 x sin2 x cos2 x 3 2 cos x cos2 x
1 sin2 x cos2 x cos2 x
1 3
sec2 x tan2 x 3 sec2 x 4 tan2 x 3
Section 5.2 43. (a)
44. (a)
5
Verifying Trigonometric Identities
1
−2 −2
455
2
2 −1
−1
Let y1
1 2 1 1 and y2 . sin x4 sin x2 tan x4
Identity (b)
Identity (b) Identity (c) sin4 2 sin2 1 cos sin2 12 cos Identity
cos2 2 cos
(c) csc4 x 2 csc2 x 1 csc2 x 12
cot2
45. (a)
x 2
cot4
cos5 x 46. (a)
3
−2
y2
y1
−2
2
−3
Let y1
3
2
−5
cos x 1 sin x . and y2 1 sin x cos x
Not an identity (b)
Not an identity (b) Not an identity (c) Not an identity (c)
cos x cos x 1 sin x 1 sin x
1 sin x
1 sin x
cos x1 sin x 1 sin2 x
cos x1 sin x 1 sin x cos2 x cos x
47. tan3 x sec2 x tan3 x tan3 x sec2 x 1
tan3 x
tan5 x
cot csc 1 . is the reciprocal of csc 1 cot They will only be equivalent at isolated points in their respective domains. Hence, not an identity.
48. tan2 x tan4 x sec 2 x
tan2 x
sin x sin x 1 cos x cos x cos x 2
4
2
4
2
1 sin4 x sin2 x 4 cos x cos2 x
1 sin2 x cos2 x sin4 x 4 cos x cos2 x
1 sin2 xcos2 x sin2 x 4 cos x cos2 x
sin2 x 1 4 cos x cos2 x
1 sec4 x tan2 x
456
Chapter 5
Analytic Trigonometry
49. sin2 x sin4 x cos x sin2 x 1 sin2 x cos x
50. sin4 x cos4 x sin2 x sin2 x cos 4 x
sin2 x cos2 x cos x
1 cos2 x 1 cos2 x cos 4 x
sin2 x cos3 x
1 2 cos2 x cos4 x cos 4 x 1 2 cos2 x 2 cos4 x
51. sin2 25 sin2 65 sin2 25 cos290 65
52. cos2 55 cos2 35 cos2 55 sin290 35
sin2 25 cos2 25
cos2 55 sin2 55
1
1
53. cos2 20 cos2 52 cos2 38 cos2 70 cos2 20 cos2 52 sin290 38 sin290 70 cos2 20 cos2 52 sin2 52 sin2 20
cos2 20 sin2 20 cos2 52 sin2 52 11 2 54. sin2 12 sin2 40 sin2 50 sin2 78 sin2 12 sin2 78 sin2 40 sin2 50
cos290 12 sin2 78 cos290 40 sin2 50
cos2 78 sin2 78 cos2 50 sin2 50
112
55. cos x csc x cot x cos x
1 cos x sin x sin x
cos x 1
1 sin2 x
cos x1 csc2 x cos xcsc2 x 1 cos x cot2 x
56. (a) (b)
h sin90 h cos h cot sin sin
(c) Greatest: 10 , Least: 90
(d) Noon
10
20
30
40
50
60
70
80
90
s
28.36
13.74
8.66
5.96
4.20
2.89
1.82
0.88
0
57. False. For the equation to be an identity, it must be true for all values of in the domain.
58. True. An identity is an equation that is true for all real values in the domain of the variable.
59. Since sin2 1 cos2 , then sin ± 1 cos2 ;
60. tan sec2 1
sin 1 cos2 if lies in Quadrant III or IV. One 7 such angle is . 4
True identity: tan ± sec2 1 tan sec2 1 is not true for 2 < < or 32 < < 2. Thus, the equation is not true for 34.
Section 5.2 61. 2 3i 26 2 3i 26i
Verifying Trigonometric Identities
62. 2 5i2 2 5i2 5i
2 3 26 i
4 20i 25i2 4 20i 25 21 20i
63. 16 1 4 4i1 2i
64. 3 2i3 3 2i3 2i3 2i
4i 8i2
9 12i 4i23 2i
4i 8
5 12i3 2i
8 4i
15 10i 36i 24i2 9 46i
65. x2 6x 12 0 a 1, b 6, c 12
66. x2 5x 7 0 a 1, b 5, c 7
6 ± 62 4112 21
x
5 ± 52 417 21
6 ± 36 48 2
x
5 ± 53 2
6 ± 84 2
6 ± 221 2
x
3 ± 21 67. 3x2 6x 12 0 3x2 2x 4 0 x2
2x 4 0
a 1, b 2, c 4 x
2 ± 22 414 21
68. 8x2 4x 3 0 a 8, b 4, c 3 x
4 ± 42 483 28 4 ± 112 16
2 ± 4 16 2
x
4 ± 47 16
2 ± 20 2
x
1 1 ± 7 4
2 ± 25 2
1 ± 5
457
458
Chapter 5
Analytic Trigonometry
Section 5.3
Solving Trigonometric Equations
■
You should be able to identify and solve trigonometric equations.
■
A trigonometric equation is a conditional equation. It is true for a specific set of values.
■
To solve trigonometric equations, use algebraic techniques such as collecting like terms, extracting square roots, factoring, squaring, converting to quadratic type, using formulas, and using inverse functions. Study the examples in this section.
Vocabulary Check 1. general
2. quadratic
3. extraneous
2. sec x 2 0
1. 2 cos x 1 0 (a) 2 cos
1 12 10 3 2
(b) 2 cos
1 5 12 10 3 2
3
(a) x
sec
1 2 2 3 cos3
(b) x sec
1 2220 12
5 3
5 1 2 2 3 cos53
4. 2 cos2 4x 1 0
3. 3 tan2 2x 1 0
(a) 3 tan 2
12
2
1 6
1 3 tan2
1 1 3
3
2
(a) x
16
161 2 cos
2 cos2 4
0 5 (b) 3 tan 2 12
2
1 2220 12
5 1 3 tan2 1 6
3 0
1 3
1 2
(b) x
2
1 4
2
2
2
2
2 1 1 1 0
2
1
1
3 16 3
16 1 2 cos
2 cos2 4
2
2 2
3 1 4
2
2
2
1
2 1 0 1
Section 5.3
sin 1 212 1 1 2 2
(a) x
0 (b) 2 sin2
459
6. csc4 x 4 csc2 x 0
5. 2 sin2 x sin x 1 0 (a) 2 sin2
Solving Trigonometric Equations
csc4
7 7 1 sin 12 6 6 2
2 1
2
1
6 4 1 4 csc2 4 6 6 sin 6 sin26
1 1 1 2 2
1 4 124 122
16 16 0
0
(b) x csc4
5 6 5 4 5 1 4 csc 4 6 6 sin 56 sin256
1 4 124 122
16 16 0 7. 2 cos x 1 0 2 cos x 1 cos x
sin x
1 2
x
x
2 2n 3
or x
4 2n 3
3 csc x 2
1 2
csc x
7 2n 6
x
11 or x 2n 6
or x
11. 3 sec2 x 4 0
10. tan x 3 0 tan x 3 x
9. 3 csc x 2 0
8. 2 sin x 1 0
sec2 x
2 n 3
cot2 x
cot x ±
n 6 5 or x n 6
sin x 0 x n
or
2 2n 3
1 3
2 3
x
13. sin xsin x 1 0
2n 3
12. 3 cot2 x 1 0
4 3
sec x ±
2 3
x or x
1 3
n 3 2 n 3
14. 3 tan2 x 1tan2 x 3 0
sin x 1 x
3 2n 2
3 tan2 x 1 0 tan x ±
or 1 3
x n 6 5 or x n 6
tan2 x 3 0 tan x ± 3 x or x
n 3 2 n 3
460
Chapter 5
Analytic Trigonometry
15. 4 cos 2 x 1 0 cos2 x
sin2 x 31 sin2 x 0
1 4
cos2 x ±
4 sin2 x 3 1 2
sin x ±
x n or 3
2 x n 3
17. 2 sin2 2x 1 sin 2x ±
x
3
2
n or 3
x
2 n 3
18. tan2 3x 3 1 2
±
2
tan 3x ± 3
2
3 2x 2n, 2x 2n, 4 4
3x
n n ⇒ x 3 9 3
3x
2 2 n n ⇒ x 3 9 3
or
5 7 2x 2n, 2x 2n 4 4 Thus, x
sin2 x 3 cos2 x
16.
3 5 7 n , n, n, n. 8 8 8 8
We can combine these as follows: x
n 3 n ,x 8 2 8 2
19. tan 3xtan x 1 0 tan 3x 0
or
20. cos 2x2 cos x 1 0
tan x 1 0
3x n
cos 2x 0
tan x 1
n x 3
x n 4
2 cos x 1 0
or
2x
n 2
cos x
x
n 4 2
x or
cos3 x cos x
21.
22. sec2 x 1 0
cos3 x cos x 0
sec2 x 1
cos xcos x 1 0 2
cos x 0, 3 or cos2 x 1 0
3 or 2 1cos x ± 1 x , 2 2 cos x 0, 3 or cos2 1x 0, 23.
3 tan3 x tan x 0 tan x3 tan2 x 1 0 tan x 0, or 3 tan2 x 1 0 3 tan x 0, or 32 1tan x ± 3
tan x 0, or 32 1tan x
5 7 11 , , , 6 6 6 6
sec x ± 1 x 0 or x
1 2
2 2n 3
x
4 2n 3
Section 5.3 2 sin2 x 2 cos x
24.
25.
2 2 cos2 x 2 cos x
sec x 2 0
cos x2 cos x 1 0 2 cos x 1 0
or
3 x , 2 2
x
2 cos x 1 cos x x
sec x csc x 2 csc x
x
sin2 x
5 , 3 3
sec x tan x 1
29.
1 ⇒ No solution 2
2 cos2 x cos x 1 0
2 cos x 1cos x 1 0
1 sin x 1 cos x cos x
2 cos x 1 0
1 sin x cos x
cos x
1 sin x2 cos2 x 1 2 sin x sin2 x cos2 x 1 2 sin x
1 0 sin x
2 sin2 x 1 0
sec x 2 0 sec x 2
sin2
x
27. 2 sin x csc x 0
csc xsec x 2 0
28.
5 , 3 3
2 4 , 3 3
2 sin x
No solution
sec x 1
1 2
sec x csc x 2 csc x 0
or
or sec x 1 0
sec x 2
cos x 0
csc x 0
sec2 x sec x 2 0
sec x 2sec x 1 0
2 cos2 x cos x 0
26.
Solving Trigonometric Equations
x1
x
sin2
x
or cos x 1 0
1 2
cos x 1
5 , 3 3
x
2 sin2 x 2 sin x 0 2 sin xsin x 1 0 sin x 0
or
sin x 1 0
x 0,
sin x 1
is extraneous.
x
3 2
32 is extraneous. x 0 is the only solution. 30.
2 sin2 x 3 sin x 1 0
31.
2 sin x 1sin x 1 0 2 sin x 1 0 sin x x
or 1 2
7 11 , 6 6
2 sec2 x tan2 x 3 0 2tan2 x 1 tan2 x 3 0
sin x 1 0 sin x 1 x
3 2
3 tan2 x 1 0 tan x ± x
3
3
5 7 11 , , , 6 6 6 6
461
462 32.
Chapter 5
Analytic Trigonometry
cos x sin x tan x 2 cos x sin x
csc x cot x 1
33.
csc x cot x2 12
cos x 2 sin x
csc2 x 2 csc x cot x cot2 x 1
cos2 x sin2 x 2 cos x
cot2 x 1 2 csc x cot x cot2 x 1 2 cot2 x 2 csc x cot x 0
1 2 cos x cos x
2 cot xcot x csc x 0
1 2
2 cot x 0 x
5 x , 3 3
or
cot x csc x 0
3 , 2 2
cos x 1 sin x sin x cos x 1 x
By checking in the original equation, we find that x and x 32 are extraneous. The only solution to the equation in the interval 0, 2 is x 2. 34. sin x 2 cos x 2
35. cos 2x
sin x cos x 2x
sin x 1 cos x
x
tan x 1
1 2
5 2n or 2x 2n 3 3 n 6
x
5 n 6
x tan1 1 x
36. sin 2x 2x
5 , 4 4
3
37. tan 3x 1
2
4 2n 3
or
2x
5 2n 3
5 n x 6
2 x n 3
3x
2n 4
x
2n 12 3
or
These can be combined as x
38. sec 4x 2
4x 2n 3 x
n 12 2
39. cos or
5 4x 2n 3 x
5 n 12 2
2x
5 2n 4
x
5 2n 12 3
n . 12 3
2
2
x 2n 2 4 x
3x
4n 2
or
x 7 2n 2 4 x
7 4n 2
Section 5.3
40. sin
3 x 2 2
x 1 2 From the graph in the textbook we see that the curve has x-intercepts at x 1 and at x 3.
41. y sin
x 4 2n 2 3 x
Solving Trigonometric Equations
or
5 x 2n 2 3
8 4n 3
x
10 4n 3
In general, we have: sin
2x 1 x 3 2n 2 2 x 3 4n
y sin x cos x
42.
sin x cos x 0 sin x cos x
x
n 4
43. y tan2
x
63
From the graph in the textbook we see that the curve has x-intercepts at x ± 2. In general, we have: tan2
1 x n 4
tan
6x 3 6x ± 3 x ± n 6 3
1 3 7 11 For 1 < x < 3 the intercepts are , , , . 4 4 4 4
x ± 2 6n
y sec4
44. sec4
8x 4
45. Graph y1 2 sin x cos x. 4
x 40 8
sec4
2
0
x 4 8
x sec ± 2 8
x n 8 4 2
−4
The x-intercepts occur at x 2.678 and x 5.820.
x 2 4n For 3 < x < 3 the intercepts are 2 and 2.
46. 4 sin3 x 2 sin2 x 2 sin x 1 0
47. Graph y1
1 sin x cos x 4. cos x 1 sin x
4 10
0
2
−2
x 0.785, 2.356, 3.665, 3.927, 5.498, 5.760
0
2
− 10
The x-intercepts occur at 5 x 1.047 and x
5.236. 3 3
463
464
Chapter 5
48.
cos x cot x 3 1 sin x y1
Analytic Trigonometry
49. x tan x 1 0
4
cos x tan x 1 sin x
Graph y1 x tan x 1.
