Download C H A P T E R 6 Trigonometry

C H A P T E R
Trigonometry
6
Section 6.1
Angles and Their Measure . . . . . . . . . . . . . . . . . 532
Section 6.2
Right Triangle Trigonometry . . . . . . . . . . . . . . . . 541
Section 6.3
Trigonometric Functions of Any Angle
Section 6.4
Graphs of Sine and Cosine Functions . . . . . . . . . . . 567
Section 6.5
Graphs of Other Trigonometric Functions . . . . . . . . . 579
Section 6.6
Inverse Trigonometric Functions . . . . . . . . . . . . . . 591
Section 6.7
Applications and Models . . . . . . . . . . . . . . . . . . 604
. . . . . . . . . . 552
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
Practice Test
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
C H A P T E R
Trigonometry
Section 6.1
6
Angles and Their Measure
You should know the following basic facts about angles, their measurement, and their applications.
■
Types of Angles:
(a) Acute: Measure between 0 and 90.
(b) Right: Measure 90.
(c) Obtuse: Measure between 90 and 180º.
(d) Straight: Measure 180.
■
Two positive angles, and are complementary if 90. They are supplementary if 180.
■
Two angles in standard position that have the same terminal side are called coterminal angles.
■
To convert degrees to radians, use 1 180 radians.
■
To convert radians to degrees, use 1 radian 180.
■
1 one minute 160 of 1.
■
1 one second 160 of 1 13600 of 1.
■
The length of a circular arc is s r where is measured in radians.
■
Speed distancetime
■
Angular speed t srt
Vocabulary Check
1. Trigonometry
2. angle
3. coterminal
4. degree
5. acute; obtuse
6. complementary; supplementary
7. radian
8. linear
1
10. A 2r 2
9. angular
1.
3.
The angle shown is
approximately 210.
The angle shown is
approximately 60.
5. (a) Since 90 < 130 < 180; 130 lies in Quadrant II.
(b) Since 270 < 285 < 360; 285 lies in Quadrant IV.
532
2.
4.
The angle shown is
approximately 120.
The angle shown is
approximately 330.
6. (a) Since 0 < 8.3 < 90; 8.3 lies in Quadrant I.
(b) Since 180 < 257 30 < 270; 257 30 lies in
Quadrant III.
Section 6.1
7. (a) Since 180 < 13250 < 90; 13250 lies in
Quadrant III.
8. (a) Since 270 < 260 < 180; 260 lies in
Quadrant II.
(b) Since 90 < 3.4 < 0; 3.4 lies in
Quadrant IV.
(b) Since 360 < 336 < 270; 336 lies in
Quadrant I.
9. (a)
y
Angles and Their Measure
(b)
y
150°
30°
x
10. (a) 270
x
(b) 120
y
y
x
x
−270°
11. (a)
− 120°
(b)
y
y
480°
405°
x
x
12. (a) 750
(b) 600
y
y
− 600°
x
x
−750°
13. (a) Coterminal angles for 45
45 360 405
45 360 315
(b) Coterminal angles for 36
14. (a) 120 360 480
120 360 240
(b) 420 720 300
420 360 60
15. (a) Coterminal angles for 300
300 360 660
300 360 60
(b) Coterminal angles for 740
36 360 324
740° 2360° 20°
36 360 396
20 360 340
533
534
Chapter 6
Trigonometry
16. (a) 520 720 200
45
17. (a) 5445 54 60 54.75
30
(b) 12830 128 60 128.5
520 360 160
(b) 230 360 590
230 360 130
10
18. (a) 24510 245 60 30 19. (a) 8518 30 85 18
60 3600 85.308
25 (b) 33025 330 3600
330.007
245 0.167
245.167
12
(b) 212 2 60 2 0.2 2.2
36
20. (a) 13536 135 3600 21. (a) 240.6 240 0.660 24036
(b) 145.8 145 0.860 14548
135 0.01 135.01
16
20
(b) 40816 20 408 60 3600 408 0.2667 0.0056
408.272
22. (a) 345.12 345 0.1260 23. (a) 2.5 2 0.5(60) 230
(b) 3.58 [3 0.58(60)]
345 7 0.260 [334.8]
345 7 12
[334 0.8(60)]
(b) 0.45 0 0.4560
0 27
33448
027
24. (a) 0.355 0 0.35560
25. (a) Complement: 90 24 66
Supplement: 180 24 156
0 21 0.360
0 21 18
(b) Complement: Not possible. 126 is greater than 90.
Supplement: 180 126 54
02118
(b) 0.7865 0 0.786560
0 47 0.1960
0 47 11.4
04711.4
26. (a) Complement: 90 87 3
27.
Supplement: 180 87 93
The angle shown is
approximately 2 radians.
(b) Complement: Not possible. 166 > 90
Supplement: 180 166 14
28.
The angle shown is
approximately 5.5 radians.
29.
The angle shown is
approximately 3 radians.
Section 6.1
30.
The angle shown is
approximately 4 radians.
31. (a) Since 0 <
(b) Since <
32. (a) Since (b) Since 34. (a) Since
35. (a)
< < 0; lies in Quadrant IV.
2
12
12
33. (a) Since 3
11
11
< < ; lies in Quadrant II.
2
9
9
3
< 6.02 < 2 ; 6.02 lies in Quadrant IV.
2
5
4
y
Angles and Their Measure
lies in Quadrant I.
< ;
5
2 5
7 3 7
lies in Quadrant III.
<
;
5
2
5
< 1 < 0; 1 lies in Quadrant IV.
2
(b) Since < 2 < ; 2 lies in Quadrant III.
2
(b) Since
(b) 2
3
< 2.25 < , 2.25 lies in Quadrant II.
2
y
5π
4
x
x
−
36. (a) 7
4
y
(b)
5
2
2π
3
y
5π
2
x
x
−
37. (a)
11
6
7π
4
y
(b) 3
y
11π
6
x
x
−3
38. (a) 4
(b) 7
y
y
7π
4
x
x
535
536
Chapter 6
Trigonometry
39. (a) Coterminal angles for
6
40. (a)
13
2 6
6
11
2 6
6
5
6
41. (a) Coterminal angles for 7
5
2 6
6
9
7
4 4
4
11
2 6
6
9
2 4
4
(b) (b) Coterminal angles for
42. (a)
7
19
2 6
6
23
11
2 6
6
5
17
2 6
6
2
28
2 15
15
5
7
2 6
6
2
32
2 15
15
26
8
2 9
9
Supplement: 98
8
2 45
45
Supplement: Supplement: (b) 144 144
(b)
>
2
180 5
6
7
4
2
3
180 180
9
4
(b) 240 240
180 3
47. (a) 20 20
3
180 2
180 49. (a)
3 3 180 270
2
2 (b)
7 7 180 210
6
6 4
5
180
20
9
9 11
12
12
180 6
(b) 150 150
7
7 180
105
12
12 53. 115 115
11
is greater than
12
2
3 4
4
48. (a) 270 270
50. (a) 3
4
180 (b) 120 120
45. (a) 30 30
2
3
3
(b) Complement: none
46. (a) 315 315
Supplement: 2
3
6
44. (a) Complement:
11
12
12
(b) Complement: Not possible;
82
8
2 45
45
2
15
5
2
12
12
43. (a) Complement:
10
8
2 9
9
(b)
(b) Coterminal angles for 9
4
180 2.007 radians
51. (a)
7 7 180 420
3
3 (b) 11 180 11
66
30
30 54. 87.4 87.4
52. (a)
11 11 180
330
6
6
(b)
34 34 180
408
15
15 180 1.525 radians
Section 6.1
55. 216.35 216.35
57. 0.83 0.83
59.
180 3.776 radians
62. 4.8 4.8
56. 48.27 48.27
180 0.014 radian
180 25.714
7
7 864.000
180
65. s r
58. 0.54 0.54
60.
6 5
180 s r
radians
29
10
15 15 180 337.500
8
8
61.
114.592
64. 0.57 0.57
67. s r
29 10
6
5
180 0.842 radians
radians
68.
32 7
32
7
537
180 0.009 radians
5 5 180
81.818
11
11 63. 2 2
66.
Angles and Their Measure
32.659
180
s r
60 75
radians
4
60
75 5 radian
Because the angle
represented is clockwise, this angle is 45
radian.
69. s r
70. r 14 feet, s 8 feet
6 27
6
2
radian
27
9
s
8
4
radian
r 14 7
73. s r, in radians
s 15180
180 15 inches
3s 3(1) 3 meters
1
77. A r 2
2
s 160 kilometers
25 14.5
3 25
50
radians
14.5 29
74. r 9 feet, 60 s r 9
47.12 inches
75. 3s r, in radians
72. r 80 kilometers,
71. 3s r
1
79. A r 2
2
3
76. r 20 centimeters, s r 20
4
4 5 centimeters 15.71 centimeters
4
1
1
A r 2 122
2
2
4
8.38 square inches
18 mm2 56.5 mm2
80. r 1.4 miles, 330
1
12.27 square feet
A 2.52225
2
180
1
330
A 1.42
2
180
s 160
2 radians
r
80
3 3 feet 9.42 feet
78. r 12 mm, 8
1
A 42
square inches
2
3
3
21.56
5.6 square miles
12
538
Chapter 6
Trigonometry
81. 411550 3247 39 8.46972 0.14782 radian
s r 40000.14782 591 miles
82. 473718 3747 36 94942 9.82833 0.17154 radian
r 4000 miles
s r 40000.17154 686 miles
450
s
0.071 radian 4.04°
r 6378
83. 84. r 6378 kilometers
400
s
0.062716 radian
r 6378
0.062716
3.59
180
The difference in latitude is about 3.59.
85. s 2.5 25
5
radian
r
6
60 12
87. (a) 65 miles per hour 65528060
5720 feet per minute
The circumference of the tire is C 2.5 feet.
The number of revolutions per minute is
r 57202.5 728.3 rev/minute.
88. (a) 2-inch diameter pulley
86. s 24
180
4.8 radians 4.8
275
r
5
(b) The angular speed is t.
5720
(2) 4576 radians
2.5
Angular speed 89. (a) Angular speed 1700 rpm 1700 2 radiansminute
(b) Linear speed 1 ft
in.
52002 feet
7.25
2
12 in. On the 4-inch diameter pulley:
1 minute
2
3141 feet per minute
3
r2
s 10681.4 2 52002 radians
1 minute
10,400 radians per minute
10681.4 radiansminute
Since r 1, the belt moves 10681.4 inchesminute.
4576 radians
4576 radiansminute
1 minute
164.5 feet per second
10681.4
5340.7
2
This pulley is turning at 5340.7 radiansminute.
(b)
5340.7
850 rpm
2
90. (a) 4 rpm 42 radiansminute
8
25.13274 radiansminute
(b) r 25 ft
r
200 ftminute
t
Linear speed 2525.13274 ftminute
628.32 ftminute
Section 6.1
Angles and Their Measure
91. (a) 2002 ≤ Angular speed ≤ 5002 radians per minute
Interval: 400, 1000 radians per minute
(b) 62002 ≤ Linear speed ≤ 65002 centimeters per minute
Interval: 2400, 6000 centimeters per minute
1
92. A R 2 r 2
2
1
93. A r 2
2
R 25 14 39
25
125°
r 14
A
r
1 125
2 180
14
1
352140
2
180
140°
35
476.39 square meters
39 2 14 2 460.069
1496.62 square meters
1445 in.
2
94. (a) Arc length of larger sprocket in feet:
s r
2
1
feet
s 2 3
3
Therefore, the chain moves
2
feet as does the smaller rear sprocket.
3
Thus, the angle of the smaller sprocket is
2
ft
s
3
4
r
2
ft
12
r 2 inches 12 feet
2
and the arc length of the tire in feet is:
s r
s 4
12 14
14
feet
3
14
s
14
3
Speed feet per second
t
1 sec
3
3600 seconds
1 mile
14 feet
10 miles per hour
3 seconds
1 hour
5280 feet
(b) Since the arc length of the tire is
Distance 14
feet and the cyclist is pedaling at a rate of one revolution per second, we have:
3
1 mile
feet
n revolutions
143 revolutions
5280
feet 7
n miles
7920
(c) Distance Rate Time
1 mile
t seconds
143 feetsecond5280
feet 7
t miles
7920
(d) The functions are both linear.
539
540
Chapter 6
Trigonometry
95. False. A measurement of 4 radians corresponds to two
complete revolutions from the initial to the terminal side
of an angle.
96. True. If and are coterminal angles, then
n360 or n2, where n is an
integer. The difference between and is
n360, or n2 if expressed in
radians.
97. False. The terminal side of 1260 lies on the negative x-axis.
98. (a) An angle is in standard position if its vertex is at the origin and its initial side is on the positive x-axis.
(b) A negative angle is generated by a clockwise rotation.
(c) Two angles with the same initial and terminal sides are coterminal.
(d) An obtuse angle measures between 90° and 180°.
100. 1 radian 99. The speed increases, since the linear speed is
proportional to the radius.
180 57.3, so one radian is much
larger than one degree.
101. Since the arc length s is given by s r, if the central angle is fixed while the radius r increases,
then s increases in proportion to r .
102. Area of circle r 2
θ
Area of sector Measure of central angle of sector
Area of circle
Measure of central angle of circle
r
Area of sector
r2
2
Area of sector r 2
103.
4
4
42 42
105.
23
2
6
2
2
2 21 r 2
42 2
8
2
36 212 2
2
104.
2
2
107. 22 62 4 36 40 4
2
3
2
3
55
5
210 2
3
3
23
3
105 25 12 2 5 2 2
106.
10 210
108. 182 122 324 144 468
36
109. 182 62 324 36 288
144
111. Horizontal shift two units
to the right
y
2
2
52
4
13 613
110. 172 92 289 81 208
2 122
16
13 413
112. f (x) x5 4
y = x5
3
y
4
Vertical shift four units
downward
2
2
1
−2
x
2
3
−1
−3
x
−2
4
y = x5
1
3
−2
−2
−3
2
y = x5 − 4
y = (x − 2)5
−6
Section 6.2
113. Reflection in the x-axis
and a vertical shift two
units upward
114. f x x 35
Reflection in the x-axis
and a horizontal shift
three units to the left
y
6
5
4
Right Triangle Trigonometry
y = x5
3
1
−3
−2
y = −(x + 3)5
2
1
2
x
1
−1
3
−2
y = 2 − x5
−3
−3
Section 6.2
■
Right Triangle Trigonometry
(d) csc hyp
opp
(e) sec hyp
adj
(f) cot adj
opp
You should know the following identities.
(a) sin 1
csc (b) csc 1
sin (c) cos 1
sec (d) sec 1
cos (e) tan 1
cot (f) cot 1
tan (g) tan sin cos (h) cot cos sin (i) sin2 cos2 1
( j) 1 tan2 sec2 θ
Adjacent side
(k) 1 cot2 csc2 ■
You should know that two acute angles and are complementary if 90, and that the cofunctions of
complementary angles are equal.
■
You should know the trigonometric function values of 30, 45, and 60, or be able to construct triangles from which
you can determine them.
Vocabulary Check
hypotenuse
sec adjacent
(e)
2. opposite; adjacent; hypotenuse
3. elevation; depression
(ii)
adjacent
cot opposite
(f)
(iii)
hypotenuse
csc opposite
(d)
(iv)
adjacent
cos hypotenuse
(b)
(v)
opposite
sin hypotenuse
(a)
(vi)
opposite
tan adjacent
(c)
Opposite side
opp
adj
us
(c) tan en
adj
hyp
ot
(b) cos yp
opp
hyp
H
(a) sin e
You should know the right triangle definition of trigonometric functions.
■
1. (i)
y = x5
1
−5 −4 −3
−2
y
3
x
−1
541
2
542
Chapter 6
Trigonometry
1. hyp 62 82 36 64 100 10
6
θ
8
sin opp
6
3
hyp 10 5
csc hyp 10 5
opp
6
3
cos 8
4
adj
hyp 10 5
sec hyp 10 5
adj
8
4
tan opp 6 3
adj
8 4
cot 8 4
adj
opp 6 3
2. adj 132 52 169 25 12
13
sin 5
opp
hyp 13
csc hyp 13
opp
5
cos adj
12
hyp 13
sec hyp 13
adj
12
tan opp
5
adj
12
cot adj
12
opp
5
5
θ
b
3. adj 412 92 1681 81 1600 40
41
9
θ
sin opp
9
hyp 41
csc hyp 41
opp
9
cos adj
40
hyp 41
sec hyp 41
adj
40
tan opp
9
adj
40
cot adj
40
opp
9
4. hyp 42 42 32 42
sin 2
opp
4
1
hyp 42 2
2
csc hyp 42
2
opp
4
cos 2
4
adj
1
hyp 42 2
2
sec hyp 42
2
adj
4
tan opp 4
1
adj
4
cot adj
4
1
opp 4
4
θ
4
5. adj 32 12 8 22
θ
3
1
sin opp 1
hyp 3
csc hyp
3
opp
cos adj
22
hyp
3
sec hyp
3
32
adj
22
4
tan 2
opp
1
adj
22
4
cot adj
22
opp
adj 62 22 32 42
6
2
sin opp 2 1
hyp 6 3
csc hyp 6
3
opp 2
cos adj
42 22
hyp
6
3
sec 6
hyp
3
32
adj
42 22
4
tan opp
2
2 1
5 adj42 22
4
cot adj
42
22
opp
2
θ
The function values are the same since the triangles are similar and the corresponding sides are proportional.
Section 6.2
Right Triangle Trigonometry
543
6.
θ
7.5
8
4
θ
15
hyp 7.52 42 hyp 152 82 289 17
sin opp
8
hyp 17
csc hyp 17
opp
8
15
adj
cos hyp 17
hyp 17
sec adj
15
8
opp
tan adj
15
15
adj
cot opp
8
17
2
sin opp
4
8
hyp 172 17
csc hyp 172 17
opp
4
8
cos 7.5
adj
15
hyp 172 17
sec hyp 172 17
adj
7.5
15
tan 4
8
opp
adj
7.5 15
cot 7.5 15
adj
opp
4
8
The function values are the same because the triangles are similar, and corresponding sides are proportional.
7. opp 52 42 3
5
θ
sin opp
3
hyp
5
csc hyp
5
opp
3
cos adj
4
hyp
5
sec hyp
5
adj
4
tan opp
3
adj
4
cot adj
4
opp
3
csc hyp 1.25 5
opp 0.75 3
sec hyp 1.25 5
adj
1
4
cot adj
1
4
opp 0.75 3
4
opp 1.252 12 0.75
opp 0.75 3
sin hyp 1.25 5
1.25
θ
1
adj
1
4
cos hyp 1.25 5
opp 0.75 3
tan adj
1
4
The function values are the same since the triangles are similar and the corresponding sides are proportional.
8.
θ
1
3
2
hyp 12
22
θ
5
6
5
opp
1
sin hyp 5
5
hyp 5
5
csc opp
1
adj
2
25
cos hyp 5
5
hyp 5
sec adj
2
tan opp 1
adj
2
cot adj
2
2
opp 1
hyp 32 62 35
sin 5
opp
1
3
hyp 35 5
5
csc hyp 35
5
opp
3
cos adj
6
2
25
hyp 35 5
5
sec hyp 35 5
adj
6
2
tan opp 3 1
adj
6 2
cot adj
6
2
opp 3
The function values are the same because the triangles are similar, and corresponding sides are proportional.
