# C H A P T E R 4 Trigonometry

```C H A P T E R
Trigonometry
4
Section 4.1
Radian and Degree Measure . . . . . . . . . . . . . . . . 335
Section 4.2
Trigonometric Functions: The Unit Circle . . . . . . . . . 344
Section 4.3
Right Triangle Trigonometry . . . . . . . . . . . . . . . . 350
Section 4.4
Trigonometric Functions of Any Angle
Section 4.5
Graphs of Sine and Cosine Functions . . . . . . . . . . . 373
Section 4.6
Graphs of Other Trigonometric Functions . . . . . . . . . 386
Section 4.7
Inverse Trigonometric Functions . . . . . . . . . . . . . . 397
Section 4.8
Applications and Models . . . . . . . . . . . . . . . . . . 410
. . . . . . . . . . 360
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Practice Test
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
C H A P T E R
Trigonometry
Section 4.1
4
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 180.
(d) Straight: Measure 180.
■
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.
■
■
■
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.
■
Linear speed ■
Angular speed t srt
arc length s
time
t
Vocabulary Check
1. Trigonometry
2. angle
3. coterminal
5. acute; obtuse
6. complementary; supplementary
7. degree
8. linear
10. A 12r 2
9. angular
1.
2.
3.
The angle shown is approximately
The angle shown is approximately
4.
The angle shown is approximately
6.
5.
The angle shown is approximately
The angle shown is approximately
The angle shown is approximately
335
336
Chapter 4
7. (a) Since 0 <
(b) Since <
Trigonometry
<
5
2 5
8. (a) Since <
7 3 7
<
;
5
2
5
(b) Since <
< < 0; 2
12
12
10. (a) Since (b) Since < 2 < ; 2 lies in Quadrant III.
2
(b) Since 9. (a) Since 11. (a) Since < 3.5 <
(b) Since
13. (a)
3
; 3.5 lies in Quadrant III.
2
12. (a) Since
< 2.25 < ; 2.25 lies in Quadrant II.
2
5
4
14. (a) y
9 3 9
<
;
8
2 8
< 1 < 0; 1 lies in Quadrant IV.
2
11
3
11
< < , 2
9
9
3
< 6.02 < 2 ; 6.02 lies in Quadrant IV.
2
(b) Since 7
4
11 3 11
<
;
8
2 8
3
< 4.25 < ; 4.25 lies in Quadrant II.
2
15. (a)
y
11
6
y
5π
4
11π
6
x
x
x
7π
−
4
(b) 2
3
(b)
5
2
y
(b) 3
y
y
5π
2
x
x
x
−3
2π
−
3
16. (a) 4
(b) 7
y
y
7π
4
x
x
Section 4.1
17. (a) Coterminal angles for
6
18. (a)
13
2 6
6
7
19
2 6
6
19. (a) Coterminal angles for
(b) (b) Coterminal angles for
5
6
2
4
2 3
3
11
2 6
6
11
23
2 6
6
(b) Coterminal angles for
5
17
2 6
6
25
2 12
12
5
7
2 6
6
23
2 12
12
20. (a) (b) 9
2 4
4
21. (a) Complement:
9
7
4 4
4
22. (a) Complement:
Supplement: 5
2
12
12
Supplement: 2
3
3
(b) Complement: Not possible,
2
32
2 15
15
23. (a) Complement:
12
2
3
6
Supplement: 2
28
2 15
15
2
3
2
8
2 3
3
7
5
2 6
6
11
2 6
6
337
3
is greater than .
4
2
3 4
4
1 0.57
2
Supplement: 1 2.14
11
12
12
(b) Complement: Not possible, 2 is greater than .
2
11
(b) Complement: Not possible,
is greater than .
12
2
Supplement: 2 1.14
11
Supplement: 12
12
24. (a) Complement: Not possible, 3 is greater than
.
2
25.
Supplement: 3 0.14
(b) Complement:
1.5 0.07
2
The angle shown is approximately 210º .
Supplement: 1.5 1.64
27.
26.
The angle shown is approximately
120.
28.
The angle shown is approximately
60º.
The angle shown is approximately
330.
338
Chapter 4
Trigonometry
29.
30.
The angle shown is approximately 165.
31. (a) Since 90 < 130 < 180, 130 lies in Quadrant II.
(b) Since 270 < 285 < 360, 285 lies in Quadrant IV.
33. (a) Since 180 < 132 50 < 90, 132 50
(b) Since 360 < 336 < 270, 336 lies in
The angle shown is approximately 10.
32. (a) Since 0 < 8.3 < 90, 8.3 lies in Quadrant I.
(b) Since 180 < 257 30 < 270, 257 30 lies in
34. (a) Since 270 < 260 < 180, 260 lies in
(b) Since 90 < 3.4 < 0, 3.4 lies in Quadrant IV.
36. (a) 270
35. (a) 30
y
y
30°
x
x
− 270°
(b) 120
(b) 150
y
y
150°
x
x
− 120°
38. (a) 750
37. (a) 405
y
y
405°
x
x
− 750°
(b) 600
(b) 480
y
y
480°
− 600°
x
x
Section 4.1
39. (a) Coterminal angles for 45
40. (a) 120 360 480
45 360 405
41. (a) Coterminal angles for 240
120 360 240
45 360 315
(b) Coterminal angles for 36
240 360 600
(b) 420 720 300
240 360 120
420 360 60
36 360 324
339
(b) Coterminal angles for 180
180 360 180
36 360 396
180 360 540
43. (a) Complement: 90 18 72
42. (a) 420 720 300
Supplement: 180 18 162
420 360 60
(b) Complement: Not possible, 115 is greater than 90 .
(b) 230 360 590
Supplement: 1180 115 65
230 360 130
44. (a) Complement: 90 3 87
45. (a) Complement: 90 79 11
Supplement: 180 3 177
Supplement: 180 79 101
(b) Complement: 90 64 26
(b) Complement: Not possible, 150 is greater than 90.
Supplement: 180 64 116
Supplement: 180 150 30
47. (a) 30 30
46. (a) Complement: Not possible, 130 is greater than 90.
Supplement: 180 130 50
(b) 150 150
(b) Complement: Not possible, 170 is greater than 90.
180 6
180 5
6
Supplement: 180 170 10
48. (a) 315 315
7
4
2
3
180 (b) 120 120
180 49. (a) 20 20
(b) 240 240
51. (a)
3 3 180 270
2
2 52. (a) (b)
7 7 180 210
6
6 (b)
54. (a)
11 11 180
330
6
6
(b)
34 34 180
408
15
15 59. 532 532
50. (a) 270 270
4
180 3
(b) 144 144
7
7 180
105
12
12 53. (a)
180
20
9
9 55. 115 115
57. 216.35 216.35
180 9
4
5
11
11 180 66
30
30 180
58. 48.27 48.27
60. 345 345
180 56. 87.4 87.4
3
7 7 180 420
3
3 (b) 180
180 2
340
Chapter 4
61. 0.83 0.83
Trigonometry
62. 0.54 0.54
5 5 180
81.818
11
11 63.
180 25.714
7
7 64.
66.
13 13 180
1170.000
2
2
67. 4.2 4.2
65.
180 15 15 180 337.500
8
8
68. 4.8 4.8
864.000
180
756.000
69. 2 2
180 70. 0.57 0.57
114.592
71. (a) 54 45 54 45
60 54.75
32.659
180
72. (a) 24510 245 10
60 245 0.167 245.167
(b) 212 2 12
60 2 0.2 2.2
(b) 128 30 128 30
60 128.5
18
30
73. (a) 85 18 30 85 60 3600 85.308
36 74. (a) 135 36 135 3600
25
(b) 330 25 330 3600 330.007
135 0.01 135.01
16
20
(b) 408 16 20 408 60 3600 408 0.2667 0.0056
408.272
76. (a) 345.12 345 0.1260 75. (a) 240.6 240 0.660 240 36
345 7 0.260 (b) 145.8 145 0.860 145 48
345 7 12
(b) 0.45 0 0.4560 0 27 0 27
78. (a) 0.355 0 0.35560 77. (a) 2.5 2 30
0 21 0.360 (b) 3.58 3 0.5860 0 21 18 0 21 18
3 34 0.860 (b) 0.7865 0 0.786560
3 34 48
0 47 0.1960 0 47 11.4 0 47 11.4
79. s r
6 5
6
5
82.
s r
60 75
60 4
75 5
Because the angle represented is
clockwise, this angle is 45 radians.