2
0
3
The x-intercepts occur at x 0.860 and x 3.426.
−6
x 0.524, 2.618 50. x cos x 1 0
4
0
2
−4
51. sec2 x 0.5 tan x 1 0
5
x 4.917 2
0
Graph y1
1 0.5 tan x 1. cos x2
The x-intercepts occur at
−5
10
x 0, x 2.678, x 3.142, and 0
x 5.820.
53. Graph y1 2 tan2 x 7 tan x 15.
52. csc2 x 0.5 cot x 5 0 y1
2
−2
sin x 1
2
1 5 2 tan x
10
2
0
x 0.515, 2.726, 3.657, 5.868 5 − 30
The x-intercepts occur at x 0.983, x 1.768, x 4.124 and x 4.910.
2
0
−5
54. 6 sin2 x 7 sin x 2 0
2
x 0.524, 0.730, 2.412, 2.618 2
0
−2
55. 12 sin2 x 13 sin x 3 0 sin x sin x
1 3
or
x 0.3398, 2.8018
Graph y1 12 sin2 x 13 sin x 3.
13 ± 132 4123 212 13 ± 5 24 sin x
30
0
3 4
x 0.8481, 2.2935
2
− 10
The x-intercepts occur at x 0.3398, x 0.8481, x 2.2935, and x 2.8018.
Section 5.3
Solving Trigonometric Equations
56. 3 tan2 x 4 tan x 4 0 tan x
465
50
4 ± 42 434 4 ± 64 2 2, 23 6 3
tan x 2
tan x
x arctan2 n
−10
x arctan
1.1071 n
2
0
2 3
23 n
0.5880 n
The values of x in 0, 2 are 0.5880, 3.7296, 2.0344, 5.1760. Graph y1 tan2 x 3 tan x 1.
57. tan2 x 3 tan x 1 0 tan x tan x
3 5 2
3 ± 32 411 3 ± 5 21 2 tan x
or
x 1.9357, 5.0773
3 5 2
4 ± 42 441 24
1 2 2
cos x
1 2 2
2
Solutions in 0, 2 are arccos 2 arccos
−3
2
> 1
1 2 2 and
1 2 2: 1.7794, 4.5038.
7
0
1 2 2
No solution
1 2
1.7794
The x-intercepts occur at x 1.9357, x 2.7767, x 5.0773, and x 5.9183. 59.
4 ± 32 1 ± 2 8 2
x arccos
−5
x 2.7767, 5.9183
cos x
2
0
58. 4 cos2 x 4 cos x 1 0
cos x
10
tan2 x 6 tan x 5 0
tan x 1tan x 5 0 tan x 1 0
or tan x 5 0
tan x 1
tan x 5
x
5 , 4 4
x arctan 5, arctan 5
466 60.
Chapter 5
Analytic Trigonometry
sec2 x tan x 3 0 1 tan2 x tan x 3 0 tan2 x tan x 2 0
tan x 2tan x 1 0 tan x 2 0
tan x 1 0
tan x 2
tan x 1
x arctan2 n
x arctan1 n
1.1071 n
n 4
5 Solutions in 0, 2 are arctan2 , arctan2 2, , . 4 4 61.
2 cos2 x 5 cos x 2 0
62.
2 sin2 x 7 sin x 3 0
2 cos x 1cos x 2 0
sin x 32 sin x 1 0
2 cos x 1 0
or cos x 2 0
sin x 3 0
1 cos x 2
cos x 2
x
5 , 3 3
No solution
No solution
2 sin x 1 0 sin x x
1 2
5 , 6 6
5 Solutions in 0, 2 are , . 6 6 63. (a) f x sin x cos x
3
Maximum:
4 , 2
Minimum:
4 , 2
0
2
5
−3
(b) cos x sin x 0 cos x sin x 1
sin x cos x
tan x 1 x
5 , 4 4
f
4 sin 4 cos 4
f
4 sin
5
2
2
2
2
2
2 2 5 5 cos sin cos 2 4 4 4 4 2 2
Therefore, the maximum point in the interval 0, 2 is 4, 2 and the minimum point is 54, 2 .
Section 5.3 64. (a) f x 2 sin x cos 2x
Solving Trigonometric Equations
3
max: 0.5240, 1.5
min: 1.5708, 1.0
max: 2.6180, 1.5
min: 4.7124, 3.0
2
0
−3
(b) 2 cos x 4 sin x cos x 0 2 cos x1 2 sin x 0 2 cos x 0 x
1 2 sin x 0
3 , 2 2
sin x
1.5708, 4.7124
x
1 2
5 , 6 6
0.5240, 2.6180
x 4 Since tan 4 1, x 1 is the smallest nonnegative fixed point.
65. f x tan
66. Graph y cos x and y x on the same set of axes. Their point of intersection gives the value of c such that f c c ⇒ cos c c. 2
−3
(0.739, 0.739)
3
−2
c 0.739 1 x (a) The domain of f x is all real numbers x except x 0.
67. f x cos
(b) The graph has y-axis symmetry and a horizontal asymptote at y 1. (c) As x → 0, f x oscillates between 1 and 1. (d) There are infinitely many solutions in the interval 1, 1 . They occur at x
2 where n is any integer. 2n 1
(e) The greatest solution appears to occur at x 0.6366. sin x x (a) Domain: all real numbers except x 0.
68. f x
(b) The graph has y-axis symmetry. (c) As x → 0, f x → 1. (d)
sin x 0 has four solutions in the interval 8, 8 . x sin x
x 0 1
sin x 0 x 2, , , 2
69.
y
1 cos 8t 3 sin 8t 12
1 cos 8t 3 sin 8t 0 12 cos 8t 3 sin 8t 1 tan 8t 3 8t 0.32175 n t 0.04
n 8
In the interval 0 ≤ t ≤ 1, t 0.04, 0.43, and 0.83.
467
468
Chapter 5
Analytic Trigonometry
70. y1 1.56e0.22t cos 4.9t
4
Right-most point of intersection: 1.96, 1 10
0
The displacement does not exceed one foot from equilibrium after t 1.96 seconds.
71. S 74.50 43.75 sin
−4
t 6
t
1
2
3
4
5
6
7
8
9
10
11
12
S
96.4
112.4
118.3
112.4
96.4
74.5
52.6
36.6
30.8
36.6
52.6
74.5
Sales exceed 100,000 units during February, March, and April.
6t
S
y2 75. Left point of intersection: 1.95, 75 Right point of intersection: 10.05, 75 So, sales exceed 7500 in January, November, and December.
Monthly sales (in thousands of dollars)
72. Graph y1 58.3 32 cos
100 75 50 25 x 2
4
6
8 10 12
Month (1 ↔ January)
Range 300 feet
73.
74. Range 1000 yards 3000 feet
v0 100 feet per second
v0 1200 feet per second
1 r 32 v02 sin 2 1 2 32 100
1 f 32 v02 sin 2
sin 2 300
1 3000 32 12002 sin 2
sin 2 0.96
sin 2 0.066667
2 arcsin0.96 73.74
2 3.8
36.9
1.9
or 2 180 arcsin0.96 106.26
53.1
75. ht 53 50 sin
16 t 2
(a) ht 53 when 50 sin
t 0 16 2
16 t 2 0.
or
t 16 2
t 16 2
3 t 16 2
t8
t 24
The Ferris wheel will be 53 feet above ground at 8 seconds and at 24 seconds. —CONTINUED—
Section 5.3
Solving Trigonometric Equations
75. —CONTINUED— (b) The person will be at the top of the Ferris wheel when sin
16 t 2 1 t 16 2 2 t 16 t 16.
2 32. During 16 160 seconds, 5 cycles will take place and the person will be at the top of the ride 5 times, spaced 32 seconds apart. The times are: 16 seconds, 48 seconds, 80 seconds, 112 seconds, and 144 seconds. The first time this occurs is after 16 seconds. The period of this function is
76. (a)
77. A 2x cos x , 0 < x <
Unemployment rate
y 8
(a)
2
2
6 4
2
0
2 −2
t 2
4
6
8 10 12 14
Year (0 ↔ 1990)
(b) By graphing the curves, we see that 1 r 1.24 sin0.47t 0.40 5.45 best fits the data.
The maximum area of A 1.12 occurs when x 0.86. (b) A ≥ 1 for 0.6 < x < 1.1
(c) The constant term gives the average unemployment rate of 5.45%. (d) Period:
2 13.4 years 0.47
(e) r 5 when t 20 which corresponds to the year 2010. 78. f x 3 sin0.6x 2 (b) gx 0.45x2 5.52x 13.70
(a) Zero: sin0.6x 2 0 0.6x 2 0
4
0.6x 2
0
2 10 x 0.6 3 (c) 0.45x2 5.52x 13.70 0 x
5.52 ± 5.522 40.4513.70 20.45
6
f
g
−4
For 3.5 ≤ x ≤ 6 the approximation appears to be good.
x 3.46, 8.81 The zero of g on 0, 6 is 3.46. The zero is close to the zero 10 3 3.33 of f.
and the period of 2 sin t 1 is 2. 2 In the interval 0, 2 the first equation has four cycles whereas the second equation has only one cycle, thus the first equation has four times the x-intercepts (solutions) as the second equation.
79. True. The period of 2 sin 4t 1 is
469
470
Chapter 5
Analytic Trigonometry 81. y1 2 sin x
80. False. sin x 3.4 has no solution since 3.4 is outside the range of sin. Also, 3.4 is outside the domain of arcsin, so x arcsin3.4 is an invalid equation. C 90 66 24
83.
cos 66
y2 3x 1 From the graph we see that there is only one point of intersection.
84. Given: A 90, B 71, b 14.6
B
22.3 a
82. By inspecting the graphs of y1 and y2, it appears they intersect at three points.
22.3
66°
A
a
b
C 90 71 19 C
sin 71
a cos 66 22.3
b 22.3
tan 71
b 22.3 tan 66 50.1
c
85. 390, 390 360 30, is in Quadrant I. sin 390 sin 30 cos 390 cos 30
71° A
C
14.6
14.6 c 14.6
5.0 tan 71
86. 600 600 360 240, Quadrant III
1 2
Reference angle: 60
3
tan 390 tan 30
B
14.6 a
15.4 sin 71
22.3 a
54.8 cos 66 tan 66
14.6 a
sin 600 sin 60
2 3 1 3 3
3
2
cos 600 cos 60
1 2
tan 600 tan 60 3 87. 1845, 45, is in Quadrant IV. sin1845 sin 45 cos1845 cos 45
88. 1410 1410 4360 30,
2
2 sin1410 sin 30
2
cos1410 cos 30 tan1410 tan 30
250 feet 1 mile
0.02367 2 miles 5280 feet
90.
θ
3
2 3
3
hyx h
tan 39.75
1.36
1 2
2
tan1845 tan 45 1
89. tan
Quadrant I
y 100
y x
100 tan 39.75 y θ
250 ft 2 mi
Not drawn to scale
tan 28
x 100
28˚
39˚45' 100 ft
100 tan 28 x h 100 tan 39.75 100 tan 28 h 30 feet 91. Answers will vary.
Section 5.4
Section 5.4
■
Sum and Difference Formulas
471
Sum and Difference Formulas
You should know the sum and difference formulas. sinu ± v sin u cos v ± cos u sin v cosu ± v cos u cos v sin u sin v tan u ± tan v 1 tan u tan v You should be able to use these formulas to find the values of the trigonometric functions of angles whose sums or differences are special angles. tanu ± v
■ ■
You should be able to use these formulas to solve trigonometric equations.
Vocabulary Check 1. sin u cos v cos u sin v 3.
2. cos u cos v sin u sin v
tan u tan v 1 tan u tan v
4. sin u cos v cos u sin v
5. cos u cos v sin u sin v
6.
1. (a) cos120 45 cos 120 cos 45 sin 120 sin 45
2 6 4
1 2 1 2 (b) cos 120 cos 45 2 2 2
3. (a) cos
(b) cos
5. (a) sin (b) sin
4 3 cos 4 cos 3 sin 4 sin 3
2
2
1
2
2
2
7
4. (a) sin
(b) sin
6 2
4 2
2
3
2
2 3
2
5 3 5 3 5 sin cos cos sin 6 4 6 4 6
22 23 22 12
4
7 1 3 1 3 sin 6 3 2 2 2
2 6
3
4
2
5 1 sin sin 3 6 6 2
22 23 22 12
(b) sin 135 cos 30
3
2 1 2 1 cos 4 3 2 2 2
6
2. (a) sin135 30 sin 135 cos 30 cos 135 sin 30
21 22 23 22
tan u tan v 1 tan u tan v
6 2
4
5 2 1 2 1 3 sin 4 6 2 2 2
6. (a) sin315 60 sin 315 cos 60 cos 315 sin 60
2 2
(b) sin 315 sin 60
2
4
1
2
2
2
3
2
6
4
2 3 2 3 2 2 2
472
Chapter 5
Analytic Trigonometry
7. sin 105 sin60 45
8. 165 135 30
sin 60 cos 45 cos 60 sin 45
3
2 2
4
2
2
1 2
sin 135 cos 30 sin 30 cos 135
2
sin 45 cos 30 sin 30 cos 45
2
3 1
cos 105 cos60 45
cos 60 cos 45 sin 60 sin 45 1 2
2
2
2
4
3
2
2
1 3
2
4
3
2
1 2
2
2
3 1
cos 165 cos135 30
tan 60 tan 45 1 tan 60 tan 45
2
cos 45 cos 30 sin 45 sin 30
1 3
3 1
2
cos 135 cos 30 sin 135 sin 30
2
tan 105 tan60 45
sin 165 sin135 30
3 1
1 3
1 3 1 3
4 23 2 3 2
2
2 2
4
3
2
2
1
2
2
3 1
tan 165 tan135 30
tan 135 tan 30 1 tan 135 tan 30
tan 45 tan 30 1 tan 45 tan 30 1
1
3
3
3
3
2 3 tan 195 tan225 30
9. sin 195 sin225 30 sin 225 cos 30 cos 225 sin 30 sin 45 cos 30 cos 45 sin 30
2
2
2
4
3
2
2
2
1
2
cos 45 cos 30 sin 45 sin 30
2 2
4
3
2
tan 45 tan 30 1 tan 45 tan 30
3 3 3 3 3 1 3
2
3 1
cos 225 cos 30 sin 225 sin 30
2
tan 225 tan 30 1 tan 225 tan 30
1
1 3
cos 195 cos225 30
2
1
2
3 3
12 63 2 3 6
3 3 3 3
Section 5.4
10. 255 300 45
11. sin
sin 255 sin300 45 sin 60 cos 45 sin 45 cos 60
2 2
4
2
2
3 1
2
2
1
2
cos
cos 255 cos300 45
11 3 sin 12 4 6 sin
sin 300 cos 45 sin 45 cos 300
3
Sum and Difference Formulas
cos 300 cos 45 sin 300 sin 45
1 2
2
2
2
4
3
2
2
tan
tan 300 tan 45 1 tan 300 tan 45
tan 60 tan 45 1 tan 60 tan 45
7 12 3 4
sin cos
2 2
4
2
2
1 2
2
4
2
1
2
cos sin sin 3 4 3 4 2
2
2
2
3
2
11 3 tan 4 4 6
3
2
1 3
2
2
1
2
3
2
2
3
1 1
3
3
3 3 3 3
12 63 2 3 6
7 tan 12 3 4
3 3
3 3
tan 3 4 1 tan tan 3 4
3 1
1 3
2 3
3 1
7 cos 12 3 4 cos
2
3 3 cos sin sin 4 6 4 6
tan
cos sin cos 3 4 4 3
3
3 1
1
tan
7 sin 12 3 4
2
tan
3 1 2 3 1 3
sin
2
2 1
3 tan 4 6 3 1 tan tan 4 6
tan 255 tan300 45
12.
