C H A P T E R 4
Trigonometric Functions
Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 272
Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . 281
Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 289
Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 300
Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 317
Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 329
Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 339
Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 350
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
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C H A P T E R 4
Trigonometric Functions
Section 4.1 Radian and Degree Measure
272
You should know the following basic facts about angles, their measurement, and their applications.
n Types of Angles:
(a) Acute: Measure between and
(b) Right: Measure
(c) Obtuse: Measure between and
(d) Straight: Measure
n and are complementary if They are supplementary if
n Two angles in standard position that have the same terminal side are called coterminal angles.
n To convert degrees to radians, use radians.
n To convert radians to degrees, use 1 radian
n one minute of
n one second of
n The length of a circular arc is where is measured in radians.
n Speed
n Angular speed 5 uyt 5 syrt
5 distanceytime
us 5 ru
185 1y60 of 19 5 1y360010 5
185 1y6019 5
5 s180ypd8.18 5 py180
a 1 b 5 1808.a 1 b 5 908.ba
1808.
1808.908
908.
908.08
1. The angle shown is
approximately 2 radians.
Vocabulary Check
1. Trigonometry 2. angle 3. standard position 4. coterminal
5. radian 6. complementary 7. supplementary 8. degree
9. linear 10. angular
2. The angle shown is
approximately radians.24
3. (a) Since lies in Quadrant IV.
(b) Since lies in
Quadrant II.
5p
2<
11p
4< 3p,
11p
4
3p
2<
7p
4< 2p,
7p
44. (a) Since lies in
Quadrant IV.
(b) Since lies in
Quadrant II.
23p
2< 2
13p
9< 2p,
13p
9
2p
2< 2
5p
12< 0,
5p
2
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Section 4.1 Radian and Degree Measure 273
9. (a)
(b)
x
y
π
32
2p
3
x
116π
y11p
6
7. (a)
(b)
x
y
π
34
4p
3
43π
x
y13p
4
5. (a) Since lies in Quadrant IV.
(b) Since lies in
Quadrant III.
2p < 22 < 2p
2; 22
2p
2< 21 < 0; 21 6. (a) Since 3.5 lies in Quadrant III.
(b) Since 2.25 lies in Quadrant II.p
2< 2.25 < p,
p < 3.5 <
3p
2,
8. (a)
(b)
x
y
52π
−
25p
2
x
74
−π
y
27p
4
10. (a) 4
(b)
x
y
−3
23
x
4
y
11. (a) Coterminal angles for
p
62 2p 5 2
11p
6
p
61 2p 5
13p
6
p
6: (b) Coterminal angles for
2p
32 2p 5 2
4p
3
2p
31 2p 5
8p
3
2p
3:
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274 Chapter 4 Trigonometric Functions
12. (a) Coterminal angles for
(b) Coterminal angles for
5p
42 2p 5 2
3p
4
5p
41 2p 5
13p
4
5p
4:
7p
62 2p 5 2
5p
6
7p
61 2p 5
19p
6
7p
6: 13. (a) Coterminal angles for
(b) Coterminal angles for
22p
152 2p 5 2
32p
15
22p
151 2p 5
28p
15
22p
15:
29p
41 4p 5
7p
4
29p
41 2p 5 2
p
4
29p
4: 14. (a) Coterminal angles for
(b) Coterminal angles for
8p
452 2p 5 2
82p
45
8p
451 2p 5
98p
45
8p
45:
7p
82 2p 5 2
9p
8
7p
81 2p 5
23p
8
7p
8:
15. Complement:
Supplement: p 2p
35
2p
3
p
22
p
35
p
6
17. Complement:
Supplement: p 2p
65
5p
6
p
22
p
65
p
3
16. Complement: Not possible; is greater than
(a)Supplement: p 23p
45
p
4
p
2.
3p
4
18. Complement: Not possible; is greater than
Supplement: p 22p
35
p
3
p
2.
2p
3
19. Complement:
Supplement: p 2 1 < 2.14
p
22 1 < 0.57 20. Complement: None
Supplement: p 2 2 < 1.14
12 >
p
22
21.
The angle shown is approximately 2108.
22.
The angle shown is approximately 2458.
23. (a) Since lies in
Quadrant II.
(b) Since lies in
Quadrant IV.
28282708 < 2828 < 3608,
1508908 < 1508 < 1808,
25. (a) Since
lies in Quadrant III.
(b) Since
lies in Quadrant I.23368 309
23608 < 23368 309 < 22708,
21328 509
21808 < 21328 509 < 2908,
24. (a) Since lies in
Quadrant I.
(b) Since lies in Quadrant I.8.5808 < 8.58 < 908,
87.9808 < 87.98 < 908,
26. (a) Since
lies in Quadrant II.
(b) Since lies
in Quadrant IV.
212.3582908 < 212.358 < 208,
2245.25822708 < 2245.258 < 21808,
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Section 4.1 Radian and Degree Measure 275
27. (a)
308x
y308 (b)
150°
x
y1508
28. (a)
(b)
x
−120°
y
21208
x
−270°
y
22708 29. (a)
(b)
780°
x
y
7808
x
405°
y
4058 30. (a)
(b)
x
y
−6008
26008
− °450
x
y
24508
31. (a) Coterminal angles for
(b) Coterminal angles for
2368 2 3608 5 23968
2368 1 3608 5 3248
2368:
528 2 3608 5 23088
528 1 3608 5 4128
528: 33. (a) Coterminal angles for
(b) Coterminal angles for
2308 2 3608 5 21308
2308 1 3608 5 5908
2308:
3008 2 3608 5 2608
3008 1 3608 5 6608
3008:32. (a) Coterminal angles for
(b) Coterminal angles for
23908 1 3608 5 2308
23908 1 7208 5 3308
23908:
1148 2 3608 5 22468
1148 1 3608 5 4748
1148:
34. (a) Coterminal angles for
(b) Coterminal angles for
27408 1 7208 5 2208
27408 1 10808 5 3408
27408:
24458 1 3608 5 2858
24458 1 7208 5 2758
24458: 35. Complement:
Supplement: 1808 2 248 5 1568
908 2 248 5 668 36. Complement: Not possible
Supplement: 1808 2 1298 5 518
37. Complement:
Supplement: 1808 2 878 5 938
908 2 878 5 38 38. Complement: Not possible
Supplement: 1808 2 1678 5 138
39. (a)
(b) 1508 5 15081 p
18082 55p
6
308 5 3081 p
18082 5p
6
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276 Chapter 4 Trigonometric Functions
44. (a)
(b) 3p 5 3p11808
p 2 5 5408
24p 5 24p11808
p 2 5 27208
46. (a)
(b)28p
155
28p
15 11808
p 2 5 3368
215p
65 2
15p
6 11808
p 2 5 2450845. (a)
(b) 213p
605 2
13p
60 11808
p 2 5 2398
7p
35
7p
3 11808
p 2 5 4208
47. 1158 5 1151p
18082 < 2.007 radians 48. radians83.78 5 83.781 p
18082 < 1.461
50. radians246.528 5 246.5281 p
18082 < 20.81249. 2216.358 5 2216.351p
18082 < 23.776 radians
51. 20.788 5 20.781 p
18082 < 20.014 radians
53.p
75
p
711808
p 2 < 25.7148 55. 6.5p 5 6.5p11808
p 2 5 11708
61. 858 189 300 5 858 1 s1860d8 1 s 30
3600d8 < 85.3088
63. 21258 360 5 21258 2 s 363600d8 5 2125.018
65. 280.68 5 2808 1 0.6s60d9 5 2808 369
67. 2345.128 5 23458 79 120
57. 22 5 2211808
p 2 < 2114.5928
59. 648 459 5 648 1 145
6028
5 64.758
52. radians3958 5 39581 p
18082 < 6.894
54.8p
135
8p
1311808
p 2 < 110.7698
56. 24.2p 5 24.2p11808
p 2 5 27568
58. 20.48 5 20.4811808
p 2 < 227.5028
60. 21248 309 5 2124.58
62. 24088 169 250 < 2408.2748
64. 3308 250 < 330.0078
66. 2115.88 5 21158 489 68. 310.758 5 3108 459
43. (a)
(b) 27p
65 2
7p
6 11808
p 2 5 22108
3p
25
3p
2 11808
p 2 5 2708
40. (a)
(b) 1208 5 12081 p
18082 52p
3
3158 5 31581 p
18082 57p
4
42. (a) (b) 1448 5 14481 p
18082 54p
522708 5 227081 p
18082 5 23p
2
41. (a)
(b) 22408 5 224081 p
18082 5 24p
3
2208 5 22081 p
18082 5 2p
9
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Section 4.1 Radian and Degree Measure 277
69.
< 220.348 5 2208 209 240
20.355 5 20.35511808
p 2 70.
< 458 39 470
5 458 1 39 1 0.786s600 d
5 458 1 s0.0631ds609d
< 45.06318
0.7865 5 0.786511808
p 2
72.
radians u 53112 5 2
712
31 5 12u
s 5 ru71.
radiansu 565
6 5 5u
s 5 ru 73.
radians3u 5327 5 4
47
32 5 7u
3s 5 ru
74.
Because the angle represented is clockwise, this angle is radians.245
u 56075 5
45 radians
60 5 75u
s 5 ru
78. radianu 5s
r5
10
225
5
11
75. The angles in radians are:
908 5p
2
608 5p
3
458 5p
4
308 5p
6
08 5 0
3308 511p
6
2708 53p
2
2108 57p
6
1808 5 p
1358 53p
4
76. The angles in degrees are:
5p
65 1508
2p
35 1208
p
35 608
p
45 458
p
65 308
p 5 1808
7p
45 3158
5p
35 3008
4p
35 2408
5p
45 2258
77.
radians u 58
15
8 5 15u
s 5 ru 79.
radians u 570
29< 2.414
35 5 14.5u
s 5 ru
80. kilometers, kilometers
u 5s
r5
160
805 2 radians
s 5 160r 5 80
82. feet,
s 5 ru 5 91p
32 5 3p feet
u 5 608 5p
3r 5 9
81. in radians
inches s 5 14s180d1 p
1802 5 14p < 43.982
s 5 ru, u
83. in radians
meters meters< 56.55 s 5 2712p
3 2 5 18p
s 5 ru, u
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278 Chapter 4 Trigonometric Functions
90.
s 5 ru 5 4000s1.0105d < 4042.0 miles
u 5 318 469 1 268 89 5 578 549 < 1.0105 rad
r 5 4000 miles
89.
s 5 ru 5 4000s0.28537d < 1141.48 miles
5 168 219 10 < 0.28537 radian
u 5 428 79 330 2 258 469 32088. r 5s
u5
8
3308spy1808d5
48
11p inches < 1.39 inches
91.
< 48 29 33.020
u 5s
r5
450
6378< 0.07056 radian < 4.048
93. < 23.878u 5s
r5
2.5
65
25
605
5
12 radian
95. (a) single axel:
(b) double axel:
(c) triple axel: 5 7p radians 312 revolutions 5 12608
5 4p 1 p 5 5p radians
212 revolutions 5 7208 1 1808 5 9008
5 2p 1 p 5 3p radians
112 revolutions 5 3608 1 1808 5 5408
96. Linear speed 5s
t5
ru
t5
s6400 1 1250d2p
110< 436.967 kmymin
92.
u 5s
r5
400
6378< 0.0627 rad < 3.598
r 5 6378, s 5 400
94. u 5s
r5
24
55 4.8 rad < 275.028
97. (a)
Angular speed
(b) Radius of saw blade
Radius in feet
Speed
5 0.3125s80pd 5 78.54 ftysec
5s
t5
ru
t5 r
u
t5 rsangular speedd
53.75
125 0.3125 ft
57.5
25 3.75 in.
5 s2pds40d 5 80p radysec
Revolutions
Second5
2400
605 40 revysec 98. (a)
Angular speed
(b) Radius of saw blade
Radius in feet
Speed
5 0.3021s160pd < 151.84 ftysec
5s
t5
ru
t5 r
u
t5 rsangular speedd
53.625
12 < 0.3021 ft
57.25
25 3.625 in.
5 s2pds80d 5 160p radysec
Revolutions
Second5
4800
605 80 revysec
86. r 5s
u5
3
4py35
9
4p meters < 0.72 meter 87. r 5
s
u5
82
1358spy1808d5
328
3p miles < 34.80 miles
84. centimeters,
centimeters cm< 28.27s 5 ru 5 1213p
4 2 5 9p
u 53p
4r 5 12 85. r 5
s
u5
36
py25
72
p feet < 22.92 feet
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Section 4.1 Radian and Degree Measure 279
99. (a)
Angular speed
Radius of wheel
Speed
(b) Let spin balance machine rate.
70 5 rsangular speedd 5 r12p ?x
s1y60d2 55
25,344120px ⇒ x < 941.18 revymin
x 5
55
25,344? 57,600p 5
125p
11< 35.70 milesyhr
5s
t5
ru
t5 r
u
t5 rsangular speedd
525y2
s12 in.yftds5280 ftymid5
5
25,344 miles
5 2p s28,800d 5 57,600p radyhr
Revolutions
Hour5
480
s1y60d5 28,800 revyhr
100. (a)
(b) Speed
For the outermost track, 6000p cmymin
6s400pd ≤ linear speed ≤ 6s1000pd
5s
t5
ru
t5 r
u
t5 rsangular speedd 5 6sangular speedd
400p radymin ≤ angular speed ≤ 1000p radymin
2ps200d ≤ angular speed ≤ 2ps500d
200 ≤revolutions
minute≤ 500
101. False, 1 radian so one radian
is much larger than one degree.