80.
s r
29 10 29
83. s r
6 27
6
2
27 9
81.
s r
32 7
4
32
84. r 14 feet, s 8 feet
s
8
4
r 14 7
Section 4.1
85.
86. r 80 kilometers,
s 160 kilometers
s r
25 14.5
88. r 9 feet, 60 3
s 15180
90. r 20 centimeters, s 31 3 meters
180 15 inches
47.12 inches
3 3 feet
341
s 160
r
80
25
50
14.5 29
s r 9
s r 20
9.42 feet
4
4 5 centimeters
15.71 centimeters
1
91. A r 2
2
92. r 12 mm, 8
1
square inches
A 42
2
3
3
4
1
93. A r 2
2
1
A 2.52225
2
180
1
1
A r 2 122
2
2
4
8.38 square inches
12.27 square feet
18 mm2
56.55 mm2
95. 41 15 50 32 47 39
94. r 1.4 miles, 330
21.56
1
330
A 1.42
5.64 square miles
2
180
12
96. r 4000 miles
47 37 18 37 47 36 9 49 42
s r 40000.14782 591.3 miles
97. 450
s
r 6378
99. s 2.5 25
5
r
6
60 12
s r 40000.1715 686.2 miles
98. r 3189 kilometers
s r
400 6378
400
6378
100. s 24
180
275
r
5
101. (a) 65 miles per hour 655280
5720 feet per minute
60
The circumference of the tire is C 2.5 feet.
The number of revolutions per minute is
5720
728.3 revolutions per minute
r
2.5
(b) The angular speed is .
t
5720
2.5
1 minute
342
Chapter 4
Trigonometry
102. Linear velocity for either pulley: 17002 3400 inches per minute
(a) Angular speed of motor pulley: v 3400
r
1
v 3400
r
2
Angular speed of the saw arbor: (b) Revolutions per minute of the saw arbor:
103. (a) Angular speed 1700
850 revolutions per minute
2
1 minute
104. (a) 4 rpm 42 radians per minute
(b) Linear speed (b) r 25 ft
1 ft
in.
52002
7.25
2
12 in. r
200 feet per minute
t
1 minute
Linear speed 2525.13274 feet per minute
2
3141 feet per minute
3
628.32 feet per minute
9869.84 feet per minute
105. (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
106. A R 2 r 2
2
R 25
25
r 25 14 11
A
1
107. A r 2
2
1 125
2 180
1
352140
2
180
125°
r
14
252 112 175 549.8 square inches
476.39 square meters
1496.62 square meters
108. (a) Arc length of larger sprocket in feet: s r
2
1
feet
s 2 3
3
Therefore, the chain moves 23 feet, as does the smaller rear sprocket. Thus, the angle of the
smaller sprocket is r 2 inches 212 feet.
s 23 feet
4 and the arc length of the tire in feet is:
r
212 feet
s r
s 4
Speed 14
feet
14
12 3
s
143
14
feet per second
t
1 second
3
14 feet
3600 seconds
1 mile
10 miles per hour
3 seconds
1 hour
5280 feet
—CONTINUED—
140°
35
Section 4.1
343
108. —CONTINUED—
(b) Since the arc length of the tire is 143 feet and the cyclist is pedaling at a rate of one revolution per second,
we have:
Distance feet
1 mile
7
n revolutions n miles
143 revolutions
5280
feet 7920
(c) Distance Rate Time
7
1 mile
t seconds t miles
143 feet per second5280
feet 7920
(d) The functions are both linear.
109. False. An angle measure of 4 radians corresponds to
two complete revolutions from the initial to the terminal
side of an angle.
110. True. If and are coterminal angles, then
n360 where n is an integer. The difference
between and is n360 2n.
111. False. The terminal side of 1260 lies on the negative x-axis.
112. (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 of the terminal side.
(c) Two angles in standard position with the same terminal sides are coterminal.
(d) An obtuse angle measures between 90° and 180°.
113. Increases, since the linear speed is proportional to the radius.
so one radian is much larger than one degree.
116. The area of a circle is A r 2 ⇒ C2
C
115. The arc length is increasing. In order for the angle to
remain constant as the radius r increases, the arc length
s must increase in proportion to r, as can be seen from
the formula s r .
A
. The circumference of a circle is C 2 r.
r2
r r
A
2
2A
r
Cr
A
2
For a sector, C s r. Thus, A 117.
4
4
42 42
2
2
r r 1 2
r for a sector.
2
2
42 2
8
2
119. 22 62 4 36 40 4
118.
10 210
5
55
210 2
105 25 12 2 5 2 2
2
120. 172 92 289 81
208 16
13 413
52
4
344
Chapter 4
Trigonometry
121. f x x 25
y
Graph of y x 5 shifted
to the right by two units
122. f x x5 4
y = x5
y
4
Vertical shift four units
downward
3
2
2
1
−3
x
−2
2
3
4
x
−2
1
y = x5 − 4
−2
−6
y = (x − 2)5
−3
123. f x 2 x 5
124. f x x 35
y
Graph of y x 5 reflected
in x-axis and shifted
upward by two units
6
4
y = x5
3
x
−1
1
−2
2
3
■
■
■
y = x5
1
x
1
2
−1
−3
Trigonometric Functions: The Unit Circle
You should know the definition of the trigonometric functions in terms of the unit circle. Let t be a real number and x, y
the point on the unit circle corresponding to t.
sin t y
csc t 1
,
y
y0
cos t x
sec t 1
,
x
x0
y
tan t , x 0
x
cot t x
,
y
y0
The cosine and secant functions are even.
cost cos t
■
2
−2
y = 2 − x5
−3
Section 4.2
y
3
−5 −4 −3
1
−2
y = −(x + 3)5
Reflection in the x-axis
and a horizontal shift
three units to the left
5
−3
3
−2
y = x5
−1
2
sect sec t
The other four trigonometric functions are odd.
sint sin t
csct csc t
tan t tan t
cott cot t
Be able to evaluate the trigonometric functions with a calculator.
Vocabulary Check
1. unit circle
2. periodic
3. period
4. odd; even
Section 4.2
1. x 8
15
,y
17
17
sin y 5
13
csc 1 13
y
5
8
17
sec 17
1
x
8
cos x 12
13
sec 1 13
x
12
y
15
x
8
cot x
8
y
15
tan y
5
x 12
cot x 12
y
5
4
3
4. x , y 5
5
12
5
,y
13
13
sin y cos x tan 12
5
,y
13
13
1 17
y
15
15
17
cos x 3. x 2. x csc sin y tan Trigonometric Functions: The Unit Circle
5
13
12
13
5
y
x
12
3
5
csc 1
5
y
3
4
5
sec 1
5
x
4
cot x 4
y 3
csc 13
1
y
5
sin y sec 1 13
x
12
cos x cot 12
x
y
5
tan 5. t 2 2
,
.
corresponds to
4
2
2
7. t 3
7
1
, .
corresponds to 6
2
2
9. t y 3
x 4
1 3
, x y ,
3
2 2
2
2
5
, x, y ,
4
2
2
8. t 3
4
1
.
corresponds to , 3
2
2
10. t 11. t 3
corresponds to 0, 1.