2
2
1 3
3
11 3 cos 12 4 6 cos
2
2
cos 60 cos 45 sin 60 sin 45
3 3 cos cos sin 4 6 4 6
2
4
473
2
4
3 1
474
Chapter 5
13. sin
17 9 5 sin 12 4 6
sin
2
4
cos
2
1 3
4
1 2
cos sin cos 6 4 4 6 2
2
2
1 33
3
2 2
4
2
2
12 63 2 3 6
3 3
3 3
3
3 1
1 3
3
2 3 15. 285 225 60 sin 285 sin225 60 sin 225 cos 60 cos 225 sin 60
2 1
2
2
2 42
2 3
2
3 1
cos 285 cos225 60 cos 225 cos 60 sin 225 sin 60
2 2 2
2 1
2
2
1 2
3 1
3
2
4
3 1
tan 285 tan225 60
tan 225 tan 60 1 3 1 tan 225 tan 60 1 3
1 3
4 23 2 3 2 3 2
1 3
3
2
tan 12 6 4
tan 6 4 1 tan tan 6 4
1 33 3 3 3 3
2
cos sin sin 6 4 6 4
tan
2
cos
tan
1 3
4
tan94 tan56 1 tan94 tan56
cos 12 6 4
3 2 1 2 2 2 2
2
sin 12 6 4 sin
9 5 9 5 cos sin sin 4 6 4 6
17 9 5 tan 12 4 6
sin
3 1
12 6 4
17 9 5 cos 12 4 6
14.
23 22 12
2
cos
tan
5 9 5 9 cos cos sin 4 6 4 6
2
cos
Analytic Trigonometry
2
2
Section 5.4
17. 165 120 45
16. 105 30 135
sin165 sin 120 45
sin30 135 sin 30 cos 135 cos 30 sin 135
sin120 45
sin 30cos 45 cos 30 sin 45
2 2 2 2 2
1
2
4
3
sin 120 cos 45 cos 120 sin 45
2
1 3
cos30 135 cos 30 cos 135 sin 30 sin 135
23 22 12 22
2
4
1
2
4
tan 30 tan 45 1 tan 30tan 45 3
2
2
1 2
2
2
1 2
2
4
2
2
3
2
2
2
1 3
tan 120 tan 45 1 tan 120 tan 45 3 1
1 3 1
1 3 1 3
4 23 2
1 3 3
1
2 3 18. 15 45 30 sin 15 sin45 30 sin 45 cos 30 cos 45 sin 30
22 23 22 12
2 3 1
4
23 1 4
cos 15 cos45 30 cos 45 cos 30 sin 45 sin 30 tan 15 tan45 30
22 23 22 12
2 3 1
4
23 1 4
tan 45 tan 30 1 tan 45 tan 30 3
1
3
3
1 1
3
3 1
tan120 tan 45
2 3
tan165 tan 120 45
1
33 1
2
cos 120 cos 45 sin 120 sin 45
1 3
3
3
cos120 45
tan 30 tan 135 tan30 135 1 tan 30 tan 135
cos165 cos 120 45
cos 30cos 45 sin 30 sin 45
Sum and Difference Formulas
3 3 3 3 3 3 3 3 3 3
3 3
3 3
12 63 2 3 6
475
476
Chapter 5
19.
13 3 12 4 3 sin
2
2 2
4
1 3
13 3 cos 12 4 3
1
2
2
2
2
3
2
2
4
1 3
7 12 3 4
sin
7 sin sin cos cos sin 12 3 4 3 4 3 4
cos
3
2
22 12 22 42
12 22 23 22
3 1
2
4
1 3
3 tan4 1 tan tan 3 4
7 tan tan 12 3 4
7 cos cos cos sin sin 12 3 4 3 4 3 4
1 3
1 13
1 3 1 3
4 23 2
2 3
3 3 cos sin sin 4 3 4 3
2
tan
22 23
1
2
13 3 tan 12 4 3
34 tan3 3 1 tan tan 4 3
3 3 cos cos sin 4 3 4 3
cos
20.
tan
13 3 sin 12 4 3 sin
cos
Analytic Trigonometry
tan
3 1 2 3 1 3 1
1 3 1 3
Section 5.4
21.
3 13 12 4 3
3
4
sin
3
3
sin 4
3
4
cos
3
3
4
tan
3
3
3
3
2
4
3 1
3 3 cos sin sin 4 3 4 3
2 2 2 4
2 1
2
2 3
3
tan 4 tan
3
1 3 1 3
2
3 1
3 tan 4 3
1 tan
22.
2
1 3
cos 4 cos
2 2 2
2
4
2 1
2
3
3 3 cos cos sin 4 3 4 3
sin
3 tan 4 3 1 3
1 3
1 3
1 3
4 23 2 3 2
5 12 4 6 sin
cos
4 6 sin 4 cos 6 cos 4 sin 6
22 23 22 12
2
4
3 1
4 6 cos 4 cos 6 sin 4 sin 6
tan 4 6
22 23 22 12
2
3 tan 1 4 6 3 3 1 tan tan 1 1 4 6 3
4
3 1
tan
3 2
23. cos 25 cos 15 sin 25 sin 15 cos25 15 cos 40 24. sin 140 cos 50 cos 140 sin 50 sin140 50 sin 190
Sum and Difference Formulas
477
478
25.
Chapter 5
Analytic Trigonometry
tan 325 tan 86 tan325 86 tan 239 1 tan 325 tan 86
26.
27. sin 3 cos 1.2 cos 3 sin 1.2 sin3 1.2 sin 1.8
tan 140 tan 60 tan140 60 tan 80 1 tan 140 tan 60
28. cos
cos sin sin cos 7 5 7 5 7 5
cos
29.
tan 2x tan x tan2x x tan 3x 1 tan 2x tan x
32. cos 15 cos 60 sin 15 sin 60 cos15 60
sin 300
33. sin
2
cos45
3
cos cos sin sin 12 4 12 4 12 4 sin
12 35
30. cos 3x cos 2y sin 3x sin 2y cos3x 2y
31. sin 330 cos 30 cos 330 sin 30 sin330 30
34. cos
3 3 3 cos sin sin cos 16 16 16 16 16 16
3
cos
2 4 2
3
2
54 tan12 5 tan 36. 5 4 12 1 tan tan 4 12 tan
tan 25 tan 110 tan25 110 35. 1 tan 25 tan 110 tan 135 1
tan
76
tan
6
For Exercises 37– 44, we have: 5 5 sin u 13 , u in Quadrant II ⇒ cos u 12 13 , tan u 12
cos v 35, v in Quadrant II ⇒ sin v 45, tan v 43, y
y
(−3, 4) (−12, 5)
5 13
u x
v x
Figures for Exercises 37– 44
3
3
2
2
Section 5.4 37. sinu v sin u cos v cos u sin v
38. cosu v cos u cos v sin u sin v
13 5 135 5
3
12
4
63 65
39. cosu v cos u cos v sin u sin v
41. tanu v
12 13
5
4
16 65
1 tan u tan v
5 4 12 3
1
5 12
43
21
12 1 59
42. cscu v
7 4
43. secv u
9 63 4 16
1 1 cosv u cos v cos u sin v sin u 1
4 5 35 12 13 5 13
53 135 45
36 20 56 65 65 65
12 45 13 53135
tan u tan v
12 13
40. sinv u sin v cos u cos v sin u
5 135 3
Sum and Difference Formulas
44. tanu v
1 1 20 56 36 65 65 65
65 56
cotu v
33 48 15 65 65 65
1 1 sinu v sinv u 1 65 33 33 65
125 43 tan u tan v 5 1 tan u tan v 1 12 43 74 4 9
63 16
1 1 16 tanu v 63 63 16
For Exercises 45–50, we have: 7 7 sin u 25 , u in Quadrant III ⇒ cos u 24 25 , tan u 24
cos v 45, v in Quadrant III ⇒ sin v 35, tan v 34 y
y
v
u x
x
25
5
(−24, −7) (− 4, −3)
Figures for Exercises 45– 50 45. cosu v cos u cos v sin u sin v
46. sinu v sin u cos v cos u sin v
4 7 3 24 25 5 25 5
7 25 45 2425 35
3
5
28 72 4 125 125 100 125 5
479
480
Chapter 5
47. tanu v
Analytic Trigonometry
tan u tan v 1 tan u tan v 7 24
34
1
7 24
11
3 4
34 247 tan v tan u 7 1 tan v tan u 1 34 24
48. tanv u
24
39 32
44 117
11 24 39 32
1 1 5 3 cosu v 3 5
Use Exercise 45 for cosu v.
50. cosu v cos u cos v sin u sin v
24 25
54 257 53
21 117 96 125 125 125
51. sinarcsin x arccos x sinarcsin x cosarccos x sinarccos x cosarcsin x x x 1 x2
1 x2
x2 1 x2 1
1
1
x
θ
x
θ = arcsin x
θ = arccos x
52. Let u arctan 2x and v arccos x tan u 2x
cos v x.
4x 2 + 1
1
2x
u
1 − x2
v x
1
sinarctan 2x arccos x sinu v sin u cos v cos u sin v
1 − x2
θ 1 − x2
2x 4x2 1
x
2x2 1 x2 4x2 1
1 4x2 1
1 x2
44 117
1 1 117 44 tanv u 117 44
cotv u
49. secu v
Section 5.4
Sum and Difference Formulas
53. cosarccos x arcsin x cosarccos x cosarcsin x sinarccos x sinarcsin x x 1 x2 1 x2
x
0 (Use the triangles in Exercise 51.) 54. Let u arccos x
v arctan x
and
cos u x
tan v x.
1
1 + x2
1 − x2
x
u v
x
1
cosarcos x arctan x cosu v cos u cos v sin u sin v x
1 1 x 1 x 1 x x 2
2
2
x x1 x2 1 x2
55. sin3 x sin 3 cos x sin x cos 3
56. sin
0cos x 1sin x
1cos x sin x0
sin x
57. sin
cos x
6 x sin 6 cos x cos 6 sin x
58. cos
5
4
x cos
1 cos x 3 sin x 2
59. cos sin
2 cos cos sin sin sin 2 cos cos 2 sin 1cos 0sin 1cos sin 0 cos cos 0
60. tan 4
tan 4 1 tan 1 tan 1 tan tan 4 tan
2 x sin 2 cos x sin x cos 2
5 5 cos x sin sin x 4 4
2
2
cos x sin x
481
482
Chapter 5
Analytic Trigonometry
61. cosx y cosx y cos x cos y sin x sin ycos x cos y sin x sin y cos2 x cos2 y sin2 x sin2 y cos2 x1 sin2 y sin2 x sin2 y cos2 x cos2 x sin2 y sin2 x sin2 y cos2 x sin2 ycos2 x sin2 x cos2 x sin2 y 62. sinx y sinx y sin x cos y sin y cos xsin x cos y sin y cos x sin2 x cos2 y sin2 y cos2 x sin2 x1 sin2 y sin2 y cos2 x sin2 x sin2 x sin2 y sin2 y cos2 x sin2 x sin2 ysin2 x cos2 x sin2 x sin2 y 63. sinx y sinx y sin x cos y cos x sin y sin x cos y cos x sin y 2 sin x cos y 64. cosx y cosx y cos x cos y sin x sin y cos x cos y sin x sin y 2 cos x cos y
65. cos
3
2
x cos
3 3 cos x sin sin x 2 2
66. cos x cos cos x sin sin x 1 cos x 0 sin x
0cos x 1sin x
cos x
sin x
2
2
− 2
2
−6
−2
−2
67. sin
3
2
6
sin
3 3 cos cos sin 2 2
68. tan
1cos 0sin
cos
tan tan 1 tan tan 0 tan 1 0 tan
tan
2
3
− 2
2 −6
6
−2 −3
Section 5.4
sin x
69. sin x cos
sin x 1 3 3
cos x sin sin x cos cos x sin 1 3 3 3 3 2 sin x0.5 1 sin x 1 x
sin x
70. sin x cos
2
1 sin x 6 6 2
1 cos x sin sin x cos cos x sin 6 6 6 6 2
2 cos x0.5
1 2
cos x
1 2
x
cos x
71. cos x cos
5 , 3 3
cos x 1 4 4
sin x sin cos x cos sin x sin 1 4 4 4 4
22 1
2 sin x
2 sin x 1 sin x sin x x
1 2 2
2
5 7 , 4 4
Sum and Difference Formulas
483
484
Chapter 5
Analytic Trigonometry tanx 2 sinx 0
72.
tan x tan 2sin x cos cos x sin 0 1 tan x tan tan x 0 2sin x1 cos x0 0 1 tan x0 tan x 2 sin x 0 1 sin x 2 sin x cos x sin x 2 sin x cos x sin x1 2 cos x 0 sin x 0
cos x
or
x 0,
Analytically: cos x
73. cos x cos
x
1 2
5 , 3 3
cos x 1 4 4
sin x sin cos x cos sin x sin 1 4 4 4 4
22 1
2 cos x
2 cos x 1
1
cos x
2 2
cos x
2
7 , 4 4
x
2
Graphically: Graph y1 cos x cos x 4 4
The points of intersection occur at x
74. tanx cos x
and y
2
1. 0
7 and x . 4 4
0 2
Answers: 0, 0, 3.14, 0 ⇒ x 0,
−2
4
0
−4
2
2
Section 5.4
75. y
1 1 sin 2t cos 2t 3 4
1 1 (a) a , b , B 2 3 4 C arctan
76.