544
Chapter 6
9. Given: sin Trigonometry
3 opp
4 hyp
10. opp 72 52 24 26
32 adj2 42
4
adj 7
7
adj
cos hyp
4
opp 26
hyp
7
tan opp 26
adj
5
csc 7
hyp
76
opp 26
12
3
θ
7
opp 37
tan adj
7
cot hyp 47
sec adj
7
2 6
θ
5
adj
5
56
opp 26
12
hyp 4
opp 3
11. Given: sec 2 2 hyp
1
adj
12. hyp 52 12 26
opp2 12 22
sin opp 3
2
opp 3
sin hyp
2
3
1
26
opp
1
hyp 26
26
cos adj
5
526
hyp
26
26
tan opp 1
adj
5
θ
adj
1
cos hyp 2
tan opp
3
adj
csc hyp 26
26
opp
1
cot 3
adj
opp
3
sec hyp 26
adj
5
csc hyp 23
opp
3
13. Given: tan 3 3 opp
1
adj
hyp 10
opp 310
hyp
10
10
adj
cos hyp
10
θ
26
5
14. opp 62 12 35
32 12 hyp2
sin 7
hyp 7
sec adj
5
7
adj
cot opp
3
csc sin sin 10
opp 35
hyp
6
3
6
adj
1
cos hyp 6
θ
1
tan opp 35
35
adj
1
cot adj
1
opp 3
csc hyp
6
635
opp 35
35
sec hyp
10
adj
cot 35
adj
1
opp 35
35
csc hyp 10
opp
3
θ
1
35
1
Section 6.2
15. Given: cot adj
3
2 opp
Right Triangle Trigonometry
16. adj 172 42 273
22 32 hyp2
13
hyp 13
2
θ
2
opp
213
sin hyp 13
13
3
sin opp
4
hyp 17
17
cos 273
adj
hyp
17
273
adj
3
313
hyp 13
13
tan 4
opp
4273
273
adj
273
tan opp 2
adj
3
sec hyp
17
17273
adj
273
273
csc hyp 13
opp
2
cot 273
adj
opp
4
sec hyp 13
adj
3
30 30
180
6 radian
sin 30 opp 1
hyp 2
60°
2
1
30°
cos 45 180
60
3
3 19.
π
6
20. degree
opp 3
3
tan 3
adj
1
3
radian
value
4
2
cos 45
3
2
18. degree
2
1
2
2
45°
2
1
radian
value
4
2
sec 45
sec
π
3
1
2
4
θ
cos 17.
545
2
1
π
4
2
2
4
1
1
1
cot 21.
30°
3
3
1
3
adj
opp
60 radian
3
3
22. degree
radian
value
4
2
csc 45
csc 45 60°
2
2
1
45°
1
1
1
23.
π
3
2
180
30
6
6 1
cos
π
6
3
adj
6
hyp
2
3
cot 1 45°
1
45°
1
1
adj
1 opp
45 45
180
4
radian
value
4
2
sin 45
sin
25.
2
24. degree
1
radian
value
6
1
3
tan 30
1
tan 30 3
1
π
4
2
1
4
2
2
26. degree
2
2
60°
2
1
30°
3
546
Chapter 6
27. sin 60 3
2
(a) tan 60 Trigonometry
1
2
, cos 60 sin 60
3
cos 60
(b) sin 30 cos 60 (c) cos 30 sin 60 13
2
, sec (a) csc 30 1
2
sin 30
(b) cot 60 tan90 60 tan 30 3
1
3
3
sin 30
2
(c) cos 30 tan 30
23
2
3
3
3
33
1
3
(d) cot 30 tan 30 3
3
2
13
3
1
2
213
csc 13
13
(a) cos 1
1
sec 5
(b) cos 1
3
313
sec 13
13
(b) cot 6
1
1
tan 26
12
sin 21313
2
cos 31313
3
13
(d) sec90 csc 2
(c) tan 1
3
(a) sec (b) sin2 cos2 1
sin2 (c) cot90º tan 26
(d) sin tan cos 26 5 1
26
5
32. tan 5
1
3
cos 3
1
2
8
9
sin 22
3
1
2
cos 3
1
(c) cot sin 4
22
22
3
1
(d) sin90 cos 3
33. tan cot tan 35. tan cos (a) cot (b) cos 1
sin2 tan 1
sin 1
cos cos sin 3
30. sec 5, tan 26
(a) sin 31. cos 3
1
2
3
cos 60
1
(d) cot 60 3
sin 60
3
29. csc 3
1
28. sin 30 , tan 30 2
3
1
1
tan 5
1
1 tan2 (c) tan90º cot 1
1 52
1
26
26
26
1
5
(d) csc 1 cot 2 1 15
1 251 2625 34. cos sec cos 36. cot sin 2
1
1
cos cos sin cos sin 26
5
Section 6.2
37. 1 cos 1 cos 1 cos2 Right Triangle Trigonometry
38. 1 sin 1 sin 1 sin2 cos2 sin2 cos 2 cos 2 sin2 39. sec tan sec tan sec2 tan2 41.
1 tan2 tan2 sin2 1 sin2 1
2 sin2 1
sin cos sin2 cos2 cos sin sin cos 1
sin cos 1
sin 40. sin2 cos2 sin2 1 sin2 42.
tan cot tan cot tan tan tan 1
1
cos cot 1
cot 1 cot2 csc2 csc sec 43. (a) tan 23.5 0.4348
(b) cot 66.5 44. (a) sin 16.35 0.2815
1
0.4348
tan 66.5
18 0.9598
60
56 (b) sin 7356 sin 73 0.9609
60
45. (a) cos 1618 cos 16 47. Make sure that your calculator is in radian mode.
(a) cot
(b)
16
tan
1
tan
16
5.0273
1
3.5523
sin 16.35
46. (a) sec 4212 sec 42.2 (b) csc 487 48. (a) sec 0.75 1
1.3499
cos 42.2
1
7 sin 48 60
1.3432
1
1.3667
cos 0.75
(Note: 0.75 is in radians)
(b) cos 0.75 0.7317
0.1989
16
49. Make sure that your calculator is in radian mode.
(a) csc 1 (b)
(b) csc 16.35 1
1.1884
sin 1
(b) cot
1
tan 0.5463
2
51. (a) sin 50. (a) sec
1
⇒ 30 2
6
(b) csc 2 ⇒ 30 6
2 1 2 2 52. (a) cos 1
2
2
1
1.1884
cos
1
2
1
0.5463
1
tan
2
2
⇒ 45 (b) tan 1 ⇒ 45º 4
4
547
548
Chapter 6
Trigonometry
53. (a) sec 2 ⇒ 60 3
(b) cot 1 ⇒ 45 4
54. (a) tan 3 ⇒ 60 (b) cos 23
⇒ 60 3
3
55. (a) csc 56. (a) cot ⇒ 45 (b) sin 2
4
2
tan 3
1
⇒ 60 2
3
3
3
3
3 ⇒ 60 3
3
(b) sec 2
cos tan 30 57.
1
30
3
30 °
32
58. sin 60 30
x
tan 60 32
x
3 32
x
x
x
tan 82 61.
x
82°
45 m
2
⇒ 45 60. sin 45 r
3
2
20
r
20
20
202
2
sin 45
2
323
32
3
3
x
45
(b) tan 62. (a)
h 270 feet
Distance between friends:
45
45
cos 82 ⇒ y
y
cos 82
6
132
3
Not drawn to scale
323.34 meters
sin 3000 ft
θ
1500 1
3000 2
1500 ft
30 6
6
h
3 135
(c) 2135 h
h
Height of the building:
123 45 tan 82 443.2 meters
63.
4
y
18
x 45 tan 82
y
2
93
3 x 32
60°
y 18 sin 60 18
x 303
x
59.
30
x
1
2
64.
tan opp
adj
tan 54 w
100
w 100 tan 54 137.6 feet
Section 6.2
65. (a) sin 23 x
(b) tan 23 y
150
x
x
150
383.9 feet
sin 23
5 ft
1
23°
y
150
y
150
353.4 feet
tan 23
(c) Moving down the line:
Dropping vertically:
150sin 23
63.98 feet per second
6
150
25 feet per second
6
66. Let h the height of the mountain.
67.
(x1, y1)
Let x the horizontal distance from where the 9 angle of
elevation is sighted to the point at that level directly below
the mountain peak.
Then tan 3.5 tan 9 h
h
and tan 9 .
x 13
x
h
h
⇒ x
x
tan 9
Substitute x Right Triangle Trigonometry
tan 3.5 30°
sin 30 y1
56
1256 28
y1 sin 3056 h
into the expression for tan 3.5.
tan 9
tan 3.5 56
h
h
13
tan 9
h tan 9
h 13 tan 9
cos 30 x1
56
x1 cos 3056 3
2
56 283
x1,y1 283, 28
(x2, y2)
h tan 3.5 13 tan 9 tan 3.5 h tan 9
13 tan 9 tan 3.5 htan 9 tan 3.5
13 tan 9 tan 3.5
h
tan 9 tan 3.5
1.2953 h
56
60°
The mountain is about 1.3 miles high.
sin 60 y2
56
y2 sin 60°56 cos 60° 2356 283
x2
56
x2 cos 60°56 x2, y2 28, 283
1256 28
549
550
Chapter 6
68. tan 3 Trigonometry
x
15
x 15 tan 3
d 5 2x 5 215 tan 3 6.57 centimeters
69. (a)
(e)
20
Angle, h
85°
80
19.7
70
18.8
60
17.3
50
15.3
h
(b) sin 85 20
40
12.9
(c) h 20 sin 85 19.9 meters
30
10.0
20
6.8
10
3.5
(d) The side of the triangle labeled h
will become shorter.
(f) The height of the balloon
decreases.
Height (in meters)
20
h
θ
70. x 9.397, y 3.420
sin 20 y
0.34
10
cos 20 x
0.94
10
tan 20 y
0.36
x
cot 20 x
2.75
y
sec 20 10
1.06
x
csc 20 10
2.92
y
71. sin 60 csc 60 1
True, csc x 1
1
⇒ sin 60 csc 60 sin 60
1
sin x
sin 60
72. True, because sec90 csc .
73. sin 45 sin 45 1
False,
74. True, because
75.
1 cot 2 csc 2 cot 2 csc 2 1
cot 2 csc 2 1.
2
2
2
2
2 1
sin 60
sin 2
sin 30
False,
sin 60 cos 30
cot 30 1.7321
sin 30
sin 30
sin 2 0.0349
Section 6.2
76. tan0.8 2 tan20.8
Right Triangle Trigonometry
77. This is true because the corresponding sides of similar
triangles are proportional.
False.
tan0.82 tan 0.64 0.745
tan 20.8 tan 0.8 2 1.060
78. Yes. Given tan , sec can be found from the identity 1 tan2 sec2 .
79. (a)
80. (a)
0.1
0.2
0.3
0.4
0.5
(b) In the interval 0, 0.5, > sin .
sin 0.0998
0.1987
0.2955
0.3894
0.4794
(c) As → 0, sin → 0, and
0
0.3
0.6
0.9
1.2
1.5
sin 0
0.2955
0.5646
0.7833
0.9320
0.9975
cos 1
0.9553
0.8253
0.6216
0.3624
0.0707
(b) On 0, 1.5, sin is an increasing function.
(c) On 0, 1.5, cos is a decreasing function.
(d) As the angle increases the length of the side opposite the angle increases relative to the
length of the hypotenuse and the length of the side adjacent to the angle decreases relative
to the length of the hypotenuse. Thus the sine increases and the cosine decreases.
81.
x2 6x
x2 4x 12
x2 12x 36
xx 6
x2 36
x 6x 2
82.
83.
x 6x 6
x 6x 6
x
, x ±6
x2
2t 2 5t 12
2t 2 5t 12
t 2 16
2
2
9 4t 2
9 4t
4t 12t 9
4t 2 12t 9
t 2 16
2t 3t 4 2t 32t 3
2t 3
3 2t3 2t t 4t 4
t 4
2t 3
3
, t ± , 4
4t
2
3
2
x
3x 2x 2 2x 22 xx 2
x 2 x 2 x2 4x 4
x 2x 22
3x2 4 2x2 4x 4 x2 2x
x 2x 22
2x2 10x 20 2x2 5x 10
x 2x 22
x 2x 22
12 x
x 4 4x 12 x x 1
84.
12 x 4,
12
12 x
4x
1
x x
3
1
x 0, 12
→ 1.
sin 551
552
Chapter 6
Trigonometry
Section 6.3
■
Trigonometric Functions of Any Angle
Know the Definitions of Trigonometric Functions of Any Angle.
If is in standard position, x, y a point on the terminal side and r x2 y2 0, then
sin y
r
r
csc , y 0
y
cos x
r
r
sec , x 0
x
y
tan , x 0
x
x
cot , y 0
y
■
You should know the signs of the trigonometric functions in each quadrant.
■
3
You should know the trigonometric function values of the quadrant angles 0, , , and .
2
2
■
You should be able to find reference angles.
■
You should be able to evaluate trigonometric functions of any angle. (Use reference angles.)
■
You should know that the period of sine and cosine is 2.
■
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
Vocabulary Check
1. reference
2. periodic
3. period
4. even; odd
1. (a) x, y 4, 3
r 16 9 5
sin y 3
r
5
csc r
5
y 3
cos x 4
r
5
sec r
5
x 4
tan y 3
x 4
cot x 4
y 3
(b) x, y 8, 15
r 64 225 17
sin y
15
r
17
csc r
17
y
15
cos x
8
r
17
sec r
17
x
8
tan y
15
x
8
cot x
8
y
15
Section 6.3
r 12 12 2
r 122 52 13
y
5
r
13
12
x
cos r
13
r
13
y
5
13
r
sec x
12
sin csc y
5
5
x 12 12
cot x 12 12
y
5
5
3. (a) x, y 3, 1
sin 2
y
1
2
r
2
csc 2
r
2
y
1
cos 2
x
1
2
r
2
sec 2
r
2
x
1
tan 1
y
1
x 1
cot x 1
1
y
1
(b) x, y 4, 1
r 16 1 17
r 3 1 2
sin y
1
r
2
csc r
2
y
sin y 17
r
17
csc r
17
y
cos 3
x
r
2
sec r
23
x
3
cos x
417
r
17
sec 17
r
x
4
tan y 3
x
3
cot x
3
y
tan y
1
x
4
cot x
4
y
4. (a) x 3, y 1
r
32
12
(b) x 4, y 4
10
r 42 42 42
sin 10
1
y
10
r
10
sin 2
y
4
r
4 2
2
cos x
3
310
10
r
10
cos 2
x
4
r
42
2
tan y 1
x 3
tan y 4
1
x
4
csc 10
r
10
y
1
csc r
42
2
y
4
sec 10
r
x
3
sec r
42
2
x
4
cot x 3
3
y 1
cot x
4
1
y 4
5. x, y 7, 24
6. x 8, y 15
r 82 152 17
r 49 576 25
y 24
r
25
x
7
cos r
25
y 24
tan x
7
sin 553
(b) x 1, y 1
2. (a) x 12, y 5
tan Trigonometric Functions of Any Angle
r
25
y 24
r
25
sec x
7
x
7
cot y 24
csc sin y 15
r
17
csc r
17
y 15
cos x
8
r
17
sec r
17
x
8
tan y 15
x
8
cot x
8
y 15
554
Chapter 6
Trigonometry
7. x, y 4, 10
8. x 5, y 2
r 52 22 29
r 16 100 229
sin y 529
r
29
sin 2
y
229
29
r
29
cos 229
x
r
29
cos 5
x
529
29
r
29
tan 5
y
x
2
tan y 2 2
x 5 5
csc 29
r
y
5
csc 29
29
r
y
2
2
sec 29
r
x
2
sec 29
29
r
x
5
5
cot x
2
y
5
cot x 5 5
y 2 2
9. x, y 3.5, 6.8
r 3.52 6.82 58.49
y
6.8
6.858.49
r
58.49
58.49
x
3.5
3.558.49
cos r
58.49
58.49
y
6.8
tan x
3.5
sin csc 58.49
r
y
6.8
sec 58.49
r
x
3.5
cot x
3.5
y
6.8
1 7
3
31
10. x 3 , y 7 2 2
4
4
r
72 314 2
2
1157
4
31
311157
y
4
sin r
1157
1157
4
7
141157
x
2
cos r
1157
1157
4
31
y
4
31
3
tan 2
x
7
14
14
2
1157
r
csc y
1157
4
31
31
4
1157
r
sec x
x
cot y
11. sin < 0 ⇒ lies in Quadrant III or in Quadrant IV.
4
7
2
1157
14
7
2
14
31
31
4
12. sin > 0 ⇒ lies in Quadrant I or in Quadrant II.
cos < 0 ⇒ lies in Quadrant II or in Quadrant III.
cos > 0 ⇒ lies in Quadrant I or in Quadrant IV.
sin < 0 and cos < 0 ⇒ lies in Quadrant III.
sin > 0 and cos > 0 ⇒ lies in Quadrant I.
13. sin > 0 ⇒ lies in Quadrant I or in Quadrant II.
14. sec > 0 ⇒ lies in Quadrant I or in Quadrant IV.
tan < 0 ⇒ lies in Quadrant II or in Quadrant IV.
cot < 0 ⇒ lies in Quadrant II or in Quadrant IV.
sin > 0 and tan < 0 ⇒ lies in Quadrant II.
sec > 0 and cot < 0 ⇒ lies in Quadrant IV.
Section 6.3
15. sin y 3
⇒ x2 25 9 16
r
5
in Quadrant II ⇒ x 4
y 3
5
r
sin csc r
5
y 3
4
x
5
r
cos sec r
5
x
4
3
y
4
x
tan cot x
4
y
3
17. sin < 0 and tan < 0 ⇒ is in Quadrant IV ⇒
y < 0 and x > 0.
15
y
tan ⇒ r 17
x
8
y
15
r
17
x
8
cos r
17
y
15
tan x
8
csc sin 19. cot r
17
y
15
17
r
x
8
x
8
cot y
15
sec 3
3
x
y
1 1
x 3, y 1, r 10
10
y
sin r
10
r
csc 10
y
x 310
r
10
1
y
tan x
3
10
r
x
3
x
cot 3
y
21. sec sec r
2
⇒ y2 4 1 3
x 1
sin > 0 ⇒ is in Quadrant II ⇒ y 3
sin y 3
r
2
csc x
1
cos r
2
tan r
sec 2
x
y
3
x
23. cot is undefined,
sin 0
r
23
y
3
cot 3
x
y
3
3
≤ ≤
⇒ y0 ⇒ 2
2
csc is undefined.
cos 1
sec 1
tan 0
cot is undefined.
x 4
⇒ y 3
r
5
16. cos in Quadrant III ⇒ y 3
sin 3
y
r
5
csc 5
3
cos 4
x
r
5
sec 5
4
tan y 3
x 4
cot 18. cos 4
3
8
x
⇒ y 15
r
17
tan < 0 ⇒ y 15
sin y 15
15
r
17
17
csc cos 8
x
r
17
sec tan y 15
15
x
8
8
cot 20. csc cos > 0 ⇒ is in Quadrant IV ⇒ x is positive;
cos Trigonometric Functions of Any Angle
17
15
17
8
8
15
4
r
⇒ x 15
y 1
cot < 0 ⇒ x 15
sin y 1
r
4
csc 4
cos 15
x
r
4
sec tan 15
y
x
15
cot 15
415
15
22. sin 0 ⇒ 0 n
sec 1 ⇒ 2n
y 0, x r
sin 0
cos tan x
r
1
r
r
y 0
r
r
csc is undefined.
sec r
r
1
x r
cot is undefined.
555
556
Chapter 6
Trigonometry
24. tan is undefined ⇒ n 2
25. To find a point on the terminal side of , use any point on
the line y x that lies in Quadrant II. 1, 1 is one
such point.
3
, x 0, y r
2
r
y r
csc 1
1
sin y
r
r
sec is undefined.
x 0
cos 0
x 0
r
r
cot 0
y
y
tan is undefined.