5 1180
p 28< 57.38,
103. True:2p
31
p
41
p
125
8p 1 3p 1 p
125 p 5 1808
102. No, is coterminal with 1808.212608
104. (a) An angle is in standard position when the origin is the vertex and the initial side
coincides with the positive -axis.
(b) A negative angle is generated by a clockwise rotation.
(c) Angles that have the same initial and terminal sides are coterminal angles.
(d) An obtuse angle is between and 1808.908
x
105. If is constant, the length of the arc is
proportional to the radius and
hence increasing.
ss 5 rud,u 106. Let A be the area of a circular sector of radius
and central angle Then
A
pr25
u
2p ⇒ A 5
1
2 r2u.
u.
r
107. square metersA 51
2r2u 5
1
2s10d2 ?
p
35
50
3p 108. Because
Hence, A 51
2r2u 5
1
2152112
152 5 90 ft2.
s 5 ru, u 512
15.
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280 Chapter 4 Trigonometric Functions
109.
(a) Domain:
Domain:
The area function changes more rapidly for because
it is quadratic and the arc length function is linear.
(b) Domain:
Domain: 0 < u < 2p s 5 ru 5 10u
0
0
2
320
A
s
0 < u < 2p r 5 10 ⇒ A 512s102du 5 50u
r > 1
r > 0 s 5 ru 5 rs0.8d
r > 0 u 5 0.8 ⇒ A 512r2s0.8d 5 0.4r2
0
0
12
8
As
A 512r2u, s 5 ru
113.
−2 2 3 4 5 6
−2
−3
−4
1
2
3
4
x
y
115.
x
y
−2−3−4 1 2 3 4
−3
1
3
4
5
117.
x
y
−2−3−4 1 2 3 4
−2
−4
−5
1
2
3
110. If a fan of greater diameter is installed, the angular speed does not change.
111. Answers will vary. 112. Answers will vary.
114.
x
y
−2−3−4 1 2 3 4
−2
−3
−6
1
2
116.
x
y
−3−4−5 1 2 3
−2
−3
−4
1
2
3
4
118.
x
y
−2 1 2 3 5 6
−2
−3
−4
1
2
3
4
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Section 4.2 Trigonometric Functions: The Unit Circle
Section 4.2 Trigonometric Functions: The Unit Circle 281
2.
cot u 5x
y5
12y13
5y135
12
5
sec u 51
x5
1
12y135
13
12
csc u 51
y5
1
5y135
13
5
tan u 5y
x5
5y13
12y135
5
12
cos u 5 x 512
13
sin u 5 y 55
13
sx, yd 5 112
13,
5
132
4.
cot u 5x
y5
24y5
23y55
4
3
sec u 51
x5
1
24y55 2
5
4
csc u 51
y5
1
23y55 2
5
3
tan u 5y
x5
23y5
24y55
3
4
cos u 5 x 5 24
5
sin u 5 y 5 23
5
sx, yd 5 124
5, 2
3
52
6. t 5p
3 ⇒ 11
2, !3
2 2
1.
csc u 51
y5
17
15
sec u 51
x5 2
17
8
cot u 5x
y5 2
8
15
tan u 5y
x5 2
15
8
cos u 5 x 5 28
17
sin u 5 y 515
17
3.
csc u 51
y5 2
13
5
sec u 51
x5
13
12
cot u 5x
y5 2
12
5
tan u 5y
x5 2
5
12
cos u 5 x 512
13
sin u 5 y 5 25
13
Vocabulary Check
1. unit circle 2. periodic 3. odd, even
5. corresponds to 1!2
2, !2
2 2.t 5p
4
n You should know how to evaluate trigonometric functions using the unit circle.
n You should know the definition of a periodic function.
n You should be able to recognize even and odd trigonometric functions.
n You should be able to evaluate trigonometric functions with a calculator in both radian and degree mode.
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282 Chapter 4 Trigonometric Functions
10. corresponds to 11
2, 2
!3
2 2.t 55p
3
12. t 5 p ⇒ s21, 0d
18. corresponds to
tan p
35
y
x5
!3y2
1y25 !3
cos p
35 x 5
1
2
sin p
35 y 5
!3
2
11
2, !3
2 2.t 5p
3
9. corresponds to 121
2, !3
2 2.t 52p
3
11. corresponds to s0, 21d.t 53p
2
13. corresponds to 1!2
2, !2
2 2.t 5 27p
414. corresponds to 12
1
2, !3
2 2.t 5 24p
3
15. corresponds to s0, 1d.t 5 23p
216. corresponds to s1, 0d.t 5 22p
17. corresponds to
tan t 5y
x5 1
cos t 5 x 5!2
2
sin t 5 y 5!2
2
1!2
2, !2
2 2.t 5p
4
19. corresponds to
tan t 5y
x5
1
!35
!3
3
cos t 5 x 5 2!3
2
sin t 5 y 5 21
2
12!3
2, 2
1
22.t 57p
620. corresponds to
tan t 5y
x5 21
cos t 5 x 5 2!2
2
sin t 5 y 5!2
2
12!2
2, !2
2 2.t 5 25p
4
21. corresponds to
tan t 5y
x5 2!3
cos t 5 x 5 21
2
sin t 5 y 5!3
2
121
2, !3
2 2.t 52p
322. corresponds to
tan t 5y
x5 2!3
cos t 5 x 51
2
sin t 5 y 5 2!3
2
11
2, 2
!3
2 2.t 55p
3
8. t 55p
4 ⇒ 12
!2
2, 2
!2
2 27. corresponds to 12!3
2, 2
1
22.t 57p
6
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Section 4.2 Trigonometric Functions: The Unit Circle 283
24. corresponds to
tan t 5y
x5 2
!3
3
cos t 5 x 5!3
2
sin t 5 y 5 21
2
1!3
2, 2
1
22.t 511p
623. corresponds to
tan t 5y
x5 !3
cos t 5 x 51
2
sin t 5 y 5!3
2
11
2, !3
2 2.t 5 25p
3
25. corresponds to
tan t 5y
x5 2
!3
3
cos t 5 x 5!3
2
sin t 5 y 5 21
2
1!3
2, 2
1
22.t 5 2p
626. corresponds to
tan123p
4 2 5y
x5 1
cos123p
4 2 5 x 5 2!2
2
sin123p
4 2 5 y 5 2!2
2
12!2
2, 2
!2
2 2.t 5 23p
4
27. corresponds to
tan t 5y
x5 1
cos t 5 x 5!2
2
sin t 5 y 5!2
2
1!2
2, !2
2 2.t 5 27p
428. corresponds to
tan t 5y
x5 2!3
cos t 5 x 5 21
2
sin t 5 y 5!3
2
121
2, !3
2 2.t 5 24p
3
29. corresponds to
is undefined.tan t 5y
x
cos t 5 x 5 0
sin t 5 y 5 1
s0, 1d.t 5 23p
2
31. corresponds to
cot t 5x
y5 21tan t 5
y
x5 21
sec t 51
x5 2!2cos t 5 x 5 2
!2
2
csc t 51
y5 !2sin t 5 y 5
!2
2
12!2
2, !2
2 2 .t 53p
4
30. corresponds to
tans22pd 5y
x5
0
15 0
coss22pd 5 x 5 1
sins22pd 5 y 5 0
s1, 0d.t 5 22p
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284 Chapter 4 Trigonometric Functions
32. corresponds to
cot t 5x
y5 2!3
sec t 51
x5 2
2
!35 2
2!3
3
csc t 51
y5 2
tan t 5y
x5 2
1
!35 2
!3
3
cos t 5 x 5 2!3
2
sin t 5 y 51
2
12!3
2,
1
22.t 55p
633. corresponds to
is undefined.
is undefined. cot t 5x
y5 0tan t 5
y
x
sec t 51
xcos t 5 x 5 0
csc t 51
y5 1sin t 5 y 5 1
s0, 1d.t 5p
2
35. corresponds to
cot u 5!3
3tan t 5
y
x5 !3
sec u 5 22cos t 5 x 5 21
2
csc t 5 22!3
3sin t 5 y 5 2
!3
2
121
2, 2
!3
2 2.t 5 22p
334. corresponds to
undefined
undefined
cot 3p
25
x
y5
0
215 0
sec 3p
25
1
x5
1
0 ⇒
csc 3p
25
1
y5
1
215 21
tan 3p
25
y
x5
21
0 ⇒
cos 3p
25 x 5 0
sin 3p
25 y 5 21
s0, 21d.t 53p
2
36. corresponds to
tan127p
4 2 5y
x5 1
cos127p
4 2 5 x 5!2
2
sin127p
4 2 5 y 5!2
2
1!2
2, !2
2 2.t 5 27p
4
cot127p
4 2 5x
y5 1
sec127p
4 2 51
x5 !2
csc127p
4 2 51
y5 !2
37. sin 5p 5 sin p 5 0
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Section 4.2 Trigonometric Functions: The Unit Circle 285
39. cos 8p
35 cos
2p
35 2
1
238. Because
cos 7p 5 coss6p 1 pd 5 cos p 5 21
7p 5 6p 1 p :
40. Because
sin 9p
45 sin12p 1
p
42 5 sin p
45
!2
2
9p
45 2p 1
p
4:
42. Because
5 sin15p
6 2 51
2
sin1219p
6 2 5 sin124p 15p
6 2
219p
65 24p 1
5p
6:
44. Because :
5 cos 4p
35 2
1
2
cos128p
3 2 5 cos124p 14p
3 2
28p
35 24p 1
4p
3
41. cos1213p
6 2 5 cos12p
62 5 cos111p
6 2 5!3
2
43. sin129p
4 2 5 sin12p
42 5 2!2
2
45.
(a)
(b) cscs2td 5 2csc t 5 23
sins2td 5 2sin t 5 21
3
sin t 51
3
47.
(a)
(b) secs2td 51
coss2td5 25
cos t 5 coss2td 5 21
5
coss2td 5 21
546. cos
(a)
(b) secs2td 51
coss2td5
1
cos t5 2
4
3
coss2td 5 cos t 5 23
4
t 5 23
4
48.
(a)
(b) csc t 51
sinstd5
1
2sins2td5 2
8
3
sin t 5 2sins2td 5 23
8
sins2td 53
849.
(a)
(b) sinst 1 pd 5 2sin t 5 24
5
sinsp 2 td 5 sin t 54
5
sin t 54
5
50.
(a)
(b) cosst 1 pd 5 2cos t 5 24
5
cossp 2 td 5 2cos t 5 24
5
cos t 54
551. sin
7p
9< 0.6428
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286 Chapter 4 Trigonometric Functions
56. cot 3.7 51
tan 3.7< 1.6007
52. tan 2p
5< 3.0777 53. cos
11p
5< 0.8090 54. sin
11p
9< 20.6428
55. csc 1.3 < 1.0378 57. coss21.7d < 20.1288
59. csc 0.8 51
sin 0.8< 1.3940
76.
(a)
(b)
ys0d 51
4e20 coss0d 5
1
4 foot
ystd 51
4e2t cos 6t
70. (a)
(b) cos 2.5 5 x < 20.8
sin 0.75 5 y < 0.7
72. (a)
(b)
t < 0.72 or t < 5.56
cos t 5 0.75
t < 4.0 or t < 5.4
sin t 5 20.75
58. coss22.5d < 20.8011 60. sec 1.8 51
cos 1.8< 24.4014
62. sins213.4d < 20.740461. sec 22.8 51
cos 22.8< 21.4486
69. (a)
(b) cos 2 < 20.4
sin 5 < 21
63. cot 2.5 51
tan 2.5< 21.3386
64. tan 1.75 < 25.5204 65. cscs21.5d 51
sins21.5d < 21.0025
66. tans22.25d < 1.2386 67. secs24.5d 51
coss24.5d < 24.7439
68. cscs25.2d 51
sins25.2d < 1.1319
71. (a)
(b)
t < 1.82 or 4.46
cos t 5 20.25
t < 0.25 or 2.89
sin t 5 0.25 73.
amperes < 0.79
Is0.7d 5 5e21.4 sin 0.7
I 5 5e22t sin t
74. At
I < 5e22s1.4d sin 1.4 < 0.30 amperes.
t 5 1.4, 75.
(a)
(b)
(c) ys12d 5
14 cos 3 < 20.2475 ft
ys14d 5
14 cos
32 < 0.0177 ft
ys0d 514 cos 0 5 0.2500 ft
ystd 514 cos 6t
(c) The maximum displacements are decreasing
because of friction, which is modeled by the
term.
(d)
t 5p
12,
p
4
6t 5p
2,
3p
2
cos 6t 5 0
1
4e2t cos 6t 5 0
e2t
0.50 1.02 1.54 2.07 2.59
0.09 0.03 20.0220.0520.15y
t
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Section 4.2 Trigonometric Functions: The Unit Circle 287
78. True
80. True
82. (a) The points and are symmetric
about the origin.