Sum and Difference Formulas
3 b arctan 0.6435 a 4
y
13 14 sin2t 0.6435
y
5 sin2t 0.6435 12
2
(b) Amplitude:
5 feet 12
(c) Frequency:
1 B 2 1 cycle per second period 2 2
2
y1 A cos 2
y2 A cos 2
y1 y2 A cos 2
t
x
t
x
T cos 2
y1 y2 A cos 2 2A cos 2
t
x
t
x
x t x t t x t x cos 2 sin 2 sin 2 A cos 2 cos 2 sin 2 sin 2
t x cos 2
78. False.
77. False.
cosu ± v cos u cos v sin u sin v
sinu ± v sin u cos v ± cos u sin v
79. False. cos x
cos x cos sin x sin 2 2 2
80. True.
cos x0 sin x1
sin x
sin x 81. cosn cos n cos sin n sin
x cos x sin 2 2
1 cos 0sin
0cos sin 1n
1ncos , where n is an integer.
1n sin , where n is an integer.
b b a ⇒ sin C , cos C a2 b2 a2 b2 a
a2 b2 sinB C a2 b2 sin B
84. C arctan
82. sinn sin n cos sin cos n
n
83. C arctan
a2 b2 a2 b2 cos B a sin B b cos B a
b
a a b ⇒ sin C , cos C a2 b2 a2 b2 b
a2 b2 cosB C a2 b2 cos B
a2 b2 sin B a2 b2
b cos B a sin B a sin B b cos B
b
a
485
486
Chapter 5
Analytic Trigonometry
85. sin cos
86. 3 sin 2 4 cos 2
a 1, b 1, B 1 (a) C arctan
a 3, b 4, B 2
b arctan 1 a 4
(a) C arctan
sin cos a2 b2 sinB C
4
2 sin
b 4 arctan 0.9273 a 3
3 sin 2 4 cos 2 a2 b2 sinB C 5 sin2 0.9273
a 3 arctan 0.6435 b 4
(b) C arctan
a (b) C arctan arctan 1 b 4
3 sin 2 4 cos 2 a2 b2 cosB C
sin cos a2 b2 cosB C
2 cos 4
88. sin 2 cos 2
87. 12 sin 3 5 cos 3
a 1, b 1, B 2
a 12, b 5, B 3 (a) C arctan
5 cos2 0.6435
b 5 arctan 0.3948 a 12
12 sin 3 5 cos 3 a2 b2 sinB C
(a) C arctan
sin 2 cos 2 a2 b2 sinB C
13 sin3 0.3948 (b) C arctan
a 12 arctan 1.1760 b 5
12 sin 3 5 cos 3
a2
b2
cosB C
13 cos3 1.1760
b arctan1 a 4
2 sin 2 (b) C arctan
sin 2 cos 2 a2 b2 cosB C
b ⇒ a0 a 2
a2 b2 2 ⇒ b 2
B1
2 sin
91.
0sin 2cos 2 cos 2
cosx h cos x cos x cos h sin x sin h cos x h h
cos x cos h cos x sin x sin h h
cos xcos h 1 sin x sin h h
cos xcos h 1 sin x sin h h h
a arctan1 b 4
2 cos 2
89. C arctan
4
90. C arctan
4
a 3 ⇒ a b, a < 0, b < 0 b 4
a2 b2 5 ⇒ a b
52 2
B1
5 cos
3 52 52 sin cos 4 2 2
Section 5.4 92. (a) The domains of f and g are the sets of real numbers, h 0. (b)
Sum and Difference Formulas
(c) The graphs are the same. 2
h
0.01
0.02
0.05
0.1
0.2
0.5
f h
0.504
0.509
0.521
0.542
0.583
0.691
g h
0.504
0.509
0.521
0.542
0.583
0.691
−3
3
−2
(d) As h → 0, f h approaches 0.5. As h → 0, gh approaches 0.5. y
93.
m1 tan and m2 tan
90 ⇒ 90
y 1 = m 1x + b 1
θ
90 ⇒ 90 90 ⇒
δ
Therefore, arctan m2 arctan m1.
β
α
x
For y x and y 3x we have m1 1 and m2 3.
y 2 = m 2x + b2
arctan3 arctan 1 60 45 15
94. For m2 > m1 > 0, the angle between the lines is:
arctan
m2 m1
1 m m 1
2
m2 1 m1
1 3
arctan
95.
1
1
1
3
1 3
arctan2 3 15
3
−2
2
Conjecture: sin2
sin2 1 4 4
−3
sin2
sin2 sin cos cos sin 4 4 4 4
2
2
sin2 cos2 sin2 cos2 sin cos sin cos 2 2 2 2
sin cos 2 2
sin cos 4 cos sin 4
sin2 cos2 1
2
sin cos 2 2
2
487
488
Chapter 5
Analytic Trigonometry
96. (a) To prove the identity for sinu v we first need to prove the identityfor cosu v. Assume 0 < v < u < 2 and locate u, v, and u v on the unit circle. y
C
u−v
1
B D u −1
A
v
O
x
1
1
The coordinates of the points on the circle are: A 1, 0, B cos v, sin v, C cosu v, sinu v, and D cos u, sin u. Since DOB COA, chords AC and BD are equal. By the distance formula we have: cosu v 12 sinu v 02 cos u cos v2 sin u sin v2
cos2u v 2 cosu v 1 sin2u v cos2 u 2 cos u cos v cos2 v sin2 u 2 sin u sin v sin2 v
cos2u v sin2u v 1 2 cosu v cos2 u sin2 u cos2 v sin2 v 2 cos u cos v 2 sin u sin v 2 2 cosu v 2 2 cos u cos v 2 sin u sin v 2 cosu v 2cos u cos v sin u sin v cosu v cos u cos v sin u sin v Now, to prove the identity for sinu v, use cofunction identities.
2 u v cos 2 u v
sinu v cos
cos
2 u cos v sin2 u sin v
sin u cos v cos u sin v (b) First, prove cosu v cos u cos v sin u sin v using the figure containing points
y 1
A1, 0
u−v
D
C
Bcosu v, sinu v
u
B
u−v
1
v
A
Ccos v, sin v
−1
Dcos u, sin u on the unit circle.
−1
Since chords AB and CD are each subtended by angle u v, their lengths are equal. Equating dA, B2 dC, D2 we have cosu v 12 sin2u v cos u cos v2 sin u sin v2. Simplifying and solving for cosu v, we have cosu v cos u cos v sin u sin v. Using sin cos
2 we have 2 u v cos 2 u v
sinu v cos
cos
2 u cosv sin2 u sinv
sin u cos v cos u sin v.
x
Section 5.4
97.
f x 5x 3
7x 8 7x y 8 8y 7 x
f x
98.
y 5x 3 y x3 5
x 7 8y ⇒ f 1x 8x 7
y 3x 5
7 f1x 8 7 8x 7 8 x
f f 1x
x 3y 5 f 1x
x 15 5
f f 1x f
Sum and Difference Formulas
x 15 x 15 5 3 5 5
f 1 f x 8
7 8 x 7
x
x 15 5 53 5
x 15 15 x f 1 f x f 15x 3
5x 15 15 5
5x 5
5x 3 15 5
x 99. f x x2 8 f is not one-to-one so
100. f 1
does not exist.
f x x 16, x ≥ 16 y x 16 y 2 x 16 x y 2 16 ⇒ f 1x x2 16, x ≥ 0 f f 1x x2 16 16 x f 1 f x x 16 16 x 2
101. log3 34x3 4x 3
102. log8 83x 3x2
103. eln6x3 6x 3
104. 12x eln xx2 12x xx 2
2
12x x2 2x x2 10x
489
490
Chapter 5
Analytic Trigonometry
Section 5.5
■
Multiple-Angle and Product-to-Sum Formulas
You should know the following double-angle formulas. (a) sin 2u 2 sin u cos u (b) cos 2u cos2 u sin2 u (b)
2 cos2 u 1
(b)
1 2 sin2 u
(c) tan 2u ■
■
2 tan u 1 tan2 u
You should be able to reduce the power of a trigonometric function. (a) sin2 u
1 cos 2u 2
(b) cos2 u
1 cos 2u 2
(c) tan2 u
1 cos 2u 1 cos 2u
You should be able to use the half-angle formulas. The signs of sin
u u u and cos depend on the quadrant in which lies. 2 2 2
1 2cos u u 1 cos u (b) cos ± 2 2 ■
■
(a) sin
u ± 2
(c) tan
u 1 cos u sin u 2 sin u 1 cos u
You should be able to use the product-sum formulas. 1 (a) sin u sin v cosu v cosu v 2
1 (b) cos u cos v cosu v cosu v 2
1 (c) sin u cos v sinu v sinu v 2
1 (d) cos u sin v sinu v sinu v 2
You should be able to use the sum-product formulas. (a) sin x sin y 2 sin
(c) cos x cos y 2 cos
xy xy cos 2 2
xy xy cos 2 2
(b) sin x sin y 2 cos
xy xy sin 2 2
(d) cos x cos y 2 sin
xy xy sin 2 2
Section 5.5
Multiple-Angle and Product-to-Sum Formulas
491
Vocabulary Check 1. 2 sin u cos u
2. cos2 u
3. cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u
4. tan2 u
5. ±
1 2cos u
sin u 1 cos u sin u 1 cos u 1 8. sinu v sinu v 2 uv uv 10. 2 sin sin 2 2 6.
1 cosu v cosu v 2 uv uv 9. 2 sin cos 2 2 7.
Figure for Exercises 1–8 sin
17
17 417 cos 17 1 tan 4
1
θ 4
1. sin
17
2. tan
17
17
1 4
3. cos 2 2 cos2 1
4 1717 1
2
4. sin 2 2 sin cos 2
5. tan 2
117417
2 tan 1 tan2
4 1 1 4
8 17
1
1 1 sin 2 2 sin cos 17 8
1 2
8 15
1
17
2
17
417 17
32 1 17
15 17
1 cos 2 1 cos2 sin2
2
1 2
7. csc 2
6. sec 2
1
2
2
1 16
8. cot 2
15 8
117
1 16 1 17 17
17 15
1 1 tan tan 2 2 tan 2
2
16
15
1 4 17
14 1 2 4
1
2
2
492
Chapter 5
Analytic Trigonometry
sin 2x sin x 0
9.
10.
sin 2x cos x 0
2 sin x cos x sin x 0
2 sin x cos x cos x 0
sin x2 cos x 1 0
cos x2 sin x 1 0
sin x 0, or 2 cos x 1 0
cos x 0
1 sin x 0, or 2 1cos x 2 sin x 0, or 2 1cos x
x
5 , 3 3
or
2 sin x 1 0
3 , 2 2
sin x x
5 x 0, , , 3 3 11. 4 sin x cos x 1
2 sin x cos x sin x cos x 0
2 sin 2x 1 1 sin 2x 2 2x 2n or 6 x x
13.
cos x2 sin2 x 1 0 cos x 0 2x
5 2n 6
x
5 n or 2 x n 12 12 13 , 12 12
or
2 x
cos 2x cos x x sin2 x cos x
cos2 x 1 cos2 x cos x 0 2 cos2 x cos x 1 0
2 cos x 1cos x 1 0 2 cos x 1 0, 4
or
cos x 1 0
1 cos x , 4 2
or
1cos x 1
2 4 , 3 3
or
1cos x 0
cos 2x sin x 0 1 2 sin2 x sin x 0 2 sin2 x sin x 1 0
2 sin x 1sin x 1 0 2 sin x 1 0 sin x x
or 1 2
7 11 , 6 6
sin2 x
1 2 2
2
3 5 7 x , , , 4 4 4 4
5 17 , 12 12
x
3 , 2 2
or 2 sin2 x 1 0
sin x ±
cos 2x cos x 0
cos2
14.
sin 2x sin x cos x
12.
sin x 1 0 sin x 1 x
2
1 2
7 11 , 6 6
Section 5.5 15.
tan 2x cot x 0
Multiple-Angle and Product-to-Sum Formulas tan 2x 2 cos x 0
16.
2 tan x 2 cos x 1 tan2 x
2 tan x cot x 1 tan2 x 2 tan x cot x1 tan2 x
2 tan x 2 cos x1 tan2 x
2 tan x cot x cot x tan2 x
2 tan x 2 cos x 2 cos x tan2 x
2 tan x cot x tan x
2 tan x 2 cos x 2 cos x
3 tan x cot x 3 tan x cot x 0 3 tan x
2 tan x 2 cos x 2
1 0 tan x
tan x cos x
3 tan2 x 1 0 tan x
sin2 x cos x
sin2 x cos x
sin x sin2 x cos x 0 cos x cos x
cot x3 tan2 x 1 0 cot x 0, 3 or
3 tan2 x 1 0
sin x sin2 x cos2 x 0 cos x
1 3 , or 3 1tan2 x 2 2 3 tan x ± x
1 sin x sin2 x 1 sin2 x 0 cos x
3
sec x2 sin2 x sin x 1 0
3
5 7 11 , , , 6 6 6 6
5 7 3 11 x , , , , , 6 2 6 6 2 6
sec x2 sin x 1sin x 1 0 sec x 0
or 2 sin x 1 0
No solution
sin x x
1 2
or
sin x 1 0 sin x 1
5 , 6 6
Also, values for which cos x 0 need to be checked.