≤ ≤ 2 ⇒ x 1, y 1, r 2
sin 1
2
cos 1
2
2
csc 2
2
2
sec 2
2
tan 1
cot 1
26. Let x > 0.
x, 3 x, Quadrant III
1
r
x
2
1 2 10 x
x 9
3
10x
3 x
1
10
y
r
10
10x
3
310
x
x
cos r
10
10x
3
1
x
y
3
1
tan x
x
3
sin csc r
y
3
10
1
x
3
10x
sec 10
r
3
x
x
3
cot x
y
x
3
1
x
3
27. To find a point on the terminal side of , use any point on
the line y 2x that lies in Quadrant III. 1, 2 is one
such point.
28. Let x > 0.
x 1, y 2, r 5
2
25
sin 5
5
5
1
cos 5
5
2
tan 2
1
4
4x 3y 0 ⇒ y x
3
x, 3 x, Quadrant IV
4
csc 5
5
2
2
5
sec 5
1
1 1
cot 2 2
r
x
2
16 2 5
x x
9
3
3 x
4
sin y
r
cos x
x
3
r
5
5
x
3
5
x
3
3 x
4
5
csc sec 5
4
5
3
4
y
tan x
29. x, y 1, 0, r 1
sin y 0
0
r
1
30. x, y 0, 1, r 1
csc
1
3
1
2
1
x
4
3
cot 3
4
31. x, y 0, 1, r 1
sec
r
1
3
⇒ undefined
2
x 0
Section 6.3
r
1
1
x 1
sin
35. x, y 1, 0, r 1
csc y 1
1
2
r
1
cot 1
undefined.
0
37. 203
36. x, y 0, 1
x 0
cot 0
2
y 1
1
r
⇒ undefined
y 0
557
34. x, y 1, 0, r 1
33. x, y 0, 1, r 1
32. x, y 1, 0, r 1
sec Trigonometric Functions of Any Angle
203 180 23
y
203°
x
θ′
38. 309
245
39.
360 245 115 coterminal angle
360 309 51
180 115 65
y
y
309°
θ′
x
θ′
x
− 245°
145
40.
41. y
360 145 215
2
3
(coterminal angle)
y
2π
3
2 3
3
θ′
x
215 180 35
42. 7
4
2 x
θ′
−145°
y
y
43. 3.5
7 4
4
3.5 7π
4
3.5
x
θ′
θ′
x
558
Chapter 6
44. Trigonometry
11
5
is coterminal with .
3
3
2 5 3
3
45. 225, 45, Quadrant III
y
sin 225 sin 45 11π
3
θ′
sin 750 sin 30 2
1
2
tan 750 tan 30 48. 405, 405 360 45 in Quadrant IV.
3
2
1
2
cos(150) cos 30 2
tan(150) tan 30 50. 840 is coterminal with 240.
51. 240 180 60 in Quadrant III.
3
2
1
cos(840) cos 60 2
tan(840) tan 60 3
, in Quadrant I.
4
4
3
sin(150) sin 30 2
2
sin(840) sin 60 2
3
49. 150, 30, Quadrant III
tan405 tan 45 1
52. 1
2
cos 750 cos 30 tan 300 tan 60 3
cos405 cos 45 2
47. 750, 30, Quadrant I
3
sin405 sin 45 2
tan 225 tan 45 1
46. 300, 360 300 60 in Quadrant IV.
cos 300 cos 60 2
cos 225 cos 45 x
sin 300 sin 60 2
sin cos
2
4
2
cos tan
1
4
tan 3
3
4
sin 3
3
2
cos
4
1
cos 3
3
2
tan
4
tan 3
3
3
6 cos 6 3
sin
6 sin 6 2
2
4
2
2
4
, , Quadrant III
3
3
53. , , Quadrant IV
6
6
sin
3
1
3
54. 3
is coterminal with
.
2
2
3
1
2
3
0
2
3
is undefined.
2
2 sin
sin 2 cos
cos 2
3
6 tan 6 3
2 tan
tan Section 6.3
55. 11
, , Quadrant II
4
4
Trigonometric Functions of Any Angle
56. 10
4
is coterminal with
.
3
3
4
in Quadrant III.
3
3
sin
2
11
sin 4
4
2
cos
2
11
cos 4
4
2
sin
3
10
sin 3
3
2
tan
11
tan 1
4
4
cos
10
1
cos 3
3
2
tan
10
tan 3
3
3
57. 3
, 2
2
58. 3
sin 1
2
2
2 3
cos 0
2
2
sin 3
tan which is undefined.
2
2
cos sin cos tan sin 59.
3
5
60.
2
25
sin
4
4
2
2
25
cos
4
4
2
25
tan
1
4
4
5
cos2 1 9
25
16
cos2 25
cos > 0 in Quadrant IV.
4
5
csc 2
1 cot 2 csc 2 cot 2 csc 2 1
cot 2 22 1
cot 2 3
cot < 0 in Quadrant IV.
cot 3
cot 3
61. tan 3
3
2
sec2 1 tan2 1 32 csc 2 cos2 1 sin2 cos 1 cot 2 csc 2 sin2 cos2 1
cos2 1 7 in Quadrant IV.
4
4
tan 62.
25
7
is coterminal with .
4
4
10 2
csc 2
csc > 0 in Quadrant II.
10 csc csc 5
8
1
1
⇒ sec sec cos sec 1
5
8
8
5
sec2 1 9
4
3
2
13
4
sec < 0 in Quadrant III.
sec 64.
cos 2
sec2 1
sin 10
1
1
sin csc 10
10
63. cos sec2 1 13
2
sec 9
4
1 tan2 sec 2 tan2 sec 2 1
4
tan2 tan 2 9
2
1
65
16
tan > 0 in Quadrant III.
tan 65
4
559
560
Chapter 6
Trigonometry
65. sin 10 0.1736
68. csc330 71. sec 72 66. sec 225 1
2.0000
sin330
1
3.2361
cos 72
70. cot 178 28.6363
72. tan188 0.1405
73. tan 4.5 4.6373
75. cos
80. csc 1
⇒ reference angle is 30 or and is in
2
6
Quadrant I or Quadrant II.
81. (a) sin 5
,
6 6
1
⇒ reference angle is 30 or and is
2
6
in Quadrant III or Quadrant IV.
7 11
,
Values in radians:
6
6
23
⇒ reference angle is 60 or and 3
3
is in Quadrant I or Quadrant II.
2
7
,
4 4
⇒ reference angle is 45 or and 2
4
is in Quadrant II or III.
(b) cos 2
3 5
,
4 4
84. (a) sec 2 ⇒ reference angle is 60 or
and is in
3
Quadrant I or IV.
Values in degrees: 60, 300
Values in degrees: 60, 120
2
,
3 3
3 7
Values in radians:
,
4 4
⇒ reference angle is 45 or and is
2
4
in Quadrant I or IV.
82. (a) cos Values in radians:
83. (a) csc Values in degrees: 135, 315
Values in degrees: 135, 225
Values in degrees: 210, 330
in Quadrant II or Quadrant IV.
15
1
4.4940
14
sin1514
Values in radians:
(b) sin (b) cot 1 ⇒ reference angle is 45 or
1
1.0436
cos 0.29
Values in degrees: 45, 315
Values in degrees: 30, 150
Values in radians:
76. tan 78. sec 0.29 11
1
0.4142
8
tan118
Values in radian:
9 0.3640
9 0.9397
77. sin0.65 0.6052
79. cot 67. cos110 0.3420
69. tan 304 1.4826
1
0.2245
tan 1.35
74. cot 1.35 1
1.4142
cos 225
Values in radians:
and is
4
5
,
3 3
(b) sec 2 ⇒ reference angle is 60 or
in Quadrant II or III.
Values in degrees: 120, 240
Values in radians:
2 4
,
3 3
and is
3
Section 6.3
Trigonometric Functions of Any Angle
Quadrant I or Quadrant III.
⇒ reference angle is 60 or and is
2
3
in Quadrant I or II.
Values in degrees: 45, 225
Values in degrees: 60, 120
85. (a) tan 1 ⇒ reference angle is 45 or
Values in radians:
and is in
4
86. (a) sin 5
,
4 4
and 6
89.
91.
3
Values in degrees: 240, 300
Values in degrees: 150, 330
5 11
,
6
6
Values in radians:
22, 22 corresponds to t 4 on the unit circle.
2
,
3 3
⇒ reference angle is 60 or and 2
3
is in Quadrant III or IV.
(b) sin is in Quadrant II or Quadrant IV.
Values in radians:
3
Values in radians:
(b) cot 3 ⇒ reference angle is 30 or
87.
561
4 5
,
3 3
1 3
, x, y ,
3
2 2
88. t sin
2
4
2
since
sin t y
sin
cos
2
4
2
3
since sin t y.
3
2
since
cos t x
cos
tan
1
4
1
since cos t x.
3
2
since
tan t y
x
3
y
2
tan 3 since tan t .
3
1
x
2
23, 12 corresponds to t 56 on the unit circle.
90. t 2
2
5
, x, y ,
4
2
2
sin
5 1
6
2
since
sin t y
sin
2
5
since sin t y.
4
2
cos
3
5
6
2
since
cos t x
cos
2
5
since cos t x.
4
2
tan
3
5
6
3
since
tan t y
x
21, 23 corresponds to t 43 on the unit circle.
5
tan
4
92. t 2
2
2
y
1 since tan t .
x
2
3
1
11
, x, y ,
6
2
2
sin
3
4
3
2
since
sin t y.
sin
1
11
since sin t y.
6
2
cos
1
4
3
2
since
cos t x.
cos
11 3
since cos t x.
6
2
tan
4
3
3
since
y
tan t .
x
1
2
11
tan
6
3
2
1
3
3
3
y
since tan t .
x
562
Chapter 6
Trigonometry
93. 0, 1 corresponds to t 3
on the unit circle.
2
3
1
2
since
sin t y
3
0
cos
2
since
cos t x
since
tan t sin
tan
3
is undefined
2
95. (a) sin 5 1
94. t , x, y 1, 0
sin 0 since sin t y.
cos 1 since cos t x.
tan 0
y
0 since tan t .
1
x
y
x
96. (a) sin 0.75 y 0.7
(b) cos 2 0.4
97. (a) sin t 0.25
98. (a) sin t 0.75
t 0.25 or 2.89
t 4.0 or t 5.4
(b) cos t 0.25
(b) cos t 0.75
t 1.82 or 4.46
t 0.72 or t 5.56
99. (a) New York City: N 22.1 sin0.52t 2.22 55.01
Fairbanks: F 36.6 sin0.50t 1.83 25.61
(b)
Month
New York City
Fairbanks
February
35
1
March
41
14
May
63
48
June
72
59
August
76
56
September
69
42
November
47
7
(c) The periods are about the same for both models, approximately 12 months.
100. S 23.1 0.442t 4.3 cos
t
6
(a) t 2; S 23.1 0.4422 4.3 cos
2
26,134 snowboards
6
(b) t 14; S 23.1 0.44214 4.3 cos
(c) t 6; S 23.1 0.4426 4.3 cos
6
21,452 snowboards
6
(d) t 18; S 23.1 0.44218 4.3 cos
101. yt 2 cos 6t
(a) y0 2 cos 0 2 centimeters
1
3
(b) y4 2 cos2 0.14 centimeter
1
(c) y2 2 cos 3 1.98 centimeters
14
31,438 snowboards
6
18
26,756 snowboards
6
(b) cos 2.5 x 0.8
Section 6.3
Trigonometric Functions of Any Angle
563
103. I 5e2t sin t
102. yt 2et cos 6t
I0.7 5e1.4 sin 0.7 0.79
(a) t 0
y0 2e0 cos 0 2 centimeters
1
(b) t 4
y14 2e14 cos6
14 0.11 centimeter
1
(c) t 2
y
1
2
2e12 cos6 12 1.2 centimeters
104. sin 6
6
⇒ d
d
sin d
(c) 120
(b) 90
(a) 30
6
6
6 miles
d
sin 90º 1
6
sin 30
d
6
12 miles
12
105. False. In each of the four quadrants, the sign of the
secant function and the cosine function will be the same
since they are reciprocals of each other.
6
sin 120
6
32
6.9 miles
106. False. The reference angle is always acute by definition.
For in Quadrant II, 180 . For in
Quadrant III, 180. For in Quadrant IV,
360 .
107.
ht) f tgt
108. As increases from 0 to 90, x decreases from 12 cm to
y
0 cm, y increases from 0 cm to 12 cm, sin 12
x
increases from 0 to 1, cos decreases from 1 to 0,
12
y
and tan increases without bound (and is undefined
x
at 90).
ht f tgt
f tgt
ht
Therefore, ht is odd.
109. (a)
0
20
40
60
80
sin 0
0.342
0.643
0.866
0.985
sin180 0
0.342
0.643
0.866
0.985
(b) Conjecture: sin sin180 110. (a)
cos
32 sin (b) cos
0
0.3
0.6
0.9
1.2
1.5
0
0.2955
0.5646
0.7833
0.9320
0.9975
0
0.2955
0.5646
0.7833
0.9320
0.9975
32 sin 564
Chapter 6
Trigonometry
111. If is obtuse, then 90 < < 180. The reference angle is 180 and we have the following:
sin sin csc csc cos cos sec sec tan tan cot cot 112. Domain: All real numbers x
Domain: All real numbers x
3
Range: 1, 1
Range: 1, 1
−␲
Period: 2
2␲
Zeros: n
Period: 2
2
The function is even.
The function is odd.
Domain: All real numbers x except x n Range: , −␲
Zeros: n −3
2
−3
Range: , 1
1, 3
3
Period: 2
−␲
−␲
2␲
−3
Domain: All real numbers x except x n Range: , 1
1, 2
3
Period: 2
−
2
Zeros: none
The function is even.
−3
Domain: All real numbers x except x n
Range: , 3
Period: 2
−␲
2␲
−3
The function is odd.
The secant function is similar to the tangent function because they both have vertical asymptotes at
x n 2␲
Zeros: none
The function is odd.
The function is odd.
Zeros: n 2␲
Domain: All real numbers x except x n
Period: Zeros: n
3
.
2
The cotangent function and the cosecant function both have vertical asymptotes at x n.
A maximum point on the sine curve corresponds to a relative minimum on the cosecant curve.
The maximum points of sine and cosine are interchanged with the minimum points of cosecant
and secant. The x-intercepts of the sine and cosine functions become vertical asymptotes of the
cosecant and secant functions.
The graphs of sine and cosine may be translated left or right (respectively) to 2 to coincide
with each other.
−3
Section 6.3
2
4x
113. f x Trigonometric Functions of Any Angle
114. gx 1
x2 4
Vertical asymptote: x 4
Vertical asymptotes: x 2, x 2
Horizontal asymptote: y 0
Horizontal asymptotes: y 0
x
0
1
2
3
5
6
x
y
1
2
2
3
1
2
2
1
gx
565
±3
± 2.25
± 1.75
±1
0
0.2
0.94
1.07
0.33
0.25
y
y
8
4
6
3
4
2
2
1
x
−2
2
4
8 10 12 14
−4 −3
x
−1
−4
1
2
3
4
−2
−6
−3
−8
−4
115. hx 2x2 12x 2xx 6
2x, x 6
x6
x6
y
12
No asymptotes
9
6
x
2
0
2
4
8
y
4
0
4
8
16
3
6
9
12
− 12
x5
x5
x2 2x 15
x 5x 3
3
−9
The graph has a hole at 2, 12.
116. f x x
−9 − 6 − 3
117. y 2x1
y
5
x5
x
1
0
1
2
3
undefined, x 5
y
1
4
1
2
1
2
4
1
,
x3
Domain: all real numbers x
Horizontal asymptote: y 0
Range: y > 0
x
4
2
1
0
1
2
1
y-intercept: 0, 2 f x
1
1
1
2
1
3
1
4
1
5
Horizontal asymptote: y 0
y
6
4
2
−8 −6
x
−2
−4
−6
−8
2
4
6
3
2
Vertical asymptote: x 3
8
4
)0, 12 ) 1
−2
−1
x
−1
1
2
3
4
566
Chapter 6
Trigonometry
118. y 3x1 2
119. y 3x2
x
3
2
y
19
9
7
3
1
0
3
5
4
x
1
y
11
y
2
9
3
0
2
4
1
1
3
1
9
7
6
5
4
3
Domain: all real numbers x
Domain: all real numbers x
y
Range: y > 2
7
y-intercept: 0, 5
5
Horizontal asymptote:
y2
3
2
Range: y > 0
6
−5 −4 −3 −2 −1
y-intercept: 0, 1
(0, 5)
Horizontal asymptote: y 0
2
1
− 5 − 4 −3 − 2 − 1
x
1
2
3
120. y 3x12
y
7
x
3
2
y
1
3
3
3
0.578
1
0
1
3 1.73
1
6
2
5
4
33 5.20
3
3
2
1
Domain: all real numbers x
(0, 3 )
x
−5 −4 −3 −2 −1
1
2
3
Range: y > 0
y-intercept: 0, 3 0, 1.73
Horizontal asymptote: y 0
121. y lnx 1
y
3
x
1.1
1.5
2
3
4
y
2.30
0.69
0
0.69
1.10
2
1
(2, 0)
x
Domain: x 1 > 0 ⇒ x > 1
−1
1
2
3
4
5
6
−2
Range: all real numbers
−3
x-intercept: 2, 0
Vertical asymptote: x 1
122. y ln x 4
y
12
x
6
3
1
12
1
2
1
3
6
y
7.17
4.39
0
2.77
2.77
0
4.39
7.17
9
6
(−1, 0)
− 12 − 9 − 6 −3
Domain: all real numbers x
Range: all real numbers y
x-intercepts: 1, 0, 1, 0
Vertical asymptote: x 0
(1, 0)
x
3
6
9 12
(0, 1)
x
1
2
3
Section 6.4
Graphs of Sine and Cosine Functions
567
124. y log103x
123. y log10 x 2
x
1.5
1
0
1
2
x
3
2
1
13
16
y
0.301
0
0.301
0.477
0.602
y
0.95
0.78
0.48
0
0.30
Domain: x 2 > 0 ⇒ x > 2
Domain: 3x > 0 ⇒ x < 0
y
Range: all real numbers
Range: all real numbers y
3
x-intercept: 1, 0
2
y-intercept: 0, 0.301
(0, 0.301)
(−1, 0)
−3
−1
y
x-intercept: 13, 0
2
Vertical asymptote: x 0
1
x
1
2
3
−3
−1
−2
x
−1
1
−1
−2
−3
Section 6.4
(− 13, 0(
−2
Graphs of Sine and Cosine Functions
■
You should be able to graph y a sinbx c and y a cosbx c. Assume b > 0.
■
Amplitude: a
■
Period:
■
Shift: Solve bx c 0 and bx c 2.
■
Key increments:
2
b
1
(period)
4
Vocabulary Check
1. cycle
2. amplitude
2
b
5. vertical shift
4. phase shift
3.
2. y 2 cos 3x
1. y 3 sin 2x
Period:
2
2
Amplitude: 3 3
Period:
3. y 2 2
b
3
5
x
cos
2
2
Period:
Amplitude: a 2
2
4
1
2
Amplitude:
4. y 3 sin
Period:
x
3
2 2
1 6
b
3
Amplitude: a 3 3
5. y 1
x
sin
2
3
Period:
6. y 2
6
3
Amplitude:
1
1
2
2
5
5
2
2
3
x
cos
2
2
Period:
2 2
4
b
2
Amplitude: a 3
2
568
Chapter 6
Trigonometry
8. y cos
7. y 2 sin x
Period:
2
2
1
Period:
Amplitude: 2 2
10. y 2 2 b
8
4
13. y 1
3
14. y 12. y 1
1
2
2
Amplitude: a The graph of g is a horizontal shift
to the right units of the graph of
f a phase shift.