(b) Because of the symmetry of the points, you can
make the conjecture that
(c) Because of the symmetry of the points, you can
make the conjecture that cosst1 1 pd 5 2cos t1.
sinst1 1 pd 5 2sin t1.
sx2, y2dsx1, y1d
84.
Therefore, sin t1 1 sin t2 Þ sinst1 1 t2d.
sin 1 < 0.8415
sins0.25d 1 sins0.75d < 0.2474 1 0.6816 5 0.9290
77. False. sin124p
3 2 5!3
2> 0
79. False. 0 corresponds to s1, 0d.
81. (a) The points have -axis symmetry.
(b) since they have the same
-value.
(c) since the -values have
opposite signs.
x2cos t1 5 cos sp 2 t1d
y
sin t1 5 sin sp 2 t1d
y
83.
Thus, cos 2t Þ 2 cos t.
2 cos 0.75 < 1.4634cos 1.5 < 0.0707,
85.
θ
θ−
(x, y)
(x, −y)
x
y
sec u 51
cos u5
1
coss2ud 5 secs2ud
cos u 5 x 5 coss2ud
87. is odd.
hs2td 5 f s2tdgs2td 5 2f stdgstd 5 2hstd
hstd 5 f stdgstd 88. and
Both and are odd functions.
The function is even.hstd 5 fstdgstd
5 sin t tan t 5 hstd
5 s2sin tds2tan td
hs2td 5 sins2td tans2td
hstd 5 fstdgstd 5 sin t tan t
gf
gstd 5 tan tfstd 5 sin t
86.
cot u 5x
y5 2cots2ud
tan u 5y
x5 2tans2ud
csc u 51
y5 2cscs2ud
sin u 5 y 5 2sins2ud
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288 Chapter 4 Trigonometric Functions
90.
f21sxd 5 3!4sx 2 1d
y 5 3!4sx 2 1d
4sx 2 1d 5 y3
x 2 1 514 y3
x 514 y3 1 1
y 514x3 1 1
−9 9
−6
6
ff−1
fsxd 514 x3 1 189.
f 21sxd 523sx 1 1d
23sx 1 1d 5 y
2x 1 2 5 3y
2x 5 3y 2 2
x 512s3y 2 2d
y 512s3x 2 2d
−9 9
−6
6
ff−1
f sxd 512s3x 2 2d
92.
f21sxd 5x
2 2 x, x < 2
x
2 2 x5 y, x < 2
x 5 ys2 2 xd
x 5 2y 2 xy
xy 1 x 5 2y
x 52y
y 1 1
y 52x
x 1 1, x > 21
−3 9
−4
4
fsxd 52x
x 1 1, x > 2191.
f21sxd 5 !x2 1 4, x ≥ 0
!x2 1 4 5 y, x ≥ 0
x2 1 4 5 y2
x2 5 y2 2 4
x 5 !y2 2 4
−2 10
−1
7
ff−1
y 5 !x2 2 4
f sxd 5 !x2 2 4, x ≥ 2, y ≥ 0
94.
Asymptotes:
10864
4
6
8
10
−6
−8
x
y
x 5 23, x 5 2, y 5 0
f sxd 55x
x2 1 x 2 65
5x
sx 1 3dsx 2 2d93.
Asymptotes:
2 4 5 6 7 8−2 −1
1
3
4
5
6
7
8
−2
−3
−4
−3−4x
y
x 5 3, y 5 2
f sxd 52x
x 2 3
95.
Asymptotes: x 5 22, y 51
2
x Þ 2 5x 1 5
2sx 1 2d,
f sxd 5x2 1 3x 2 10
2x2 2 85
sx 2 2dsx 1 5d2sx 2 2dsx 1 2d
x
y
−1−5−6−7 1−1
−2
−3
−4
1
2
3
4
−2
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Section 4.3 Right Triangle Trigonometry 289
96.
Slant asymptote:
Vertical asymptotes: x < 3.608, x < 21.108
y 5x
22
7
4 4
6
8
10
−8
−10
−2−4−6−8 6 8 10
2
y
x
f sxd 5x3 2 6x2 1 x 2 1
2x2 2 5x 2 85
x
22
7
42
15sx 1 4d4s2x2 2 5x 2 8d
97.
Domain: all real numbers
Intercepts:
No asymptotes
s0, 24d, s1, 0d, s24, 0d
0 5 x2 1 3x 2 4 5 sx 1 4dsx 2 1d ⇒ x 5 1, 24
y 5 x2 1 3x 2 4 98.
Domain: all
Intercepts:
Asymptote: x 5 0
s1, 0d, s21, 0d
ln x4 5 0 ⇒ x4 5 1 ⇒ x 5 ±1
x Þ 0
y 5 ln x4
99.
Domain: all real numbers
Intercept:
Asymptote: y 5 2
s0, 5d
fsxd 5 3x11 1 2 100.
Domain: all real numbers
Intercepts:
Asymptotes: x 5 22, y 5 0
s7, 0d, 10, 27
42x Þ 22
fsxd 5x 2 7
sx2 1 4x 1 4d 5x 2 7
sx 1 2d2
Section 4.3 Right Triangle Trigonometry
n You should know the right triangle definition of trigonometric functions.
(a) (b) (c)
(d) (e) (f)
n You should know the following identities.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
( j) (k)
n You should know that two acute angles are complementary if and cofunctions of
complementary angles are equal.
n You should know the trigonometric function values of and or be able to construct triangles from
which you can determine them.
608,308, 458,
a 1 b 5 908,a and b
1 1 cot2 u 5 csc2 u1 1 tan2 u 5 sec2 u
sin2 u 1 cos2 u 5 1cot u 5cos u
sin utan u 5
sin u
cos u
cot u 51
tan utan u 5
1
cot usec u 5
1
cos u
cos u 51
sec ucsc u 5
1
sin usin u 5
1
csc u
cot u 5adj
oppsec u 5
hyp
adjcsc u 5
hyp
opp
Adjacent side
Opposi
te s
ide
Hyp
oten
use
θ
tan u 5opp
adjcos u 5
adj
hypsin u 5
opp
hyp
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290 Chapter 4 Trigonometric Functions
Vocabulary Check
1. (a) iii (b) vi (c) ii (d) v (e) i (f) iv
2. hypotenuse, opposite, adjacent 3. elevation, depression
1.
cot u 5adj
opp5
4
3
sec u 5hyp
adj5
5
4
csc u 5hyp
opp5
5
3
tan u 5opp
adj5
3
4
cos u 5adj
hyp5
4
5
sin u 5opp
hyp5
3
5
adj 5 !52 2 32 5 !16 5 4
35
θ
3.
cot u 5adj
opp5
15
8
sec u 5hyp
adj5
17
15
csc u 5hyp
opp5
17
8
tan u 5opp
adj5
8
15
cos u 5adj
hyp5
15
17
sin u 5opp
hyp5
8
17
hyp 5 !82 1 152 5 17
θ
15
8
2.
sin
cos
tan
csc
sec
cot u 5adj
opp5
12
5
u 5hyp
adj5
13
12
u 5hyp
opp5
13
5
u 5opp
adj5
5
12
u 5adj
hyp5
12
13
u 5opp
hyp5
5
13
b 5 !132 2 52 5 !169 2 25 5 12
513
b
θ
4.
sin
cos
tan
csc
sec
cot u 52
3
u 5!13
2
u 5!13
3
u 518
125
3
2
u 512
6!135
2!13
13
u 518
6!135
3!13
13
C 5 !182 1 122 5 !468 5 6!13
12
18 c
θ
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Section 4.3 Right Triangle Trigonometry 291
6.
sin
cos
tan
csc
sec
cot u 5adj
opp5
!161
8
u 5hyp
adj5
15
!1615
15!161
161
u 5hyp
opp5
15
8
u 5opp
adj5
8
!1615
8!161
161
u 5adj
hyp5
!161
15
u 5opp
hyp5
8
15
158
θ
adj 5 !152 2 82 5 !161
sin
cos
tan
csc
sec
cot u 5adj
opp5
!161
2 ? 45
!161
8
u 5hyp
adj5
7.5
s!161y2d 515
!1615
15!161
161
u 5hyp
opp5
7.5
45
15
8
u 5opp
adj5
4
s!161y2d 58
!1615
8!161
161
u 5adj
hyp5
!161
2 ? 7.55
!161
15
u 5opp
hyp5
4
7.55
8
15
7.54
θ
adj 5 !7.52 2 42 5!161
2
The function values are the same because the triangles are similar, and corresponding sides are proportional.
5.
cot u 5adj
opp5
8
65
4
3
sec u 5hyp
adj5
10
85
5
4
csc u 5hyp
opp5
10
65
5
3
tan u 5opp
adj5
6
85
3
4
cos u 5adj
hyp5
8
105
4
5
sin u 5opp
hyp5
6
105
3
5
θ
8
10
opp 5 !102 2 82 5 6
cot u 5adj
opp5
2
1.55
4
3
sec u 5hyp
adj5
2.5
25
5
4
csc u 5hyp
opp5
2.5
1.55
5
3
tan u 5opp
adj5
1.5
25
3
4
cos u 5adj
hyp5
2
2.55
4
5
sin u 5opp
hyp5
1.5
2.55
3
5
θ
2
2.5opp 5 !2.52 2 22 5 1.5
The function values are the same since the triangles are similar and the corresponding sides are proportional.
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292 Chapter 4 Trigonometric Functions
7.
cot u 5adj
opp5 2!2
sec u 5hyp
adj5
3
2!25
3!2
4
csc u 5hyp
opp5 3
tan u 5opp
adj5
1
2!25
!2
4
cos u 5adj
hyp5
2!2
3
θ3
1sin u 5
opp
hyp5
1
3
adj 5 !32 2 12 5 !8 5 2!2
8.
sin
cos
tan
csc
sec
cot u 5adj
opp5
2
15 2
u 5hyp
adj5
!5
2
u 5hyp
opp5
!5
15 !5
u 5opp
adj5
1
2
u 5adj
hyp5
2
!55
2!5
5
u 5opp
hyp5
1
!55
!5
5
1
2
θhyp 5 !12 1 22 5 !5
sin
cos
tan
csc
sec
cot u 56
35 2
u 53!5
65
!5
2
u 53!5
35 !5
u 53
65
1
2
u 56
3!55
2
!55
2!5
5
u 53
3!55
1
!55
!5
5
3
6
θ
hyp 5 !32 1 62 5 3!5
The function values are the same because the triangles are similar, and corresponding sides are proportional.
cot u 5adj
opp5
4!2
25 2!2
sec u 5hyp
adj5
6
4!25
3
2!25
3!2
4
csc u 5hyp
opp5
6
25 3
tan u 5opp
adj5
2
4!25
1
2!25
!2
4
cos u 5adj
hyp5
4!2
65
2!2
3
θ
62
sin u 5opp
hyp5
2
65
1
3
adj 5 !62 2 22 5 !32 5 4!2
The function values are the same since the triangles are similar and the corresponding sides are proportional.
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Section 4.3 Right Triangle Trigonometry 293
10.
sin
cos
tan
csc
sec u 5hyp
adj5
!26
5
u 5hyp
opp5
!26
15 !26
u 5opp
adj5
1
5
u 5adj
hyp5
5
!265
5!26
26
u 5opp
hyp5
1
!265
!26
26
1
5
26θ
hyp 5 !52 1 12 5 !26
12.
sin
tan
csc
sec
cot u 53
2!105
3!10
20
u 57
3
u 57
2!105
7!10
20
u 52!10
3
u 52!10
7
θ
102
3
7
opp 5 !72 2 32 5 !40 5 2!10
9. Given:
csc u 5hyp
opp5
6
5
sec u 5hyp
adj5
6
!115
6!11
11
cot u 5adj
opp5
!11
5
tan u 5opp
adj5
5
!115
5!11
11
cos u 5adj
hyp5
!11
6
adj 5 !11
52 1 sadjd2 5 62
56
11
θ
sin u 55
65
opp
hyp
11. Given:
csc u 54
!155
4!15
15
cot u 51
!155
!15
15
tan u 5 !15
cos u 51
4
sin u 5!15
4
opp 5 !15
soppd2 1 12 5 42
15
θ
4
1
sec u 5 4 54
15
hyp
adj
14.
sin
cos
tan
sec
cot u 51
tan u5
!273
4
u 51
cos u5
17
!2735
17!273
273
u 5opp
adj5
4
!2735
4!273
273
u 5adj
hyp5
!273
17
u 5opp
hyp5
4
17
417
273
θ
adj 5 !172 2 42 5 !27313. Given:
csc u 5!10
3
cot u 51
3
sec u 5 !10
cos u 5!10
10
sin u 53!10
10
hyp 5 !10
32 1 12 5 shypd2
3
1
10
θ
tan u 5 3 53
15
opp
adj
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294 Chapter 4 Trigonometric Functions294 Chapter 4 Trigonometric Functions
16.
cos
tan
csc
sec
cot u 51
tan u5
!55
3
u 51
cos u5
8
!555
8!55
55
u 51
sin u5
8
3
u 5opp
adj5
3
!555
3!55
55
u 5adj
hyp5
!55
8
adj 5 !82 2 32 5 !55
38
55
θ
sin u 53
815. Given:
csc u 5!97
4
sec u 5!97
9
tan u 54
9
cos u 59
!975
9!97
97
sin u 54
!975
4!97
97
hyp 5 !97
42 1 92 5 shypd2
97
θ
9
4
cot u 59
45
adj
opp
Function (deg) (rad) Function Valueuu
26. tan1
!3
p
6308
24. sin!2
2
p
4458
22. csc !2p
4458
20. sec !2p
4458
18. cos!2
2
p
4458
Function (deg) (rad) Function Value
17. sin
19. tan
21. cot
23. cos
25. cot 1p
4458
!3
2
p
6308
!3
3
p
3608
!3p
3608
1
2
p
6308
uu
27. sin u 51
csc u28. cos u 5
1
sec u29. tan u 5
1
cot u30. csc u 5
1
sin u
31. sec u 51
cos u32. cot u 5
1
tan u 33. tan u 5sin u
cos u34. cot u 5
cos u
sin u
35. sin2 u 1 cos2 u 5 1 36. 1 1 tan2 u 5 sec2 u 37. sins908 2 ud 5 cos u 38. coss908 2 ud 5 sin u
39. tans908 2 ud 5 cot u 40. cots908 2 ud 5 tan u 41. secs908 2 ud 5 csc u 42. cscs908 2 ud 5 sec u
43.