3 , are solutions. 2 2 x
sin 4x 2 sin 2x
17.
sin 4x 2 sin 2x 0 2 sin 2x cos 2x 2 sin 2x 0 2 sin 2xcos 2x 1 0 2 sin 2x 0
or
sin 2x 0
2x 2n
n x 2 x 0,
cos 2x 1 0 cos 2x 1
2x n
3 , , 2 2
sin2 x cos2 x
sin x sin2 x cos x cos x cos x
1 3 tan2 x 1 0 tan x
x
493
x
n 2
x
3 , 2 2
5 3 , , , 6 2 6 2
x
3 2
494
Chapter 5
Analytic Trigonometry
sin 2x cos 2x2 1
18.
sin2 2x 2 sin 2x cos 2x cos2 2x 1 2 sin 2x cos 2x 0 sin 4x 0 4x n x
n 4
3 5 3 7 x 0, , , , , , , 4 2 4 4 2 4 19. 6 sin x cos x 32 sin x cos x
20. 6 cos2 x 3 32 cos2 x 1
3 sin 2x
3 cos 2x 22. cos x sin xcos x sin x cos2 x sin2 x
21. 4 8 sin2 x 41 2 sin2 x
cos 2x
4 cos 2x 4 3 3 ⇒ cos u 23. sin u , < u < 5 2 5
2 24. cos u , < u < 3 2
sin 2u 2 sin u cos u 2
24 54 53 25
sin 2u 2 sin u cos u 2
9 16 7 cos 2u cos2 u sin2 u 25 25 25
cos 2u cos2 u sin2 u
2 3 2 tan u 8 9 24 2 1 tan u 1 16 3 7 7 9 4
tan 2u
2 tan u tan 2u 1 tan2 u
2
3
u x
−2
csc u 3,
3 3 4 ⇒ sin u and cos u 25. tan u , 0 < u < 4 2 5 5 sin 2u 2 sin u cos u 2
24 3545 25
cos 2u cos2 u sin2 u
2 4 3 16 24 2 tan u 9 2 7 1 tan2 u 1 16 7 3
tan 2u
9 7 16 25 25 25
< u < 2
3
3 2
4 5 1 9 9 9 5
2 5 1 4
y
5
5
4
5
45 9
Section 5.5
26. cot u 4,
Multiple-Angle and Product-to-Sum Formulas
3 < u < 2 2
y
sin 2u 2 sin u cos u 2
1
17 17 4
17
8
u 4 −1
cos 2u cos2 u sin2 u
17 17 4
2
1
2 tan u tan 2u 1 tan2 u
x
17 2
4 1 1 4 2
15 17
1
2
8 15
21 5 2 and cos u 27. sec u , < u < ⇒ sin u 2 2 5 5
521 52 4 2521 2 21 17 cos 2u cos u sin u 5 5 25 21 2 2 tan u 2 tan 2u 1 tan u 21 1 2
sin 2u 2 sin u cos u 2
2
2
2
2
2
421 21 21 17 1 4 1 22 42 3 3 9
28. sin 2u 2 sin u cos u 2
cos 2u cos2 u sin2 u
tan 2u
2
2 tan u 1 tan2 u
22 3
3
2 1 4 2
29. cos4 x cos2 xcos2 x
1
2
7 9 1
2
3
u x
−2 2
4
2
y
2
1 cos 2x 2
42 7
1 cos 2x 1 2 cos 2x cos2 2x 2 4
1 2 cos 2x
1 cos 4x 2
4
2 4 cos 2x 1 cos 4x 8
3 4 cos 2x cos 4x 8
1 3 4 cos 2x cos 4x 8
495
496
Chapter 5
Analytic Trigonometry
30. sin8 x sin4 x sin4 x sin2 x sin2 xsin2 x sin2 x sin2 x
1 cos 2x 2
sin4 x
2x 1 cos 2x 1 cos 2 2
1 1 2 cos 2x cos2 2x 4
1 1 cos 4x 1 2 cos 2x 4 2
1 3 4 cos 2x cos 4x 8 sin8 x sin4 x sin4 x
1 3 4 cos 2x cos 4x3 4 cos 2x cos 4x 64
1 9 24 cos 2x 16 cos2 2x 6 cos 4x 8 cos 2x cos 4x cos2 4x 64
1 1 cos 4x 1 1 cos 8x 9 24 cos 2x 16 6 cos 4x 8 cos 6x cos 2x 64 2 2 2
1 35 1 28 cos 2x 14 cos 4x 4 cos 6x cos 8x 64 2 2
1 35 56 cos 2x 28 cos 4x 8 cos 6x cos 8x 128
1 In the above, we used cos 2x cos 4x cos 6x cos 2x. 2
31. sin2 xcos2 x
1 cos 2x 2
1
cos2
1 cos 2x 2
2x
4
32. sin4 x cos4 x sin2 x sin2 x cos2 x cos2 x sin2 x cos2 xsin2 x cos2 x
14 sin 2x14 sin 2x
4x 4x 14 1 cos 14 1 cos 2 2
2
2
1 1 cos 4x 1 4 2
1 2 1 cos 4x 8
1 1 2 cos 4x cos2 4x 64
1 1 cos 4x 8
1 1 cos 8x 1 2 cos 4x 64 2
1 3 1 2 cos 4x cos 8x 64 2 2
1 3 4 cos 4x cos 8x 128
Section 5.5
33. sin2 x cos4 x sin2 x cos2 x cos2 x
1 cos 2x 2
1 cos 2x 2
Multiple-Angle and Product-to-Sum Formulas
1 cos 2x 2
1 1 cos 2x1 cos 2x1 cos 2x 8 1 1 cos2 2x1 cos 2x 8 1 1 cos 2x cos2 2x cos3 2x 8
1 cos 4x 1 cos 4x 1 1 cos 2x cos 2x 8 2 2
1 2 2 cos 2x 1 cos 4x cos 2x cos 2x cos 4x 16
1 1 cos 2x cos 4x cos 2x cos 4x 16
34. sin4 x cos2 x sin2 x sin2 x cos2 x
1 cos 2x 2
1 cos 2x 2
1 cos 2x 2
1 1 cos 2x1 cos2 2x 8 1 1 cos 2x cos2 2x cos3 2x 8
1 1 cos 4x 1 cos 4x 1 cos 2x cos 2x 8 2 2
1 2 2 cos 2x 1 cos 4x cos 2x cos 2x cos 4x 16
1 1 1 1 cos 2x cos 4x cos 2x cos 6x 16 2 2
1 2 2 cos 2x 2 cos 4x cos 2x cos 6x 32
1 2 cos 2x 2 cos 4x cos 6x 32
Figure for Exercises 35– 40
17 8 sin 17
8
θ
15 cos 17
15
35. cos
36. sin
2
2
1 2cos 1 2
1 2cos 1 2
15 17
15 17
3234 1617 4 1717
2 2 17
1 17
17
17
497
498
Chapter 5
Analytic Trigonometry
37. tan
sin 8 17 8 2 1 cos 1 15 17 17
1
17
32 4
38. sec
1 1 2 cos 2 1 cos 2
39. csc
1 1 2 sin 2 1 cos 2
1 1 15 17 2
41. sin 75 sin
sin 8 17 1 2 tan 2 1 cos 1 15 17
17
8 17 4 2 17
12 150 1 cos2 150 1 2 3 2
1 2 3 2
12 150 1 sincos150150 1 1 2 3 2 2 3
1 2 3
42. sin 165 sin
2 3
2 330 1
cos 165 cos tan 165 tan
2 3 2 3 43
1 cos 330 2
2 330 1
1 2 3 2 212
1 cos 330 2
3
1 2 3 2 212
3
1 2
2 330 1 cos 330 1 3 2 2 13 3 2 1
sin 330
43. sin 112 30 sin
2 225 1
cos 112 30 cos tan 112 30 tan
44. sin 67 30 sin
2 225 1
tan 67 30 tan
1 2 2 2 212
1 cos 225 2
2
1 2 2 2 212
2 2
2 225 1 cos 225 1 2 2 1 2 1
sin 225
2 135 1
cos 67 30 cos
1 cos 225 2
1 cos 135 2
2 135 1
1 cos 135 2
1 2 2 2 212
2
1 2 2 2 212
2
2 2
2 135 1 cos 135 1 2 2 1 2 1
sin 135
1 16 17
4
1 2 3 2
tan 75 tan
1 1 17
17
12 150 1 cos2 150 1 2 3 2
cos 75 cos
40. cot
1 1 15 17 2
2
Section 5.5
1 45. sin sin 8 2 4
cos
tan
1 cos 8 2 4
1 tan 8 2 4
3 1 sin 47. sin 8 2
3 1 cos cos 8 2
tan
1 cos
1 3 tan 8 2
1 cos
1 cos
1 tan tan 12 2 6
2
3 4
2
2
sin
1 cos
2
6
1 cos
6
1
6
1 2
1
3
2
1 cos7 6 2
1 cos
7 6
2
1
2
1 2 2 2
3
2
2 1
1 2 3 2
3
2
2
1 2 3 2
1 7 6 2 2 3 7 3 1 cos 1 6 2 sin
3 50. cos u , 0 < u < 5 2
1 cos u 2
1 2
5 13
2 1 cos u 1
526 1 12 13 26 2
12 13
5
12 13
26
26
sin
u2 1 2cos u 1 2
3 5
u2 1 2cos u 1 2
3 5
cos
2
2
2 3
2
2
3
1 2 3 2
sin
1 cos u 2
sin u
2
1
1 2 2 2
2
2
6
2
1
5 12 , < u < ⇒ cos u 13 2 13
u
u
2
2 2 3 2 3 4 2 2 2 1 2 4 2 2 2 2 3 1 cos 1 2 4 2
tan
3 4
2 1
1 cos 12 2 6
7 1 7 tan tan 12 2 6
u
1
2
1 cos
499
1 2 3 2
3 4
cos
cos
2
1 cos
7 1 7 cos cos 12 2 6
2
2
1 cos 4
sin
4
1 2 2 2
2
7 1 7 sin 48. sin 12 2 6
49. sin u
4
1 46. sin sin 12 2 6
1 2 2 2
2
sin
3 4
4
Multiple-Angle and Product-to-Sum Formulas
5
5 25 5
500
Chapter 5
Analytic Trigonometry
5 3 5 8 < u < 2 ⇒ sin u and cos u 51. tan u , 89 89 8 2
u sin 2
u cos 2
52. cot u 3, < u <
u cos 2
2
89 8
289
89 1788
89
8
1
89
2
89 8
289
89 1788
89
89
5
8 89 5
89
3 2
y
1 cos u 2
89
8
1
8
1
1 cos u 2
u 1 cos u tan 2 sin u
u sin 2
1 cos u 2
1
10 203
10
3
10
2
10 203
1 2
10 53
10
10
1 2
10
1 10
−1
10 53
10
10 3
5 3 3 4 ⇒ sin u and cos u 53. csc u , < u < 3 2 5 5 sin
2 u
cos
tan
2 u
2 u
1 cos u 2
310 1 45 10 2
1 cos u 2
1 2
4 5
10
10
4 5
1 cos u 1 3 sin u 35
7 54. sec u , < u < 2 2
u sin 2
cos
2 u
1 cos u 2
1 cos u u tan 2 sin u
y
1 cos u 2
1 3
1
2 7
5
7
2 7
2
314 14
2 1 70 7 2 14
35 5
3 5
7 u −2
u
−3
3
1
10
2
1 cos u 2
u 1 cos u tan 2 sin u
3
1
x
x 10
Section 5.5
55.
6x sin 3x 1 cos 2
57.
1 cos 8x 1 cos 8x
Multiple-Angle and Product-to-Sum Formulas
56.
8x 1 cos 2 8x 1 cos 2
4x 4x cos 1 cos 2 2
cos 2x
1 cos2x 1 sinx 2 1
58.
sin 4x cos 4x
tan 4x
59.
sin ±
x cos x 0 2
hx sin
60.
1 cos x cos x 2
sin
1 cos x cos2 x 2
x cos x 1 0 2
±
1 2cos x 1 cos x 1 cos x 1 2 cos x cos2 x 2
0 2 cos2 x cos x 1 2 cos x 1cos x 1
1 cos x 2 4 cos x 2 cos2 x
1 cos x , 5or cos x 1 2 x
5 , or cos x 3 3
2 cos2 x 3 cos x 1 0
2 cos x 1cos x 1 0 2 cos x 1 0
2
0
cos x 2
−2
By checking these values in the original equation, we see that x 3 and x 53 are extraneous, and x is the only solution.
x cos x 1 2
x
or
1 2
cos x 1 0 cos x 1
5 , 3 3
x0
5 are all solutions to the equation. 0, , and 3 3 1
0
−2
2
501
502
Chapter 5
61.
Analytic Trigonometry
cos ±
x sin x 0 2
gx tan
62.