2
3
18. f x sin 3x, gx sin3x
17. f x cos 2x
gx cos 2x
g is a reflection of f about the
y-axis.
The graph of g is a reflection in
the x-axis of the graph of f.
19. f x cos x
20. f x sin x, gx sin 3x
The period of g is one-third the
period of f.
gx cos 2x
21. f x sin 2x
f x 3 sin 2x
The period of f is twice that of g.
The graph of g is a vertical shift
three units upward of the graph of f.
22. f x cos 4x, gx 2 cos 4x
g is a vertical shift of f two units
downward.
23. The graph of g has twice the
amplitude as the graph of f. The
period is the same.
25. The graph of g is a horizontal shift units to the right of
the graph of f.
24. The period of g is one-third the
period of f.
26. Shift the graph of f two units upward to obtain the graph
of g.
27. f x 2 sin x
Period:
y
2 2
2
b
1
5
4
3
g
f
Amplitude: 2
−π
2
Symmetry: origin
Key points: Intercept
0, 0
5
2
gx sinx 2
2
20
b
10
Amplitude: a 16. f x cos x, gx cosx g is a horizontal shift of f units
to the left.
2
2
8
b
14
15. f x sin x
1
1
4
4
x
5
cos
2
4
Period:
2
x
cos
3
10
Period:
2 10
5
Amplitude: 3 3
2
3
23
Amplitude:
2
1
2
Amplitude:
Period:
2x
1
cos
2
3
Period:
1
sin 2x
4
Period:
2
2
3
b
23
11. y Amplitude: a 9. y 3 sin 10x
Amplitude: a 1 1
1
sin 8x
3
Period:
2x
3
Minimum
, 2
2
Intercept
Maximum
, 0
3
,0
2
Intercept
−5
2, 0
Since gx 4 sin x 2 f x, generate key points for the graph of gx by multiplying
the y-coordinate of each key point of f x by 2.
3π
2
x
Section 6.4
Graphs of Sine and Cosine Functions
28. f x sin x
Period:
y
2 2
2
b
1
2
g f
Amplitude: 1
6π
x
Symmetry: origin
Key points: Intercept
0, 0
Since gx sin
Maximum
Intercept
Minimum
Intercept
2 , 1
, 0
32, 1
2, 0
−2
3x f 3x , the graph of gx is the graph of f x, but stretched horizontally by a factor of 3.
Generate key points for the graph of gx by multiplying the x-coordinate of each key point of f x by 3.
29. f x cos x
Period:
y
2 2
2
b
1
g
Amplitude: 1
Symmetry: y-axis
Key points: Maximum
0, 1
π
Intercept
Minimum
Intercept
Maximum
2 , 0
, 1
32, 0
2, 1
2π
x
f
−1
Since gx 1 cosx f x 1, the graph of gx is the graph of f x, but translated upward by one unit.
Generate key points for the graph of gx by adding 1 to the y-coordinate of each key point of f x.
30. f x 2 cos 2x
Period:
y
2 2
b
2
2
f
g
Amplitude: 2
π
Symmetry: y-axis
Key points: Maximum
0, 2
Intercept
Minimum
Intercept
Maximum
,0
4
, 2
2
3
,0
4
, 2
Since gx cos 4x 2 f 2x, the graph of gx is the graph of f x, but
1
i) shrunk horizontally by a factor of 2,
1
ii) shrunk vertically by a factor of 2, and
iii) reflected about the x-axis.
Generate key points for the graph of gx by
i) dividing the x-coordinate of each key point of f x by 2, and
ii) dividing the y-coordinate of each key point of f x by 2.
−2
x
569
570
Chapter 6
Trigonometry
1
x
31. f x sin
2
2
Period:
y
5
2
2
4
b
12
g
4
3
Amplitude:
1
2
2
1
Symmetry: origin
Key points: Intercept
0, 0
Minimum
Intercept
Maximum
Intercept
, 21
2, 0
3, 12
4, 0
f
−π
−1
x
3π
1
x
sin 3 f x, the graph of gx is the graph of f x, but translated upward by three units.
2
2
Generate key points for the graph of gx by adding 3 to the y-coordinate of each key point of f x.
Since gx 3 32. f x 4 sin x
Period:
y
2 2
2
b
4
f
2
1
x
Amplitude: 4
g
Symmetry: origin
Key points: Intercept
0, 0
Maximum
1
,2
2
Intercept
1, 0
Minimum
3
, 2
2
Intercept
−8
2, 0
Since gx 4 sin x 3 f x 3, the graph of gx is the graph of f x, but translated downward by three units.
Generate key points for the graph of gx by subtracting 3 from the y-coordinate of each key point of f x.
33. f x 2 cos x
Period:
y
2 2
2
b
1
3
f
Amplitude: 2
π
2π
x
Symmetry: y-axis
g
Key points: Maximum
0, 2
Intercept
Minimum
Intercept
Maximum
2 , 0
, 2
32, 0
2, 2
−3
Since gx 2 cosx f x , the graph of gx is the graph of f x, but with a phase shift (horizontal translation)
of . Generate key points for the graph of gx by shifting each key point of f x units to the left.
34. f x cos x
Period:
y
2 2
2
b
1
2
g
Amplitude: 1
π
Symmetry: y-axis
Key points: Minimum
0, 1
2π
x
f
Intercept
Maximum
Intercept
Minimum
2 , 0
, 1
32, 0
2, 1
−2
Since gx cosx f x , the graph of gx is the graph of f x, but with a phase shift (horizontal translation)
of . Generate key points for the graph of gx by shifting each key point of f x units to the right.
Section 6.4
35. y 3 sin x
Period: 2
4
Amplitude: 3
2
0, 0, , 3 , , 0,
2
−π
2
0, 0,
2
3
1
3
π
2
−3
Key points:
π
2π
−π
π
2π
x
−1
1
2 , 4, , 0,
−2
2
−3
0, 4,
− 43
x
2
1
y
4
Key points:
x
−1
1
1
0, , , 0 , , ,
3
2
3
39. y cos
− 2π
Amplitude: 4
1
1
1
4
Period: 2
1
3
1
, 0 , 2,
2
3
2
38. y 4 cos x
y
4
3
y
Key points:
3
Period: 2
x
3π
2
571
2 , 4, 2, 0
1
cos x
3
Amplitude:
π
2
−4
3
, 3 , 2, 0
2
37. y Amplitude:
1
3
− π
2
1
sin x
4
Period: 2
3
Key points:
36. y y
Graphs of Sine and Cosine Functions
− 2π
y
2
2
Period:
Amplitude: 1
Amplitude: 1
2π
−1
3, 0, 4, 1
−2
−4
3
, 0 , 2, 4
2
Period: 4
− 2π
4π
x
2π
−2
40. y sin 4x
y
0, 1, , 0, 2, 1,
π
2 , 0, , 4,
2
Key points:
−π
1
x
x
π
4
Key points:
0, 0,
8 , 1, 4 , 0,
3
−2
8 , 1, 2 , 0
41. y cos 2x
Period:
2
1
2
42. y sin
y
2
Period:
1
Amplitude: 1
x
1
Key points:
0, 1,
14, 0, 12, 1, 34, 0
2
x
4
y
2
2
8
4
Amplitude: 1
1
−6
−2
Key points:
−2
0, 0, 2, 1, 4, 0,
6, 1, 8, 0
−2
x
2
6
572
Chapter 6
43. y sin
Period:
Trigonometry
2
2x
; a 1, b , c0
3
3
44. y 10 cos
2
3
23
Period:
x
6
2
12
6
Amplitude: 10
Amplitude: 1
Key points: 0, 0,
Key points:
4, 1, 2, 0, 4, 1, 3, 0
3
3
9
0, 10, 3, 0, 6, 10, 9, 0, 12, 10
y
y
3
12
2
8
4
x
−1
2
3
− 12
x
−4
4
8
12
−2
−3
− 12
45. y sin x ; a 1, b 1, c 4
4
y
Period: 2
3
Amplitude: 1
1
2
Shift: Set x 0 and x 2
4
4
x
4
−π
−2
9
x
4
3
5
x
π
−3
7
9
4 , 0, 4 , 1, 4 , 0, 4 , 1, 4 , 0
Key points:
47. y 3 cosx 46. y sinx Period: 2
Period: 2
Amplitude: 1
Amplitude: 3
Shift: Set x 0
and
x
Key points: , 0,
x 2
Shift: Set x 0
x x 3
32, 1, 2, 0, 52, 1, 3, 0
x
Key points: , 3, , 0 , 0, 3, , 0 , , 3
2
2
y
y
6
2
4
2
−π
2
−1
−2
3π
2
x
x 2
and
−π
π
−4
−6
x
Section 6.4
48. y 4 cos x 4
Graphs of Sine and Cosine Functions
y
6
Period: 2
Amplitude: 4
2
0
4
Shift: Set x and
x
x
4
2
4
7
x
4
−π
−2
π
49. y 2 sin
−4
−6
2x
3
50. y 3 5 cos
Period: 3
Period:
Amplitude: 1
Key points: 0, 2,
3
3
9
, 1 , , 2 , , 3 , 3, 2
4
2
4
Amplitude: 5
Key points: 0, 2, 6, 3, 12, 8, 18, 3, 24, 2
y
16
12
8
4
5
4
x
Period:
1
−1
2
4
3
1
cos 60x
10
52. y 2 cos x 3
Period: 2
1
2
60 30
Amplitude: 2
1
Amplitude:
10
Key points:
Vertical shift two units upward
0, 1,
2 , 3, , 5, 32, 3, 2, 1
Key points:
y
1
1
1
1
,2 ,
, 1.9 ,
,2 ,
, 2.1
0, 2.1,
120
60
40
30
1
−π
π
y
2.2
−4
−5
−6
−7
1.8
− 0.1
12
−8
− 12
− 16
− 20
− 24
1
51. y 2 t
− 12
2
–1
t
12
2
24
12
y
–2
x
2π
4 , 4, 4 , 0, 34, 4, 54, 0, 74, 4
Key points:
–3
573
x
0
0.1
0.2
2π
x
574
Chapter 6
Trigonometry
53. y 3 cosx 3
y
Period: 2
4
2
Amplitude: 3
Shift: Set x 0
π
x 2
and
x x
Key points: , 0, , 3 , 0, 6, , 3 , , 0
2
2
54. y 4 cos x x
2π
−8
4
4
y
10
Period: 2
6
Amplitude: 4
4
Shift: Set x 0
4
x
Key points:
55. y and
x 2
4
4
x
2
− 2π − π
7
4
y
4
Period: 4
3
2
2
Amplitude:
3
1
x
2
2
−3
−4
5 2
7
9 2
2 , 3 , 2 , 0, 2 , 3 , 2 , 0, 2 , 3 56. y 3 cos6x y
2 6
3
3
2
Amplitude: 3
Shift: Set 6x 0
x
Key points:
x
4π
−2
9
x
2
3
π
−1
x
x
0 and
2
2
4
2
4
Period:
x
3π
4 , 8, 4 , 4, 34, 0, 54, 4, 74, 8
Key points:
2π
−4
2
x
2
1
cos ; a , b ,c
3
2
4
3
2
4
Shift:
π
and
π
6x 2
6
x
x
6
6 , 3, 12 , 0, 0, 3, 12 , 0, 6 , 3
57. y 2 sin4x 58. y 4 sin
4
−6
6
−4
3x 3 2
8
−12
12
−8
Section 6.4
59. y cos 2x 1
2
Graphs of Sine and Cosine Functions
60. y 3 cos
3
x
2
2
2
2
−6
−3
6
3
−6
−1
61. y 0.1 sin
x
10 62. y 0.12
− 20
1
sin 120 t
100
0.02
− 0.03
20
Amplitude:
64. f x a cos x d
1
3 1 2 ⇒ a 2
2
Amplitude:
Vertical shift one unit upward of
gx 2 cos x ⇒ d 1. Thus, f x 2 cos x 1.
1 3
2
2
1 2 cos 0 d
d 1 2 1
a 2, d 1
65. f x a cos x d
Amplitude:
66. f x a cos x d
1
8 0 4
2
Amplitude:
Since f x is the graph of gx 4 cos x reflected in the
x-axis and shifted vertically four units upward, we have
a 4 and d 4. Thus, f x 4 cos x 4.
2 4
1
2
Reflected in the x-axis: a 1
4 1 cos 0 d
d 3
a 1, d 3
68. y a sinbx c
67. y a sinbx c
Amplitude: a 3 Since the graph is reflected in the
x-axis, we have a 3.
2
Period:
⇒ b2
b
Amplitude: 2 ⇒ a 2
Period: 4
2
1
4 ⇒ b b
2
Phase shift: c 0
Phase shift: c 0
Thus, y 3 sin 2x.
1
a 2, b , c 0
2
69. y a sinbx c
70. y a sinbx c
Amplitude: a 2
Amplitude: 2 ⇒ a 2
Period: 2 ⇒ b 1
Period: 2
Phase shift: bx c 0 when x 1
4 c0 ⇒
Thus, y 2 sin x .
4
4
c
2
4 ⇒ b
b
2
4
0.03
− 0.02
− 0.12
63. f x a cos x d
575
Phase shift:
c
1 ⇒ c b
2
a 2, b ,c
2
2
576
Chapter 6
Trigonometry
72. y1 cos x
2
71. y1 sin x
−2
1
y2 2
y2 1
2
In the interval 2, 2,
2
y1 y2 when x , −2␲
−2
2␲
−2
5 7 11
1
sin x when x , , ,
.
2
6
6 6 6
73. y 0.85 sin
t
3
v
1.00
(a) Time for one cycle 2
6 sec
3
0.75
0.50
0.25
t
60
10 cycles per min
(b) Cycles per min 6
(c) Amplitude: 0.85; Period: 6
3
Key points: 0, 0, , 0.85 , 3, 0,
2
74. v 1.75 sin
2
− 0.25
4
8
10
− 1.00
9
, 0.85 , 6, 0
2
t
2
(a)
Period 2
4 seconds
2
(b)
1 cycle
4 seconds
60 seconds
15 cycles per minute
1 minute
(c)
v
3
2
1
t
1
3
5
7
−2
−3
76. P 100 20 cos
75. y 0.001 sin 880t
(a) Period:
(b) f 1
2
seconds
880 440
1
440 cycles per second
p
1
1
77. (a) a high low 83.5 29.6 26.95
2
2
(a) Period:
(b)
(b)
5 t
3
6
2
seconds
53 5
1 heartbeat
65 seconds
60 seconds
50 heartbeats per minute
1 minute
100
p 2high time low time 27 1 12
b
2 2 p
12
6
0
c
7 ⇒ c7
3.67
b
6
1
1
d high low 83.5 29.6 56.55
2
2
Ct 56.55 26.95 cos
—CONTINUED—
12
0
6t 3.67
The model is a good fit.
Section 6.4
77. —CONTINUED—
(c)
Graphs of Sine and Cosine Functions
577
2
12 months.
6
This is as we expected since one full period is one year.
(e) The period for both models is
100
0
(f) Chicago has the greater variability in temperature throughout the year. The amplitude, a, determines this variability
since it is 12high temp low temp.
12
0
The model is a good fit.
(d) Tallahassee average maximum: 77.90
Chicago average maximum: 56.55
The constant term, d, gives the average maximum
temperature.
78. (a) and (c)
(b) Vertical shift:
y
Amplitude:
Percent of moon’s
face illuminated
1.0
1
1
⇒ a
2
2
0.8
0.6
Period:
0.4
0.2
20
30
88768
7.4 average length of
5
interval in data
2
47.4 29.6
b
x
10
40
Day of the year
b
Reasonably good fit
(d) Period is 29.6 days.
2
0.21
29.6
Horizontal shift: 0.213 7.4 C 0
(e) March 12 ⇒ x 71. y 0.44 44%
C 0.92
The Naval observatory says that 50% of the moon’s
face will be illuminated on March 12, 2007.
79. C 30.3 21.6 sin
2 t
365 10.9
2
365
Period 2
365
(a)
1
1
⇒ d
2
2
y
80. (a) Period The average daily fuel consumption is given by the
amount of the vertical shift (from 0) which is given
by the constant 30.3.
(c)
60
2
12 minutes
6
The wheel takes 12 minutes to revolve once.
(b) Amplitude: 50 feet
Yes, this is what is expected because there are
365 days in a year.
(b)
1 1
sin0.21x 0.92
2 2
The radius of the wheel is 50 feet.
(c)
110
0
20
0
0
365
0
The consumption exceeds 40 gallons per day when
124 < x < 252.
81. False. The graph of sin(x 2) is the graph of sin(x)
translated to the left by one period, and the graphs are
indeed identical.
1
cos 2x has an amplitude that is half that
2
of y cos x. For y a cos bx, the amplitude is a .
82. False. y 578
Chapter 6
Trigonometry
83. True.
Since cos x sin x , y cos x sin x , and so is a reflection in the x-axis of y sin x .
2
2
2
84. Answers will vary.
85.
Since the graphs are the
same, the conjecture is that
y
2
sinx cos x f=g
1
− 3π
2
x
3π
2
π
2
.
2
−2
86. f x sin x, gx cos x x
2
0
1
0
1
0
sin x
cos x 87. (a)
2
2
y
2
Conjecture: sin x cos x f=g
1
3
2
2
0
1
0
0
1
0
− 3π
2
3π
2
π
2
2
x
−2
x3
x5
x7
3! 5! 7!
x2
x4
x6
cos x 1 2! 4! 6!
(c) sin x x 2
−2␲
2␲
2
2
−2
The graphs are nearly the same for (b)
< x < .
2
2
−2␲
2␲
−2
2
−2␲
−2␲
2␲
−2
The graphs now agree over a wider range, 2␲
3
3
.
< x <
4
4
−2
The graphs are nearly the same for 88. (a) sin
sin
< x < .
2
2
1 1 12 3 125
0.4794
2 2
3!
5!
1
0.4794 (by calculator)
2
63 65
(c) sin 1 0.5000
6
3!
5!
sin
0.5 (by calculator)
6
(e) cos 1 1 1
1
0.5417
2! 4!
cos 1 0.5403 (by calculator)
(b) sin 1 1 1
1
0.8417
3! 5!
sin 1 0.8415 (by calculator)
0.5 2 0.54
0.8776
2!
4!
cos0.5 0.8776 (by calculator)
(d) cos0.5 1 (f) cos
cos
42 42
1
0.7074
4
2!
4!
0.7071 (by calculator)
4
The error in the approximation is not the same in each case. The error appears to increase as x moves farther away from 0.
Section 6.5
89. log10 x 2 log10x 212 91. ln
1
log10x 2
2
t3
ln t 3 lnt 1 3 ln t lnt 1
t1
Graphs of Other Trigonometric Functions
90. log2x2x 3 log2 x2 log2x 3
2 log2 x log2x 3
z
92. ln
2
z
1
z
1
ln 2
ln z lnz2 1
1 2
z 1
2
93.
1
1
log10 x log10 y log10xy
2
2
log2 x2(xy)
log10 xy
ln
1
1
ln z lnz2 1
2
2
94. 2 log2 x log2xy log2 x2 log2xy
log2 x3y
95. ln 3x 4 ln y ln 3x ln y4
96.
3xy
1
1
ln 2x 2 ln x 3 ln x ln 2x ln x2 ln x3
2
2
4
1
2x
ln 2 ln x3
2
x
2xx ln x
2x
lnx x
ln
3
2
3
2
lnx22x 97. Answers will vary.
Section 6.5
■
■
Graphs of Other Trigonometric Functions
You should be able to graph
y a tanbx c
y a cotbx c
y a secbx c
y a cscbx c
When graphing y a secbx c or y a cscbx c you should first graph y a cosbx c or
y a sinbx c because
(a) The x-intercepts of sine and cosine are the vertical asymptotes of cosecant and secant.