(a)
(c) cos 308 5 sin 608 5!3
2
tan 608 5sin 608
cos 6085 !3
sin 608 5!3
2, cos 608 5
1
2
(b)
(d) cot 608 5cos 608
sin 6085
1
!35
!3
3
sin 308 5 cos 608 51
2
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Section 4.3 Right Triangle Trigonometry 295
44.
(a)
(b)
(c)
(d) cot 308 51
tan 3085
3
!35
3!3
35 !3
cos 308 5sin 308
tan 3085
s1y2ds!3y3d 5
3
2!35
!3
2
cot 608 5 tans908 2 608d 5 tan 308 5!3
3
csc 308 51
sin 3085 2
sin 308 51
2, tan 308 5
!3
3
46.
(a)
(b) cot u 51
tan u5
1
2!65
!6
12
cos u 51
sec u5
1
5
sec u 5 5, tan u 5 2!6
45.
(a)
(b)
(c)
(d) secs90º 2 ud 5 csc u 5 3
tan u 5sin u
cos u5
1y3
s2!2 dy35
!2
4
cos u 51
sec u5
2!2
3
sin u 51
csc u5
1
3
csc u 5 3, sec u 53!2
4
47.
(a)
(b)
sin a 5!15
4
sin2 a 515
16
sin2 a 1 11
422
5 1
sin2 a 1 cos2 a 5 1
sec a 51
cos a5 4
cos a 51
4
(c)
(d) sins908 2 ad 5 cos a 51
4
5!15
15 5
1
!15
51y4
!15y4
cot a 5cos a
sin a
(c)
(d) sin u 5 tan u cos u 5 s2!6 d11
52 52!6
5
cots908 2 ud 5 tan u 5 2!6
48.
(a)
(b)
(c)
(d) csc b 5 !1 1 cot2 b 5!1 11
255
!26
5
tans908 2 bd 5 cot b 51
5
sec2 b 5 1 1 tan2 b ⇒ cos b 5 1
!1 1 tan2 b5
1
!1 1 255
1
!265
!26
26
cot b 51
tan b5
1
5
sb lies in Quadrant I or III.dtan b 5 5
50. csc u tan u 51
sin u?
sin u
cos u5
1
cos u5 sec u49. tan u cot u 5 tan u1 1
tan u2 5 1
51. tan u cos u 5 1 sin u
cos u2 cos u 5 sin u 52. cot u sin u 5cos u
sin u sin u 5 cos u
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296 Chapter 4 Trigonometric Functions296 Chapter 4 Trigonometric Functions
53.
5 sin2 u
5 ssin2 u 1 cos2 ud 2 cos2 u
s1 1 cos uds1 2 cos ud 5 1 2 cos2 u
55.
5 csc u sec u 51
sin u?
1
cos u
51
sin u cos u
sin u
cos u1
cos u
sin u5
sin2 u 1 cos2 u
sin u cos u
54.
5 1
scsc u 1 cot udscsc u 2 cot ud 5 csc2 u 2 cot2 u
56.
5 1 1 cot2 u 5 csc2 u
5 1 1cot u
s1ycot ud
tan u 1 cot u
tan u5
tan u
tan u1
cot u
tan u
57. (a)
(b) cos 878 < 0.0523
sin 418 < 0.6561
59. (a)
(b) csc 48º 79 51
sins48 1760dº < 1.3432
sec 428 129 5 sec 42.28 51
cos 42.28< 1.3499
58. (a)
(b) cot 71.58 51
tan 71.58< 0.3346
tan 18.58 < 0.3346
60. (a)
(b) secs88 509 250 d 51
coss88 509 250 d < 1.0120
< coss8.840278d < 0.9881
coss88 509 250 d 5 cos18 150
601
25
36002
61. Make sure that your calculator is in radian mode.
(a)
(b) tan p
8< 0.4142
cot p
165
1
tanspy16d< 5.0273
63. (a)
(b) csc u 5 2 ⇒ u 5 308 5p
6
sin u 51
2 ⇒ u 5 308 5
p
6
62. (a)
(b) coss1.25d < 0.3153
secs1.54d 51
coss1.54d < 32.4765
64. (a)
(b) tan u 5 1 ⇒ u 5 458 5p
4
cos u 5!2
2 ⇒ u 5 458 5
p
4
65. (a)
(b) cot u 5 1 ⇒ u 5 458 5p
4
sec u 5 2 ⇒ u 5 608 5p
3
67. (a)
(b) sin u 5!2
2 ⇒ u 5 458 5
p
4
csc u 52!3
3 ⇒ u 5 608 5
p
3
66. (a)
(b) cos u 51
2 ⇒ u 5 608 5
p
3
tan u 5 !3 ⇒ u 5 608 5p
3
68. (a)
(b)
cos u 51
!25
!2
2 ⇒ u 5 458 5
p
4
sec u 5 !2
tan u 53
!35 !3 ⇒ u 5 608 5
p
3
cot u 5!3
3
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Section 4.3 Right Triangle Trigonometry 297
69.
cos 308 5105
r ⇒ r 5
105
cos 3085
105
!3y25
210
!35 70!3
tan 308 5y
105 ⇒ y 5 105 tan 308 5 105 ?
!3
35 35!3
70.
sin 308 5y
15 ⇒ y 5 15 sin 308 5 1511
22 515
2
cos 308 5x
15 ⇒ x 5 15 cos 308 5 15 ?
!3
25
15!3
2
71.
sin 608 5y
16 ⇒ y 5 16 sin 608 5 161!3
2 2 5 8!3
cos 608 5x
16 ⇒ x 5 16 cos 608 5 1611
22 5 8
72.
sin 608 538
r ⇒ r 5
38
sin 6085
38
!3y25
76!3
3
cot 608 5x
38 ⇒ x 5 38 cot 608 5 38 ?
1
!35
38!3
3
73.
r2 5 202 1 202 ⇒ r 5 20!2
tan 458 520
x ⇒ 1 5
20
x ⇒ x 5 20 74.
r2 5 102 1 102 ⇒ r 5 10!2
tan 458 5y
10 ⇒ 1 5
y
10 ⇒ y 5 10
75.
r2 5 s2!5d2 1 s2!5d2 5 20 1 20 5 40 ⇒ r 5 2!10
tan 458 52!5
x ⇒ 1 5
2!5
x ⇒ x 5 2!5
76.
r2 5 s4!6d2 1 s4!6d2 5 96 1 96 5 192 ⇒ r 5 8!3
tan 458 5y
4!6 ⇒ 1 5
y
4!6 ⇒ y 5 4!6
77. (a)
(b) and Thus,
(c) feeth 56s21d
55 25.2
6
55
h
21.tan u 5
h
21tan u 5
6
5
6
h
516
θ
78. (a)
(b)
(c) x 5 30 sin 758 < 28.98 meters
sin 758 5x
30
x
75°
30
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298 Chapter 4 Trigonometric Functions298 Chapter 4 Trigonometric Functions
80.
Subtracting,
< 1.295 < 1.3 miles. <13
16.3499 2 6.3138
h 513
cot 3.58 2 cot 98
13
h5 cot 3.58 2 cot 98
cot 3.58 513 1 c
h
not drawn to scale
9°3.5°
13 c
h
cot 98 5c
h79.
feetw 5 100 tan 588 < 160
tan 588 5w
100
tan u 5opp
adj
81. (a)
(b)
(c) rate down the zip line
vertical rate50
65
25
3 ftysec
50!2
65
25!2
3 ftysec
L2 5 502 1 502 5 2 ? 502 ⇒ L 5 50!2 feet
tan u 550
505 1 ⇒ u 5 458 82. (a)
(b) feet above sea level
(c) minutes to reach top
Vertical rate
feet per minute < 173.8
5519.3
s896.5y300d
896.5
300
1693.5 2 519.3 5 1174.2
x 5 896.5 sin 35.48 < 519.3 feet
sin 35.48 5x
896.5
83.
sx1, y1d 5 s28!3, 28d
x1 5 cos 308s56d 5!3
2s56d 5 28!3
cos 308 5x1
56
y1 5 ssin 308ds56d 5 11
22s56d 5 28
sin 308 5y1
56
30°
56
( , )x y1 1
sx2, y2d 5 s28, 28!3 d
x2 5 scos 608ds56d 5 11
22s56d 5 28
cos 608 5x2
56
y2 5 sin 608s56d 5 1!3
2 2s56d 5 28!3
sin 60º 5y2
56
60°
56
( , )x y2 2
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Section 4.3 Right Triangle Trigonometry 299
84.
csc 208 510
y< 2.92sec 208 5
10
x< 1.06
cot 208 5x
y< 2.75tan 208 5
y
x< 0.36
cos 208 5x
10< 0.94sin 208 5
y
10< 0.34
x < 9.397, y < 3.420
86. False
5 !2 Þ 1 sin 458 1 cos 458 5!2
21
!2
2
87. True
for all u1 1 cot2 u 5 csc2 u
85. True
sin 608 csc 608 5 sin 608 1
sin 6085 1
88. No. so you can find ± sec u.tan2 u 1 1 5 sec2 u,
(b) Sine and tangent are increasing, cosine is
decreasing.
(c) In each case, tan u 5sin u
cos u.
89. (a)
0 0.3420 0.6428 0.8660 0.9848
1 0.9397 0.7660 0.5000 0.1736
0 0.3640 0.8391 1.7321 5.6713tan u
cos u
sin u
80860840820808u
90.
1 0.9397 0.7660 0.5000 0.1736
1 0.9397 0.7660 0.5000 0.1736sins908 2 ud
cos u
80860840820808u
are complementary angles.u and 908 2 u
cos u 5 sins908 2 ud
91.
−6 6
−1
7 93.
−6 6
−1
7
95.
−2 10
−4
4 97.
−2 10
−4
4
92.
−3 3
−3
1 94.
−6 6
−6
2
96.
−2 10
−4
4 98.
−2 10
−4
4
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Section 4.4 Trigonometric Functions of Any Angle
300 Chapter 4 Trigonometric Functions
n Know the Definitions of Trigonometric Functions of Any Angle.
If is in standard position, a point on the terminal side and then:
n You should know the signs of the trigonometric functions in each quadrant.
n You should know the trigonometric function values of the quadrant angles
n You should be able to find reference angles.
n You should be able to evaluate trigonometric functions of any angle. (Use reference angles.)
n You should know that the period of sine and cosine is
n You should know which trigonometric functions are odd and even.
Even: cos x and sec x
Odd: sin x, tan x, cot x, csc x
2p.
0, p
2, p, and
3p
2.
cot u 5x
y, y Þ 0tan u 5
y
x, x Þ 0
sec u 5r
x, x Þ 0cos u 5
x
r
csc u 5r
y, y Þ 0sin u 5
y
r
r 5 !x2 1 y2 Þ 0,sx, ydu
Vocabulary Check
1. 2. 3. 4.