2
12. t , x, y 1, 0
13. t 2 2
,
.
corresponds to
4
2
2
sin t y cos t x tan t 15. t 2
3
1
, .
corresponds to
6
2
2
tan t cos
1
3
2
tan
32
3
3
12
4 22
3
cos 2
3
1
y
x
3
3
sin 4 2
2
22
1
4 22
tan 2
2
16. t , x, y ,
4
2
2
1
2
sin t y cos t x 1 3
, x, y ,
3
2 2
3
3
2
2
y
1
x
5
1 3
, x, y , 3
2
2
sin
2
14. t 2
6. t 345
346
Chapter 4
17. t 2 2
7
,
.
corresponds to
4
2
2
2
y
1
x
tan t 21. t 3
3
1
y
x
3
3
y
is undefined.
x
25. t 5
1 3
, x, y , 3
2
2
sin
3
5
3
2
cos
5 1
3
2
tan
5 32
3
3
12
tan2 2
2
cos t x cos2 1
2 2
3
,
.
corresponds to 4
2
2
tan t 32
4
3
3
12
sin2 0
cos t x 0
sin t y 22. t 2, x, y 1, 0
sin t y 1
23. t 4
1
3
2
2
3
corresponds to 0, 1.
2
tan t 20. t 1
2
3
4
3
2
tan 3
1
11
corresponds to
, .
6
2
2
cos t x cos 2
sin t y 4
1 3
, x, y ,
3
2 2
sin 2
cos t x tan t 18. t 2
sin t y 19. t Trigonometry
2
2
y
1
x
24. t 0
0
1
3 1
5
, x, y ,
6
2 2
csc t 1
2
y
sin
5 1
6
2
csc
5
1
2
6
sin t
sec t 1
2
x
cos
3
5
6
2
sec
5
1
23
6
cos t
3
cot t x
1
y
tan
3
12
5
6
3
32
cot
5
1
3
6
tan t
corresponds to 0, 1.
2
26. t 3
, x, y 0, 1
2
sin t y 1
csc t 1
1
y
sin
3
1
2
csc
3
1
1
2
sin t
cos t x 0
sec t 1
is undefined.
x
cos
3
0
2
sec
3
is undefined.
2
cot t x
0
y
tan
3
is undefined.
2
cot
0
3
0
2
1
tan t y
is undefined.
x
Section 4.2
27. t 3
4
1
.
corresponds to , 3
2
2
sin
2
7
4
2
csc
7
1
2
4
sin t
1
2
sec t 1
2
x
cos
7 2
4
2
sec
7
1
2
4
cos t
y
3
x
cot t x 3
y
3
tan
7 22
1
4
22
cot
1
7
1
4
tan t
2
30. cos 5 cos 1
29. sin 5 sin 0
9
2
sin 4
4
2
9
7
2
31. cos
2
1
8
cos
3
3
2
7
1
19
sin
6
6
2
15
cos 0
2
2
34. sin
8
4
1
cos
3
3
2
37. sin t 33. cos 4 sin 4 2
35. sin 2
2
7
, x, y ,
4
2
2
1
23
y
3
3
cos t x 32. sin
28. t 347
csc t sin t y tan t Trigonometric Functions: The Unit Circle
36. cos 1
3
(a) sint sin t 1
3
(b) csc t csc t 3
38. sint 3
8
39. cos t (a) sin t sint (b) csc t 41. sin t 3
8
(b) sec t 4
5
42. cos t 4
5
(b) sint sin t 44. tan
1.7321
3
47. cos1.7 0.1288
50. sec 1.8 1
4.4014
0051.8
53. (a) sin 5 y 1
(b) cos 2 x 0.4
40. cos t (a) cos t cost 8
1
sin t
3
(a) sin t sin t 1
5
4
5
1
5
(a) cost cos t 1
5
cos t
(b) sect sec t 4
5
43. sin
(a) cos t cos t 4
5
(b) cost cos t 4
5
45. csc 1.3 1
1.0378
sin 1.3
1
1.4486
cos 22.8
46. cot 1 (b) cos 2.5 x 0.8
1
4
cos t
3
1
0.6421
tan 1
1
1.3940
sin 0.8
52. sin0.9 0.7833
54. (a) sin 0.75 y 0.7
3
4
0.7071
4
49. csc 0.8 48. cos2.5 0.8011
51. sec 22.8 3
4
348
Chapter 4
Trigonometry
56. (a) sin t 0.75
55. (a) sin t 0.25
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 0.7 or t 5.6
57. yt 14 et cos 6t
(a)
t
0
1
4
1
2
3
4
1
y
0.25
0.0138
0.1501
0.0249
0.0883
(b) From the table feature of a graphing utility we see that y 0 when t 5 seconds.
(c) As t increases, the displacement oscillates but decreases in amplitude.
58. y t 1
cos 6t
4
(a) y 0 (b) y
(c) y
59. False. sint sin t means the function is odd, not
that the sine of a negative angle is a negative number.
1
cos 0 0.2500 feet
4
For example: sin 4 4 cos 2 0.0177 feet
1
1
3
3
3
sin
1 1.
2
2
Even though the angle is negative, the sine value is positive.
2 4 cos 3 0.2475 feet
1
1
60. True. tan a tana 6 since the period of tan is .
61. (a) The points have y-axis symmetry.
(b) sin t1 sin t1 since they have the same y-value.
(c) cos t1 cos t1 since the x-values have the
opposite signs.
62. cos x cos sec 63.
1
sec x
1
y 3x 2
2
So sec and cos are even.
1
x 3y 2
2
sin y
sin y sin csc 1
y
2x 3y 2
y
2
2
x y
3
3
1
csc csc y
(x, y)
2
2 2
f 1x x x 1
3
3 3
So sin and csc are odd.
tan tan y
x
y
tan x
x
cot y
cot x
cot y
So tan and cot are odd.
1
f x 3x 2
2
θ
−θ
x
(x, −y)
Section 4.2
1
64. f x x3 1
4
y x2 4
x y2 4
x 2 y2 4
1
x y3 1
4
± x2 4 y
y
x2
x4
x
y2
y4
xy 4x y 2
xy y 4x 2
y3
x 1
yx 1 4x 2
3 4
3 4x 1
x f1
67. f x x2
x4
xy 4 y 2
f 1x x2 4, x ≥ 0
1
x 1 y3
4
y
66. f x f x x2 4, x ≥ 2
65.
1
y x3 1
4
4x 1 Trigonometric Functions: The Unit Circle
2x
x3
y
22x 1
x1
f 1x 22x 1
x1
y
8
Intercept: 0, 0
6
Vertical asymptote: x 3
2
4
Horizontal asymptote: y 2
x
−6 − 4 − 2
−2
2
4
6
8 10
2
4
6
−4
x
1
0
1
2
4
5
6
y
1
2
0
1
4
8
5
4
68. f x −6
−8
5x
5x
, x 3, 2
x2 x 6 x 3x 2
y
8
Horizontal asymptote: x 0
6
4
Vertical asymptote: x 3, x 2
2
Intercept: 0, 0
−4 −2
−2
x
8
−4
x
6
4
2
y
5
4
10
3
5
2
69. f x 0
1
3
5
0
5
4
5
2
25
24
−6
−8
x 5x 2
x2 3x 10
x5
,x2
2x2 8
2x 2x 2 2x 2
5
Intercepts: 5, 0, 0,
4
x
Hole in the graph at 2,
7
8
4
0
1
4
3
4
3
1
0
2
5
4
1
3
1
4
5
2
1
Vertical asymptote: x 2
Horizontal asymptote: y 5
y
1
2
y
1
−6 −5
−2 −1
−1
−2
−3
−4
x
1
2
349
350
Chapter 4
70. f x Trigonometry
x3 6x2 x 1 1
7
15x 4
x 2x2 5x 8
2
4 42x2 5x 8
Vertical asymptote: 2x 2 5x 8 0
x
5 ± 52 428
22
x
5 ± 89
; x 1.11, x 3.61
4
4
3
32
y
15
4
17
5
155
32
1
0
2
9
1
8
3
2
3
4
7
5
29
4
1
y
−6 −4 −2
−2
1
7
Slant asymptote: y x 2
4
y-intercept:
x
x
2
6
8
4
0, 18
x-intercept: 5.86, 0
Section 4.3
■
Right Triangle Trigonometry
You should know the right triangle definition of trigonometric functions.