(b) The maximums of sine and cosine are the local minimums of cosecant and secant.
(c) The minimums of sine and cosine are the local maximums of cosecant and secant.
■
You should be able to graph using a damping factor.
Vocabulary Check
1. vertical
2. reciprocal
3. damping
4. 5. x n
6. , 1 1, 7. 2
579
580
Chapter 6
Trigonometry
1. y sec 2x
2. y tan
2
2
Period:
Period:
Matches graph (e).
x
2
3. y 2
b
1
2
1
cot x
2
Period:
1
Matches graph (a).
Asymptotes: x , x Matches graph (c).
1
x
sec
2
2
2 2
Period:
4
b
2
4. y csc x
5. y Period: 2
Matches graph (d).
6. y 2 sec
Period:
x
2
2 2
4
b
2
Asymptotes: x 1, x 1
Asymptotes: x 1, x 1
Matches graph (f).
Reflected in x-axis
Matches graph (b).
7. y 1
tan x
3
Period: 3
Two consecutive
asymptotes:
1
x
4
0
4
y
1
3
0
1
3
−π
π
9. y tan 3x
3x 12
y
1
0
12
0
1
1
−π
x
0
4
y
1
4
0
1
4
Period:
π
3
x
x
π
−3
y
10. y 3 tan x
1
⇒ x
2
6
2
4
2
⇒ x
2
6
3
x ,x
2
2
3
−π
3
y
Two consecutive
asymptotes:
4
3
x
x
y
Two consecutive
asymptotes:
3x 1
tan x
4
Period: 2
x and x 2
2
Period:
8. y y
1
Two consecutive
asymptotes:
x
2
−4
1
1
x ,x
2
2
x
y
3
1
4
−8
0
1
4
0
3
Section 6.5
1
11. y sec x
2
y
12. y 3
Period: 2
1
−π
x
π
3
y
1
0
1
2
x
1
y
1
2
4
Two consecutive
asymptotes:
1
2
x
−1
x 0, x 1
1
2
1
6
1
2
5
6
y
2
1
2
2
2
−2
−1
x
1
−1
2
3
y
1
17. y csc
Period:
1
3
y
8
6
4
2
−π
4
4
24
8
5
24
y
6
3
6
2 4
2
6
4
Two consecutive
asymptotes:
0
1
3
x
0
1
y
2
x
2
18. y csc
y
2
4
12
Two consecutive
asymptotes:
6
Period:
4
2
π
x
12
2
−π
4
0
12
0
2
π
4
x
3
π
2
x
y
2
6
13
Two consecutive
asymptotes:
6
4
2
π 2π
x 0, x 3
x 0, x 2
x
π
4
−2
y
x ,x
8
8
1
1
x ,x
2
2
x
1
2
x
Period:
−3
1
4
16. y 2 sec 4x 2
y
Two consecutive
asymptotes:
3
Two consecutive
asymptotes:
−4
15. y sec x 1
0
2
4
2
x 0, x −3
x
Period:
Period:
3
−2
3
x
2π
14. y 3 csc 4x
y
2
2
Period:
3
Two consecutive
asymptotes:
3
13. y csc x
y
x ,x
2
2
x ,x
2
2
x
1
sec x
4
581
Period: 2
2
Two consecutive
asymptotes:
Graphs of Other Trigonometric Functions
x
3
5
3
x
2
3
2
5
2
y
2
1
2
y
2
1
2
x
582
Chapter 6
19. y cot
Period:
Trigonometry
x
2
y
20. y 3 cot
3
2
12
1
− 2π
x
2π
x
y
2
1
3
2
0
1
1
sec 2x
2
y
1
23. y tan
Period:
6
2
x
−2
2
3
0
6
1
2
1
π
x
4
2
x
−4
4
x
1
0
1
y
1
0
1
25. y csc x
Period: 2
4
Two consecutive
asymptotes:
2
1
− 3π
2
6
2
5
6
y
2
1
2
0
3
y
3
2
Two consecutive
asymptotes:
x
y
1
2
−π
4
0
4
0
π
2
x
x
π
1
2
y
Period: 4
Two consecutive
asymptotes:
2
3
1
x
π
x
4
0
4
y
1
0
1
Period:
3
x
3
26. y csc2x y
x 0, x y
x ,x
2
2
x
⇒ x 2
4
2
x ⇒ x2
4
2
3
2
24. y tanx 6
Two consecutive asymptotes:
1
x ,x
2
2
y
4
4
1
2
Period: −π
x
4
x
1
22. y tan x
2
y
2
Period:
2
4
x 0, x 2
x
⇒ x 2
2
x
6
Two consecutive
asymptotes:
x
0 ⇒ x0
2
21. y y
2
Period:
2
2
Two consecutive
asymptotes:
x
2
y
2
2
Two consecutive
asymptotes:
x 0, x 2
−1
−2
x
12
4
5
12
y
2
1
2
π
2
π
3π
2
2π
x
Section 6.5
27. y 2 secx 4
Two consecutive
asymptotes:
2
y
29. y Period:
3
1
x ,x
2
2
x
28. y sec x 1
y
Period: 2
3
−π
4
2
1
csc x 4
4
π
−1
2π
3π
x
2
2π
y
1
2
31. y tan
7
12
1
4
1
2
x
x
3
1
1
3
0
1
3
y
1
0
1
2
x
y
2
4
0
4
0
2
−␲
2
␲
2
−4
4
36. y 2
3
−3
1
cot x 4
2
− 3␲
2
−2
3␲
2
1
4 tan x 3
2
cos 4x
4
3␲
4
35. y tan x x
−2
−3
1
cos x
3π
2
33. y 2 sec 4x − 3␲
4
−5
34. y sec x ⇒ y 4
− 3π
2
3
5
5␲
3
y
x ,x
2
2
32. y tan 2x
−5␲
2
Two consecutive
asymptotes:
x
3
x ,x
4
4
4
1
Period: Two consecutive
asymptotes:
12
2
1
3
x
30. y 2 cot x y
Period: 2
x
2
2
1
1
x ,x
2
2
4
583
y
Two consecutive
asymptotes:
3
0
Graphs of Other Trigonometric Functions
2
3
−3
− 3␲
2
3␲
2
−3
37. y csc4x y
1
sin4x 38. y 2 sec(2x ) ⇒
3
−␲
2
␲
2
−3
y
2
cos2x 4
−␲
␲
−4
584
Chapter 6
39. y 0.1 tan
Trigonometry
4x 4 40. y 0.6
−6
x 1
sec
3
2
2
⇒
y
1
x 3 cos
2
2
6
2
−0.6
−6
6
−2
41. tan x 1
43. cot x 42. tan x 3
7
3 5
x , , ,
4
4 4 4
5 2 4
x , , ,
3
3 3 3
y
x
3
2 5
4
, ,
,
3
3 3 3
y
2
3
y
2
3
2
1
1
π
2π
x
π
2π
π
2
x
3π
2
x
−3
x
46. sec x 2
45. sec x 2
44. cot x 1
7 3 5
, , ,
4
4 4 4
x±
2
4
, ±
3
3
x
5 5
, , ,
3
3 3 3
y
y
y
3
2
− 3π
2
1
− π2
π
2
x
3π
2
−2π
−π
1
π
2π
x
−2π
−3
47. csc x 2
x
48. csc x 7
5 3
, , ,
4
4 4 4
x
23
3
2 4 5 , , ,
3
3 3 3
y
y
3
3
2
2
1
− 3π
2
−π
2
−1
π
2
1
x
3π
2
− 3π
2
−π
2
π
2
x
3π
2
−π
π
2π
x
Section 6.5
f x sec x 49.
1
cos x
50. f x tan x
y
4
f x secx
Graphs of Other Trigonometric Functions
y
tanx tan x
3
3
2
1
cosx
−π
π
2π
Thus, the function is odd
and the graph of y tan x
is symmetric about the origin.
x
− 3π
2
− π2
π
2
1
cos x
−3
f x
Thus, f x sec x is an even function and the graph has
y-axis symmetry.
51. f x 2 sin x
gx (a)
52. f x tan
1
csc x
2
x
1
x
, gx sec
2
2
2
(a)
3
g
y
−1
1
f
3
1
−1
−3
f
2
g
π
4
π
2
3π
4
x
π
(b) f > g on the interval,
(b)
1
The interval in which f < g is 1, 3 .
(c)
1
The interval in which 2f < 2g is 1, 3 ,
which is the same interval as part (b).
5
< x <
6
6
(c) As x → , f x 2 sin x → 0 and
1
gx csc x → ± since g x is the reciprocal
2
of f x.
54. y1 sin x sec x, y2 tan x
53. y1 sin x csc x and y2 1
2
−3
4
−2␲
3
−2
sin x csc x sin x
sin x 1, sin x 0
1
sin x sec x sin x
cos x
1
and y2 cot x sin x
tan x
cot x 2␲
−4
The expressions are equivalent except when sin x 0
and y1 is undefined.
55. y1 585
cos x
sin x
The expressions are equivalent.
sin x
1
tan x
cos x cos x
The expressions are equivalent.
4
−2␲
2␲
−4
x
3π
2
586
Chapter 6
Trigonometry
56. y1 sec 2 x 1, y2 tan2 x
1
57. f x x cos x
3
As x → 0, f x → 0 and f x > 0.
tan2
x sec x
tan2
x sec x 1
2
2
− 3␲
2
3␲
2
The expressions are equivalent.
Matches graph (d).
−1
58. f x x sin x
59. gx x sin x
Matches graph (a) as x → 0, f x → 0.
As x → 0, gx → 0 and gx is odd.
Matches graph (b).
61. f x sin x cos x 60. gx x cos x
Matches graph (c) as x → 0, gx → 0.
, gx 0
2
f x gx, The graph is the line y 0.
y
3
2
1
−3
−2
−1
x
1
−1
2
3
−2
−3
62. f x sin x cos x 2
1
63. f x sin2 x, gx 1 cos 2x
2
f x gx
gx 2 sin x
y
y
4
3
2
2
−π
x
π
−π
−4
x
π
–1
It appears that f x gx. That is,
sin x cos x 64. f x cos2
2 sin x.
2
x
2
65. gx ex 2 sin x
2
y
ex 2 ≤ gx ≤ ex 2
2
3
1
gx 1 cos x
2
−6
x
−3
3
−1
2
The damping factor is
2
y ex 2.
2
It appears that f x gx.
That is,
x 1
1 cos x.
cos2
2
2
1
6
As x →
, gx →0.
−8
8
−1
Section 6.5
Graphs of Other Trigonometric Functions
67. f x 2x4 cos x
66. f x ex cos x
68. hx 2x 4 sin x
2
2x4 ≤ f x ≤ 2x4
Damping factor: ex
Damping factor: 2x 4
2
Damping factor: y 2x4.
3
1
6
−3
−8
6
−9
8
9
−1
−3
As x →
, f x → 0.
−6
As x →
, h x → 0.
As x→ , f x → 0.
69. y 6
cos x, x > 0
x
70. y 6
4
sin 2x, x > 0
x
71. gx sin x
x
2
6
8␲
0
−2
−6␲
6␲
0
6␲
−1
−2
As x → 0, y →
72. f (x) .
As x → 0, y →
1 cos x
x
73. f x sin
As x → 0, gx → 1.
.
1
x
74. h(x) x sin
2
1
−6␲
−␲
6␲
2
␲
−␲
d
−1
As x → 0, f x oscillates
between 1 and 1.
As x → 0, f (x) → 0.
75. tan x ␲
−2
−1
7
d
As x → 0, h(x) oscillates.
76. cos x 7
7 cot x
tan x
d
27
d
27
27 sec x, < x <
cos x
2
2
d
d
14
80
2
−2
−6
π
4
π
2
3π
4
π
x
60
Distance
Ground distance
10
6
40
20
−π
2
− 10
− 14
Angle of elevation
1
x
−π
4
0
π
4
Angle of camera
π
2
x
587
588
Chapter 6
Trigonometry
77. C 5000 2000 sin
(a)
t
t
, R 25,000 15,000 cos
12
12
50,000
R
C
0
100
0
(b) As the predator population increases, the number of prey decreases. When the number of prey is small,
the number of predators decreases.
(c) The period for both C and R is:
p
2
24 months
12
When the prey population is highest, the predator population is increasing most rapidly.
When the prey population is lowest, the predator population is decreasing most rapidly.
When the predator population is lowest, the prey population is increasing most rapidly.
When the predator population is highest, the prey population is decreasing most rapidly.
In addition, weather, food sources for the prey, hunting, all affect the populations of both the predator and the prey.
t
6
S
Lawn mower sales
(in thousands of units)
78. S 74 3t 40 cos
150
135
120
105
90
75
60
45
30
15
t
2
4
6
8 10 12
Month (1 ↔ January)
79. Ht 54.33 20.38 cos
t
t
15.69 sin
6
6
Lt 39.36 15.70 cos
t
t
14.16 sin
6
6
(a) Period of cos
t 2
:
12
6 6
Period of sin
t 2
:
12
6 6
(b) From the graph, it appears that the greatest difference
between high and low temperatures occurs in summer.
The smallest difference occurs in winter.
(c) The highest high and low temperatures appear to
occur around the middle of July, roughly one month
after the time when the sun is northernmost in the sky.
Period of Ht : 12 months
Period of Lt : 12 months
80. (a)
1
y et4 cos 4t
2
0.6
0
81. True. Since
y csc x 1
,
sin x
4␲
−0.6
(b) The displacement is a damped sine wave.
y → 0 as t increases.
for a given value of x, the y-coordinate of csc x is the
reciprocal of the y-coordinate of sin x.
Section 6.5
Graphs of Other Trigonometric Functions
83. As x →
82. True.
y sec x 1
cos x
As x →
589
from the left, f x tan x → .
2
from the right, f x tan x → .
2
If the reciprocal of y sin x is translated 2 units to the
left, we have
y
1
sin x 2
1
sec x.
cos x
84. As x → from the left, f (x) csc x →
.
As x → from the right, f (x) csc x → .
85. f x x cos x
(a)
(b) xn cosxn1
2
x0 1
−3
3
x1 cos 1 0.5403
x2 cos 0.5403 0.8576
−2
The zero between 0 and 1 occurs at x 0.7391.
x3 cos 0.8576 0.6543
x4 cos 0.6543 0.7935
x5 cos 0.7935 0.7014
x6 cos 0.7014 0.7640
x7 cos 0.7640 0.7221
x8 cos 0.7221 0.7504
x9 cos 0.7504 0.7314
This sequence appears to be approaching the zero
of f : x 0.7391.
86. y tan x
yx
2x3 16x5
3!
5!
− 3␲
2
The graphs are nearly the
same for 1.1 < x < 1.1.
88. (a) y1 87. y1 sec x
6
4
1
sin x sin 3x
3
y2 1 3␲
2
y2 2
4
1
1
sin x sin 3x sin 5x
3
5
2
−3
3
−3
3
y2
y1
−2
—CONTINUED—
x2
5x4
2!
4!
− 3␲
2
The graph appears to
coincide on the interval
1.1 ≤ x ≤ 1.1.
−6
6
−2
3␲
2
−6
590
Chapter 6
Trigonometry
88. —CONTINUED—
(b) y3 4
1
1
1
sin x sin 3x sin 5x sin 7x
3
5
7
2
−3
3
−2
(c) y4 4
1
1
1
1
sin x sin 3x sin 5x sin 7x sin 9x
3
5
7
9
89. e2x 54
90. 83x 98
2x ln 54
91.
3x log8 98
ln 54
x
1.994
2
x
300
100
1 e x
300
1 e x
100
ln 98
0.735
3 ln 8
3 1 e x
2 e x
ln 2 x
x ln 2 0.693
92.
1 0.15
365 365t
93. ln3x 2 73
5
3x 2 e73
0.15
1
1.00041096
365
3x 2 e73
1.00041096365t 5
x
365t log1.00041096 5
t
2 e73
3
1.684 1031
1
log10 5
10.732
365 log10 1.00041096
94. ln(14 2x) 68
95. lnx2 1 3.2
14 2x e68
x2 1 e3.2
14 e68 2x
x
x2 e3.2 1
14 e68
1.702 1029
2
x ± e3.2 1 ± 4.851
96. lnx 4 5
97. log8 x log8x 1 1
3
1
ln(x 4) 5
2
log8xx 1 1
3
xx 1 8
ln(x 4) 10
x4
e10
x
e10
13
x x2
2
4
22,022.466
x2
x20
x 2x 1 0
x 2, 1
x 1 is extraneous (not in the
domain of log8 x) so only x 2 is
a solution.
98. log6 x log6(x2 1) log6(64x)
log6x(x2 1 log6(64x)
x(x2 1) 64x
x2 1 64
x ± 65
Since 65 is not in the domain
of log6 x, the only solution is
x 65 8.062.
Section 6.6
Section 6.6
Inverse Trigonometric Functions
591
Inverse Trigonometric Functions
You should know the definitions, domains, and ranges of y arcsin x, y arccos x, and y arctan x.
■
■
Function
Domain
Range
y arcsin x ⇒ x sin y
1 ≤ x ≤ 1
y arccos x ⇒ x cos y
1 ≤ x ≤ 1
0 ≤ y ≤ y arctan x ⇒ x tan y
< x <
≤ y ≤
2
2
< x <
2
2
You should know the inverse properties of the inverse trigonometric functions.
sinarcsin x x and arcsinsin y y, ≤ y ≤
2
2
cosarccos x x and arccoscos y y, 0 ≤ y ≤ tanarctan x x and arctantan y y, ■
< y <
2
2
You should be able to use the triangle technique to convert trigonometric functions of inverse trigonometric functions
into algebraic expressions.
Vocabulary Check
Function
Alternative
Notation
Domain
1. y arcsin x
y sin1 x
1 ≤ x ≤ 1
2. y arccos x
y cos1 x
1 ≤ x ≤ 1
3. y arctan x
y tan1 x
< x <
Range
≤ y ≤
2
2
0 ≤ y ≤ < y <
2
2
1. y arcsin
1
1
⇒ sin y for ≤ y ≤
⇒ y
2
2
2
2
6
2. y arcsin 0 ⇒ sin y 0 for 3. y arccos
1
1
⇒ cos y for 0 ≤ y ≤ ⇒ y 2
2
3
4. y arccos 0 ⇒ cos y 0 for 0 ≤ y ≤ ⇒ y 5. y arctan
3
3
⇒ tan y < y <
⇒ y
2
2
6
3
3
for
≤ y ≤
⇒ y0
2
2
6. y arctan1 ⇒ tan y 1 for
< y <
⇒ y
2
2
4
2
592
Chapter 6
7. y arccos Trigonometry
3
2
⇒ cos y 23 for
0 ≤ y ≤ ⇒ y
5
6
9. y arctan 3 ⇒ tan y 3 for
2 ⇒ cos y 2 for
1
1
0 ≤ y ≤ ⇒ y
13. y arcsin
3
3
2
< y <
⇒ y0
2
2
2
⇒ sin y 1.5
f
3
3
⇒ tan y 33 for
< y <
⇒ y
2
2
6
f
−2
19. arccos 0.28 cos1 0.28 1.29
20. arcsin 0.45 0.47
21. arcsin0.75 sin10.75 0.85
22. arccos0.7 2.35
24. arctan 15 1.50
34. arctan 0.75 0.85
1
95
1.50
7
25. arcsin 0.31 sin1 0.31 0.32
27. arccos0.41 cos10.41 1.99
26. arccos 0.26 1.31
3
␲
2
g
Graph y3 x.
−1
28. arcsin0.125 0.13
2
−␲
2
Graph y2 tan1 x.
g
23. arctan3 tan13 1.25
for
≤ y ≤
⇒ y
2
2
4
Graph y1 tan x.