1 cos x sin x 2
tan
x sin x 0 2
1 cos x sin2 x 2
1 cos x sin x sin x
1 cos x 2 sin2 x
1 cos x sin2 x
1 cos x 2 2 cos2 x
1 cos x 1 cos2 x
2 cos2 x cos x 1 0
cos2 x cos x 0
2 cos x 1cos x 1) 0
cos xcos x 1 0 cos x 0
2 cos x 1 0, 5or cos x 1 0 1 cos x , 5or 2 x x
x sin x 2
x
1cos x 1
or
cos x 1 0
3 , 2 2
cos x 1 x0
5 , or cos 1x 3 3
3 0, , and are all solutions to the equation. 2 2
5 , , 3 3
3
3, , and 53 are all solutions to the equation. 0
2
2
0
−3
2
−2
63. 6 sin
1 cos 6 sin sin 4 4 2 4 4 4 4
64. 4 cos
5 1 5 5 sin 4 sin sin 3 6 2 3 6 3 6
3sin 2 sin 0 2sin76 sin 2 2sin76 sin2
65. 10 cos 75 cos 15 1012 cos75 15 cos75 15 5 cos 60 cos 90 66. 6 sin 45 cos 15 612 sin 60 sin 30 3sin 60 sin 30 67. cos 4 sin 6 12 sin4 6 sin4 6 12 sin 10 sin2 12sin 10 sin 2 68. 3 sin 2 sin 3 3 12 cos2 3 cos2 3 32 cos cos 5 32 cos cos 5 1 5 69. 5 cos5 cos 3 5 2 cos5 3 cos5 3 2 cos8 cos2
52cos 8 cos 2 1 1 70. cos 2 cos 4 12 cos2 4 cos2 4 2 cos2 cos 6 2cos 2 cos 6
71. sinx y sinx y 2cos 2y cos 2x 1
72. sinx y cosx y 2sin 2x sin 2y 1
Section 5.5
Multiple-Angle and Product-to-Sum Formulas 74. sin sin 12 cos 2 cos 2
1 73. cos sin 2 sin 2 sin2 1 2sin 2 sin 2
75. sin 5 sin 3 2 cos
5 2 3 sin5 2 3
76. sin 3 sin 2 sin
2 cos 4 sin
77. cos 6x cos 2x 2 cos
78. sin x sin 5x 2 sin
2 sin 2 cos
6x 2x 6x 2x cos 2 cos 4x cos 2x 2 2
80. cos 2 cos 2 cos
sin 2 cos sin 2 2
sin x 2 sin 82. sin x 2 2
83. sin 60 sin 30 2 sin sin 60 sin 30
3
2
22 cos 22 2 cos cos
81. cos cos 2 sin 2 2
x 5x x 5x cos 2 sin 3x cos2x 2 sin 3x cos 2x 2 2
79. sin sin 2 cos
2 2 2
x
x 2 2 2
sin
x
cos
2 2 2
x 2 2 2
1 3 1 2 2
120 30 120 30 2 cos 75 cos 45 cos 2 2
1 3 3 1 cos 120 cos 30 2 2 2
3 85. cos cos 2 sin 4 4
3 4 4 2
3 4 4 sin 2
2 sin
sin
sin 2 4
2 2 3 cos 2 4 4 2 2
5 3 sin 2 cos 86. sin 4 4
60 30 60 30 cos 2 sin 45 cos 15 2 2
84. cos 120 cos 30 2 cos
cos
3 2 cos3 2
5 3 4 4 2
5 3 4 4 sin 2
2 2 3 5 sin 2 4 4 2 2
2 cos sin
4
2 sin sin
2 sin x cos
2
2
503
504
Chapter 5
Analytic Trigonometry sin 6x sin 2x 0
87. 2 sin
6x 2x 6x 2x cos 0 2 2
2sin 4x cos 2x 0 sin 4x 0
or
cos 2x 0
4x n
or
cos 2x
n 2
or cos 2x
n 4 2
x
n 4
2
In the interval 0, 2 we have x 0,
2
0
3 5 3 7 , , , , , , . 4 2 4 4 2 4
−2
hx cos 2x cos 6x
88.
cos 2x cos 6x 0 2 sin 4x sin2x 0 2 sin 4x sin 2x 0 sin 4x 0
or
n 4
x
3 5 3 7 x 0, , , , , , , 4 2 4 4 2 4
89.
2
2x n
4x n x
sin 2x 0
cos 2x 10 sin 3x sin x cos 2x 1 sin 3x sin x cos 2x 1 2 cos 2x sin x 2 sin x 1 sin x x
3 x 0, , , 2 2
−2
f x sin2 3x sin2 x
90.
sin2 3x sin2 x 0
sin 3x sin xsin 3x sin x 0 2 sin 2x cos x2 cos 2x sin x 0 3 sin 2x 0 ⇒ x 0, , , 2 2 cos x 0 ⇒ x
1 2
5 , 6 6
2
0
n 2
cos 2x 0 ⇒ x
3 , 2 2
1 0
2 0
−2 −1
or
3 5 7 , , , 4 4 4 4
sin x 0 ⇒ x 0,
2
2
or
or
Section 5.5
3
5
β
Multiple-Angle and Product-to-Sum Formulas
13 5
α4 12
Figure for Exercises 91–94
91. sin2
13
2
5
25 169
92. cos2 cos 2
sin2 1 cos2 1 1
5
sin cos cos
95. csc 2
12
2
4
4
1
13
1
144 25 169 169
94. cos sin
2 sin 2 4
4
1 sin 2
96. sec 2
1 2 sin cos
1 sin
csc 2 cos
2
144 169
2
135 65 12
3
36
2 cos 2
135 65 12
3
36
1 1 cos 2 cos2 sin2 1 cos2 sin2 1 cos2
1
2 cos
97. cos2 2 sin2 2 cos 22
5
cos sin sin
135 13 5
12
cos2 1 sin2
25 144 169 169
135 13
93. sin cos
13
13
sec2 1 tan2
sec2 1 sec2 1
sec2 2 sec2
98. cos4 x sin4 x cos2 x sin2 xcos2 x sin2 x cos 2x1
cos 4
cos 2x
99. sin x cos x2 sin2 x 2 sin x cos x cos2 x sin2 x cos2 x 2 sin x cos x 1 sin 2x 101. 1 cos 10y 1 cos2 5y sin2 5y 1 cos2 5y 1 cos2 5y 2 cos2 5y
100. sin
3 cos 3 2 2sin 3 cos 3 1
1 2 sin 2 3
505
506
102.
Chapter 5
Analytic Trigonometry
cos 3 cos2 cos cos
103. sec
u 2
cos 2 cos sin 2 sin cos
1 2
sin2
cos 2 cos sin sin cos
1 2 sin2 2 sin2
1 cos
±
1 2cos u
±
sin u21sin ucos u
±
sin u 2 sinsin uu cos u
±
±
tan 2u tan sinu u
1 4 sin2
104. tan
cos u 1 sin u sin u
xy x y cos 2 2 xy xy 2 sin sin 2 2 2 sin
xy 2
cot
cos t cos 3t 108. sin 3t sin t
cos 4x cos 2x 107. sin 4x sin 2x
2 cos 2 4t 2t 2 cos sin 2 2
2 cos
4t
cost sint
cost cot t sint
x±y 2
4x 2x
4x 2x
2 cos 2 4x 2x 4x 2x 2 sin cos 2 2
2 cos
2 cos 3x cos x cot 3x 2 sin 3x cos x
2t
109. sin
6 x sin 6 x 2 sin 6 cos x 2
1 cos x 2
cos x
x x x x 3 3 3 3 x cos x 2 cos cos 110. cos 3 3 2 2
x y
x±y
2 cos 2 xy xy 2 cos cos 2 2 2 sin
tan
csc u cot u
sin x sin y 106. cos x cos y
2 sin u cos u sin u sin u cos u cos u cos u
sin x ± sin y 105. cos x cos y
u 1 cos u 2 sin u
u 2
2 cos 2
3 cosx
12 cos x cos x
Section 5.5
111.
Multiple-Angle and Product-to-Sum Formulas
507
3
Let y1 cos3x and −2
2
y2 cos x3 3sin x2 cos x.
−3
cos 3 cos2 cos 2 cos sin 2 sin cos2 sin2 cos 2 sin cos sin cos3 sin2 cos 2 sin2 cos cos3 3 sin2 cos 112. sin 4 2 sin 2 cos 2
113.
22 sin cos 1 2 sin2
3
−2
4 sin cos 1 2 sin2
Let y1
2
3
and y2 sin x.
−3
−2
2
cos 4x cos 2x 2 sin 3x
−3
cos 3x cos x 114. sin 3x sin x
3x x
2 sin
4x 2x 4x 2x sin 2 2 2 sin 3x
2 sin 3x sin x sin x 2 sin 3x
3x x
2 sin 2 3x x 3x x 2 cos sin 2 2
2 sin
cos 4x cos 2x 2 sin 3x
115. sin2 x
1 cos 2x 1 cos 2x 2 2 2
y
2 sin 2x sin x 2 cos 2x sin x
2 1
tan 2x
π
3
2π
x
−1
−
−2
−3
116. f x cos2 x
1 cos 2x 1 cos 2x 2 2 2
Shifted upward by
1 unit. 2
1 Amplitude: a 2
Period:
2 2
117. sin2 arcsin x 2 sinarcsin x cosarcsin x 2x1 x2
y 2 1
π −1 −2
2π
x
508
Chapter 5
Analytic Trigonometry
118. cos2 arccos x cos2arccos x sin2arccos x
119.
1 752 sin 2 130 32
x2 1 x2 2x2 1
13032 752
sin 2
1 1 13032 sin 2 752
23.85 1 A bh 2
120. (a)
cos
(b) A 100 sin θ
h ⇒ h 10 cos 2 10 2
10 m
cos 2 2
(b) A 50 2 sin
10 m h
cos 2 2
(b) A 50 sin sin
12b 1 ⇒ b 10 sin 2 10 2 2
(b) When 2, sin 1 ⇒ the area is a maximum.
b
A 10 sin 10 cos ⇒ A 100 sin cos 2 2 2 2
121. sin
(b) A 50 sin
1 2 M
(a) sin
1 2
(b) sin
1 2 4.5
arcsin 1 2
1 arcsin 2 4.5
2 2
2 arcsin
0.4482
S (c) 760 1
(d) sin
2 arcsin
S 3420 miles per hour
r 1 cos So, x 2r1 cos .
1 2 M
S 4.5 760
x 1 cos 2r sin2 2r 2 2 2
4.51
1 arcsin 2 M
S 760 miles per hour
122.
501 50 square feet 2
123. False. For u < 0, sin 2u sin2u
M1 124. False. If 90 < u < 180, u is in the first quadrant and 2
2 sinu cosu 2sin u cos u 2 sin u cos u.
sin
u 2
1 2cos u.
Section 5.5
125. (a) y 4 sin
x cos x 2
Multiple-Angle and Product-to-Sum Formulas
126. f x cos 2x 2 sin x (a)
4
2
) 2
0
−3
2
0 0
Maximum points: 3.6652, 1.5, 5.7596, 1.5
Maximum: , 3 2 cos
(b)
Minimum points: 1.5708, 3, 4.7124, 1
x sin x 0 2
(b) 2 cos x2 sin x 1 0 2 cos x 0
1 2cos x sin x
2 ±
cos x 0
1 cos x 4 sin2 x 2
x
21 cos x 1 cos x 2
cos2
or
3 4.7124 2
11 5.7596 2
cos x 1 0
127. f x sin4 x cos4 x (a) sin4 x cos4 x sin2 x2 cos2 x2
1 cos 2x 2
2
1 cos 2x 2
2
1 1 cos 2x2 1 cos 2x2 4 1 1 2 cos 2x cos2 2x 1 2 cos 2x cos2 2x 4 1 2 2 cos2 2x 4 1 1 cos 22x 22 4 2 1 3 cos 4x 4 (b) sin4 x cos4 x sin2 x2 cos4 x
1 cos2 x2 cos4 x 1 2 cos2 x cos4 x cos4 x 2 cos4 x 2 cos2 x 1 (c) sin4 x cos4 x sin4 x 2 sin2 x cos2 x cos4 x 2 sin2 x cos2 x sin2 x cos2 x2 2 sin2 x cos2 x 1 2 sin2 x cos2 x (d) 1 2 sin2 x cos2 x 1 2 sin x cos xsin x cos x 1 sin 2x
2 sin 2x 1
1 2 sin 2x 2 (e) No, it does not mean that one of you is wrong. There is often more than one way to rewrite a trigonometric expression. 1
x 7 3.6652 6
2
x
sin x
1.5708 2
x 2 cos x 1 0 cos x 1
3 , 2 2
2 sin x 1 0 1 2
7 11 , 6 6
509
510
Chapter 5
128. (a)
Analytic Trigonometry 129. (a)
2
y 6
−2
2
(−1, 4)
5
3
(5, 2)
2
−2
1
(b) The graph appears to be that of sin 2x.
(c) 2 sin x 2 cos2
x
−3 − 2 − 1 −1
2x 1 2 sin x cos x
1
4
5
(b) d 1 52 4 22 62 22 40 210 (c) Midpoint: 131. (a)
y 12
3
−2
sin 2x
130. (a)
2
5 21, 2 2 4 2, 3
y
(6, 10)
3
10
( 43 , 52)
8 2
6 4 2 −6 −4
1 x
−2
2
4
6
(0, 12 )
8 10 −1
(− 4, − 3)
42 6, 3 2 10 1, 72
(c) Midpoint:
5
2 2 3 , 2 3 2
Supplement: 180 162 18
(− 1, − 32 ) − 2
13 1 23 23 43 136 16 169 233 1 233 9 36 36 6 2
2
2
(c) Midpoint:
,
(b) Complement: Not possible. 162 > 90
x 1 −1
1 3
4 1 3 2
Supplement: 180 55 125
( 13 , 23)
−1
(b) d
0 2
133. (a) Complement: 90 55 35
y
−2
2
100 169 269
1
2
(b) d
102 132
132. (a)
2
43 0 52 21 169 4 52 2 13 9 3
(b) d 4 62 3 102
(c) Midpoint:
x 1
2 3 23 56 1 3 2 1 5 , , , 2 2 2 2 3 12
2
Review Exercises for Chapter 5
134. (a) The supplement is 180 109 71.
135. (a) Complement:
4 2 18 9
There is no complement. Supplement:
(b) The supplement is 180 78 102. The complement is 90 78 12.
(b) Complement:
17 18 18
9 2 20 20
Supplement:
9 11 20 20
137. Let x profit for September, then x 0.16x profit for October.
136. (a) The supplement is 0.95 2.19. The complement is
511
0.95 0.62. 2
x x 0.16x 507,600
(b) The supplement is 2.76 0.38.
2.16x 507,600 x 235,000
There is no complement.
x 0.16x 272,600 Profit for September: $235,000 Profit for October: $272,600 138. Let x number of gallons of 100% concentrate. 0.3055 x 1.00x 0.5055
139.
d 2 902 902
Second base 90 ft
16,200
90 ft
d 16,200
16.50 0.30x x 27.50 d
0.70x 11 x 15.7 gallons
902 127 feet
90 ft
90 ft Home plate
Review Exercises for Chapter 5 1.
1 sec x cos x
2.
1 csc x sin x
3.
4.
1 cot x tan x
5.
cos x cot x sin x
6. 1 tan2 x sec2 x sec x
sin x cos x
3 5 4 5
1 4 cot x tan x 3 1 5 sec x cos x 4 5 1 csc x sin x 3
13 2 8. tan , sec 3 3
3 4 7. sin x , cos x 5 5 tan x
1 cos x sec x
3 4
is in Quadrant I. cos
1 3 313 sec 13 13
sin 1 cos2 csc
13 1 sin 2
cot
1 3 tan 2
1 139 134 2 1313
512
Chapter 5
9. sin
2 x
sin x
Analytic Trigonometry 2
2
⇒ cos x
1
2
2
10. csc
2
2 sec 9, sin 4 9 5
is in Quadrant I.