■
(a) sin opp
hyp
hyp
(c) tan opp
(d) csc hyp
opp
(e) sec hyp
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 (k) 1 cot2 csc2 ■
You should know that two acute angles and are complementary if 90, and that 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
1. (i)
(iv)
hypotenuse
cos hypotenuse
(v)
(ii)
cot opposite
(iv)
(iii)
hypotenuse
csc opposite
(iii)
(v)
opposite
sin hypotenuse
(i)
(vi)
opposite
3. elevation; depression
(vi)
(ii)
Section 4.3
Right Triangle Trigonometry
1. hyp 62 82 36 64 100 10
6
sin 6
3
opp
hyp 10 5
csc hyp 10 5
opp
6
3
8
4
hyp 10 5
sec hyp 10 5
8
4
tan opp 6 3
8 4
8
4
opp 6
3
θ
8
adj 132 52 169 25 12
2.
13
5
θ
sin 5
opp
hyp 13
csc hyp 13
opp
5
cos 12
hyp 13
sec hyp 13
12
tan 5
opp
12
cot 12
opp
5
b
3. adj 412 92 1681 81 1600 40
41
θ
9
sin opp
9
hyp 41
csc hyp 41
opp
9
cos 40
hyp 41
sec hyp 41
40
tan opp
9
40
40
opp
9
hyp 42 42 32 42
4.
4
θ
4
sin 2
opp
4
1
hyp 42 2
2
csc hyp 42
2
opp
4
cos 2
4
1
hyp 42 2
2
sec hyp 42
2
4
tan opp 4
1
4
4
1
opp 4
5. adj 32 12 8 22
θ
3
1
sin opp 1
hyp 3
csc hyp
3
opp
22
hyp
3
sec hyp
3
32
22
4
tan 2
1
opp
22
4
22
opp
6
2
sin opp 2 1
hyp 6 3
csc hyp 6
3
opp 2
42 22
hyp
6
3
sec hyp
6
3
32
42 22
4
tan 2
opp
2
1
42 22
4
42
22
opp
2
θ
The function values are the same since the triangles are similar and the corresponding sides are proportional.
351
352
Chapter 4
Trigonometry
6.
4
θ
8
7.5
θ
hyp 7.52 42 15
hyp 152 82 289 17
sin opp
8
hyp 17
csc hyp 17
opp
8
cos 15
hyp 17
sec hyp 17
15
tan 8
opp
15
cot 15
opp
8
17
2
sin opp
4
8
hyp 172 17
csc hyp 172 17
opp
4
8
cos 7.5
15
hyp 172 17
sec hyp 172 17
7.5
15
tan 4
8
opp
7.5 15
cot 7.5 15
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
4
sin opp
3
hyp
5
csc hyp
5
opp
3
4
hyp
5
sec hyp
5
4
tan opp
3
4
4
opp
3
opp 1.252 12 0.75
1.25
θ
1
sin opp 0.75 3
hyp 1.25 5
csc hyp 1.25 5
opp 0.75 3
1
4
hyp 1.25 5
sec hyp 1.25 5
1
4
tan opp 0.75 3
1
4
cot 1
4
opp 0.75 3
The function values are the same since the triangles are similar and the corresponding sides are proportional.
8.
θ
1
3
2
θ
hyp 12 22 5
6
5
1
opp
sin hyp 5
5
hyp 5
5
csc opp
1
25
2
cos hyp
5
5
hyp 5
2
tan opp 1
2
2
2
opp 1
hyp 32 62 35
sin 5
3
1
35 5
5
csc 35
5
3
cos 6
2
25
35 5
5
sec 35 5
6
2
tan 3 1
6 2
cot 6
2
3
The function values are the same because the triangles are similar, and corresponding sides are proportional.
Section 4.3
9. Given: sin 3 opp
4 hyp
10. Given: cos 32 adj2 42
7
hyp
4
3
θ
hyp 4
csc opp 3
csc hyp
7
76
opp 26
12
sec hyp 47
7
sec hyp 7
5
cot 7
opp
3
cot 5
56
opp 26
12
11. Given: sec 2 opp 2
12
2 hyp
1
12. Given: cot 22
hyp opp 3
sin opp 3
hyp
2
2
1
cos hyp 2
52
12
5
526
hyp 26
26
tan opp 1
5
csc hyp 23
opp
3
csc hyp 26
26
opp
1
cot 3
opp
3
sec hyp 26
5
3 opp
1
14. Given: sec 32 12 hyp2
hyp 10
sin opp 310
hyp
10
cos 10
hyp
10
hyp 10
csc opp
3
hyp
10
1
cot opp 3
2 6
θ
5
θ
26
1
5
26
26
opp
1
hyp 26
26
θ
1
5
1 opp
sin 3
opp
3
13. Given: tan 3 7
opp 26
sin hyp
7
opp 26
5
7
opp 37
7
6 hyp
1
opp 62 12 35
10
θ
1
3
353
5
7 hyp
opp 72 52 24 26
4
cos Right Triangle Trigonometry
opp 35
hyp
6
6
1
hyp 6
θ
tan opp 35
35
1
csc 6
hyp
635
opp 35
35
cot 35
1
opp 35
35
1
35
354
Chapter 4
Trigonometry
15. Given: cot 22
32
3
2 opp
16. Given: csc hyp
2
13
hyp 13
172
17 hyp
4
opp
17
273
42
273
sin 4
opp
hyp 17
cos 273
hyp
17
tan opp
4
4273
273
273
hyp 13
csc opp
2
sec 17
hyp
17273
273
273
hyp 13
3
cot 273
opp
4
sin cos θ
2
opp
213
hyp 13
13
3
hyp 13
3
313
13
opp 2
3
sec 17.
30 30
180
sin 30 opp 1
hyp 2
60°
2
1
30°
18.
2
1
3
4
θ
45 45
180
cos 45 2
1
hyp 2
2
45°
1
180
60
3
3 19.
π
6
2
tan
3
180
45
4
4 20.
opp 3
3
3
1
hyp 2
2
4
1
2
sec
1
π
4
π
3
1
1
cot 21.
30°
3
3
60 3
1
3
opp
csc 2 22.
3
2
45 45
1
hyp
opp
180
45°
60°
1
1
23.
π
3
2
180
30
6
6 1
cos
π
6
24.
2
3
6
hyp
2
1
sin
π
4
3
180
45
4
4 2
opp
1
4
hyp 2
2
1
cot 1 25.
45°
2
1
45°
1
1
1 opp
45 45
4
tan 26.
180
3
3
60°
2
1
30 30
30°
3
6
1
3
180
opp
Section 4.3
27. sin 60 3
2
(a) tan 60 3
1
28. sin 30 , tan 30 2
3
1
(a) csc 30 2
sin 30
1
2
, cos 60 sin 60
3
cos 60
(b) sin 30 cos 60 (c) cos 30 sin 60 1
2
(b) cot 60 tan90 60 tan 30 3
2
3
cos 60
1
(d) cot 60 3
sin 60
3
29. csc 13
2
, sec Right Triangle Trigonometry
13
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
30. sec 5, tan 26
3
(a) sin 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
(c) tan 213
13
sin 2
cos 3
313
13
(d) sec90 csc 31. cos (c) cot90º tan 26
(d) sin tan cos 26 5 1
26
5
13
2
1
3
32. tan 5
1
3
cos (a) cot 1
1
tan 5
(b) sin2 cos2 1
(b) cos 1
1
sec 1 tan2 (a) sec 3
sin2 3
1
2
1
sin2 8
9
sin 22
3
1
2
cos 3
1
(c) cot sin 2
2
4
2 2
3
1
(d) sin90 cos 3
33. tan cot tan tan 1
1
1
1 52
1
26
(c) tan90º cot 26
26
1
1
tan 5
(d) csc 1 cot 2 1 15
1 251 2625 34. cos sec cos 2
cos1 1
26
5
355
356
Chapter 4
35. tan cos Trigonometry
sin cos cos sin 37. 1 cos 1 cos 1 cos2 36. cot sin cos sin cos sin 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) sin 10 0.1736
44. (a) tan 23.5 0.4348
(b) cos 80 0.1736
Note: cos 80 sin90 80 sin 10
45. (a) sin 16.35 0.2815
47. (a) sec 42 12
sec 42.2 1
1.3499
cos 42.2
1
sin 48 1
0.4348
tan 66.5
18 0.9598
60
56 0.9609
(b) sin 73 56
sin 73 60
46. (a) cos 16 18
cos 16 1
3.5523
(b) csc 16.35 sin 16.35
(b) csc 48 7
(b) cot 66.5 7
60
48. (a) cos 4 50
15 cos 4 50
15 60 3600
0.9964
1.3432
(b) sec 4 50
15 1
cos 4 50
15
1.0036
49. (a) cot 11 15
1
5.0273
tan 11.25
(b) tan 11 15
tan 11.25 0.1989
51. (a) csc 32 40
3 1
1.8527
sin 32.6675
(b) tan 44 28
16 tan 44.4711 0.9817
8
10 1.7946
60 3600
10 8
0.5572
60 3600
50. (a) sec 56 8
10 sec 56 (b) cos 56 8
10 cos 56 52. (a) sec
(b) cot
95 20 32 2.6695
95 30 32 0.0699
Section 4.3
53. (a) sin 1
⇒ 30 2
6
54. (a) cos (b) csc 2 ⇒ 30 6
58. (a) cot tan 3
3
(b) cot 1 ⇒ 45 4
23
⇒ 60 3
3
2
2
⇒ 45 3
tan 30 30
x
cos 60. sin 60 30°
1
2
2
2
⇒ 45 y
18
y 18 sin 60 18
3
x 303
x
61.