−1.5
4 sin
2
16. y arccos 1 ⇒ cos y 1 for 0 ≤ y ≤ ⇒ y 0
gx arcsin x
31. arcsin
2
18. f x tan x and gx arctan x
1
yx
2
≤ y ≤
⇒ y
2
2
3
17. f x sin x
< y <
⇒ y
2
2
3
14. y arctan for
15. y arctan 0 ⇒ tan y 0 for ⇒ sin y 22 for
≤ y ≤
⇒ y
2
2
4
12. y arcsin
2
3
⇒ sin y 2
2
10. y arctan3 ⇒ tan y 3 for
< y <
⇒ y
2
2
3
11. y arccos 2
8. y arcsin 30. arctan 2.8 1.23
29. arctan 0.92 tan1 0.92 0.74
3 1.91
32. arccos 1
33. arctan
2 tan
7
1
3.5 1.29
35. This is the graph of y arctan x. The coordinates are
3, 3 , 3 , 6 , and 1, 4 .
3
Section 6.6
36. arccos1 2
3
2 arccos 37. tan 1
Inverse Trigonometric Functions
x
4
tan arctan
3
cos
6
2
x
x
4
θ
38. cos 4
4
x
39. sin arccos
593
4
x
x2
5
sin arcsin
x2
5
5
x+2
θ
40. tan x1
10
41. cos x1
arctan
10
x3
2x
arccos
x3
2x
2x
θ
x+3
42. tan x1
1
2
x 1 x1
arctan
43. sinarcsin 0.3 0.3
44. tanarctan 25 25
46. sinarcsin0.2 0.2
47. arcsinsin 3 arcsin0 0
1
x1
x1
45. cosarccos0.1 0.1
Note: 3 is not in the range of
the arcsine function.
48. arccos cos
Note:
7
arccos 0 2
2
3
49. Let y arctan .
4
7
is not in the range of the arccosine function.
2
y
3
tan y , 0 < y <
4
2
3
and sin y .
5
5
3
y
x
4
4
50. Let u arcsin ,
5
4
sin u , 0 < u < .
5
2
4
5
sec arcsin
sec u 5
3
51. Let y arctan 2.
5
4
u
3
y
2
tan y 2 , 0 < y <
1
2
and cos y 1
5
5
5
.
5
2
y
1
x
594
Chapter 6
Trigonometry
5
52. Let u arccos
cos u 5
5
sin arccos
5
,0 < u <
5
5
5
.
13
53. Let y arcsin
,
.
2
sin u sin y 2
5
25
y
5
, 0 < y <
13
2
and cos y 5
12
.
13
13
5
y
x
12
5
2
u
1
54. Let u arctan 5 .
5
,
12
55. Let y arctan 5
tan u , < u < 0.
12
2
5
csc arctan 12
13
csc u 5
12
u
13
and sec y 34
5
5
x
y
−3
34
.
3 .
3
57. Let y arccos 3 sin u , < u < 0.
4
2
3
tan arcsin 4
3
tan y , < y < 0
5
2
y
−5
4 ,
56. Let u arcsin 3
37
3
tan u 7
7
y
2
2 < y < cos y ,
3 2
and sin y 5
3
5
3
.
y
−2
7
u
4
−3
x
Section 6.6
5
58. Let u arctan ,
8
595
59. Let y arctan x.
5
tan u , 0 < u < .
8
2
cot arctan
Inverse Trigonometric Functions
89
tan y x 5
u
x
1
and cot y .
x
8
5
8
cot u 8
5
x2 + 1
x
1
y
1
61. Let y arcsin2x.
60. Let u arctan x,
x
tan u x .
1
sin y 2x sinarctan x sin u 1
2x
and cos y 1 x
x2
2x
1
4x2.
y
1
1 − 4x 2
x2 + 1
x
u
1
62. Let u arctan 3x,
tan u 3x 63. Let y arccos x.
3x
.
1
cos y x 9x 2 + 1
secarctan 3x sec u 9x2 1
3x
x
1
1
and sin y 1 x2.
1 − x2
y
x
u
1
64. Let u arcsinx 1,
sin u x 1 65. Let y arccos
x1
.
1
secarcsinx 1 sec u 1
cos y 1
x
x
3
and tan y 2x x2
3 .
3
9 x2
x
.
9 − x2
y
x
x −1
u
2x − x 2
1
66. Let u arctan ,
x
67. Let y arctan
x2 + 1
1
tan u .
x
cot arctan
1
1
cot u x
x
tan y x
.
x2 + 2
x
x
2
and csc y u
x
2
x2 2
x
.
y
2
596
Chapter 6
68. Let u arcsin
sin u Trigonometry
xh
,
r
xh
.
r
cos arcsin
r
xh
cos u r
r 2
x h
r
x−h
2
u
r 2 − (x − h) 2
69. f x sinarctan 2x, gx 2x
1 4x2
2
They are equal. Let y arctan 2x.
tan y 2x −2
1 4x2
.
1 + 4x 2
2x
2x
f x
gx 1 4x2
y
The graph has horizontal asymptotes at y ± 1.
x
2
70. f x tan arccos
gx 1
2
−3
4 x 2
9
71. Let y arctan .
x
3
tan y x
−2
arcsin y These are equal because:
x
Let u arccos .
2
2
x
tan u
2
f x tan arccos
4 x
2
x
4 − x2
arcsin y arcsin
, x > 0;
, x < 0.
9
gx
y
6
x
u,
36 x 2
36 x2
6
9
x2 81
x
36 x 2
then sin u 9
x2 81
x 2 + 81
u
Thus, f x gx.
72. If arcsin
9
9
9
and sin y , x > 0;
, x < 0.
x2 81
x2 81
x
Thus,
Asymptote: x 0
3
2x
1
2x
and sin y −3
6
73. Let y arccos
.
6
x
arccos
6
cos y 36 − x 2
3
x2 2x 10
and sin y u
x
3
x2 2x 10
x 1
x 12 9
Thus, y arcsin
.
x 1
x − 1
3
3
x 12 9
x2 2x 10
(x − 1) 2 + 9
y
. Then,
.
Section 6.6
74. If arccos
x2
u,
2
then cos u arccos
Inverse Trigonometric Functions
597
75. y 2 arccos x
Domain: 1 ≤ x ≤ 1
x2
.
2
Range: 0 ≤ y ≤ 2
x2
arctan
2
4x x 2
This is the graph of f x arccos x with a factor of 2.
x2
y
2π
2
4x − x 2
π
u
−2
x−2
76. y arcsin
x
2
Range: ≤ y ≤
2
2
This is the graph of
f x arcsin x with a
horizontal stretch of a
factor of 2.
Range: x
−2
1
π
≤ y ≤
2
2
π
Range: This is the graph of
y arccos t shifted
two units to the left.
arctan x
2
2
4
π
< y <
2
2
y
81. hv tanarccos v Domain: all real numbers
3
y
−4
This is the graph of
gx arctanx with a
horizontal stretch of a
factor of 2.
t
−1
2
−π
Domain: all real numbers
Range: 0 ≤ y ≤ −2
1
79. f x arctan 2x
y
−3
x
−1
This is the graph of
gx arcsinx shifted
one unit to the right.
Domain: 3 ≤ t ≤ 1
x
−2
−π
1 v2
v
Domain: 1 ≤ v ≤ 1, v 0
π
Range: 0 < y ≤ This is the graph of
y arctan x shifted
upward 2 units.
y
2
−π
−4
2
Domain: 0 ≤ x ≤ 2
π
78. gt arccost 2
80. f x 1
77. f x arcsinx 1
y
Domain: 2 ≤ x ≤ 2
x
−1
Range: all real numbers
−4
−2
y
x
2
4
π
1
1 − v2
y
v
−2
v
1
2
598
Chapter 6
82. f x arccos
Trigonometry
x
4
83. f x 2 arccos2x
y
2␲
Domain: 4 ≤ x ≤ 4
π
Range: 0 ≤ y ≤ −4
x
−2
2
−1
4
1
0
85. f x arctan2x 3
84. f x arcsin4x
2␲
86. f x 3 arctan x
␲
2
␲
−4
−0.5
−2
0.5
−␲
−2␲
87. f x arcsin
4
4
−2␲
23 2.412
88. f x 1
arccos
2.82
2
4
4
−4
−4
5
5
−2
−2
89. f t 3 cos 2t 3 sin 2t 32 32 sin 2t arctan
3
3
32 sin2t arctan 1
32 sin 2t 4
6
−2␲
2␲
−6
The graphs are the same and implies that the identity is true.
90. f t 4 cos t 3 sin t
91. (a) sin 42 32 sin t arctan
5 sin t arctan
4
3
4
3
sin arcsin
6
−6
6
−6
The graph implies that
A cos t B sin t A2 B2 sin t arctan
is true.
5
s
A
B
5
s
(b) s 40: arcsin
5
0.13
40
s 20: arcsin
5
0.25
20
Section 6.6
92. (a) tan s
750
arctan
93. arctan
(a)
s
750
Inverse Trigonometric Functions
599
3x
x2 4
1.5
(b) When s 300,
arctan
0
300
0.38 21.8.
750
6
− 0.5
When s 1200,
(b) is maximum when x 2 feet.
1200
arctan
1.01 58.0.
750
(c) The graph has a horizontal asymptote at 0. As x
increases, decreases.
94. (a) tan 11
17
arctan
95.
11
0.5743 32.9
17
1
(b) r 40 20
2
tan 41 ft
(a) tan h
h
r
20
h 20 tan 20 20 ft
θ
20
41
tan arctan
11
12.94 feet
17
(b) tan 26 41 26.0
20
h
50
tanh 50 tan 26 24.39 feet
96. (a) tan 6
x
arctan
97. (a) tan 6
x
tan arctan
(b) x 7 miles
6
arctan 0.71 40.6
7
(b) x 5: arctan
5
14.0
20
12
31.0
20
6
1.41 80.5
1
99. False.
98. False.
5
is not in the range of arcsinx.
6
arcsin
x
20
x 12: arctan
x 1 mile
arctan
x
20
1 2
6
5
is not in the range of the arctangent function.
4
arctan 1 100. False.
arctan x is defined for all real x, but arcsin x and arccos x require 1 ≤ x ≤ 1.
Also, for example,
arctan 1 arcsin 1
.
arccos 1
Since arctan 1 2
arcsin 1
undefined.
, but
4
arccos 1
0
4
600
Chapter 6
Trigonometry
101. y arccot x if and only if cot y x.
Domain: < x <
102. y arcsec x if and only if sec y x where
x ≤ 1 x ≥ 1 and 0 ≤ y < and < y ≤ . The
2
2
domain of y arcsec x is , 1 1, and the
y
Range: 0 < x < π
2 2 , .
range is 0,
π
2
y
−2
−1
x
1
2
π
π
2
−2
103. y arccsc x if and only if csc y x.
x
1
2
y
Domain: , 1 1, Range:
−1
π
2
2 , 0 0, 2 −2
x
−1
1
2
−π
2
104. (a) y arcsec 2 ⇒ sec y 2 and 0 ≤ y <
(b) y arcsec 1 ⇒ sec y 1 and 0 ≤ y <
< y ≤ ⇒ y
2 2
4
< y ≤ ⇒ y0
2 2
(c) y arccot 3 ⇒ cot y 3 and 0 < y < ⇒ y (d) y arccsc 2 ⇒ csc y 2 and 5
6
≤ y < 00 < y ≤
⇒ y
2
2
6
105. Area arctan b arctan a
106. f x x
(a) a 0, b 1
gx 6 arctan x
Area arctan 1 arctan 0 0 4
4
12
g
(b) a 1, b 1
Area arctan 1 arctan1
4
4
2
(c) a 0, b 3
Area arctan 3 arctan 0
1.25 0 1.25
(d) a 1, b 3
Area arctan 3 arctan1
4 2.03
1.25 f
0
6
0
As x increases to infinity, g approaches 3, but f has
no maximum. Using the solve feature of the graphing
utility, you find a 87.54.
Section 6.6
Inverse Trigonometric Functions
601
107. f x sinx, f 1x arcsinx
(a) f f 1 sinarcsin x
f 1
f arcsinsin x
2
2
−␲
−␲
␲
␲
−2
−2
(b) The graphs coincide with the graph of y x only for certain values of x.
f f 1 x over its entire domain, 1 ≤ x ≤ 1.
f 1
f x over the region 2
≤ x ≤
, corresponding to the region where sin x is
2
one-to-one and thus has an inverse.
(b) Let y arctanx. Then,
108. (a) Let y arcsinx. Then,
sin y x
tan y x, sin y x
tan y x
siny x
tany x, y arcsin x
y arcsin x.
< y <
2
2
arctantany arctan x
Therefore, arcsinx arcsin x.
y arctan x
(c) Let y2 y1.
2
arctan x arctan
< y <
2
2
y arctan x
Thus, arctanx arctanx.
1
y1 y2
x
y1 2 y 2
1
y2
(d) Let arcsin x and arccos x, then sin x and
cos x. Thus, sin cos which implies that and
are complementary angles and we have
2
arcsin x arccos x .
2
1
y1
x
(e) arcsin x arcsin
x
x
arctan
1 x2
1
y2
1
x
y1
1 − x2
109. 8.23.4 1279.284
110. 10(14)2 10
10
0.051
142 196
602
Chapter 6
Trigonometry
111. 1.150 117.391
113.
sin 112. 162 3
opp
4
hyp
114. tan 2
adj2 32 42
adj2 9 16
adj2 7
4
3
θ
adj 7
cos 4
7
37
7
3
4
47
7
7
csc 4
3
cos 5
adj
6 hyp
opp2 25 36
opp 11
2
opp 11
tan cot sec csc sin 2
5
cot 1
2
6
6
11
1
8
θ
5
3
22
cos 1
3
sin 22
3
θ
1
tan 2 2
5
5
511
11
11
6
5
6
θ
opp 32 12
11
11
2
116. sec 3
opp2 52 62
sin 1
5
1
csc 5
2
sec 115.
cos sec 5
7
cot hyp 12 22 5
7
3
tan 1
2.718 108
162
611
11
sec 3
csc cot 3
22
1
22
32
4
2
4
Section 6.6
117. Let x the number of people presently in the group. Each person’s share is now
If two more join the group, each person’s share would then be 250,000.
x2
Inverse Trigonometric Functions
250,000
.
x
Share per person with
Original share
6250
two more people
per person
250,000 250,000
6250
x2
x
250,000x 250,000x 2 6250xx 2
250,000x 250,000x 500,000 6250x2 12500x
6250x2 12500x 500,000 0
6250x2 2x 80 0
6250x 10x 8 0
x 10 or x 8
x 10 is not possible.
There were 8 people in the original group.
118. Rate downstream: 18 x
Rate upstream: 18 x
d
r
rate time distance ⇒ t 35
35
4
18 x 18 x
3518 x 3518 x 418 x18 x
630 35x 630 35x 4324 x2
1260 4324 x2
315 324 x2
x2 9
x ±3
The speed of the current is 3 miles per hour.
120. Data: 2, 742,000
4, 632,000
To find: 8, y
P P0
Assume:
ert
742,000 P0er 2
632,000 P0er 4
Then: er 2 P0er 4 632
P0er 2 742
y P0er 8 P0er 4
632,000 er 22
632,000 632
742 458,504.31
0.035
4
0.035
12
0.035
365
(b) A 15,000 1 (Time to go upstream) (Time to go downstream) 4
2
er 4
410
119. (a) A 15,000 1 (c) A 15,000 1 $21,253.63
1210
36510
$21,275.17
$21,285.66
(d) A 15,000e0.03510 $21,286.01
603
604
Chapter 6
Section 6.7
Trigonometry
Applications and Models
■
You should be able to solve right triangles.
■
You should be able to solve right triangle applications.
■
You should be able to solve applications of simple harmonic motion.
Vocabulary Check
1. elevation; depression
2. bearing
3. harmonic motion
2. Given: B 54, c 15
1. Given: A 20, b 10
tan A a
⇒ a b tan A 10 tan 20 3.64
b
cos A b
10
b
⇒ c
10.64
c
cos A cos 20
B 90 20 70
sin B b
⇒ b c sin B
c
b = 10
b
b
24
⇒ a
8.26
a
tan B tan 71
b
b
24
sin B ⇒ c 25.38
c
sin B sin 71
A
cos B 4. Given: A 8.4, a 40.5
B 90 A
90 8.4 81.6
tan A a
a
⇒ b
b
tan A
B
c
sin A A
b = 24
40.5
274.27
tan 8.4
a
a
40.5
⇒ c
277.24
c
sin A sin 8.4
B
c
a = 40.5
C
5. Given: a 6, b 10
8.4°
234 11.66
6
3
a
⇒ A arctan 30.96º
tan A b 10
5
B 90 30.96 59.04
600 24.49
sin A c
cos B A
c = 35
a = 25
352 252
C
b
A
a
a
⇒ A arcsin
c
c
arcsin
a=6
B
b c2 a2
B
b = 10
A
b
6. Given: a 25, c 35
c2 a2 b2 ⇒ c 36 100
C
A
b
a
⇒ a c cos B 15 cos 54 8.82
c
A 90 71 19
C
C
c = 15
15 sin 54 12.14
3. Given: B 71, b 24
a 71°
54°
a
90 54 36
c
20°
C
tan B A 90 B
B
a
B
25
45.58
35
a
25
a
⇒ B arccos arccos
44.42
c
c
35
Section 6.7
7. Given: b 16, c 52
a
522
a
16
cos A 52
c = 52
a c2 b2 87.5601 9.36
1.32
b
b
81.97
cos A ⇒ A arccos arccos
c
c
9.45
sin B A arccos
16
72.08º
52
C
605
8. Given: b 1.32, c 9.45
B
162
2448 1217 49.48
Applications and Models
b
b
⇒ B arcsin
c
c
arcsin
b = 16 A
B 90 72.08 17.92
1.32
9.45
B
a
c = 9.45
8.03
A
C
b = 1.32
9. Given: A 1215, c 430.5
B 90 1215 7745
sin 12 15 a
430.5
a 430.5 sin 12 15 91.34
cos 12 15 10. Given: B 6512, a 14.2
A 90 B 90 6512 2448
cos B a
14.2
a
⇒ c
33.85
c
cos B cos 6512
tan B b
⇒ b a tan B 14.2 tan 6512 30.73
a
b
430.5
B
b 430.5 cos 12 15 420.70
a = 14.2
B
65°12′
c
C
c = 430.5
A
b
a
12°15′
b
C
11. tan A
h
1
⇒ h b tan 12b
2
1
h 4 tan 52 2.56 inches
2
θ
θ
1
h 10 tan 18 1.62 meters
2
θ
θ
1
b
2
1
b
2
1
b
2
b
b
13. tan h
1
⇒ h b tan 12b
2
h
h
1
b
2
12. tan h
1
⇒ h b tan 12b
2
1
h 46 tan 41 19.99 inches
2
14. tan h
1
⇒ h b tan 12b
2
1
h 11 tan 27 2.80 feet
2
h
θ
θ
1
b
2
1
b
2
b
h
θ
θ
1
b
2
1
b
2
b
606
Chapter 6
15. tan 25 Trigonometry
50
x
16. tan 20 50
50
x
tan 25
25°
x
x
107.2 feet
17. 16 sin 80 600
x
600
600
tan 20
20°
x
1648.5 feet
h
20
18. tan 33 20 sin 80 h
h
125
h 125 tan 33
20 ft
16 si 74 h 19.7 feet
h
h
81.2 feet
33°
125
80°
19. (a)
20. tan 51 h
h
100
h 100 tan 51
y
123.5 feet
x
h
47° 40′
50 ft
35°
(b) Let the height of the church x and the height of the
church and steeple y. Then,
x
y
tan 35 and tan 47 40 50
50
51°
100
x 50 tan 35 and y 50 tan 47 40
h y x 50tan 47 40 tan 35.