5. 6. 7. referencecot ucos u
r
x
y
xcsc u
y
r
1. (a)
cot u 5x
y5
4
3tan u 5
y
x5
3
4
sec u 5r
x5
5
4cos u 5
x
r5
4
5
csc u 5r
y5
5
3sin u 5
y
r5
3
5
r 5 !16 1 9 5 5
sx, yd 5 s4, 3d (b)
cot u 5x
y5
8
15tan u 5
y
x5
15
8
sec u 5r
x5 2
17
8cos u 5
x
r5 2
8
17
csc u 5r
y5 2
17
15sin u 5
y
r5 2
15
17
r 5 !64 1 225 5 17
sx, yd 5 s28, 215d
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Section 4.4 Trigonometric Functions of Any Angle 301
2. (a)
cot u 5x
y5
12
255 2
12
5
sec u 5r
x5
13
12
csc u 5r
y5
13
255 2
13
5
tan u 5y
x5
25
125 2
5
12
cos u 5x
r5
12
13
sin u 5y
r5
25
135 2
5
13
r 5 !122 1 s25d2 5 13
x 5 12, y 5 25 (b)
cot u 5x
y5
21
15 21
sec u 5r
x5
!2
215 2!2
csc u 5r
y5
!2
15 !2
tan u 5y
x5
1
215 21
cos u 5x
r5
21
!25 2
!2
2
sin u 5y
r5
1
!25
!2
2
r 5 !s21d2 1 12 5 !2
x 5 21, y 5 1
4. (a)
cot u 5x
y5
3
15 3
sec u 5r
x5
!10
3
csc u 5r
y5
!10
15 !10
tan u 5y
x5
1
3
cos u 5x
r5
3
!105
3!10
10
sin u 5y
r5
1
!105
!10
10
r 5 !32 1 12 5 !10
x 5 3, y 5 1 (b)
cot u 5x
y5
2
245 2
1
2
sec u 5r
x5
2!5
25 !5
csc u 5r
y5
2!5
245 2
!5
2
tan u 5y
x5
24
25 22
cos u 5x
r5
2
2!55
!5
5
sin u 5y
r5
24
2!55 2
2!5
5
r 5 !22 1 s24d2 5 2!5
x 5 2, y 5 24
3. (a)
cot u 5x
y5 !3tan u 5
y
x5
!3
3
sec u 5r
x5
22!3
3cos u 5
x
r5
2!3
2
csc u 5r
y5 22sin u 5
y
r5 2
1
2
r 5 !3 1 1 5 2
sx, yd 5 s2!3, 21d (b)
cot u 5x
y5 21tan u 5
y
x5 21
sec u 5r
x5 2!2cos u 5
x
r5 2
!2
2
csc u 5r
y5 !2sin u 5
y
r5
!2
2
r 5 !4 1 4 5 2!2
sx, yd 5 s22, 2d
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302 Chapter 4 Trigonometric Functions
5.
cot u 5x
y5
7
24tan u 5
y
x5
24
7
sec u 5r
x5
25
7cos u 5
x
r5
7
25
csc u 5r
y5
25
24sin u 5
y
r5
24
25
r 5 !49 1 576 5 25
sx, yd 5 s7, 24d
7.
cot u 5x
y5 2
5
12
sec u 5r
x5
13
5
csc u 5r
y5 2
13
12
tan u 5y
x5 2
12
5
cos u 5x
r5
5
13
sin u 5y
r5 2
12
13
r 5 !52 1 s212d2 5 !25 1 144 5 !169 5 13
sx, yd 5 s5, 212d
6.
sin cos
tan csc
sec cot u 5x
y5
8
15u 5
r
x5
17
8
u 5r
y5
17
15u 5
y
x5
15
8
u 5x
r5
8
17u 5
y
r5
15
17
r 5 !82 1 152 5 17
x 5 8, y 5 15,
8.
sin
cos
tan
csc
sec
cot u 5x
y5 2
12
5
u 5r
x5 2
13
12
u 5r
y5
13
5
u 5y
x5
10
2245 2
5
12
u 5x
r5
224
265
212
13
u 5y
r5
10
265
5
13
r 5 !s224d2 1 s10d2 5 26
x 5 224, y 5 10
9.
cot u 5x
y5 2
2
5
sec u 5r
x5 2
!29
2
csc u 5r
y5
!29
5
tan u 5y
x5 2
5
2
cos u 5x
r5 2
2!29
29
sin u 5y
r5
5!29
29
r 5 !16 1 100 5 2!29
sx, yd 5 s24, 10d 10.
sin
cos
tan
csc
sec
cot u 5x
y5
5
6
u 5r
x5 2
!61
5
u 5r
y5 2
!61
6
u 5y
x5
26
255
6
5
u 5x
r5
25
!615
25!61
61
u 5y
r5
26
!615
26!61
61
r 5 !s25d22 1 s26d2 5 !61
x 5 25, y 5 26
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Section 4.4 Trigonometric Functions of Any Angle 303
11.
cot u 5x
y5 2
5
4
sec u 5r
x5 2
2!41
5
csc u 5r
y5
!41
4
tan u 5y
x5
8
2105 2
4
5
cos u 5x
r5
210
2!415 2
5!41
41
sin u 5y
r5
8
2!415
4!41
41
r 5 !s210d2 1 82 5 !164 5 2!41
sx, yd 5 s210, 8d 12.
sin
cos
tan
csc
sec
cot u 5x
y5 2
1
3
u 5r
x5 !10
u 5r
y5 2
!10
3
u 5y
x5
29
35 23
u 5x
r5
3
3!105
!10
10
u 5y
r5
29
3!105
23!10
10
r 5 !32 1 s29d2 5 !90 5 3!10
x 5 3, y 5 29
13. lies in Quadrant III or Quadrant IV.
lies in Quadrant II or Quadrant III.
and lies in Quadrant III.cos u < 0 ⇒ usin u < 0
cos u < 0 ⇒ u
sin u < 0 ⇒ u
15. lies in Quadrant I or Quadrant III.
lies in Quadrant I or Quadrant IV.
and lies in Quadrant I.cos u > 0 ⇒ ucot u > 0
cos u > 0 ⇒ u
cot u > 0 ⇒ u
17.
in Quadrant II
cot u 5x
y5 2
4
3tan u 5
y
x5 2
3
4
sec u 5r
x5 2
5
4cos u 5
x
r5 2
4
5
csc u 5r
y5
5
3sin u 5
y
r5
3
5
⇒ x 5 24u
sin u 5y
r5
3
5 ⇒ x2 5 25 2 9 5 16
14. and
and
Quadrant IV
x
y< 0
r
x> 0
cot u < 0sec u > 0
16. and
and
Quadrant III
r
y< 0
y
x> 0
csc u < 0tan u > 0
18.
in Quadrant III
sin csc
cos sec
tan cot u 54
3u 5
y
x5
3
4
u 5 25
4u 5
x
r5 2
4
5
u 5 25
3u 5
y
r5 2
3
5
⇒ y 5 23u
cos u 5x
r5
24
5 ⇒ |y| 5 3
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304 Chapter 4 Trigonometric Functions
19.
cot u 5x
y5 2
8
15
sec u 5r
x5
17
8cos u 5
x
r5
8
17
csc u 5r
y5 2
17
15sin u 5
y
r5 2
15
17
tan u 5y
x5
215
8 ⇒ r 5 17
sin u < 0 ⇒ y < 0 20. csc
cot
sin csc
cos sec
tan cot u 5 2!15u 5y
x5 2
!15
15
u 5 24!15
15u 5
x
r5 2
!15
4
u 5 4u 5y
r5
1
4
u < 0 ⇒ x 5 2!15
u 5r
y5
4
1 ⇒ x 5 ±!15
21.
cot u 5x
y5 2
!3
3tan u 5
y
x5 2!3
sec u 5r
x5 22cos u 5
x
r5 2
1
2
csc u 5r
y5
2!3
3sin u 5
y
r5
!3
2
sin u ≥ 0 ⇒ y 5 !3
sec u 5r
x5
2
21 ⇒ y2 5 4 2 1 5 3
23. is undefined
is undefined.
is undefined.cot utan u 5y
x5
0
x5 0
sec u 5r
x5 21cos u 5
x
r5
2r
r5 21
csc u 5r
ysin u 5
y
r5 0
p
2≤ u ≤
3p
2⇒ u 5 p, y 5 0, x 5 2r
⇒ u 5 np.cot u
22. and
is undefined.
is undefined.cot u
sec u 5 21
csc u
tan u 5sin u
cos u5 0
cos u 5 cos p 5 21
p
2≤ u ≤
3p
2 ⇒ u 5 psin u 5 0
24. is undefined and
is undefined.
is undefined.
cot u 5 0
sec u
csc u 5 21
tan u
cos u 5 0
sin u 5 21
p ≤ u ≤ 2p ⇒ u 53p
2.tan u
25. To find a point on the terminal side of use any point on the line that lies in
Quadrant II. is one such point.
cot u 5 21tan u 5 21
sec u 5 2!2cos u 5 21
!25 2
!2
2
csc u 5 !2sin u 51
!25
!2
2
x 5 21, y 5 1, r 5 !2
s21, 1dy 5 2xu,
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Section 4.4 Trigonometric Functions of Any Angle 305
27. To find a point on the terminal side of use any
point on the line that lies in Quadrant III.
is one such point.
cot u 521
225
1
2
sec u 5!5
215 2!5
csc u 5!5
225 2
!5
2
tan u 522
215 2
cos u 5 21
!55 2
!5
5
sin u 5 22
!55 2
2!5
5
x 5 21, y 5 22, r 5 !5
s21, 22dy 5 2x
u,26. Quadrant III,
sin
cos
tan
csc
sec
cot u 5x
y5
2x
s21y3d x5 3
u 5r
x5
s!10xdy3
2x5 2
!10
3
u 5r
y5
s!10xdy3
s21y3dx5 2!10
u 5y
x5
s21y3dx
2x5
1
3
u 5x
r5
2x
s!10xdy35 2
3!10
10
u 5y
r5
s2xy3ds!10xdy3
5 2!10
10
r 5!x2 11
9x2 5
!10x
3
x > 012x, 21
3x2
28.
Quadrant IV,
sin csc
cos sec
tan cot u 5 23
4u 5
y
x5
s24y3d x
x5 2
4
3
u 55
3u 5
x
r5
x
s5y3dx5
3
5
u 5 25
4u 5
y
r5
s24y3dx
s5y3d x5 2
4
5
r 5!x2 116
9x2 5
5
3x
x > 01x, 24
3x2
4x 1 3y 5 0 ⇒ y 5 24
3x
29.
sec p 5r
x5
1
215 21
sx, yd 5 s21, 0d 31.
50
215 0 cot13p
2 2 5x
y
sx, yd 5 s0, 21d
33.
sec 0 5r
x5
1
15 1
sx, yd 5 s1, 0d
35.
undefined cot p 5x
y5 2
1
0 ⇒
sx, yd 5 s21, 0d
30. undefined
since corresponds to s0, 1d.p
2
tan p
25
y
x5
1
0 ⇒
32. undefinedcsc 0 5r
y5
1
0 ⇒ 34. csc
3p
25
1
sin 3p
2
51
215 21
36. csc p
25
1
sin p
2
51
15 1
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306 Chapter 4 Trigonometric Functions
37.
u9 5 1808 2 1208 5 608
x
y
1208=θ608=θ ′
u 5 1208
39. is coterminal with
x
y
−2358=θ
558=θ ′
u9 5 2258 2 1808 5 458
2258.u 5 21358
38.
x
y
2258=θ
458=θ ′
u9 5 2258 2 1808 5 458
40.
x
y
−3308=θ 308=θ ′
u9 5 23308 1 3608 5 308
41.
x
y
=
θ =
π3
π
35
θ ′
u 55p
3, u9 5 2p 2
5p
35
p
3
43. is coterminal with
x
y
=
θ =
π
6 π
65
−
θ ′
u9 57p
62 p 5
p
6
7p
6.u 5 2
5p
6
42.
x
y
=π4
θ =π
43
θ ′
u9 5 p 23p
45
p
4
44.
x
y
=π3
θ =π
32
−θ ′
u9 522p
31 p 5
p
3
45.
u9 5 2088 2 1808 5 288
= 208°
x
y
θ
θ
′ = 28°
u 5 2088 46.
u9 5 3608 2 3228 5 388
x
y
= 322°
θ
θ
′ = 38°
u 5 3228
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Section 4.4 Trigonometric Functions of Any Angle 307
47.
u9 5 3608 2 2928 5 688
x
y
= −292°θ
θ′ = 68°
u 5 22928 48. lies in
Quadrant III.
Reference angle:
1808 2 1658 5 158x
y
−1658=
θ ′ 158=
θ
u 5 21658
49. is coterminal with
x
y
π
511
π
5
=
θ
θ
′ =
u9 5p
5
p
5.u 5
11p
5
51. lies in Quadrant III.
Reference angle:
x
y
−1.8=θ ′ 1.342=
θ
p 2 1.8 < 1.342
u 5 21.8
50.
x
y
π
717
π
73
θ
θ
′ =
=
u9 517p
72
14p
75
3p
7
52. lies in Quadrant IV.