tan 60 32
x
3 32
x
32
62. sin 45 r
20
r
20
20
202
sin 45 22
3 x 32
x
60°
x
32
3
323
3
x
45
(b) tan 64. (a)
h
Height of the building:
y
h 270 feet
x
123 45 tan 82 443.2 meters
6
Distance between friends:
132
82°
45
45
cos 82 ⇒ y
y
cos 82
3
Not drawn to scale
45 m
323.34 meters
66. tan 65.
3000 ft
1500 ft
θ
1500 1
3000 2
6
6
h
3 135
(c) 2135 h
x 45 tan 82
0 4
23 9
1
30
3
x
30
sin 4
(b) sec 2
3 ⇒ 60 opp
tan 54 357
55. (a) sec 2 ⇒ 60 4
(b) sin 3
59.
63. tan 82 csc 3
3
4
57. (a)
1
⇒ 60 2
3
3
2
⇒ 45 (b) tan 1 ⇒ 45º 56. (a) tan 3 ⇒ 60 (b) cos 2
Right Triangle Trigonometry
w
100
w 100 tan 54 137.6 feet
358
Chapter 4
Trigonometry
67.
x
150 ft
23°
5 ft
y
(a) sin 23 x
145
x
(b) tan 23 145
371.1 feet
sin 23
y
145
y
(c) Moving down the line:
145sin 23
61.85 feet per second
6
145
341.6 feet
tan 23
Dropping vertically:
145
24.17 feet per second
6
68. Let h the height of the mountain.
69.
(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 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
70. tan 3 x
15
x 15 tan 3
d 5 2x 5 215 tan 3 6.57 centimeters
1256 28
Section 4.3
(e)
71. (a)
20
Angle, Height (in meters)
80
19.7
70
18.8
60
17.3
h
85°
50
15.3
h
(b) sin 85 20
40
12.9
(c) h 20 sin 85 19.9 meters
30
10.0
(d) The side of the triangle labeled
h will become shorter.
20
6.8
10
3.5
72. x 9.4, y 3.4
Right Triangle Trigonometry
359
(f) The height of the balloon
decreases as decreases.
20
h
θ
73. True,
sin 20 y
0.34
10
cot 20 x
2.75
y
cos 20 x
0.94
10
sec 20 10
1.06
x
tan 20 y
0.36
x
csc 20 10
2.92
y
csc x 74. True, sec 30 csc 60 because sec90 csc .
75. False,
76. True, cot2 10 csc2 10 1 because
77. False,
1 cot2 csc2 2
2
1
1
⇒ sin 60 csc 60 sin 60
1
sin x
sin 60
2
2
2 1
sin 60 cos 30
cot 30 1.7321;
sin 30
sin 30
sin 2 0.0349
cot 2 csc2 1
cot 2 csc2 1.
78. False, tan5 2 tan25.
79. This is true because the corresponding sides of similar
triangles are proportional.
tan5 tan 25 0.4663
2
tan2 5 tan 5tan 5 0.0077
80. Yes. Given tan , sec can be found from the identity 1 tan2 sec2 .
81. (a)
82. (a)
(b) In the interval 0, 0.5, > sin .
0.1
0.2
0.3
0.4
0.5
sin 0.0998
0.1987
0.2955
0.3894
0.4794
0
18
36
54
72
90
sin 0
0.3090
0.5878
0.8090
0.9511
1
cos 1
0.9511
0.8090
0.5878
0.3090
0
—CONTINUED—
(c) As approaches 0, sin approaches .
360
Chapter 4
Trigonometry
82. —CONTINUED—
(b) sin increases from 0 to 1 as increases from 0 to 90.
(c) cos decreases from 1 to 0 as increases from 0 to 90.
(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.
83.
x2 6x
x2 4x 12
x2 12x 36
xx 6
x2 36
x 6x 2
84.
x
, x ±6
x2
2t 2 5t 12
t 2 16
2t 2 5t 12
2
2
9 4t
4t 12t 9
9 4t 2
85.
x 6x 6
x 6x 6
4t 2 12t 9
t 2 16
2t 3t 4 2t 32t 3
2t 3 2t 3
3
, t ± , 4
3 2t3 2t t 4t 4
t 4
4t
2
3
2
x
3x 2x 2 2x 22 xx 2
2
x 2 x 2 x 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
86.
12 x 4, x 0, 12
12
12 x
4x
1
x x
3
1
Section 4.4
■
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.
■
You should know the trigonometric function values of the quadrant angles 0,
■
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.
3
, , and
.
2
2
Section 4.4
Trigonometric Functions of Any Angle
Vocabulary Check
1. sin 4.
y
r
2. csc r
x
5.
3. tan x
cos r
y
x
6. cot 7. reference
1. (a) x, y 4, 3
(b) x, y 8, 15
r 16 9 5
r 64 225 17
sin y 3
r
5
csc r
5
y 3
sin y
15
r
17
csc r
17
y
15
cos x 4
r
5
sec 5
r
x 4
cos 8
x
r
17
sec 17
r
x
8
tan y 3
x 4
cot x 4
y 3
tan y
15
x
8
cot x
8
y
15
(b) x 1, y 1
2. (a) x 12, y 5
r 12 12 2
r 122 52 13
sin y
5
r
13
sin 2
y
1
2
r
2
cos x
12
r
13
cos 2
1
x
2
r
2
tan y
5
5
x 12 12
tan y
1
1
x 1
csc 13
13
r
y 5 5
csc 2
r
2
y
1
sec r
13
13
x 12
12
r 2
sec 2
x 1
cot x 12 12
y
5
5
cot 3. (a) x, y 3, 1
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
361
362
Chapter 4
Trigonometry
4. (a) x 3, y 1
(b) x 4, y 4
r 32 12 10
r 42 42 42
sin 10
1
y
10
r
10
sin 2
y
4
r
42
2
cos x
3
310
r
10
10
cos 2
4
x
r
4 2
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 42
r
2
x
4
cot x 3
3
y 1
cot 4
x
1
y 4
5. x, y 7, 24
7. x, y 4, 10
6. x 8, y 15
r 49 576 25
r 16 100 229
r 82 152 17
sin y 24
r
25
sin y 15
r
17
sin y 529
r
29
cos x
7
r
25
cos x
8
r
17
cos 229
x
r
29
tan y 24
x
7
tan y 15
x
8
tan y
5
x
2
csc r
25
y 24
csc r
17
y 15
csc 29
r
y
5
sec r
25
x
7
sec r
17
x
8
sec 29
r
x
2
cot x
7
y 24
cot x
8
y 15
cot x
2
y
5
8. x 5, y 2
9. x, y 3.5, 6.8
r 5 2 29
2
sin 2
y
2
229
29
r
29
r 12.25 46.24 5849
10
sin y 685849
0.9
r
5849
cos 355849
x
0.5
r
5849
tan y
68
1.9
x
35
29
29
r
sec x
5
5
csc 5849
r
1.1
y
68
x 5 5
y 2 2
sec 5849
r
2.2
x
35
cot x
35
0.5
y
68
x
529
5
cos 29
r
29
y 2 2
tan x 5 5
29
29
r
csc y
2
2
cot Section 4.4
Trigonometric Functions of Any Angle
3
31
1 7
10. x 3 , y 7 2 2
4
4
r
72 314 2
2
1157
4
314
y
311157
0.9
r
1157
11574
x
72
141157
cos 0.4
r
1157
11574
31
y 314
tan 2.2
x
72
14
sin 11. sin < 0 ⇒ lies in Quadrant III or in Quadrant IV.