(c) h 19.9 feet
21. sin 34 x
4000
22. tan x 4000 sin 34º
75
50
23. (a)
arctan
2236.8 feet
3
56.3
2
1
12 2 ft
θ
1
17 3 ft
34°
(b) tan x
4000
75 ft
24. 12,500 4000 16,500
14.03
Angle of depression 90 14.03 75.97
θ
α
i
4000
16,500
16,500 mi
0m
4,00
arcsin
1212
35.8
1713
The angle of elevation of
the sum is 35.8.
50 ft
4000
16,500
1713
(c) arctan
θ
sin 1212
Not drawn to scale
Section 6.7
25. 1200 feet 150 feet 400 feet 950 feet
5 miles 5 miles
Not drawn to scale
feet
26,400 feet
5280
1 mile θ
950
feet
θ
5 miles
950
tan 26,400
artan
Applications and Models
26,400 2.06
950
(b) sin 18 26. (a) Since the airplane speed is
275sec60 min 16,500 min,
ft
sec
ft
s
after one minute its distance travelled is 16,500 feet.
a
sin 18 16,500
10,000
275(sin 18)
117.7 seconds
275s
a 16,500 sin 18 5099 ft
16500
10,000
275s
10,000
feet
18°
a
18°
x
4
27. sin 10.5 4
10.5°
x
x 4 sin 10.5
0.73 mile
θ
28.
29. The plane has traveled 1.5600 900 miles.
100x
12x = y
4 miles = 21
,120 feet
Angle of grade: tan 12x
100x
sin 38 a
⇒ a 554 miles north
900
cos 38 b
⇒ b 709 miles east
900
arctan 0.12 6.8
N
Change in elevation:
y
sin 21,120
900
52°
a
38°
b
W
y 21,120 sin 21,120 sinarctan 0.12
2516.3 feet
S
30. (a) Reno is 2472 sin 10 429 miles N of Miami.
N
Reno is 2472 cos 10 2434 miles W of Miami.
(b) The return heading is 280.
100°
Reno
W
2472 mi
80°
N
10°
W
S
10°
E
280°
S
E
Miami
E
607
608
Chapter 6
31.
Trigonometry
32.
N
N
W
E
1.4°
W
E
29°
120
428
a
88.6°
20
b
S
(a) cos 29 sin 29 a
⇒ a 104.95 nautical miles south
120
(a) t b
⇒ b 58.18 nautical miles west
120
(b) After 12 hours, the yacht will have traveled
240 nautical miles.
428
21.4 hours
20
240 sin 1.4 5.9 miles E
78.18
20 b
⇒ 36.7
a
104.95
(b) tan Not drawn to scale
S
240 cos 1.4 239.9 miles S
Bearing: S 36.7 W
(c) Bearing from N is 178.6.
Distance: d 104.952 78.182
130.9 nautical miles from port
33. 32, 68
(a) 90 32 58
32
(b)
90 22
Bearing from A to C: N 58º E
C 54
N
B
tan C d
φ
C
γ
β
α
50
β
θ
W
A
E
d
⇒ tan 54
50
d
⇒ d 68.82 meters
50
S
34. tan 14 tan 34 d
⇒ x d cot 14
x
35. tan d
d
y
30 x
cot 34 d
30 d cot 14
30 d cot 14
d
45
⇒ 56.3
30
36. Bearing 180 arctan
208.0 or 528 W
Bearing: N 56.3 W
N
N
45
Port
Plane
θ
W
30
160
E
Ship
d cot 34 30 d cot 14
d
W
30
cot 34 cot 14
tan 4 350
⇒ d 3071.91 ft
d
350
⇒ D 5005.23 ft
D
6.5°
4°
350
S1
d
S2
D
Not drawn to scale
Distance between ships: D d 1933.3 ft
85
Airport
S
S
5.46 kilometers
37. tan 6.5 85
160
E
Section 6.7
38. cot 55 d
⇒ d 7 kilometers
10
39. tan 57 D
cot 28 ⇒ D 18.8 kilometers
10
tan 16 Distance between towns:
tan 16 D d 18.8 7 11.8 kilometers
cot 16 28°
55°
10 km
T1
55°
d
28°
Applications and Models
a
⇒ x a cot 57
x
a
16°
x 55
6
550
60
x
a
a cot 57 55
6
a cot 57 55
6
a
55
⇒ a 3.23 miles
6
17,054 ft
h
x
h
2.5°
17
h
x
tan 2.5
x
9°
x − 17
41. L1: 3x 2y 5 ⇒ y 3
5
3
⇒ m1 x
2
2
2
L2: 3x y 1 ⇒ y x 1 ⇒ m2 1
Not drawn to scale
tan h
tan 9 x 17
x
a
57°
D
40. tan 2.5 P2
P1
H
a cot 16 a cot 57 T2
609
1 32
1 132 52
12
5
arctan 5 78.7
h
17
tan 9
h
h
17
tan 2.5 tan 9
h
17
1.025 miles
1
1
tan 2.5 tan 9
5410 feet
42. L1 2x y 8 ⇒ m1 2
L2 x 5y 4 ⇒ m2 tan m2 m1
1 m2m1
arctan
1
5
tan a
2a
arctan
1
m2 m1
5 2
arctan
1
1 m2m1
1 52
arctan
44. tan 43. The diagonal of the base has a length of
a2 a2 2a. Now, we have
1
2
a
1
2
θ
35.3.
2a
97 52.1
a2
2
a
45. sin 36 d
⇒ d 14.69
25
Length of side: 2d 29.4 inches
arctan 2 54.7º
a
2
θ
d
a
36°
25
610
Chapter 6
Trigonometry
46.
47.
a
48.
r
2
c
b 30°
r
30° 25
a
15°
y
c
x
sin 30 b
Length of side 2a 212.5
10
10
10
10
a
2
3r
50. tan 35°
10
12
18
arctan
b
10
cos b 10 tan 35 7
cos 35 3r
y 2b 2
b
tan 35 Distance 2a 9.06 centimeters
23r 25 inches
a
35°
10
4.53
b r cos 30
a 25 sin 30 12.5
49.
a
c
a c sin 15 17.5 sin 15
sin 15 b
cos 30 r
a
25
a
10
a
35
17.5
2
2
0.588 rad 33.7
3
18
a
f
a
18
21.6 feet
cos θ
6
c
φ
b
6
36
21.6
10.8 feet
f
2
10
12.2
cos 35
90 33.7 56.3
sin b
6
b
6
7.2 feet
sin c 10.82 7.22 13 feet
51. d 0 when t 0, a 4, Period 2
Use d a sin t since d 0 when t 0.
Thus, d 4 sin t.
d 3 sin
53. d 3 when t 0, a 3, Period 1.5
Use d a cos t since d 3 when t 0.
2
4
1.5 ⇒ 3
4
Amplitude: a 3
2
6 ⇒ Period:
3
2
2 ⇒ Thus, d 3 cos
52. Displacement at t 0 is 0 ⇒ d a sin t
54. Displacement at t 0 is 2 ⇒ d a cos t
Amplitude: a 2
Period:
4t
3 t 3 cos 3 .
t
3
2
10 ⇒ 5
d 2 cos
t
5
9
Section 6.7
55. d 4 cos 8t
56. d (a) Maximum displacement amplitude 4
(b) Frequency 8
2 2
1
cos 20 t
2
(a) Maximum displacement: a (b) Frequency:
4 cycles per unit of time
(c) d 4 cos 40 4
(d) 8 t Applications and Models
1
1
2
2
20
10 cycles per unit of time
2
2
(c) t 5 ⇒ d 1
⇒ t
2
16
611
1
1
cos 100 2
2
(d) Least positive value for t for which d 0
1
cos 20 t 0
2
cos 20 t 0
20 t arccos 0
57. d 1
sin 120t
16
58. d (a) Maximum displacement amplitude 120
(b) Frequency 2
2
60 cycles per unit of time
1
(c) d sin 600 0
16
(d) 120t ⇒ t 1
120
1
16
20 t 2
t
2
1
1
20 40
1
sin 792t
64
(a) Maximum displacement: a (b) Frequency:
792
396 cycles per unit of time
2
2
(c) t 5 ⇒ d 1
sin3960 0
64
(d) Least positive value for t for which d 0
1
sin 792 t 0
64
sin 792 t 0
792 t arcsin 0
792 t t
59.
d a sin t
Frequency 2
264 2
2264 528
1
1
64
64
1
792 792
612
Chapter 6
Trigonometry
60. At t 0, buoy is at its high point ⇒ d a cos t.
61. y Distance from high to low 2 a 3.5
1
cos 16t, t > 0
4
(a)
7
a 4
y
1
Returns to high point every 10 seconds:
2
10 ⇒ Period:
5
d
π
8
3π
8
π
2
t
−1
t
7
cos
4
5
(b) Period:
(c)
62. (a)
π
4
(b)
2 16
8
1
⇒ t
cos 16t 0 when 16t 4
2
32
L1
L2
L1 L2
0.1
2
sin 0.1
3
cos 0.1
23.0
0.5
0.2
2
sin 0.2
3
cos 0.2
13.1
0.6
0.3
2
sin 0.3
3
cos 0.3
9.9
0.4
2
sin 0.4
3
cos 0.4
8.4
L1 L2
L1
L2
2
sin 0.5
2
sin 0.6
3
cos 0.5
3
cos 0.6
0.7
2
sin 0.7
3
cos 0.7
7.0
0.8
2
sin 0.8
3
cos 0.8
7.1
7.6
7.2
The minimum length of the elevator is 7.0 meters.
(c)
2
3
L L1 L2 sin cos (d)
12
−2␲
2␲
−12
From the graph, it appears that the minimum length is
7.0 meters, which agrees with the estimate of part (b).
(c) A 8 8 16 cos 63. (a) and (b)
Base 1
Base 2
Altitude
Area
8
8 16 cos 10º
8 sin 10º
22.1
8
8 16 cos 20º
8 sin 20º
42.5
8
8 16 cos 30º
8 sin 30º
59.7
8
8 16 cos 40º
8 sin 40º
72.7
8
8 16 cos 50º
8 sin 50º
80.5
8 sin 2
16 16 cos 4 sin 641 cos sin (d)
100
0
90
0
8
8 16 cos 60º
8 sin 60º
83.1
8
8 16 cos 70º
8 sin 70º
80.7
The maximum occurs when 60 and is approximately
83.1 square feet.
The maximum of 83.1 square feet occurs when
60.
3
64. (a)
(b)
Average sales
(in millions of dollars)
S
15
1
a 14.3 1.7 6.3
2
2
12 ⇒ b b
6
12
9
Shift: d 14.3 6.3 8
6
3
S d a cos bt
t
2
4
6
8 10 12
Month (1 ↔ January)
(c) Period:
Applications and Models
S 8 6.3 cos
2
12
6
15
12
9
6
3
t
2
4
6
8 10 12
Month (1 ↔ January)
t
6
Note: Another model is S 8 6.3 sin
This corresponds to the 12 months in a year. Since the
sales of outerwear is seasonal this is reasonable.
613
S
Average sales
(in millions of dollars)
Section 6.7
6t 2 .
The model is a good fit.
(d) The amplitude represents the maximum displacement from
average sales of 8 million dollars. Sales are greatest in
December (cold weather Christmas) and least in June.
65. False. Since the tower is not exactly vertical, a right
triangle with sides 191 ft and d is not formed.
66. False. One period is the time for one complete cycle of
the motion.
67. No. N 24 E means 24 east of north.
68. Aeronautical bearings are always taken clockwise from
North (rather than the acute angle from a north-south line).
69. m 4, passes through 1, 2
70. Linear equation m 12 through 13, 0.
y
y 2 4x 1
y 12x b
7
6
b
y 2 4x 4
5
0
y 4x 6
3
0 16 b
2
y
12 13
3
2
1
b 16
1
x
−4 −3 −2 −1
−1
1
2
3
4
y
−3
1
2x
−2
1
6
−1
x
2
3
−1
−2
−3
71. Passes through 2, 6 and 3, 2
m
26
4
3 2
5
4
y 6 x 2
5
4
8
y6 x
5
5
22
4
y x
5
5
72. Linear equation through
y
m
7
6
23 1
2
y
1
4
3
1
3
4
4
3
2
1
−2 −1
−1
1
3
14, 32 and 21, 13.
x
1
2
3
4
5
2
1
4
3
−3
2
4
1
y x
3
3
4
4
1
y x
3
3
−2
−1
x
2
−1
−2
−3
3
614
Chapter 6
Trigonometry
Review Exercises for Chapter 6
1. 40
2. 269
3. (a) 70
4. (a) 280
y
y
280°
70°
x
x
(b) The angle lies in Quadrant I.
(b) The angle lies in Quadrant IV.
(c) Coterminal angles: 70 360 430
70 360 290
(c) Coterminal angles: 280° 360° 640°
280° 360° 80°
6. (a) 405
5. (a) 110
y
y
−405°
x
x
− 110°
(b) The angle lies in Quadrant III.
(b) The angle lies in Quadrant IV.
(c) Coterminal angles: 110 360 250
110 360 470
(c) Coterminal angles: 405° 720° 315°
405° 360° 45°
7. (a) 11
4
8. (a) 2
9
y
y
11π
4
2π
9
x
x
(b) The angle lies in Quadrant II.
(b) The angle lies in Quadrant I.
3
11
(c) Coterminal angles:
2 4
4
(c) Coterminal angles:
3
5
2 4
4
20
2
2 9
9
2
16
2 9
9
Review Exercises for Chapter 6
9. (a) 4
3
10. (a) 23
3
y
y
−
23π
3
x
x
−
4π
3
(b) The angle lies in Quadrant I.
(b) The angle lies in Quadrant II.
(c) Coterminal angles: 11. 480 480 2
4
2 3
3
4
10
2 3
3
rad 8
rad 8.378 rad
180
3
13. 16.5 16.5 rad
0.288 rad
180
15. 3345 33.75 33.75 16. 9825 98.416 18. 19677 197.283
20.
7 7
5
5
22. rad
1.718 rad
180
rad
3.443 rad
180
180
24. 8.3 8.3 180
rad 330
180
475.555
rad
180
rad 326.586
12. 120 120 s 11.52 m
60
180
113 rad
5
23
6 3
3
rad
2.094 rad
180
14. 127.5 127.5 rad
2.225 rad
180
17. 8415 84.25 84.25 19.
5 rad 5 rad
7
7
21. 180
27. 138 180
rad 108
23. 3.5 rad 3.5 rad 25. 1.75 rad rad
1.470 rad
180
rad 128.571
3
3 rad
rad 5
5
180
200.535
rad
1.75
180
rad 100.268
1
rad
138 23
radians
180
30
s r 20
28. s r 11 23
8 3
3
3
rad
rad 0.589 rad
180
16
rad 252
11
11
6
6
26. 5.7 5.7
(c) Coterminal angles: 2330 48.17 inches
615
616
Chapter 6
Trigonometry
29. (a) Angular speed 30. Linear speed 3313 2 radians
1 minute
66 23
Linear speed rad
272 inches5 second
1 2 1 5
r 65 2
2
2 6
A 55.31 mm2
34. opp 6, adj 6, hyp 62
666 23 inches
1 minute
400 inches per minute
radians per minute
212.1 inchessecond
32. A (b)
31. 120 120 2
radians
180
3
1
2
1
339.29 square inches
A r 2 182
2
2
3
33. opp 4, adj 5, hyp 42 52 41
sin opp
441
4
hyp 41
41
csc hyp 41
opp
4
cos adj
5
541
hyp 41
41
sec 41
hyp
adj
5
tan opp 4
adj
5
cot adj
5
opp 4
35. adj 4, hyp 8, opp 82 42 48 43
sin 2
opp
6
hyp 62
2
sin opp 43 3
hyp
8
2
cos 2
adj
6
hyp 62
2
cos adj
4 1
hyp 8 2
tan opp
1
adj
tan opp 43
3
adj
4
csc hyp 62
2
opp
6
csc 8
hyp
23
opp 43
3
sec hyp 62
2
adj
6
sec hyp
8
2
adj
4
cot adj
1
opp
cot 3
adj
4
opp 43
3
36. opp 5
adj hyp 9
92
52 214
sin opp 5
hyp 9
cos adj
214
hyp
9
tan 5
opp
514
adj
28
214
csc hyp 9
opp 5
sec hyp
9
914
adj
28
2 14
cot adj
214
opp
5
Review Exercises for Chapter 6
37. sin 1
3
38. tan 4
(a) cot 1
3
sin (a) csc (b) 1 tan2 sec2 (b) sin2 cos2 1
13
2
1
1
tan 4
sec2 1 16 17
cos2 1
sec 17
1
9
cos2 1 (c) cos 17
1
1
sec 17
17
cos2 8
9
cos 89
csc2 cos 22
3
csc (d) cot2 1 csc2 14
2
1
17
16
17
4
1
3
32
cos 22
4
1
2
1
sin 3
(d) tan cos 4
22
22
3
(c) sec 39. csc 4
40. csc 5
(a) sin 1
1
csc 4
(a) sin (b) sin2 cos2 1
1
4
2
cos2 (b) cot2 1 csc2 1
cot2 csc2 1 52 1 24
cos2 1 cot 26
1
16
cos2 15
16
cos 1516
cos 1
1
csc 5
(c) tan 6
1
1
cot 26
12
(d) sec90 csc 5
15
4
1
4
415
cos 15
15
1
15
sin 4
1
(d) tan cos 15
15
15
4
(c) sec 41. tan 33 0.6494
44. sec 79.3 1
5.3860
cos 79.3
42. csc 11 1
5.2408
sin 11
43. sin 34.2 0.5621
45. cot 1514 cot 15.2333 1
3.6722
tan 15.2333
617
618
Chapter 6
Trigonometry
46. cos 7811 58 cos 78.1994
47. cos
18 0.9848
48. tan
5
0.5774
6
0.2045
49. sin 110 x
3.5
m
3.5 k
1°10'
x 3.5 sin 1101000
tan 52 50.
x
x
Not drawn to scale
71.3 meters
25
x
25
19.5 feet
tan 52
25
52°
x
51. x 12, y 16, r 144 256 400 20
52. x 3, y 4, r 32 (4)2 5
sin y 4
r
5
csc 5
r
y 4
sin 4
y
r
5
csc r
5
y
4
cos x 3
r
5
sec r
5
x 3
cos x 3
r
5
sec r
5
x 3
tan y 4
x 3
cot x 3
y 4
tan y
4
x
3
cot x
3
y
4
5
2
53. x , y 3
2
r
54. x 2
3
2
5
2
2
241
6
5
2
y
15241
15
sin 241
r
241
241
6
2
x
4241
3
4
cos 241
r
241
241
6
5
y
2
15
tan x
2
4
3
241
csc r
y
6
5
2
2241 241
30
15
241
sec r
x
6
2
3
2
x
3
4
cot y
5
15
2
241
4
r
2
10
,y
3
3
103 32
2
2
226
3
2
26
y
3
sin r
26
226
3
10
526
x
3
cos r
26
226
3
2
y
1
3
tan x
5
10
3
r
csc y
226
3
26
2
3
226
26
3
10
5
3
10
3
x
5
cot y
2
3
r
sec x
Review Exercises for Chapter 6
1
9
55. x 0.5 , y 4.5 2
2
r
1
2
2
9
2
2
56. x 0.3, y 0.4
r (0.3)2 (0.4)2 0.5
82
2
92
y
982
9
r
82
822
82
82
x
12
1
cos r
82
822
82
y
92
9
tan x 12
sin csc 822
82
r
y
92
9
sec 822
r
82
x
12
cot x 12
1
y
92
9
57. x, 4x , x > 0
sin y 0.4 4
r
0.5 5
csc r
0.5 5
y
0.4 4
cos x 0.3 3
r
0.5 5
sec 0.5 5
r
x 0.3 3
tan y 0.4 4
x 0.3 3
cot x 0.3 3
y 0.4 4
58. x 2x, y 3x, x > 0
x x, y 4x
r (2x)2 (3x ) 2 13x
r x2 4x2 17 x
sin y
3x
313
r
13x
13
cos x
2x
213
r
13x
13
tan y 3x 3
x 2x 2
csc 13x
13
r
y
3x
3
17x
r
17
sec x
x
sec 13x
13
r
x
2x
2
x
x
1
y 4x 4
cot x 2x 2
y 3x 3
sin y
4x
17x
r
417
17
17
x
x
cos 17x
r
17
y 4x
4
tan x
x
17x
17
r
csc y
4x
4
cot 619
6
59. sec , tan < 0 ⇒ is in Quadrant IV.