2
2
1 1 sec 9
1 sin x 2 tan x 1 cos x 1 2 1 cot x 1 tan x
cos
sec x
1 2 cos x
csc
1 9 95 sin 45 20
csc x
1 2 sin x
cot
5 1 1 tan 45 20
sin tan cos
45 9 45 1 9
sin tan 1 cos 12. 1 cos2 sin2 sin cos
1 1 11. sin2 x cot2 x 1 csc2 x
csc sec 13. tan2 xcsc2 x 1 tan2 xcot2 x tan2 x
sin 15.
2 sin
tan1 x 1
14. cot2 xsin2 x
2
cot
cos cot sin
16.
17. cos2 x cos2 x cot2 x cos2 x1 cot2 x cos2 xcsc2 x cos2 x
x cot sin1 x cos sin x
2 u cos u
cos2 x 2 sin x cos2 x sin2 x
tan u tan u sec u cos u
18. tan2 csc2 tan2 tan2 csc2 1 tan2 cot2 1
2
2
2
2
x
19. tan x 12 cos x tan2 x 2 tan x 1 cos x sec2 x 2 tan x cos x sec2 x cos x 2
sin x cos x sec x 2 sin x cos x
20. sec x tan x2 sec2 x 2 sec x tan x tan2 x 1 tan2 x 2 sec x tan x tan2 x
21.
1 1 csc 1 csc 1 csc 1 csc 1 csc 1csc 1
1 2 sec x tan x 2 tan2 x
2 csc2 1
2 cot2
2 tan2
22.
cos2 x cos2 x 1 sin x 1 sin x
1 sin x
1 sin x
cos2 x1 sin x 1 sin2 x
1 sin x
23. csc2 x csc x cot x
1 1 sin2 x sin x 1 cos x sin2 x
x cos sin x
Review Exercises for Chapter 5
24. sin12 x cos x
1 sin x
sin x
cos x
sin x
sin x
cos x
25. cos xtan2 x 1 cos x sec2 x
x sin x cot x cos sin x
1 sec2 x sec x
sec x
26. sec2 x cot x cot x cot xsec2 x 1 cot x tan2 x
27. cos x
cos x cos sin x sin 2 2 2
1 tan2 x tan x tan x
28. cot
30.
sin x
2 x tan x by the Cofunction Identity
1 tan x csc x sin x
1 1 1 tan x tan x sin x sin x
cos x0 sin x1
29.
1 1 cos tan csc sin 1 cos sin
31. sin5 x cos2 x sin4 x cos2 x sin x 1 cos2 x2 cos2 x sin x 1 2 cos2 x cos4 x cos2 x sin x
cot x
cos2 x 2 cos4 x cos6 x sin x 33. sin x 3 sin x
32. cos3 x sin2 x cos x cos2 x sin2 x cos x1 sin2 x sin2 x cos x
sin2
x
sin4
x
sin2 x sin4 x cos x
34. 4 cos 1 2 cos
sin x x
36.
2 2 n, 2 n 3 3
tan u
1 2
2n or 3
u
5 2n 3
1 sec x 1 0 2
csc2 x
n 6
4 3
sin x ±
sec x 2 1 cos x 2
2n or 3
1 3
37. 3 csc2 x 4
1 sec x 1 2
x
2
35. 33 tan u 3
2 cos 1 cos
3
x 5 2n 3
3
2
2 4 5 2 n, 2 n, 2 n, 2 n 3 3 3 3
These can be combined as: x
2 n or x n 3 3
513
514
Chapter 5
Analytic Trigonometry
38. 4 tan2 u 1 tan2 u
2 cos2 x cos x 1
39.
3 tan2 u 1 0
2 cos2 x cos x 1 0
1 3
2 cos x 1cos x 1 0
tan2 u
2 cos x 1 0
3 1 tan u ± ± 3 3
u
n or 6
cos x 5 n 6
x
2 sin2 x 3 sin x 1
40.
cos x 1 0 1 2
cos x 1
2 4 , 3 3
x0
cos2 x sin x 1
41.
2 sin2 x 3 sin x 1 0
1 sin2 x sin x 1 0
2 sin x 1sin x 1 0
sin xsin x 1 0
2 sin x 1 0
or sin x 1 0
1 2
sin x 1
sin x
5 x , 6 6
sin x 0 x 0,
1
cos2
x
cos2
2
43. 2 sin 2x 2 0
x 2 cos x 2 0
sin x 1
x 2
sin2 x 2 cos x 2
42.
sin x 1 0
sin 2x
x 2 cos x 1
0 cos x 12
2x
cos x 1 0 cos x 1 x0
2
2
3 2 n, 2 n 4 4
x
3 n, n 8 8
x
3 9 11 , , , 8 8 8 8
45. cos 4xcos x 1 0
44. 3 tan 3x 0 tan 3x 0
cos 4x 0
3x 0, , 2, 3, 4, 5
4x
2 4 5 x 0, , , , , 3 3 3 3
x
cos x 1 0
3 2 n, 2 n 2 2
cos x 1
3 n, n 8 2 8 2
x0
3 5 7 9 11 13 15 x 0, , , , , , , , 8 8 8 8 8 8 8 8 46. 3 csc2 5x 4
47. sin2 x 2 sin x 0
48. 2 cos2 x 3 cos x 0
4 3
sin xsin x 2 0
cos x2 cos x 3 0
sin x 0
cos x 0
csc2 5x csc 5x ±
No real solution
4 3
x 0,
sin x 2 0 No solution
or 2 cos x 3 0
3 x , 2 2
2 cos x 3 cos x
3 2
No solution
Review Exercises for Chapter 5 49.
515
tan2 tan 12 0
tan 4tan 3 0 tan 4 0
tan 3 0
arctan4 n
arctan 3 n
arctan4 , arctan4 2, arctan 3, arctan 3 51. sin 285 sin315 30
sec2 x 6 tan x 4 0
50. 1
tan2
sin 315 cos 30 cos 315 sin 30
x 6 tan x 4 0
tan2 x 6 tan x 5 0
tan x 5tan x 1 0 tan x 5 0
tan x 1 0
or
tan x 5
tan x 1
2
2
4
23 2212
3 1
cos 285 cos315 30
3 7 x , 4 4
x arctan5
2
cos 315 cos 30 sin 315 sin 30
x arctan5 2
22 23 2212
2
3 1
4
tan 285 tan315 30
tan 315 tan 30 1 tan 315 tan 30
33 2 3 3 1 1 3 1
52. sin345 sin300 45
53. sin
25 11 11 11 sin cos cos sin sin 12 6 4 6 4 6 4
sin 300 cos 45 cos 300 sin 45
3
2
3 1
2
2
2
4
2
1 2 2
21 22 23 22
2 4
cos
1 3
23 22 21 22
1 2
2
4
2
25 11 tan tan 12 6 4
2
3 2
2
1 3
tan 300 tan 45 3 1 1 tan 300 tan 45 1 31
4 23 2 3 2
1 3
1 3
4
3 1
2
4
3 1
11 tan 6 4 11 1 tan tan 6 4 tan
33 1 2 3 3 1 1 3
tan345 tan300 45
2
cos 300 cos 45 sin 300 sin 45 2
25 11 11 11 cos cos sin sin cos 12 6 4 6 4 6 4
cos345 cos300 45
516
Chapter 5
Analytic Trigonometry
54. sin
1912 sin116 4
tan
1912 cos116 4
11 11 cos cos sin 6 4 6 4
sin
cos
1 2
2
4
2
2
3
2
cos
2
2
1 3
2
4
3 1
3
2 2
4
11 11 cos sin sin 6 4 6 4
2
2
21 22
3 1
1912 tan116 4 11 tan 6 4 11 tan 1 tan 6 4 tan
3
3
1
1
3
3
1
3 3 3 3
3 3 3 3
12 63 2 3 6
55. sin 60 cos 45 cos 60 sin 45 sin60 45 sin 15 56. cos 45 cos 120 sin 45 sin 120 cos45 120 cos 165
57.
tan 25 tan 10 tan25 10 1 tan 25 tan 10
58.
tan 68 tan 115 tan68 115 tan47 1 tan 68 tan 115
tan 35 y
y
4
3
13
u −
12 x
v
7 −5
x
Figures for Exercises 59–64
59. sinu v sin u cos v cos u sin v
34 135 4712 13
3 5 47 52
tan u tan v 60. tanu v 1 tan u tan v
37 125 3 12 1 7 5
15 127 36 57
36 57 960 5077 36 57 1121
Review Exercises for Chapter 5 61. cosu v cos u cos v sin u sin v
62. sinu v sin u cos v cos u sin v
3 12 5 4 13 4 13
7
1 57 36 52
63. cosu v cos u cos v sin u sin v
7
4
135 3412 13
67. cot
cos x cos sin x sin 2 2 2
34 135 47 12 13
15 127 127 15 52 52
tan u tan v 64. tanu v 1 tan u tan v
1 57 36 52
65. cos x
66. sin x
15 127 36 57
960 5077 1121
37 125 3 12 1 7 5
36 57 57
36
3 3 3 sin x cos cos x sin 2 2 2
cos x0 sin x1
sin x0 cos x1
sin x
cos x
2 x tan x by the cofunction identity.
68. sin x sin cos x cos sin x 0 cos x 1sin x sin x
69. cos 3x cos2x x
70.
cos 2x cos x sin 2x sin x cos2 x sin2 x cos x 2 sin x cos x sin x cos3 x sin2 x cos x 2 sin2 x cos x cos3 x 3 sin2 x cos x cos3 x 31 cos2 x cos x cos3 x 3 cos x 3 cos3 x 4 cos3 x 3 cos x
71. sin x
sin x 1 4 4
2 cos x sin
1 4
cos x x
2
2
7 , 4 4
sin sin cos cos sin cos cos cos cos
sin cos cos sin cos cos cos cos
sin sin cos cos
tan tan
517
518
Chapter 5
Analytic Trigonometry
cos x
72.
cos x 1 6 6
cos x cos 6 sin x sin 6 cos x cos 6 sin x sin 6 1 2 sin x sin 2 sin x
1 6
12 1
sin x 1 x
73. sin x
3 2
sin x 3 2 2
2 cos x sin
3 2 3
cos x x
2
11 , 6 6
cos x
74.
3 3 cos x 0 4 4
cos x cos 34 sin x sin 34 cos x cos 34 sin x sin 34 0 2 sin x sin
3 0 4
22 0
2 sin x
2 sin x 0 sin x 0 x 0, 4 3 75. sin u , < u < 5 2 cos u 1 sin2 u tan u
76. cos u 3 5
tan u
,
1 and < u < ⇒ sin u 2 5
1 2
15 25 45 2 cos 2u cos u sin u 5 15 53
sin u 4 cos u 3
sin 2u 2 sin u cos u 2 24 54 53 25
53 54 2
cos 2u cos2 u sin2 u
43 4 1 3 2
2
24 7
2
sin 2u 2 sin u cos u 2
2 tan u tan 2u 1 tan2 u
2 5
2
2
7 25
2
2
2 tan u tan 2u 1 tan2 u
21 1 1 2
2
2
1 4 3 3 4
Review Exercises for Chapter 5
77. sin 4x 2 sin 2x cos 2x
78.
2 2 sin x cos xcos2 x sin2 x
1 1 2 sin2 x 1 cos 2x 1 cos 2x 1 2 cos x2 1
4 sin x cos x2 cos2 x 1 8 cos3 x sin x 4 cos x sin x
2 sin2 x 2 cos2 x
tan2 x
2
4
−2
2 −2 −2
−1
1 cos 4x 2 2x 2 1 cos 4x sin 79. tan2 2x cos2 2x 1 cos 4x 1 cos 4x 2
81. sin2 x tan2 x sin2 x
2
4
2
2
2x 1 cos 2 1 cos 2x 2
2
1 2 cos 2x cos2 2x 4 1 cos 2x 2
1 cos 4x 2 21 cos 2x
1 2 cos 2x
3 4 cos 2x cos 4x 41 cos 2x
83. sin75
1 cos 150 2
1
3
2
2
3
2
2
1 2 3 2
1 cos 150 2
1
3
2
2
3
2
2
1 2 3 2
1 cos 150 tan75 sin 150
2 3
1 cos 6x 2
82. cos2 x tan2 x sin2 x
2 4 cos 2x 1 cos 4x 41 cos 2x
80. cos2 3x
sin x sin x cos x cos x
cos75
2
1 1 2
3
2
2 3
1 cos 2x 2
519
520
Chapter 5
Analytic Trigonometry
30 84. sin 15 sin 2
30 cos 15 cos 2
1 cos 30 2
1 cos 30 30 tan 15 tan 2 sin 30
19 85. sin 12
19 cos 12
19 tan 12
1 cos
19 6
2
1 cos
19 6
2
2
2
1 2 3 2
3
2
2 3
1
3
2
3
2
2
2
3
2
2
17 6 2
17 6 2
1 cos
17 6
2
1 cos
17 6
2
1 2
tan
1 2 3 2
1 2 3 2
10
10
u2 1 2cos u 1 245 109 3 1010
u2 1 sincosu u 1 3 54 5 31
3
2
12 2
1
3 17 1 6 2 17 1 sin 6 2
1 cos
u2 1 2cos u 1 245 101
cos
4 3 u ⇒ cos u and is in Quadrant I. 87. Given sin u , 0 < u < 5 2 5 2 sin
3
2
2
1
1 2 3 2
3
1
1 2
17 tan tan 12
1
2
2
3 19 1 6 2 2 3 19 1 sin 6 2
17 cos cos 12
1 cos
17 17 6 sin 86. sin 12 2
3
1
1 cos 30 2
2
3
2
3
12
2
2
3
3
Review Exercises for Chapter 5 5 3 88. tan u , < u < 8 2 sin u cos u
521 y
5
−8
89
u
8
−5
x
89
89
1 2cos u 1 8 2
sin
u 2
c os
u 2
89
1 2cos u 1 8 2
89
u 1 cos u t an 2 sin u
1
8 89
5 89
89 8
5
89 8
289
89 1788
89
89 8
289
89 1788
89
8 89 5
u 2 35 89. Given cos u , < u < ⇒ sin u and is in Quadrant I. 7 2 7 2 sin
u2 1 2cos u 1 227 149 314 3 1414
cos tan
u2 1 2cos u 1 227 145
70
14
3 5 97 3 u2 1 sincosu u 1 327 5 57 357 5
1 361 635, u 1 cos u 1 16 7 21 sin 2 2 2 12 6 u 1 cos u 1 16 5 15 cos 2 2 2 12 6
90. sec u 6,
< u < , 2
sin u
cos u
1 6
tan
u 1 cos u 1 16 7 2 sin u 6 356
tan
u sinu2 216 21 35 2 cosu2 5 156 15
91.