cos < 0 ⇒ lies in Quadrant II or in Quadrant III.
sin < 0 and cos < 0 ⇒ lies in Quadrant III.
csc 1157
11574
r
1.1
31
y
314
sec 11574
1157
r
2.4
x
72
14
cot 72
14
x
0.5
y 314
31
12. sin > 0 and cos > 0
y
x
> 0 and > 0
r
r
13. sin > 0 ⇒ lies in Quadrant I or in Quadrant II.
tan < 0 ⇒ lies in Quadrant II or in Quadrant IV.
sin > 0 and tan < 0 ⇒ lies in Quadrant II.
14. sec > 0 and cot < 0
r
x
> 0 and < 0
x
y
15. sin y 3
⇒ x2 25 9 16
r
5
y 3
r
5
x
4
cos r
5
y
3
tan x
4
17. tan x 4
⇒ y 2 25 16 9
r
5
in Quadrant III ⇒ y 3
in Quadrant II ⇒ x 4
sin 16. cos 5
r
y 3
5
r
sec x
4
4
x
cot y
3
csc y 15
x
8
18. cos tan < 0 ⇒ y 15
x 8, y 15, r 17
sin y
15
r
17
8
x
cos r
17
y
15
tan x
8
r
17
y
15
r
17
sec x
8
x
8
cot y
15
csc csc 8
x
⇒ y 15
r
17
sin < 0 and tan < 0 ⇒ is in Quadrant IV ⇒
y < 0 and x > 0.
sin 5
3
5
sec 4
4
cot 3
3
y
r
5
4
x
cos r
5
y 3
tan x 4
sin y 15
15
r
17
17
x
8
cos r
17
y 15
15
tan x
8
8
17
15
17
sec 8
8
cot 15
csc 363
364
Chapter 4
19. cot Trigonometry
3
3
x
y
1 1
20. csc cos > 0 ⇒ is in Quadrant IV ⇒ x is positive;
x 3, y 1, r 10
sin cos 10
y
r
10
csc x 310
r
10
sec 1
y
tan x
3
21. sec r
10
y
10
r
x
3
x
cot 3
y
2
r
⇒ y2 4 1 3
x 1
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
22. sin 0 ⇒ 0 n
sin > 0 ⇒ is in Quadrant II ⇒ y 3
sec 1 ⇒ 2n
y 0, x r
sin y 3
r
2
csc r
23
y
3
sin 0
cos x
1
r
2
sec r
2
x
cos tan y
3
x
cot 3
x
y
3
tan 23. cot is undefined,
3
≤ ≤
⇒ y0 ⇒ 2
2
sin 0
csc is undefined
cos 1
sec 1
tan 0
cot is undefined
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.
x 1, y 1, r 2
sin 1
2
cos 1
2
tan 1
csc 2
sec 2
2
2
csc r
is undefined
y
x r
1
r
r
sec r
1
x
y
0
x
cot x
is undefined
y
24. tan is undefined ⇒ n 2
2
3
, x 0, y r
2
y r
r
1
sin csc 1
r
r
y
x 0
r
cos 0
sec is undefined.
r
r
x
y
x 0
tan is undefined.
cot 0
x
y
y
≤ ≤ 2 ⇒ 26. Let x > 0.
1
r
x
2
1 2 10 x
x 9
3
10
y
13x
r
10
10 x3
310
x
x
cos r
10
10 x3
sin 2
415
15
tan csc cot 1
y 13 x 1
x
x
3
r
10 x3 10
y
13 x
sec r
10 x3 10
x
x
3
cot x
x
3
y 13 x
Section 4.4
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.
4
4x 3y 0 ⇒ y x
3
4
2
25
sin 5
5
r
5
1
cos 5
5
csc sec cot 2
2
1
5
2
5
1
32. sec since
2
16 2 5
x x
9
3
sin 4
y 43 x
r
53 x
5
csc cos x
3
x
r
53 x 5
sec tan y 43 x
4
x
x
3
tan 5
4
5
3
3
4
1 1
2 2
30. csc
y
0
r
r
1
3
1
2
y 1
since
r
1
1
x 1
31. x, y 0, 1, r 1
3
corresponds to 0, 1.
2
sec
sin
y
1
2
r
37. 203
x 0
0
2
y 1
since
corresponds to 0, 1.
2
38. 309
203 180 23
360 309 51
y
y
203°
x 1
(undefined)
y
0
since corresponds to 1, 0.
36. cot
r
1
⇒ undefined
y 0
r
1
3
⇒ undefined
2
x 0
34. cot 33. x, y 0, 1, r 1
35. x, y 1, 0, r 1
θ′
2
5
3
corresponds to 1, 0.
2
csc x
5
29. x, y 1, 0, r 1
sin 365
28. Let x > 0.
x 1, y 2, r 5
tan Trigonometric Functions of Any Angle
309°
x
x
θ′
366
Chapter 4
Trigonometry
245
39.
40. 145 is coterminal with 215.
360 245 115 coterminal angle
215 180 35
180 115 65
y
y
θ′
x
θ′
x
− 145°
−245°
41. 2
3
42. y
2 3
3
2π
3
7
4
2 θ′
y
7 4
4
7π
4
x
x
θ′
43. 3.5
11
is coterminal
3
5
with
.
3
3.5 3.5
x
θ′
45. 225, 360 225 45, Quadrant III
sin 225 sin 45 2
cos 225 cos 45 2
2
2
47. 750 is coterminal with 30.
5 3
3
x
θ′
46. 300, 360 300 60, Quadrant IV
sin 300 sin 60 3
2
1
cos 300 cos 60 2
48. 405 is coterminal with 315.
1
2
cos 750 cos 30 tan 750 tan 30 2 11π
3
tan 300 tan 60 3
tan 225 tan 45 1
sin 750 sin 30 y
44. y
sin405 sin 45 3
2
3
3
cos405 cos 45 2
2
2
2
tan405 tan 45 1
Section 4.4
1
2
cos150 cos 30 tan150 tan 30 51. 3
sin840 sin 60 3
2
cos840 cos 60 2
3
4
4
6
6
sin
3
4
sin 3
3
2
sin
2
4
2
sin cos
4
1
cos 3
3
2
cos
2
4
2
cos tan
4
tan 3
3
3
tan
1
4
tan 3
is coterminal with
.
2
2
3
2
3
6 tan 6 3
3
11
is coterminal with .
4
4
56. 10
4
is coterminal with
.
3
3
3
0
2
sin
sin
3
10
sin 3
3
2
3
is undefined.
2
2
11
sin 4
4
2
cos
2
11
cos 4
4
2
cos
10
1
cos 3
3
2
tan
11
tan 1
4
4
tan
10
tan 3
3
3
2 tan
57. 3
is coterminal with , .
2
2
2
3
4
4
58. 3
sin 1
2
2
2 3
cos 0
2
2
sin 3
tan which is undefined.
2
2
cos cos 4
3
3
25
7
is coterminal with .
4
4
tan 6 cos 6 1
2 cos
sin 3
1
2
cos tan 55. 6 sin 6 2
2 sin
sin 1
2
tan840 tan 60 3
3
4
3
3
54. 367
50. 840 is coterminal with 240.