5
3
60. csc , cos < 0 ⇒ is in Quadrant II.
2
r 6, x 5, y 36 25 11
r 3, y 2, x 32 22 5
sin 11
y
r
6
csc r
611
y
11
sin y 2
r
3
cos x 5
r
6
sec r
6
x 5
cos 5
x
r
3
tan 11
y
x
5
cot x
511
y
11
y
25
2
x
5
5
3
csc 2
tan r
3
35
x 5
5
x 5
cot y
2
sec 620
Chapter 6
Trigonometry
5
61. tan , cos < 0 ⇒ is in Quadrant III.
4
3
62. sin , cos < 0 ⇒ is in Quadrant II.
8
x 4, y 5
y 3, r 8, x 55
r (4)2 (5)2 41
y
5
541
r
41
41
4
x
441
cos r
41
41
5
tan 4
sin sin y 3
r
8
cos 55
x
r
8
tan 3
y
355
55
x
55
csc 41
41
r
y
5
5
csc 8
r
y 3
sec 41
41
r
x
4
4
sec r
8
855
x
55
55
cot x 4 4
y 5 5
cot 55
x
y
3
63. tan y
12
⇒ r 13
x
5
64. cos sin > 0 ⇒ is in Quadrant II ⇒ y 12, x 5
x 2
⇒ y2 21
r
5
sin > 0 ⇒ is in Quadrant II ⇒ y 21
sin y 12
r
13
csc r
13
y 12
sin y 21
r
5
cos x
5
r
13
sec r
13
x
5
tan 21
y
x
2
tan y
12
x
5
cot 5
x
y
12
csc 5
r
521
y 21
21
sec r
5
5
x 2
2
cot x
2
221
y 21
21
65. 264
66. 635 720 85
y
264 180 84
y
85
264°
635°
x
x
θ′
θ′
67.
6
5
6
4
2 5
5
y
68. 17 18 3
3
3
6 θ′
3
y
17π
3
x
4 5
5
−
6π
5
x
3
θ′
Review Exercises for Chapter 6
3
3
2
69. sin
70. sin
2
4
2
71. sin
621
5
1
sin 6
6
2
cos
1
3
2
cos
2
4
2
cos
3
5
cos 6
6
2
tan
3
3
tan
2 2
1
4
2
2
tan
3
5
tan 6
6
3
3
5
5
sin sin 2 3
3
3
2
72. sin
3
7
sin 3
3
2
7
1
cos 3
3
2
7
tan 3
3
3
73. sin cos
5
5
1
cos cos 2 3
3
3
2
cos tan
3
5
1
3
3
2
2
tan 74. 5
3
is coterminal with .
4
4
5
3
3
2
sin
sin sin 4
4
4
4
2
2
5
3
3
cos
cos cos 4
4
4
4
2
2
2
5
1
4
2
2
sin cos tan 75. sin 495 sin 45 2
76. sin 120 sin180 120 sin 60 2
cos 495 cos 45 2
tan 495 tan 45 1
tan 120 3
2
1
2
21 3
1
2
cos150 cos 30 tan150 tan 30 2
cos 120 cos180 120 cos 60 2
77. sin150 sin 30 3
3
2
3
3
78. 420 is coterminal with 300.
sin420 sin 300 sin360 300 sin 60 cos420 cos 300 cos360 300 cos 60 tan420 79. sin 4 0.0698
3
2
3
2
1
2
1
3
2
80. tan 231 1.2349
81. sec 2.8 1
1.0613
cos 2.8
622
Chapter 6
Trigonometry
82. cos 5.5 0.7087
85. t sin
2
1 3
, x, y ,
3
2 2
83. sin 17
0.4067
15
86. cos
3
2
y
3
2
sin
2
1
cos
x
3
2
tan
87. t sin
25
4.3813
7
2
7
4
2
y
74 22
x
7
tan
1
4
(
2 y 32
3
3
x
12
3
7
1
, x, y ,
6
2
2
88. cos
7
1
y
6
2
sin
3
7
cos
x
6
2
tan
84. tan 34 22
34 (
−
2, 2
2 2
2,
2
−
2
2
(
y
(
2
2
x
3
1
tan
4
3
12
1
7 y
6
x 32 3
3
89. y sin x
90. y cos x
y
Amplitude: 1
Period: 2
y
2
Amplitude: 1
1
Period: 2
x
π
2
− 3π
2
−π
Amplitude: 3
Period:
2
1
2
π
x
2π
−1
−2
−2
91. y 3 cos 2x
2
92. y 2 sin x
y
Amplitude: 2
3
2
Period:
1
x
1
2
2
2
y
3
1
x
1
3
−1
−2
−3
93. f x 5 sin
2x
5
Amplitude: 5
Period:
2
2
5
4x y
94. f (x) 8 cos 6
2
−2
8
Amplitude: 8
4
5
y
6π
x
Period:
2
1
4
8
4π
−4
−6
−6
−8
8π
x
Review Exercises for Chapter 6
95. y 2 sin x
96. y 4 cos x
y
Amplitude: 1
4
Amplitude: 1
Period: 2
3
Period: 2
y
−3
−2
−1
−π
π
−1
2π
−1
−5
−6
98. g(t) 3 cos(t )
y
Period: 2
3
−3
97. gt 52 sint Amplitude:
2
x
−2
5
2
x
1
−2
2
Shift the graph of y sin x
two units upward.
623
y
4
Amplitude: 3
4
3
Period: 2
3
2
1
−1
1
π
t
t
π
−2
−3
−3
−4
−4
99. y a sin bx
100. (a) st 18.09 1.41 sin
2
1
⇒ b 528
(a) a 2,
b
264
6t 4.60
22
y 2 sin528x
(b) f 1
1
264
264 cycles per second
0
12
14
2
26 12
6
12 months 1 year, so this is expected.
(b) Period:
(c) Amplitude: 1.41
The amplitude represents the maximum change in
time from the average time d 18.09 of sunset.
101. f x tan x
102. f (t) tan t 4
103. f x cot x
Graph of tant shifted to the
right by 4.
y
4
3
y
4
3
y
2
2
1
3
π
x
1
2
−π
π
1
−π
2
π
2
t
−3
−4
x
624
Chapter 6
Trigonometry
Period:
2
106. h(t) sec t Graph y cos x first.
Graph of sec t shifted to the right
by .
4
y
y
4
105. f x sec x
104. g(t) 2 cot 2t
3
y
2
1
−π
π
−π
t
x
π
−1
1
−2
−3
t
π
−4
107. f x csc x
108. f (t) 3 csc 2t Graph y sin x first.
4
109. f x x cos x
Graph y1 x cosx
y
y2 x
y
4
y3 x
2
3
2
π
1
− 3π
2
As x → , f x → .
t
6
x
π
2
−9
−3
9
−4
−6
110. g(x) ex cos x
21 arcsin 12 6
111. arcsin 300
Graph y ± ex first.
−1
7
−150
112. arcsin(1) 2
115. sin 1 0.44 0.46 radian
118. arccos
22 4
121. arccos 0.324 1.24 radians
124. tan1(8.2) 1.45 radians
113. arcsin 0.4 0.41 radian
114. arcsin(0.213) 0.21
116. sin1(0.89) 1.10
117. arccos
119. cos 11 120. cos1
122. arccos(0.888) 2.66 radians
123. tan11.5 0.98 radian
3
2
6
23 6
Review Exercises for Chapter 6
125. f x 2 arcsin x 2 sin1x
127. f x arctan
126. y 3 arccos x
625
2x tan 2x 1
3␲
␲
␲
2
−1.5
1.5
−4
−1.5
4
1.5
0
−␲
−␲
2
129. cosarctan 34 45. Use a right triangle.
128. f x arcsin 2x
Let arctan 34. Then
␲
2
5
tan 34 and cos 45.
−1.5
3
θ
1.5
4
−␲
2
130. tanarccos 35 43
12
13
131. secarctan 5 5
Use a right triangle. Let
Use a right triangle. Let
u arccos 35. Then
arctan 12
5 . Then
3
5
cos u and
5
13
12
13
tan 12
5 and sec 5 .
4
θ
tanarccos 35 tan u 43.
5
u
3
5
132. arcsin 12
13 12
5
Use a right triangle. Let u arcsin 12
13 . Then
u
sin u 12
13 and
5
cotarcsin 12
13 cot u 12 .
133. Let y arccos
Then cos y 2x .
4 x 2
x
2
x
2
arcsinx 1 ⇒ 4 − x2
≤ ≤
2
2
sin x 1
cos 12 x 12
y
.
−12
134. secarcsinx 1
x
and
2
tan y tan arccos
13
x
1
x2 x
sec 1
x2 x
θ
12 − (x − 1)2
x −1
626
Chapter 6
135. tan Trigonometry
70
30
136.
137.
r
arctan
70
66.8
30
h
1808 ft
21°
25.2°
r
25 feet
tan 21 1808
4246.33 ft
sin 25.2
h
25
h 25 tan 21 9.6 feet
138. sin 48 d1
⇒ d1 483
650
cos 25 d2
⇒ d2 734
810
d
cos 48 3 ⇒ d3 435
650
sin 25 d4
⇒ d4 342
810
N
d1 d2 1217
48°
W
d3
d3 d4 93
A
B
25°
48° 65° 810
650
D
θ
d1
d2
E
d4
C
S
tan 93
⇒ 4.4
1217
sec 4.4 D
⇒ D 1217 sec 4.4 1221
1217
The distance is 1221 miles and the bearing is 85.6.
139. cos 45 140. High point at time t 0 ⇒ d a cos bt
a
⇒
60
a
a 42.43 nautical miles north
sin 45 b
⇒
60
1
1.5 0.75 inches
2
Period 3 seconds N
b
b 42.43 nautical miles east
d 0.75 cos
a 45°
60
W
2
2
⇒ b
b
3
23t
E
S
141. False. The sine or cosine functions are often useful for
modeling simple harmonic motion.
142. True. The inverse sine, y arcsin x, cannot be defined as
a function over any interval that is greater than the
interval defined as ≤ y ≤ .
2
2
143. False. For each there corresponds exactly one
value of y.
144. False.
arctan(1) 4
3
is not in the range of the arctan function.
4
Review Exercises for Chapter 6
145. y 3 sin x
146. y 3 sin x
147. y 2 sin x
Amplitude: 3
Amplitude: 2
Amplitude: 3
Period: 2
Period: 2
Period: 2
a < 0 ⇒ graph is reflected in
the x-axis
Matches graph d.
627
Matches graph b.
Matches graph (a).
x
2
148. y 2 sin
149. f sec is undefined at the zeros of g cos 1
since sec .
cos Amplitude: 2
Period: 4
Matches graph (c).
150. (a)
tan 2
cot (b) tan 0.1
0.4
0.7
1.0
1.3
9.9666
2.3652
1.1872
0.6421
0.2776
9.9666
2.3652
1.1872
0.6421
0.2776
cot 2
151. The ranges for the other four trigonometric functions are
not bounded. For y tan x and y cot x, the range is
, . For y sec x and y csc x, the range is
, 1 1, .
1
152. y Aekt cos bt 5 et10 cos 6t
1
1
(a) If A is changed from 5 to 3, the amplitude of each
oscillation is increased.
1
1
(b) If k is changed from 10 to 3, the oscillations are
damped more quickly.
(c) If b is changed from 6 to 9, the frequency of the
oscillations increases.
153. (a) tan x
12
(b)
800
x 12 tan Area Area of triangle Area of sector
0
1
1
bh r 2
2
2
0
As ⇒
1
1
1212 tan 122
2
2
72 tan 72
72tan x
θ
12
␲
2
, ⇒ .
2
The area increases without bound as approaches .
2
628
Chapter 6
Trigonometry
1
1
(b) A 2 r 2 210 2 50, > 0
1
1
154. (a) A 2 r 2 2 r 20.8 0.4r 2, r > 0
s r 10, > 0
s r 0.8r, r > 0
30
4
A
A
s
0
s
3
0
6
0
0
The area function increases more rapidly than the
arc length function because it is a function of the
square of the radius, while the arc length function is
a function of the radius.
155. Answers will vary.
Problem Solving for Chapter 6
1. (a) 8:57 6:45 2 hours 12 minutes 132 minutes
132 11
revolutions
48
4
1142 112 radians or 990
Gear 2:
24
360 332.308 5.80 radians
26
Gear 3:
24
360 392.727 6.85 radians
22
Gear 4:
40
5
360 450 radians
32
2
Gear 5:
24
360 454.737 7.94 radians
19
(b) s r 47.255.5 816.42 feet
3. (a) sin 39 d
3000
d
3000
4767 feet
sin 39
(b) tan 39 x
24
3
radians
360 270 32
2
2. Gear 1:
3000
x
3000
3705 feet
tan 39
(c) tan 63 w 3705
3000
3000 tan 63 w 3705
w 3000 tan 63 3705 2183 feet
4. (a) ABC, ADE, and AFG are all similar triangles since they all have the same angles. A is part of
all three triangles and C E G 90. Thus, B D F.
(b) Since the triangles are similar, the ratios of corresponding sides are equal.
BC DE
FG
AB
AD AF
opp BC DE FG
sin A it does not matter which triangle is used to calculate sin A.
hyp
AB
AD
AF
Any triangle similar to these three triangles could be used to find sin A. The value of sin A would not change.
(c) Since the ratios:
(d) Since the values of all six trigonometric functions can be found by taking the ratios of the sides of a right triangle,
similar triangles would yield the same values.
Problem Solving for Chapter 6
5. (a) hx cos2 x
(b) hx sin2 x
3
3
−2␲
−2␲
2␲
2␲
−1
−1
h is even.
h is even.
6. Given: f is an even function and g is an odd function.
hx f x2
(a)
(b)
hx gx2
hx gx2
hx f x2
f x2 since f is even
gx2 since g is odd
hx
gx2
hx
Thus, h is an even function.
Thus, h is an even function.
Conjecture: The square of either an even function or an odd function is an even function.
7. If we alter the model so that h 1 when t 0, we can use either a sine or a cosine model.
1
1
a max min 101 1 50
2
2
1
1
d max min 101 1 51
2
2
b 8
For the cosine model we have: h 51 50 cos8 t
For the sine model we have: h 51 50 sin 8 t 2
Notice that we needed the horizontal shift so that the sine value was one when t 0 .
Another model would be: h 51 50 sin 8 t 3
2
Here we wanted the sine value to be 1 when t 0.
8. P 100 20 cos
(a)
83t
(b) Period 130
2
6 3
sec
8
8 4
3
This is the time between heartbeats.
0
5
70
(c) Amplitude: 20
The blood pressure ranges between 100 20 80
and 100 20 120.
(e) Period 64 60 15
sec
64 16
64
32
60
⇒ b
2 15 2b
60
(d) Pulse rate 60 secmin
80 beatsmin
3
4 secbeat
629
630
Chapter 6
Trigonometry
9. Physical (23 days): P sin
2 t
,t ≥ 0
23
Emotional (28 days): E sin
2 t
,t ≥ 0
28
Intellectual (33 days): I sin
2 t
,t ≥ 0
33
(a)
2
E
P
I
7300
7380
−2
20 years
5
leap years
11
remaining
July days
31
t 365 20
(b) Number of days since birth until September 1, 2006:
August days
1
day in
September
t 7348
2
7349
I
7379
P
E
−2
All three drop early in the month, then peak toward the middle of the month, and drop again
toward the latter part of the month.
(c) For September 22, 2006, use t 7369.
P 0.631
E 0.901
I 0.945
10. f x 2 cos 2x 3 sin 3x
gx 2 cos 2x 3 sin 4 x
(a)
(b) The graphs intersect when x 5.35 32 0.65.
6
g
−␲
11. (a) Both graphs have a period of 2 and intersect when
x 5.35. They should also intersect when
x 5.35 2 3.35 and x 5.35 2 7.35.
␲
f
−6
(b) The period of f x is 2.
The period of gx is .
(c) hx A cos x sin x is periodic since the sine
and cosine functions are periodic.
(c) Since 13.35 5.35 42 and 4.65 5.35 52
the graphs will intersect again at these values. Therefore
f 13.35 g4.65.
Problem Solving for Chapter 6
12. (a) f t 2c f t is true since this is a two period
horizontal shift.
13.
θ1
1
1
(b) f t c f t is not true.
2
2
x
(a)
1
f t is a doubling of the period of f t.
2
d
y
θ2
2 ft
1
f t c is a horizontal translation of f t.
2
631
sin 1
1.333
sin 2
sin 2 1
1
(c) f
t c f t is not true.
2
2
sin 1 sin 60
0.6497
1.333
1.333
2 40.52
1
1
1
f
t c f t c is a horizontal
2
2
2
1
translation of f t by half a period.
2
(b) tan 2 tan 1 12 2 sin12.
For example, sin
x
⇒ x 2 tan 40.52 1.71 feet
2
y
⇒ y 2 tan 60 3.46 feet
2
(c) d y x 3.46 1.71 1.75 feet
(d) As you more closer to the rock, 1 decreases, which
causes y to decrease, which in turn causes d to
decrease.
14. arctan x x (a)
x3 x5 x7
3
5
7
(b)
2
−␲
2
␲
2
−2
The graphs are nearly the same for 1 < x < 1.
2
−␲
2
␲
2
−2
The accuracy of the approximation improved slightly by
adding the next term x99.
632
Chapter 6
Chapter 6
Trigonometry
Practice Test
1. Express 350° in radian measure.
2. Express 59 in degree measure.
3. Convert 1351412 to decimal form.
4. Convert 22.569 to DMS form.
5. If cos 23, use the trigonometric identities to find tan .
6. Find given sin 0.9063.
7. Solve for x in the figure below.
8. Find the magnitude of the reference angle for 65.
35
20°
x
10. Find sec given that lies in Quadrant III and tan 6.
9. Evaluate csc 3.92.
x
11. Graph y 3 sin .
2
12. Graph y 2 cosx .
13. Graph y tan 2x.
14. Graph y csc x 15. Graph y 2x sin x, using a graphing calculator.
16. Graph y 3x cos x, using a graphing calculator.
17. Evaluate arcsin 1.
18. Evaluate arctan3.
19. Evaluate sin arccos
.
4
4
.
35
20. Write an algebraic expression for cos arcsin
For Exercises 21–23, solve the right triangle.
21. A 40, c 12
22. B 6.84, a 21.3
23. a 5, b 9
B
c
A
b
a
C
24. A 20-foot ladder leans against the side of a barn. Find the height of the top of the ladder if
the angle of elevation of the ladder is 67°.
25. An observer in a lighthouse 250 feet above sea level spots a ship off the shore. If the
angle of depression to the ship is 5°, how far out is the ship?
x
.
4