1 cos2 10x cos 10x2
93. cos
35 7 6 or 5 35 35
1 1 sin sin sin 0 sin 6 6 2 3 2 3
cos 5x
92.
sin 6x tan 3x 1 cos 6x
94. 6 sin 15 sin 45 6
12 cos15 45 cos15 45
3 cos30 cos 60
3cos 30 cos 60 1 95. cos 5 cos 3 2 cos 2 cos 8
1 96. 4 sin 3 cos 2 42 sin3 2 sin3 2
2sin 5 sin
522
Chapter 5
Analytic Trigonometry
97. sin 4 sin 2 2 cos
4 2 2 sin4 2 2
98. cos 3 cos 2 2 cos
2 cos 3 sin
2 cos
3 2 3 2 cos 2 2
5 cos 2 2
sin x 2 cos 100. sin x 4 4
r
x 4 x 4 2
sin
x 4 x 4 2
2 cos x sin
1 2 v sin 2 32 0
range 100 feet v0 80 feet per second r
1 802 sin 2 100 32
sin 2 0.5 2 30
15 or
12
102. Volume V of the trough will be the area A of the isosceles triangle times the length l of the trough. VAl (a)
cos x 2 sin x sin 6 6 6
99. cos x
101.
A cos
1 bh 2
h ⇒ h 0.5 cos 2 0.5 2
4m b
b 2 b ⇒ 0.5 sin sin 2 0.5 2 2 A 0.5 sin
0.5 cos 2 2
0.52 sin 0.25 sin
(b) V sin
cos square meters 2 2 cos cubic meters 2 2
cos cubic meters 2 2
1 1 cos 2 sin cos sin cubic meters 2 2 2 2 2 2
Volume is maximum when
0.5 m Not drawn to scale
cos 2 2
V 0.254 sin sin
h 0.5 m
. 2
4
Review Exercises for Chapter 5
103. y 1.5 sin 8t 0.5 cos 8t 2
523
1 104. y 1.5 sin 8t 0.5 cos 8t 3 sin 8t 1 cos 8t 2 Using the identity a sin B b cos B a2 b2 sinB C,
2
0
b C arctan , a > 0 a
−2
(Exercise 83, Section 5.4), we have
1 1 y 32 12 sin 8t arctan 2 3
105. Amplitude
10
2
2
sin 8t arctan
106. Frequency
feet
< < , then < < and is in 2 4 2 2 2 Quadrant I.
107. False. If
cos
10
13.
1 4 cycles per second 2 8
108. False. The correct identity is sinx y sin x cos y cos x sin y.
> 0 2
109. True. 4 sinxcosx 4sin xcos x 4 sin x cos x 22 sin x cos x 2 sin 2x
110. True. It can be verified using a product-to-sum identity.
111. Reciprocal Identities: sin
1 csc
csc
1 sin
cos
1 sec
sec
1 cos
tan
1 cot
cot
1 tan
tan
sin cos
cot
cos sin
1 4 sin 45 cos 15 4 sin 60 sin 30
2 2
23 12 3 1 Quotient Identities:
Pythagorean Identities: sin2 cos2 1 1 tan2 sec2 1 cot2 csc2 112. No. For an equation to be an identity, the equation must be true for all real numbers. sin 12 has an infinite number of solutions but is not an identity.
113. a sin x b 0 sin x
b a
If b > a , then
b > 1 and there is no solution a
since sin x ≤ 1 for all x.
524
Chapter 5
Analytic Trigonometry
3 cos 3 114. S 6hs s 2 , 0 < ≤ 90 2 sin
where h 2.4 inches, s 0.75 inch, and is the given angle. (a) For a surface area of 12 square inches,
(b) Using a graphing calculator yields the following graph:
3 cos 3 S 62.40.75 0.752 12 2 sin
10.8 0.84375
0.84375
3 cos
sin
3 cos
sin
20
12
(0.9553, 11.99) 3 4
0
1.2.
0
Using the minimum function yields
Using the solve function of a graphing calculator gives
0.9553 radians or 54.73466.
49.91479 or 59.86118.
116. y1
115. The graph of y1 is a vertical shift of the graph of y2 one unit upward so y1 y2 1.
cos 3x , cos x
y2 2 sin x2
If the graph of y2 is reflected in the x-axis and then shifted upward by one unit, it coincides with the graph of y1. Therefore, cos 3x 2 sin x2 1. cos x So, y1 1 y2.
117. y x 3 4 cos x
1 x 118. y 2 x2 3 sin 2 2
11
Zeros: x 1.8431, 2.1758, 3.9903, 8.8935, 9.8820 −4
20 −2
Approximate roots: 3.1395, 2.0000,
7
−10
0.4378, 2.0000
−7
x 1 y 2 x2 3 sin 2 2
Problem Solving for Chapter 5 1. (a) Since sin2 cos2 1 and cos2 1 sin2 : cos ± 1 sin2 tan
sin sin ± 1 sin2 cos
cot
1 sin2 1 ± tan sin
We also have the following relationships: cos sin tan
2
sin sin 2
sin
2
1 1 sec ± 1 sin2 cos
cot
1 cos sin
sec
1 sin 2
csc
1 sin
—CONTINUED—
sin
10
Problem Solving for Chapter 5 1. —CONTINUED— (b) sin ± 1 cos2
2. cos
We also have the following relationships:
tan
1 sin ± cos cos
csc
1 1 ± 1 cos2 sin
tan
cos 2 cos
sec
1 cos
csc
1 cos 2
cot
1 cos ± tan 1 cos2
sec
1 cos
cot
cos cos 2
cos2
sin cos
2n 2 1 cos2n2
cos n
2
3. sin
2
12n 1 1 sin 12n 6 6
sin
± 10 01 Thus, sin
0
4. pt
sin 2n
cos n cos sin n sin 2 2
Thus, cos
1 12n 1 for all integers n. 6 2
1 p t 30p2t p3t p5t 30p6t 4 1
1 p2t sin1048 t 2
1.4
p1
p2 p3
− 0.003
1 p3t sin1572 t 3
p5 p6 0.003
−1.4
1 p5t sin2620 t 5 1 p6t sin3144 t 6 The graph of pt
1 1 1 sin524 t 15 sin1048 t sin1572 t sin2620 t 5 sin3144 t 4 3 5 y
yields the graph shown in the text and to the right.
y = p(t) 1.4
t 0.006
—CONTINUED—
1 6 2
2n 2 1 0 for all integers n.
(a) p1t sin524 t
6
−1.4
525
526
Chapter 5
Analytic Trigonometry
4. —CONTINUED— (b) Function
Period
(c)
p1t
2 1 0.0038 524 262
p2t
2 1 0.0019 1048 524
p3t
2 1 0.0013 1572 786
p5t
2 1 0.0008 2620 1310
p6t
1 2 0.0006 3144 1572
1.4
Max
0
−1.4
0.00382
Min
Over one cycle, 0 ≤ t <
1 262 ,
we have four t-intercepts:
t 0, t 0.00096, t 0.00191, and t 0.00285 (d) The absolute maximum value of p over one cycle is p 1.1952, and the absolute minimum value of p over one cycle is p 1.1952.
The graph of p appears to be periodic with a period 1 of 262 0.0038.
5. From the figure, it appears that u v w. Assume that u, v, and w are all in Quadrant I. From the figure: tan u
s 1 3s 3
tan v
s 1 2s 2
tan w
s 1 s
tanu v
tan u tan v 1 tan u tan v
1 3 1 2 1 1 31 2
5 6 1 1 6
1 tan w. Thus, tanu v tan w. Because u, v, and w are all in Quadrant I, we have arctan tan(u v arctan tan w u v w.
6. y
16 x2 tan x h0 v02 cos2
Let h0 0 and take half of the horizontal distance:
1 1 2 1 2 1 2 v sin 2 v 2 sin cos v sin cos 2 32 0 64 0 32 0 Substitute this expression for x in the model. y
16 2 cos2
v0
321 v
2
0
sin cos
sin 1 v cos 32 2
1 2 2 1 2 2 v sin v sin 64 0 32 0
1 2 2 v sin 64 0
2
0
sin cos
Problem Solving for Chapter 5 7.
527
The hypotenuse of the larger right triangle is: 1
sin2 1 cos 2 sin2 1 2 cos cos2
θ 2
2(1 + cos θ )
2 2 cos 21 cos
1
cos θ θ
sin θ
sin
2 21sin cos 21sin cos 11 cos cos
sin 1 cos sin 1 cos 21 cos2 2 sin
1 2cos
2 211coscos 211coscos 1 2cos 2
cos tan
2 1 sincos
8. F
0.6W sin 90 sin 12
(a) F
t 0.2 182.6
9. Seward: D 12.2 6.4 cos
0.6Wsin cos 90 cos sin 90 sin 12
0.6W sin 0 cos 1 sin 12
0.6W cos sin 12
t 0.2 182.6
New Orleans: D 12.2 1.9 cos (a)
20
0
0.6185 cos x . (b) Let y1 sin 12
365 0
550
(b) The graphs intersect when t 91 and when t 274. These values correspond to April 1 and October 1, the spring equinox and the fall equinox. 0
90 0
(c) The force is maximum (533.88 pounds) when 0. The force is minimum (0 pounds) when 90.
(c) Seward has the greater variation in the number of daylight hours. This is determined by the amplitudes, 6.4 and 1.9. (d) Period:
2 365.2 days 182.6
t when t 0 corresponds to 12:00 A.M. 6.2 (a) The high tides occur when cos t 1. Solving yields t 6.2 or t 18.6. 6.2 These t-values correspond to 6:12 A.M. and 6:36 P.M.
10. d 35 28 cos
The low tide occurs when cos
t 1. Solving yields t 0 and t 12.4 which corresponds to 12:00 A.M. 6.2
and 12:24 P.M. (b) The water depth is never 3.5 feet. At low tide the depth is d 35 28 7 feet. (c)
70
0
24 0
528
Chapter 5
Analytic Trigonometry
11. (a) Let y1 sin x and y2 0.5.
(b) Let y1 cos x and y2 0.5.
2
2
2
0
−2
−2
sin x ≥ 0.5 on the interval
5
6 , 6 .
(c) Let y1 tan x and y2 sin x.
cos x ≤ 0.5 on the interval
2 4
3 , 3 .
(d) Let y1 cos x and y2 sin x.
2
2
2
0
2
0
−2
−2
tan x < sin x on the intervals
sin 12. (a) n
2
0
2 , and 32, 2.
2 2 sin
(b) For glass, n 1.50.
2
1.50
sin cos cos sin 2 2 2 2 sin 2
cos cot sin 2 2 2
sin 30 For 60, n cos 30 cot 2
n
3
2
2 1.50
3
2
3
2
1 cot 2 2
cot2
1 tan 2 3 3
2 tan1
3 3
76.52
1 . cot 2 2
13. (a) sinu v w sin u v w sinu v cos w cosu v sin w sin u cos v cos u sin v cos w cos u cos v sin u sin v sin w sin u cos v cos w cos u sin v cos w cos u cos v sin w sin u sin v sin w (b) tanu v w tan u v w
tanu v tan w 1 tanu v tan w
1tan utanutantanvv tan w 1 tan u tan v 1 tan u tan v tan u tan v 1 tan w 1 tan u tan v
tan u tan v 1 tan u tan v tan w 1 tan u tan v tan u tan v tan w
tan u tan v tan w tan u tan v tan w 1 tan u tan v tan u tan w tan v tan w
4 and 54 , 2.
cos x ≥ sin x on the intervals 0,
1
Problem Solving for Chapter 5 (b) cos4 cos2 2
14. (a) cos3 cos2 cos 2 cos sin 2 sin
cos 2 cos 2 sin 2 sin 2
1 2
cos2 2 sin2 2
sin2
cos 2 sin cos sin
cos 4 sin2 cos
1 sin2 2 sin2 2
cos 1 4 sin2
1 2 sin2 2 1 22 sin cos 2 1 8 sin2 cos2
15. h1 3.75 sin 733t 7.5
h2 3.75 sin 733 t (a)
4 7.5 3
15
0
1 0
(b) The period for h1 and h2 is
2 0.0086. 733
12
0 3
2 733
The graphs intersect twice per cycle. 1 There are 116.66 cycles in the interval 0, 1 , so the graphs intersect approximately 233.3 times. 2 733
529
530
Chapter 5
Chapter 5
Analytic Trigonometry
Practice Test sec2 x csc2 x . csc2 x1 tan2 x
1. Find the value of the other five trigonometric functions, 4 given tan x 11 , sec x < 0.
2. Simplify
3. Rewrite as a single logarithm and simplify ln tan ln cot .
4. True or false:
cos
2 x csc x 1
5. Factor and simplify: sin4 x sin2 x cos2 x
6. Multiply and simplify: csc x 1csc x 1
7. Rationalize the denominator and simplify:
8. Verify:
cos2 x 1 sin x 9. Verify: tan4 x 2 tan2 x 1 sec4 x
1 cos sin 2 csc sin 1 cos 10. Use the sum or difference formulas to determine: (a) sin 105
(b) tan 15
1 tan . 4 1 tan
11. Simplify: sin 42 cos 38 cos 42 sin 38
12. Verify tan
13. Write sinarcsin x arccos x as an algebraic expression in x.
14. Use the double-angle formulas to determine:
15. Use the half-angle formulas to determine: (a) sin 22.5 (b) tan 12
16. Given sin 45, lies in Quadrant II, find cos2.
17. Use the power-reducing identities to write sin2 x cos2 x in terms of the first power of cosine.
18. Rewrite as a sum: 6sin 5 cos 2.
19. Rewrite as a product: sinx sinx .
20. Verify
21. Verify:
22. Find all solutions in the interval 0, 2:
cos u sin v 12sinu v sinu v. 23. Find all solutions in the interval 0, 2: tan2 3 1 tan 3 0
(a) cos 120
sin 9x sin 5x cot 2x. cos 9x cos 5x
4 sin2 x 1 24. Find all solutions in the interval 0, 2: sin 2x cos x
25. Use the quadratic formula to find all solutions in the interval 0, 2: tan2 x 6 tan x 4 0
(b) tan 300