49. 150 is coterminal with 210.
sin150 sin 30 Trigonometric Functions of Any Angle
4
4
2
25
sin
4
4
2
2
25
cos
4
4
2
25
tan
1
4
4
tan 368
Chapter 4
Trigonometry
sin 59.
3
5
60. cot 3
1 cot 2 csc 2 sin2 cos2 1
cos2 1 cos2 1 3
5
10 2
4
5
1 cot 2 csc 2 cot 2 csc 2 1
2 1
3
sec2 1 2
sec2 1 9
4
2
sec2 1
sin 5
8
64. sec cos 1
1
⇒ sec sec cos sec 1
8
58 5
4
tan2 tan 2 66. sec 225 1
2.0000
sin330
1
3.2361
cos 72
1
0.2245
tan 1.35
77. sin0.65 0.6052
11
8
9
4
9
2
1
65
16
tan > 0 in Quadrant III.
65. sin 10 0.1736
79. cot 2
tan2 sec 2 1
tan 74. cot 1.35 13
1 tan2 sec 2 cot 3
71. sec 72 2
13
4
sec cot < 0 in Quadrant IV.
68. csc330 3
sec < 0 in Quadrant III.
10
1
1
sin csc 10
10
63. cos 62. csc 2
cot 2 csc 2
csc > 0 in Quadrant II.
csc 16
25
cot 2 sec2 1 tan2 10 csc 9
25
cos > 0 in Quadrant IV.
cos 3
2
1 32 csc 2 cos2 1 sin2 cos2 61. tan 4
67. cos110 0.3420
1
28.6363
tan 178
69. tan 304 1.4826
70. cot 178 72. tan188 0.1405
73. tan 4.5 4.6373
75. tan
0.3640
9
9 0.3640
76. tan 78. sec 0.29 1
0.4142
11
tan 8
1
1.4142
cos 225
65
80. csc 1
1.0436
cos 0.29
15
14
1
4.4940
15
sin 14
Section 4.4
81. (a) sin Trigonometric Functions of Any Angle
369
1
⇒ reference angle is 30 or and is in Quadrant I or Quadrant II.
2
6
Values in degrees: 30, 150
(b) sin 5
,
6 6
1
⇒ reference angle is 30 or and is in Quadrant III or Quadrant IV.
2
6
Values in degrees: 210, 330
82. (a) cos 2
2
7 11
,
6
6
and is in Quadrant I or IV.
4
⇒ reference angle is 45 or
Values in degrees: 45, 315
(b) cos 2
2
7
,
4 4
⇒ reference angle is 45 or
and is in Quadrant II or III.
4
Values in degrees: 135, 225
83. (a) csc 3 5
,
4 4
23
⇒ reference angle is 60 or and is in Quadrant I or Quadrant II.
3
3
Values in degrees: 60, 120
2
,
3 3
(b) cot 1 ⇒ reference angle is 45 or
4
Values in degrees: 135, 315
3 7
,
4 4
84. (a) sec 2 ⇒ reference angle is 60 or
and is in
3
5
,
3 3
(b) sec 2 ⇒ reference angle is 60 or
Values in degrees: 120, 240
2 4
,
3 3
and is in
4
Values in degrees: 45, 225
Values in degrees: 60, 300
85. (a) tan 1 ⇒ reference angle is 45 or
and is
3
5
,
4 4
(b) cot 3 ⇒ reference angle is 30 or
Values in degrees: 150, 330
5 11
,
6
6
and 6
370
Chapter 4
Trigonometry
⇒ reference angle is 60 or and is
2
3
86. (a) sin 3
87. (a) New York City:
N 22.099 sin0.522t 2.219 55.008
Values in degrees: 60, 120
Fairbanks:
2
,
3 3
F 36.641 sin0.502t 1.831 25.610
(b)
(b) sin ⇒ reference angle is 60 or and 2
3
is in Quadrant III or IV.
3
Month
Values in degrees: 240, 300
4 5
,
3 3
New York City
Fairbanks
February
34.6
1.4
March
41.6
13.9
May
63.4
48.6
June
72.5
59.5
August
75.5
55.6
September
68.6
41.7
November
46.8
6.5
(c) The periods are about the same for both models,
approximately 12 months.
88. S 23.1 0.442t 4.3 cos
t
6
89. yt 2 cos 6t
(a) y0 2 cos 0 2 centimeters
(a) For February 2006, t 2.
S 23.1 0.4422 4.3 cos
2
26,134 units
6
(b) For February 2007, t 14.
S 23.1 0.44214 4.3 cos
14
31,438 units
6
(b) y
4 2 cos2 0.14 centimeter
(c) y
2 2 cos 3 1.98 centimeters
1
3
1
(c) For June 2006, t 6.
S 23.1 0.4426 4.3 cos
6
21,452 units
6
(d) For June 2007, t 18.
S 23.1 0.44218 4.3 cos
18
26,756 units
6
90. yt 2et cos 6t
91.
I0.7 5e1.4 sin 0.7 0.79 ampere
(a) t 0
y0 2e0 cos 0 2 centimeters
1
(b) t 4
y14 2e14 cos6
14 0.11 centimeters
1
(c) t 2
y2 2e12 cos6
1
I 5e2t sin t
12 1.2 centimeters
Section 4.4
92. sin 6
6
⇒ d
d
sin 371
93. 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.
(a) 30
d
Trigonometric Functions of Any Angle
6
6
12 miles
sin 30 12
(b) 90
d
6
6
6 miles
sin 90º 1
(c) 120
d
6
6.9 miles
sin 120
94. False. For example, if n 1 and 225, 0 ≤ 135 ≤ 360, but 360n 135 is not the reference angle.
The reference angle would be 45. For in Quadrant II, 180 . For in Quadrant III, 180.
For in Quadrant IV, 360 .
95. As increases from 0 to 90, x decreases from 12 cm to 0 cm and y increases from 0 cm to 12 cm.
y
x
Therefore, sin increases from 0 to 1 and cos decreases from 1 to 0. Thus,
12
12
y
tan increases without bound, and when 90 the tangent is undefined.
x
96. Determine the trigonometric function of the reference angle and prefix the appropriate sign.
97. y x2 3x 4 x 4x 1
98. y 2x2 5x x2x 5
x-intercepts: 4, 0, 1, 0
y-intercept: 0, 4
8
No asymptotes
4
Domain: All real numbers x
5
x-intercepts: 0, 0, 2, 0
y
y-intercepts: 0, 0
6
−8 −6
2
No asymptotes
2
(− 4, 0)
y
1
(1, 0)
−2
−2
2
−4
4
x
6
8
Domain: All real numbers x
(52 , 0(
(0, 0)
−3 −2 −1
−1
1
2
3
4
x
5
−2
(0, −4)
−3
−4
−8
99. f x x3 8
100. gx x4 2x2 3 x2 3x2 1
y
x2 3x 1x 1
12
x-intercept: 2, 0
10
y-intercept: 0, 8
x-intercepts: 1, 0, 1, 0
(0, 8)
4
y-intercepts: 0, 3
No asymptotes
Domain: All real numbers x
y
(− 2, 0)
x
−8 − 6 − 4
2
−4
4
6
3
2
No asymptotes
8
Domain: All real numbers x
(−1, 0)
1
(1, 0)
x
−4 −3 −2
2
−3
−4
(0, − 3)
3
4
372
Chapter 4
101. f x Trigonometry
x7
x7
x2 4x 4 x 22
y
4
x-intercept: 7, 0
2
(7, 0)
x
y-intercept:
7
0, 4
−8
−2
2
4
0, − 7
4
(
6
8
(
Vertical asymptote: x 2
Horizontal asymptote: y 0
Domain: All real numbers except x 2
102. hx x2 1 x 1x 1
x5
x5
103. y 2x1
x-intercepts: 1, 0, 1, 0,
To find the y-intercept, let x 0:
y-intercept:
0, 21
y-intercept:
02 1
1
05
5
Horizontal asymptote: y 0
Domain: All real numbers x
0, 51
Vertical asymptote: x 5
To find the slant asymptote, use long division:
x
1
0
1
2
3
y
1
4
1
2
1
2
4
1
24
x5
x5
x5
x2
y
Slant asymptote: y x 5
5
Domain: All real numbers except x 5
4
3
y
8
2
) )
0, 1 1
2
(1, 0)
x
− 12
−8
(−1, 0)
−8
4
1
0, − 5
−2
−1
( (
x
1
2
3
−1
− 16
− 24
104. y 3x1 2
y
7
This is an exponential function (always positive)
translated two units upward. There are no x-intercepts.