Reference angle:
x
y
θ ′= 1.358
θ = 4.5
4.5 2 p < 1.358
u
53. Quadrant III
tan 2258 5 tan 458 5 1
cos 2258 5 2cos 458 5 2!2
2
sin 2258 5 2sin 458 5 2!2
2
u9 5 458,
55. is coterminal with Quadrant IV.
tans27508d 5 2tan 308 5 2!3
3
coss27508d 5 cos 308 5!3
2
sins27508d 5 2sin 308 5 21
2
u9 5 3608 2 3308 5 308
3308,u 5 27508
54. Quadrant IV
sin
cos
tan 3008 5 2tan 608 5 2!3
3008 5 cos 608 51
2
3008 5 2sin 608 5 2!3
2
u 5 3008, u9 5 3608 2 3008 5 608,
56. Quadrant III
tans24958d 5 tan 458 5 1
coss24958d 5 2cos 458 5 2!2
2
sins24958d 5 2sin 458 5 2!2
2
u 5 24958, u9 5 458,
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308 Chapter 4 Trigonometric Functions
57. Quadrant IV
tan15p
3 2 5 2tan1p
32 5 2!3
cos15p
3 2 5 cos1p
32 51
2
sin15p
3 2 5 2sin1p
32 5 2!3
2
u9 5 2p 25p
35
p
3
u 55p
3, 58.
sin
cos
tan 3p
45 21
3p
45 2
!2
2
3p
45
!2
2
u 53p
4, u9 5 p 2
3p
45
p
4
59. Quadrant IV
tan12p
62 5 2tan p
65 2
!3
3
cos12p
62 5 cos p
65
!3
2
sin12p
62 5 2sin p
65 2
1
2
u9 5p
6,
61. Quadrant II
tan 11p
45 2tan
p
45 21
cos 11p
45 2cos
p
45 2
!2
2
sin 11p
45 sin
p
45
!2
2
u9 5p
4,
60. ,
sin
cos
tan124p
3 2 5 2!3
124p
3 2 521
2
124p
3 2 5!3
2
u9 5p
3u 5
24p
3
62. is coterminal with
in Quadrant III.
sin
cos
tan 10p
35 tan
p
35 !3
10p
35 2cos
p
35 2
1
2
10p
35 2sin
p
35 2
!3
2
u9 54p
32 p 5
p
3
4p
3.u 5
10p
3
63. is coterminal with
Quadrant III
tan1217p
6 2 5 tan1p
62 5!3
3
cos1217p
6 2 5 2cos1p
62 5 2!3
2
sin1217p
6 2 5 2sin1p
62 5 21
2
u9 57p
62 p 5
p
6,
7p
6.u 5 2
17p
664.
sin
cos
tan1220p
3 2 5 !3
1220p
3 2 521
2
1220p
3 2 52!3
2
u 5220p
3, u9 5
p
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Section 4.4 Trigonometric Functions of Any Angle 309
65.
in Quadrant IV.
cos u 54
5
cos u > 0
cos2 u 516
25
cos2 u 5 1 29
25
cos2 u 5 1 2 123
522
cos2 u 5 1 2 sin2 u
sin2 u 1 cos2 u 5 1
sin u 5 23
566.
sin u 51
csc u5
1
!105
!10
10
csc u 51
sin u
!10 5 csc u
csc u > 0 in Quadrant II.
10 5 csc2 u
1 1 s23d2 5 csc2 u
1 1 cot2 u 5 csc2 u
cot u 5 23
67.
cot u 5 2!3
cot u < 0 in Quadrant IV.
cot2 u 5 3
cot2 u 5 s22d2 2 1
cot2 u 5 csc2 u 2 1
1 1 cot2 u 5 csc2 u
csc u 5 22
69.
tan u 5!65
4
tan u > 0 in Quadrant III.
tan2 u 565
16
tan2 u 5 129
422
2 1
tan2 u 5 sec2 u 2 1
1 1 tan2 u 5 sec2 u
sec u 5 29
4
68.
sec u 51
5y85
8
5
cos u 51
sec u ⇒ sec u 5
1
cos u
cos u 55
8
70.
Quadrant IV ⇒ csc u 5 2!41
5
csc u 5 ±!41
5
1 116
255 csc2 u
1 1 cot2 u 5 csc2 u
tan u 5 25
4 ⇒ cot u 5 2
4
5
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310 Chapter 4 Trigonometric Functions
71. is in Quadrant II.
cot u 51
tan u5
2!21
2
sec u 51
cos u5
25
!215
25!21
21
csc u 51
sin u5
5
2
tan u 5sin u
cos u5
2y5
2!21y55
22
!215
22!21
21
cos u 5 2!1 2 sin2 u 5 2!1 24
255 2
!21
5
sin u 52
5 and cos u < 0 ⇒ u
72. is in Quadrant III.
sec u 51
cos u5 2
7
3
csc u 51
sin u5
27
2!105
27!10
20
cot u 51
tan u5
3
2!105
3!10
20
tan u 5sin u
cos u5
22!10y7
23y75
2!10
3
sin u 5 2!1 2 cos2 u 5 2!1 29
495
2!40
75
22!10
7
cos u 5 23
7 and sin u < 0 ⇒ u
73. is in Quadrant II.
cot u 51
tan u5 2
1
4
csc u 51
sin u5
17
4!175
!17
4
sin u 5 tan u cos u 5 s24d12!17
17 2 54!17
17
cos u 51
sec u5
21
!175
2!17
17
sec u 5 2!1 1 tan2 u 5 2!1 1 16 5 2!17
tan u 5 24 and cos u < 0 ⇒ u 74. is in Quadrant II.
csc u 51
sin u5
26
!265 !26
sin u 5 tan u cos u 5 121
52125!26
26 2 5!26
26
cos u 51
sec u5
25
!265
25!26
26
sec u 5 2!1 1 tan2 u 5 2!1 11
255
2!26
5
tan u 51
cot u5
21
5
cot u 5 25 and sin u > 0 ⇒ u
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Section 4.4 Trigonometric Functions of Any Angle 311
78. sec 2358 51
cos 2358< 21.7434
75. is in Quadrant IV.
cot u 51
tan u5
2!5
2
tan u 5sin u
cos u5
22y3
!5y35
22
!55
22!5
5
sec u 51
cos u5
3
!55
3!5
5
cos u 5 !1 2 sin2 u 5!1 24
95
!5
3
sin u 51
csc u5 2
2
3
csc u 5 23
2 and tan u < 0 ⇒ u 76. is in Quadrant III.
cot u 51
tan u5
3
!75
3!7
7
tan u 5sin u
cos u5
2!7y4
23y45
!7
3
csc u 51
sin u5 2
4
!75
24!7
7
sin u 5 2!1 2 cos2 u 5 2!1 29
165 2
!7
4
cos u 51
sec u5 2
3
4
sec u 5 24
3 and cot u > 0 ⇒ u
77. sin 108 < 0.1736 79. tan 2458 < 2.1445
81. coss21108d < 20.342080. csc 3208 51
sin 32085 21.5557 82.
< 21.1918
cots22208d 51
tans22208d
84. csc 0.33 51
sin 0.33< 3.0860
86. tan 11p
9< 0.8391 88. cos1215p
14 2 < 20.9749
83.
< 5.7588
secs22808d 51
coss22808d 85. tan12p
9 2 < 0.8391
87.
< 22.9238
csc128p
9 2 51
sin128p
9 2
89. (a) reference angle is or
and is in Quadrant I or Quadrant II.
Values in degrees:
Values in radian:
(b) reference angle is or
and is in Quadrant III or Quadrant IV.
Values in degrees:
Values in radians:7p
6,
11p
6
2108, 3308
up
6
308sin u 5 21
2 ⇒
p
6,
5p
6
308, 1508
up
6
308sin u 51
2 ⇒ 90. (a) cos reference angle is 45º or
and is in Quadrant I or IV.
Values in degrees:
Values in radians:
(b) cos reference angle is or
and is in Quadrant II or III.
Values in degrees:
Values in radians:3p
4,
5p
4
1358, 2258
up
4
458u 5 2!2
2 ⇒
p
4,
7p
4
458, 3158
up
4
u 5!2
2 ⇒
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312 Chapter 4 Trigonometric Functions
92. (a) csc
Reference angle is
Values in degrees:
Values in radians:
(b) csc
Reference angle is
Values in degrees:
Values in radians:p
6,
5p
6
308, 1508
p
6 or 308.
sin u 51
2u 5 2 ⇒
5p
4,
7p
4
2258, 3158
458 or p
4.
sin u 521
!2u 5 2!2 ⇒ 91. (a) reference angle is or
and is in Quadrant I or Quadrant II.
Values in degrees:
Values in radians:
(b) reference angle is or
and is in Quadrant II or Quadrant IV.
Values in degrees:
Values in radians:3p
4,
7p
4
1358, 3158
up
4
458cot u 5 21 ⇒
p
3,
2p
3
608, 1208
up
3
608csc u 52!3
3 ⇒
93. (a) reference angle is or
and is in Quadrant II or Quadrant III.
Values in degrees:
Values in radians:
(b) reference angle is or
and is in Quadrant II or Quadrant III.
Values in degrees:
Values in radians:2p
3,
4p
3
1208, 2408
u
608,p
3cos u 5 2
1
2 ⇒
5p
6,
7p
6
1508, 2108
u308,
p
6sec u 5 2
2!3
3 ⇒ 94. (a)
Reference angle is
Values in degrees:
Values in radians:
(b) Values in degrees:
Values in radians:p
4 or
7p
4
458 or 3158
5p
6,
11p
6
1508, 3308
p
6 or 308.
cot u 5 2!3 ⇒ cos u
sin u5 2!3
95. (a)
(b)
(c) fcos 308g2 5 1!3
2 22
53
4
cos 308 2 sin 308 5!3 2 1
2
fsud 1 gsud 5 sin 308 1 cos 308 51
21
!3
25
1 1 !3
2
96. (a)
(b)
(c) fcos 608g2 5 11
222
51
4
cos 608 2 sin 608 51
22
!3
25
1 2 !3
2
fsud 1 gsud 5 sin 608 1 cos 608 5!3
21
1
25
!3 1 1
2
(d)
(e)
(f) coss2308d 5 cos 308 5!3
2
2 sin 308 5 211
22 5 1
sin 308 cos 308 5 11
221!3
2 2 5!3
4
(d)
(e)
(f) coss2608d 5 cos 608 51
2
2 sin 608 5 2!3
25 !3
sin 608 cos 608 5 1!3
2 211
22 5!3
4
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Section 4.4 Trigonometric Functions of Any Angle 313
97. (a)
(b)
(c) fcos 3158g2 5 1!2
2 22
51
2
cos 3158 2 sin 3158 5!2
22 12!2
2 2 5 !2
fsud 1 gsud 5 sin 3158 1 cos 3158 5 2!2
21
!2
25 0
98. (a)
(b)
(c)
(d) sin 2258 cos 2258 5 12!2
2 212!2
2 2 51
2
fcos 2258g2 5 12!2
2 22
51
2
cos 2258 2 sin 2258 52!2
22 12!2
2 2 5 0
fsud 1 gsud 5 sin 2258 1 cos 2258 5 2!2
22
!2
25 2!2
(d)
(e)
(f) coss23158d 5 coss3158d 5!2
2
2 sin 3158 5 212!2
2 2 5 2!2
sin 3158 cos 3158 5 12!2
2 21!2
2 2 5 21
2
(e)
(f) coss22258d 5 coss2258d 5 2!2
2
2 sin 2258 5 212!2
2 2 5 2!2
99. (a)
(b)
(c)
(d) sin 1508 cos 1508 51
2?
2!3
25
2!3
4
fcos 1508g2 5 12!3
2 22
53
4
cos 1508 2 sin 1508 52!3
22
1
25
21 2 !3
2
fsud 1 gsud 5 sin 1508 1 cos 1508 51
21
2!3
25
1 2 !3
2
(e)
(f ) coss21508d 5 coss1508d 52!3
2
2 sin 1508 5 211
22 5 1
100. (a)
(b)
(c)
(d) sin 3008 cos 3008 5 12!3
2 211
22 52!3
4
fcos 3008g2 5 11
222
51
4
cos 3008 2 sin 3008 51
22 12!3
2 2 51 1 !3
2
fsud 1 gsud 5 sin 3008 1 cos 3008 52!3
21
1
25
1 2 !3
2
(e)
(f) coss23008d 5 coss3008d 51
2
2 sin 3008 5 212!3
2 2 5 2!3
101. (a)
(b)
(c)
(d) sin 7p
6 cos
7p
65 12
1
2212!3
2 2 5!3
4
3cos 7p
6 42
5 12!3
2 22
53
4
cos 7p
62 sin
7p
65
2!3
22 12
1
22 51 2 !3
2
fsud 1 gsud 5 sin 7p
61 cos
7p
65 2
1
22
!3
25
21 2 !3
2
(e)
(f) cos127p
6 2 5 cos17p
6 2 52!3
2
2 sin 7p
65 212
1
22 5 21
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314 Chapter 4 Trigonometric Functions
102. (a)
(b)
(c)
(d) sin 5p
6 cos
5p
65 11
2212!3
2 2 52!3
4
3cos 5p
6 42
5 12!3
2 22
53
4
cos 5p
62 sin
5p
65
2!3
22
1
25
21 2 !3
2
fsud 1 gsud 5 sin 5p
61 cos
5p
65
1
21
2!3
25
1 2 !3
2
(e)
(f) cos125p
6 2 5 cos15p
6 2 52!3
2
2 sin 5p
65 211
22 5 1
103. (a)
(b)
(c)
(d) sin 4p
3 cos
4p
35 12!3
2 2121
22 5!3
4
3cos 4p
3 42
5 121
222
51
4
cos 4p
32 sin
4p
35 2
1
22 12!3
2 2 5!3 2 1
2
fsud 1 gsud 5 sin 4p
31 cos
4p
35
2!3
22
1
25
21 2 !3
2
(e)
(f) cos124p
3 2 5 cos14p
3 2 5 21
2
2 sin 4p
35 212!3
2 2 5 2!3
104. (a)
(b)
(c)
(d) sin 5p
3 cos
5p
35 12!3
2 211
22 52!3
4
3cos 5p
3 42
5 11
222
51
4
cos 5p
32 sin
5p
35
1
22 12!3
2 2 51 1 !3
2
fsud 1 gsud 5 sin 5p
31 cos
5p
35
2!3
21
1
25
1 2 !3
2
(e)
(f) cos125p
3 2 5 cos15p
3 2 51
2
2 sin 5p
35 212!3
2 2 5 2!3
105. (a)
(b)
(c) fcos 2708g25 02
5 0
cos 2708 2 sin 2708 5 0 2 s21d 5 1
fsud 1 gsud 5 sin 2708 1 cos 2708 5 21 1 0 5 21 (d)
(e)
(f) coss22708d 5 coss2708d 5 0
2 sin 2708 5 2s21d 5 22
sin 2708 cos 2708 5 s21ds0d 5 0
106. (a)
(b)
(c) fcos 1808g25 s21d2
5 1
cos 1808 2 sin 1808 5 21 2 0 5 21
fsud 1 gsud 5 sin 1808 1 cos 1808 5 0 2 1 5 21 (d)
(e)
(f) coss21808d 5 coss1808d 5 21
2 sin 1808 5 2s0d 5 0
sin 1808 cos 1808 5 0s21d 5 0
107. (a)
(b)
(c) 3cos 7p
2 42
5 025 0
cos 7p
22 sin
7p
25 0 2 s21d 5 1
fsud 1 gsud 5 sin 7p
21 cos
7p
25 21 1 0 5 21 (d)
(e)
(f) cos127p
2 2 5 cos17p
2 2 5 0
2 sin 7p
25 2s21d 5 22
sin 7p
2 cos
7p
25 s21ds0d 5 0
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Section 4.4 Trigonometric Functions of Any Angle 315
108. (a)
(b)
(c) 3cos 5p
2 42
5 025 0
cos 5p
22 sin
5p
25 0 2 1 5 21
fsud 1 gsud 5 sin 5p
21 cos
5p
25 1 1 0 5 1 (d)
(e)
(f) cos125p
2 2 5 cos15p
2 2 5 0
2 sin 5p
25 2s1d 5 2
sin 5p
2 cos
5p
2 5 s1ds0d 5 0
109.
(a) January:
(b) July:
(c) December: t 5 12 ⇒ T < 31.758
t 5 7 ⇒ T 5 708
t 5 1 ⇒ T 5 49.5 1 20.5 cos1p s1d6
27p
6 2 5 298
T 5 49.5 1 20.5 cos1p t
62
7p
6 2
110.
(a) thousand
(b) thousand
(c) thousand
(d) thousand
Answers will vary.
Ss6d < 25.8
Ss5d < 27.5
Ss14d < 33.0
Ss1d 5 23.1 1 0.442s1d 1 4.3 sin1p
62 < 25.7
S 5 23.1 1 0.442t 1 4.3 sin1pt
6 2, t 5 1 ↔ Jan. 2004
111. sin
(a)
miles
(b)
miles
(c)
milesd 56
sin 1208< 6.9
u 5 1208
d 56
sin 9085
6
15 6
u 5 908
d 56
sin 3085
6
s1y2d5 12
u 5 308
u 56
d ⇒ d 5
6
sin u
113. True. The reference angle for is
and sine is positive
in Quadrants I and II.
u9 5 1808 2 1518 5 298,
u 5 1518
112. As increases from to x decreases from
12 cm to 0 cm and y increases from 0 cm to
12 cm. Therefore, increases from 0
to 1, and decreases from 1 to 0.
Thus, begins at 0 and increases
without bound. When the tangent
is undefined.
u 5 908,
tan u 5 yyx
cos u 5 xy12
sin u 5 yy12
908,08u
114. False. and
cot12p
42 5 21
2cot13p
4 2 5 2s21d 5 1
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316 Chapter 4 Trigonometric Functions
115. (a)
(b) It appears that sin u 5 sins1808 2 ud.
0 0.3420 0.6428 0.8660 0.9848
0 0.3420 0.6428 0.8660 0.9848sins1808 2 ud
sin u
80860840820808u
116.
Patterns and conclusions may vary.
Function
Domain All reals except
Range
Evenness No Yes No
Oddness Yes No Yes
Period
Zeros npp
21 npnp
p2p2p
s2`, `df21, 1gf21, 1g
p
21 nps2`, `ds2`, `d
tan xcos xsin x
Function
Domain All reals except All reals except All reals except
Range
Evenness No Yes No
Oddness Yes No Yes
Period
Zeros None Nonep
21 np
p2p2p
s2`, `ds2`, 21g < f1, `ds2`, 21g < f1, `d
npp
21 npnp
cot xsec xcsc x
118.
x 5 2179 < 21.889
9x 5 217
44 2 9x 5 61
120.
x < 1.186, 21.686
x 521 ±!1 1 4s4ds2d
45
21 ±!33
4
2x2 1 x 2 4 5 0
117.
x 5 7
3x 5 21
3x 2 7 5 14
119.
x < 3.449, x < 21.449
x 52 ±!4 1 20
25 1 ±!6
x2 2 2x 2 5 5 0
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Section 4.5 Graphs of Sine and Cosine Functions
Section 4.5 Graphs of Sine and Cosine Functions 317
121.
x < 25.908, 4.908
x 521 ±!1 1 4s29d
25
21 ±!117
2
x2 1 x 2 29 5 0
27 5 sx 2 1dsx 1 2d
3
x 2 15
x 1 2
9
123.
x 5 3 2 log4 726 5 3 2ln 726
ln 4< 21.752
3 2 x 5 log4 726
432x 5 726
125.
x 5 e26 < 0.002479 < 0.002
ln x 5 26
122.
( extraneous)x 5 0x 5 6
xsx 2 6d 5 0
x2 2 6x 5 0
10x 5 x2 1 4x
5
x5
x 1 4
2x
124.
x 51
2 ln 86 < 2.227
2x 5 ln 86
86 5 e2x
90 5 4 1 e2x
4500
4 1 e2x5 50
126. ⇒ x 1 10 5 e2 ⇒ x 5 e2 2 10 < 22.611ln !x 1 10 512 lnsx 1 10d 5 1 ⇒ lnsx 1 10d 5 2
n You should be able to graph and
n Amplitude:
n Period:
n Shift: Solve and
n Key increments: (period)1
4
bx 2 c 5 2p.bx 2 c 5 0
2p
|b|
|a|y 5 a cossbx 2 cd.y 5 a sinsbx 2 cd
Vocabulary Check
1. amplitude 2. one cycle 3. 4. phase shift2p
b
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318 Chapter 4 Trigonometric Functions
1.
(a) -intercepts:
(b) -intercept:
(c) Increasing on:
Decreasing on:
(d) Relative maxima:
Relative minima: 12p
2, 212, 13p
2, 212
123p
2, 12, 1p
2, 12
123p
2, 2
p
22, 1p
2,
3p
2 2
122p, 23p
2 2, 12p
2,
p
22, 13p
2, 2p2
s0, 0dy
s22p, 0d, s2p, 0d, s0, 0d, sp, 0d, s2p, 0dx
fsxd 5 sin x
2.
(a) -intercepts:
(b) -intercept:
(c) Increasing on:
Decreasing on:
(d) Relative maxima:
Relative minima: s2p, 21d, sp, 21d
s22p, 1d, s0, 1d, s2p, 1d
s22p, 2pd, s0, pd
s2p, 0d, sp, 2pd
s0, 1dy
123p
2, 02, 12
p
2, 02, 1p
2, 02, 13p
2, 02x
fsxd 5 cos x
7.
Period:
Amplitude: |23| 52
3
2p
p5 2
y 52
3 sin px
3.
Period:
Amplitude: |3| 5 3
2p
25 p
y 5 3 sin 2x
5.
Period:
Amplitude: |52| 55
2
2p
1y25 4p
y 55
2 cos
x
2
Xmin = -2
Xmax = 2
Xscl =
Ymin = -4
Ymax = 4
Yscl = 1
py2
p
p
Xmin = -4
Xmax = 4
Xscl =
Ymin = -3
Ymax = 3
Yscl = 1
p
p
p
4.
Period:
Amplitude: |a| 5 2
2p
b5
2p
3
y 5 2 cos 3x
6.
Period:
Amplitude: |a| 5 |23| 5 3
2p
b5
2p
s1y3d5 6p
y 5 23 sin x
3
8.
Period:
Amplitude: |a| 53
2
2p
b5
2p
spy2d5 4
y 53
2 cos
px
2
Xmin = -
Xmax =
Xscl =
Ymin = -3
Ymax = 3
Yscl = 1
py4
p
p
Xmin = -
Xmax =
Xscl =
Ymin = -4
Ymax = 4
Yscl = 1
p
6p
6p
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Section 4.5 Graphs of Sine and Cosine Functions 319
9.
Period:
Amplitude: |22| 5 2
2p
15 2p
y 5 22 sin x 10.
Period:
Amplitude: |a| 5 |21| 5 1
2p
b5
2p
2y55 5p
y 5 2cos 2x
511.
Period:
Amplitude: |14| 51
4
2p
2y35 3p
y 51
4 cos
2x
3
13.
Period:
Amplitude: |13| 51
3
2p
4p5
1
2
y 51
3 sin 4px
15.
The graph of is a horizontal shift to the right
units of the graph of (a phase shift).fp
g
gsxd 5 sinsx 2 pd
f sxd 5 sin x
17.
The graph of is a reflection in the axis of the
graph of f.
x-g
gsxd 5 2cos 2x
f sxd 5 cos 2x
19.
The graph of has five times the amplitude of
and reflected in the axis.x-
f,g
gsxd 5 25 cos x
fsxd 5 cos x
21.
The graph of is a vertical shift upward of five
units of the graph of f.
g
gsxd 5 5 1 sin 2x
f sxd 5 sin 2x
12.
Period:
Amplitude: |a| 55
2
2p
b5
2p
1y45 8p
y 55
2 cos
x
414.
Amplitude: |a| 53
4
Period: 2p
b5
2p
py125 24
y 53
4 cos
p x
12
16.
g is a horizontal shift of f units to the left.p
fsxd 5 cos x, gsxd 5 cossx 1 pd
18.
g is a reflection of f about the y-axis.
(or, about the -axis)x
f sxd 5 sin 3x, gsxd 5 sins23xd
20.
The amplitude of g is one-half that of f. g is a
reflection of f in the x-axis.
fsxd 5 sin x, gsxd 5 212 sin x
22.
g is a vertical shift of f six units downward.
f sxd 5 cos 4x, gsxd 5 26 1 cos 4x
24. The period of g is one-half the period of f.23. The graph of has twice the amplitude as the graph
of The period is the same.f.
g
25. The graph of is a horizontal shift units to the
right of the graph of f.
pg 26. Shift the graph of f two units upward to obtain the
graph of g.
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320 Chapter 4 Trigonometric Functions
27.
Period:
Amplitude: 1
Period:
Amplitude:
−2
−3
−4
1
4
x
y
π
2π
23
−
f
g
|24| 5 4
2p
gsxd 5 24 sin x
2p
f sxd 5 sin x
29.
Period:
Amplitude: 1
is a vertical shift of the graph of
four units upward.
f
g
ππ−−1
−2
22
2
3
4
6
x
y
fsxdgsxd 5 4 1 cos x
2p
fsxd 5 cos x
28.
–2
2
x
π6
fg
y
gsxd 5 sin x
3
fsxd 5 sin x
30.
–2
2
xπ
f
g
y
gsxd 5 2cos 4x
fsxd 5 2 cos 2x
x 0
0 1 0 0
0 1!3
2
!3
2
1
2sin
x
3
21sin x
2p3p
2p
p
2
x 0
2 0 0 2
1 1 2121212cos 4x
222 cos 2x
p3p
4
p
2
p
4
31.
Period:
Amplitude:
is the graph of
shifted vertically three units upward.
fsxdgsxd 5 3 21
2 sin
x
2
1
2
4p
x
y
π3π2 π4ππ4 −−
f
g
−2
−1
−3
−4
1
2
3
4
fsxd 5 21
2 sin
x
2
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Section 4.5 Graphs of Sine and Cosine Functions 321
33.
Period:
Amplitude: 2
is the graph of shifted
units to the left.
–3
3
xπ π2
f
g
y
p
fsxdgsxd 5 2 cossx 1 pd
2p
fsxd 5 2 cos x
36. fsxd 5 sin x, gsxd 5 2cos1x 1p
22
x 0
0 1 0 0
0 1 0 0212cos1x 1p
22
21sin x
2p3p
2p
p
2
32.
x
y
f
g−1
−2
−2 2−1 1
2
3
4
1
gsxd 5 4 sin px 2 2
fsxd 5 4 sin px
34.
f g
ππ
ππ
−
−
1
−1
22
x
y
gsxd 5 2cos1x 2p
22fsxd 5 2cos x
x 0 1 2
0 4 0 0
2 22262222gsxd
24f sxd
3
2
1
2
x 0
0 1 0
0 0 1 0212cos1x 2p
2221212cos x
2p3p
2p
p
2
35.
Period:
Amplitude: 1
−2
−2
2
2
f = g
pp
2p
sin x 5 cos1x 2p
22
f sxd 5 sin x, gsxd 5 cos1x 2p
22
Conjecture: sin x 5 2cos1x 1p
22
−2
−2
2
2
pp
f = g
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322 Chapter 4 Trigonometric Functions
38.
Conjecture: cos x 5 2cossx 2 pd
−2
−2
2
2
pp
f = g
fsxd 5 cos x, gsxd 5 2cossx 2 pd37.
Thus,
−2
−2
2
2
f = g
pp
f sxd 5 gsxd.
gsxd 5 2sin1x 2p
22 5 sin1p
22 x2 5 cos x
fsxd 5 cos x
x 0
1 0 0 1
1 0 0 1212cossx 2 pd
21cos x
2p3p
2p
p
2
39.
Period:
Amplitude: 3
Key points:
x
y
π
2π
2π
23 π
23
− −
−4
1
2
3
4
s2p, 0d13p
2, 232,sp, 0d,1p
2, 32,s0, 0d,
2p
y 5 3 sin x
41.
Period:
Amplitude: 1
Key points:
x
y
−1
−2
−3
2
3
π2 π4
s2p, 21d, s3p, 0d, s4p, 1ds0, 1d, sp, 0d,
4p
y 5 cos x
2
40.
Period:
Amplitude:
x
y
−0.5
−1
0.5
1
π π
2π
2π2
−
1
4
2p
y 51
4 cos x
42.
Period:
Amplitude:
x
y
π
8π
8
−2
1
2
3π
83 π
85
−
1
2p
45
p
2
y 5 sin 4x
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Section 4.5 Graphs of Sine and Cosine Functions 323
43.
Period:
Amplitude: 1
Shift: Set and
Key points: 1p
4, 02, 13p
4, 12, 15p
4, 02, 17p
4, 212, 19p
4, 02
x 59p
4x 5
p
4
x 2p
45 2px 2
p
45 0
2p
–3
–2
1
2
3
xπ−π
yy 5 sin1x 2p
42; a 5 1, b 5 1, c 5p
4
44.
Horizontal shift units to the rightp
x
y
π
2π
2−
−1
−2
1
2
π
23
y 5 sinsx 2 pd
45.
Period:
Amplitude: 8
Key points: s2p, 28d, 12p
2, 02, s0, 8d, 1p
2, 02, sp, 28d
2p
x
y
−2
−4
−6
−8
−10
2
8
10
πππ2 π2− −
y 5 28 cossx 1 pd
46.
Amplitude: 3
Horizontal shift units to the left
Note: 3 cos1x 1p
22 5 23 sin x
1
3
4
−2
−3
−4
x
y
π
2
π
23π
2−
p
2
y 5 3 cos1x 1p
22 47.
Amplitude: 2
Period:
6−6
−4
4
2p
2py35 3
y 5 22 sin 2px
3
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324 Chapter 4 Trigonometric Functions
48.
Amplitude: 10
Period:
18−18
−12
12
2p
py65 12
y 5 210 cos px
649.
Amplitude: 5
Period:
30−30
−20
20
2p
py125 24
y 5 24 1 5 cos p t
1250.
Amplitude: 2
Period:
6−6
−2
6
2p
2py35 3
y 5 2 2 2 sin 2px
3
51.
Amplitude:
Period:
−2
2
4π 5π−
2p
1y25 4p
2
3
y 52
3 cos1x
22
p
42 52.
Amplitude: 3
Period:
−6
6
π
2
π
2−
2p
65
p
3
y 5 23 coss6x 1 pd 53.
Amplitude: 2
Period:
−
−4
4
pp
p
2
y 5 22 sins4x 1 pd
55.
Amplitude: 1
Period: 1
−3
−1
3
3
y 5 cos12px 2p
22 1 154.
Amplitude: 4
Period:
−8
8
12−12
3p
y 5 24 sin12
3x 2
p
32 56.
Amplitude: 3
Period: 4
−7
1
6−6
y 5 3 cos1px
21
p
22 2 3
58.
Amplitude: 5
Period:
−
0
12
pp
p
y 5 5 cossp 2 2xd 1 657.
Amplitude: 5
Period:
−
−4
20
pp
p
y 5 5 sinsp 2 2xd 1 10 59.
Amplitude:
Period:
−
−0.02
0.02
p
180
p
180
160
1100
y 51
100 sin 120p t©
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Section 4.5 Graphs of Sine and Cosine Functions 325
60.
Amplitude:
Period:
−0.04
0.04
0.06−0.06
1
25
1
100
y 521
100 coss50p td 61.
Amplitude:
Since is the graph of
reflected about the
axis and shifted vertically four
units upward, we have
and Thus,
5 4 2 4 cos x.
fsxd 5 24 cos x 1 4
d 5 4.
a 5 24
x-
gsxd 5 4 cos x
f sxd
1
2f8 2 0g 5 4
fsxd 5 a cos x 1 d
63.
Amplitude:
Graph of is the graph of
reflected about the
-axis and shifted vertically one
unit upward. Thus,
f sxd 5 26 cos x 1 1.
x
gsxd 5 6 cos x
f
1
2f7 2 s25dg 5 6
f sxd 5 a cos x 1 d 65.
Amplitude:
Since the graph is reflected
about the axis, we have
Period:
Phase shift:
Thus, f sxd 5 23 sin 2x.
c 5 0
2p
b5 p ⇒ b 5 2
a 5 23.
x-
|a| 5 3
f sxd 5 a sinsbx 2 cd
67.
Amplitude:
Period:
Phase shift: when
Thus, f sxd 5 sin1x 2p
42.
s1d1p
42 2 c 5 0 ⇒ c 5p
4
x 5p
4.bx 2 c 5 0
2p ⇒ b 5 1
a 5 1
f sxd 5 a sinsbx 2 cd
62.
Amplitude:
Reflected in the x-axis:
y 5 23 2 cos x
d 5 23
24 5 21 cos 0 1 d
a 5 21
22 2 s24d2
5 1
fsxd 5 a cos x 1 d
64.
Reflected in -axis,
y 5 24 21
2 cos x
d 5 24
a 5 21
2x
Period: 2p
Amplitude: 1
2
y 5 a cos x 1 d
66.
Amplitude:
Period:
Phase shift:
y 5 2 sin1x
22c 5 0
2p
b5 4p ⇒ b 5
1
2
4p
2 ⇒ a 5 2
y 5 a sinsbx 2 cd
68.
Amplitude:
Period:
Phase shift:
y 5 2 sin1px
21
p
22
c
b5 21 ⇒ c 5 2
p
2
2p
b5 4 ⇒ b 5
p
2
4
2 ⇒ a 5 2
y 5 a sinsbx 2 cd 69.
In the interval when
−2
−2
2
2
pp
y1
y2
x 5 25p
6, 2
p
6,
7p
6,
11p
6.
sin x 5 212f22p, 2pg,
y2 5 21
2
y1 5 sin x
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326 Chapter 4 Trigonometric Functions
70.
.
−2
−2
2
2
pp
y1
y2
y1 5 y2 when x 5 p, 2p
y2 5 21
y1 5 cos x
72.
(a)
(b) Maximum sales: December
Minimum sales: June st 5 6d
st 5 12d
1
0
12
120
S 5 74.50 1 43.75 cos p t
6
71.
(a)
(b)
(c) Cycles per min cycles per min
(d) The period would change.
560
65 10
Time for one cycle 5 one period 52p
py35 6 sec
0
−2
4
2
p
v 5 0.85 sin pt
3
73.
(a)
(b) Minimum:
Maximum: 30 1 25 5 55 feet
30 2 25 5 5 feet
0
0
135
60
h 5 25 sin p
15st 2 75d 1 30
74.
1 heartbeat
s3y4d ⇒ 4
3 heartbeatsysecond 5 80 heartbeatsyminute
Period: 2p
s8py3d 53
4
0
60
1.5
130
P 5 100 2 20 cos 8p
3t
75.
(a)
This is to be expected:
(b) The constant 30.3 gallons is the average daily
fuel consumption.
365 days 5 1 year
Period: 2p
b5
2p
s2py365d5 365 days
C 5 30.3 1 21.6 sin12pt
3651 10.92
(c)
Consumption exceeds 40 when
(Graph together with
(Beginning of May through part of September)
y 5 40.)C124 ≤ x ≤ 252.
gallonsyday
0
0
365
60
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Section 4.5 Graphs of Sine and Cosine Functions 327
76. (a) Yes, y is a function of t because for each
value of t there corresponds one and only
one value of y.
(b) The period is approximately
The amplitude is approximately12s2.35 2 1.65d 5 0.35 centimeters.
2s0.375 2 0.125d 5 0.5 seconds.
(c) One model is
(d)
0
0
0.9
4
y 5 0.35 sin 4p t 1 2.
77. (a)
(b) y 5 0.506 sins0.209x 2 1.336d 1 0.526
−10 10
1.0
2.0
0.5
1.5
20 30 40 50 60 70
−0.5
x
y (c)
The model is a good fit.
(d) The period is
(e) June 29, 2007 is day 545. Using the model,
or 27.09%.y < 0.2709
2p
0.209< 30.06.
−10 10
1.0
2.0
1.5
20 30 40 50 60 70
−0.5
x
y
78. (a)
(b)
The model somewhat fits the data.
(c)
The model is a good fit.
0
45
12
95
0
0
12
80
Astd 5 19.73 sins0.472t 2 1.74d 1 70.2 (d) Nantucket:
Athens:
The constant term (d) gives the average daily high
temperature.
(e) Period for
Period for
You would expect the period to be 12 (1 year).
(f ) Athens has greater variability. This is given by the
amplitude.
Astd 52p
0.472< 13
Nstd 52p
s2py11d 5 11
70.28
588
79. True. The period is 2p
3y105
20p
3.
81. True
80. False. The amplitude is that of y 5 cos x.1
2
82. Answers will vary.
83. The graph passes through and has period
Matches (e).
p.s0, 0d 84. The amplitude is 4 and the period Since
is on the graph, matches (a).s0, 24d2p.
85. The period is and the amplitude is 1. Since
and are on the graph, matches (c).sp, 0ds0, 1d4p 86. The period is Since is on the graph,
matches (d).
s0, 21dp.
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328 Chapter 4 Trigonometric Functions
87. (a) is even.
−
−2
2
pp
h(x) = cos2 x
hsxd 5 cos2 x (b) is even.
−
−2
2
pp
h(x) = sin2 x
hsxd 5 sin2 x (c) is odd.
−
−2
2
pp
h(x) = sin x cos x
hsxd 5 sin x cos x
88. (a) In Exercise 87, is even and we saw that is even.
Therefore, for even and we make the conjecture that is even.
(b) In Exercise 87, is odd and we saw that is even.
Therefore, for odd and we make the conjecture that is even.
(c) From part (c) of 87, we conjecture that the product of an even function and an odd
function is odd.
hsxdhsxd 5 fgsxdg2,gsxdhsxd 5 sin2xgsxd 5 sin x
hsxdhsxd 5 f fsxdg2,f sxdhsxd 5 cos2 xfsxd 5 cos x
89. (a)
0.8415 0.9983 1.0 1.0sin x
x
20.00120.0120.121x
0 0.001 0.01 0.1 1
Undef. 1.0 1.0 0.9983 0.8415sin x
x
x
(b)
As approaches 1.
(c) As approaches 0, approaches 1.sin x
xx
x → 0, f sxd 5sin x
x
−1 1
0.8
1.1
90. (a) (b)
As approaches 0.
(c) As approaches 0, approaches 0.1 2 cos x
xx
x → 0, 1 2 cos x
x
1−1
0.5
−0.5
20.000520.00520.0520.45971 2 cos x
x
20.00120.0120.121x
0 0.001 0.01 0.1 1
Undef. 0.0005 0.005 0.05 0.45971 2 cos x
x
x
91. (a)
(b)
−2π π
−2
2
2
−2π π
−2
2
2
(c) Next term for sine approximation:
Next term for cosine approximation:
−2π π
−2
2
2−2π π
−2
2
2
2x6
6!
2x7
7!
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Section 4.6 Graphs of Other Trigonometric Functions
Section 4.6 Graphs of Other Trigonometric Functions 329
92. (a) (d)
(b) (e)
(c) (f )
In all cases, the approximations are very accurate.
cos121
22 < 0.8776sin1p
82 < 0.3827
cos12p
42 < 0.7071sin11
22 < 0.4794
coss21d < 0.5403sins1d < 0.8415 93.
Slope 57 2 1
2 2 05 3
x
y
−1−2−3−4 1 2 3 4−1
7
6
5
4
3
2
1
(2, 7)
(0, 1)
95. 8.5 5 8.511808
p 2 < 487.014894.
m 522 2 4
3 1 15
26
45
23
2
−2
−1−1 1 2 3 4−2−3−4
−3
−4
1
2
3
4
x
y
(−1, 4)
(3, −2)
96. 20.48 5 20.4811808
p 2 < 227.5028 97. Answers will vary. (Make a Decision)
n You should be able to graph:
n When graphing or you should know to first graph
or since
(a) The intercepts of sine and cosine are vertical asymptotes of cosecant and secant.
(b) The maximums of sine and cosine are local minimums of cosecant and secant.
(c) The minimums of sine and cosine are local maximums of cosecant and secant.
n You should be able to graph using a damping factor.
y 5 a sinsbx 2 cdy 5 a cossbx 2 cdy 5 a cscsbx 2 cdy 5 a secsbx 2 cd
y 5 a cscsbx 2 cdy 5 a secsbx 2 cd
y 5 a cotsbx 2 cdy 5 a tansbx 2 cd
Vocabulary Check
1. vertical 2. reciprocal 3. damping
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