6
5
To find the y-intercept, let x 0:
3
y 301 2 3 2 5
y-intercepts: 0, 5
The horizontal asymptote is the horizontal asymptote of
y 3x1 translated two units upward.
Horizontal asymptote: y 2
Domain: All real numbers x
(0, 5)
2
1
−5 −4 −3 −2 −1
x
1
2
3
4
Section 4.5
105. y ln x4
Graphs of Sine and Cosine Functions
y
12
Domain: All real numbers except x 0
9
x-intercepts: ± 1, 0
6
Vertical asymptote: x 0
(−1, 0)
− 12 − 9 − 6 − 3
(1, 0)
x
3
6
9 12
106. y log10x 2
y
To find the x-intercept, let y 0:
3
2
0 log10x 2 ⇒ 10 x 2 ⇒ x 1
x-intercepts: 1, 0
To find the y-intercept, let x 0:
y log10x 2 log10 2 0.301
(−1, 0)
−3
−1
x
1
2
3
−1
−2
−3
y-intercepts: 0, 0.301
The vertical asymptote is the horizontal asymptote of y log10 x translated two units to the left.
Vertical asymptote: x 2
Domain: All real numbers x such that x > 2
Section 4.5
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
■
2
Period:
b
■
Shift: Solve bx c 0 and bx c 2.
■
Key increments:
1
(period)
4
Vocabulary Check
1. cycle
2. amplitude
2
b
4. phase shift
3.
5. vertical shift
373
374
Chapter 4
Trigonometry
2. y 2 cos 3x
1. y 3 sin 2x
Period:
2
2
Period:
Amplitude: 3 3
4. y 3 sin
Period:
x
3
Amplitude: a 3 3
2
2
1
10. y 1
sin 8x
3
Period:
Amplitude: a 13. y 1
3
1
sin 2x
4
Period:
2
1
2
Amplitude:
Period:
1
1
4
4
16. f x cos x, gx cosx g is a horizontal shift of f units
to the left.
6. y 2
6
3
Amplitude:
1
1
2
2
2x
3
2
2
3
b
23
2x
1
cos
2
3
14. y 1
1
2
2
gx cos 2x
The period of f is twice that of g.
3
2
Period:
2
2
20
b
10
Amplitude: a 2 10
5
Amplitude: 3 3
5
x
cos
2
4
2
2
8
b
14
Amplitude: a 5
2
15. f x sin x
2
3
17. f x cos 2x
gx cos 2x
The graph of g is a reflection in
the x-axis of the graph of f.
19. f x cos x
Period:
2
x
cos
3
10
Period:
2
2
4
b
2
Amplitude: a 12. y 2
3
23
Amplitude:
5
5
2
2
9. y 3 sin 10x
Period:
3
x
cos
2
2
Period:
Amplitude: a 1 1
11. y 2 2 b
8
4
2
4
12
Amplitude:
x
1
sin
2
3
Period:
Amplitude: 2 2
Period:
8. y cos
7. y 2 sin x
Period:
2 2
b
3
x
5
cos
2
2
Amplitude: a 2
5. y 2
2
6
b
13
3. y 20. f x sin x, gx sin 3x
The period of g is one-third the
period of f.
gx sinx The graph of g is a horizontal shift
to the right units of the graph of
f a phase shift.
18. f x sin 3x, gx sin3x
g is a reflection of f about the
y-axis.
21. f x sin 2x
f x 3 sin 2x
The graph of g is a vertical shift
three units upward of the graph of f.
Section 4.5
22. f x cos 4x, gx 2 cos 4x
g is a vertical shift of f two units
downward.
Graphs of Sine and Cosine Functions
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
Minimum
, 2
2
Intercept
Maximum
, 0
3
,0
2
x
3π
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.
28. f x sin x
Period:
y
2 2
2
b
1
2
g f
Amplitude: 1
6π
Symmetry: origin
Key points: Intercept
0, 0
Since gx sin
Maximum
,1
2
Intercept
Minimum
Intercept
, 0
3
, 1
2
2, 0
x
−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
375
π
Intercept
Minimum
Intercept
Maximum
2 , 0
, 1
32, 0
2, 1
−1
2π
f
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.
x
376
Chapter 4
Trigonometry
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
4 , 0
2 , 2
34, 0
, 2
x
−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
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.
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
3π
x
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.
Section 4.5
Graphs of Sine and Cosine Functions
33. f x 2 cos x
Period:
377
y
2 2
2
b
1
3
f
Amplitude: 2
π
Symmetry: y-axis
x
2π
g
Key points: Maximum
Intercept
Minimum
Intercept
Maximum
, 2
3
,0
2
2, 2
,0
2
0, 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
x
2π
f
Key points: Minimum
Intercept
Maximum
Intercept
Minimum
, 1
3
,0
2
2, 1
,0
2
0, 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.
35. y 3 sin x
Period: 2
4
Amplitude: 3
2
3
− π
2
π
2
−3
2
−3
0, 3, 2 , 0, , 3,
1
1
− 2π
−π
π
2π
x
−1
1
2 , 4, , 0,
−2
−1
− 43
1
y
Period: 2
2
3
1
3
3
1
, 0 , 2,
2
3
1
4
38. y 4 cos x
1
Key points:
0, 0,
y
2
Key points:
4
3
1
x
3π
2
3
Period: 2
1
y
2 , 4, 2, 0
1
cos x
3
Amplitude:
−π
2
−4
3
, 3 , 2, 0
2
37. y Amplitude:
1
0, 0, , 3 , , 0,
2
1
sin x
4
Period: 2
3
Key points:
36. y y
4
Amplitude: 4
π
2
π
2π
x
Key points:
0, 4,
2 , 0, , 4,
3
, 0 , 2, 4
2
− 2π
−π
π
−2
−4
2π
x
378
Chapter 4
Trigonometry
x
2
39. y cos
40. y sin 4x
y
2
y
2
2
Period: 4
Period:
Amplitude: 1
Amplitude: 1
Key points:
− 2π
2π
0, 1, , 0, 2, 1,
−1
3, 0, 4, 1
−2
4π
1
x
x
π
4
Key points:
0, 0,
8 , 1, 4 , 0,
3
−2
8 , 1, 2 , 0
41. y cos 2x
Period:
42. y sin
y
2
1
2
2
1
x
1
Period:
2
2
1
Amplitude: 1
−6
0, 0, 2, 1, 4, 0,
−2
−2
6, 1, 8, 0
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
Key points:
−2
Key points:
1
1
3
0, 1, , 0 , , 1 , , 0
4
2
4
43. y sin
y
2
8
Period:
4
Amplitude: 1
Key points:
x
4
3
−π
−2
9
x
4
5
7
π
−3
9
4 , 0, 4 , 1, 4 , 0, 4 , 1, 4 , 0
x
x
2
6
Section 4.5
46. y sinx Graphs of Sine and Cosine Functions
47. y 3 cosx Period: 2
Period: 2
Amplitude: 1
Amplitude: 3
Shift: Set x 0
and
x
Key points: , 0,
Shift: Set x 0
x 2
x 2
and
x x 3
x
Key points: , 3, , 0 , 0, 3, , 0 , , 3
2
2
32, 1, 2, 0, 52, 1, 3, 0
y
y
6
2
4
2
−π
2
−1
x
3π
2
−π
x
π
−4
−2
−6
48. y 4 cos x 4
49. y 2 sin
2x
3
Period: 2
Period: 3
Amplitude: 4
Amplitude: 1
Shift: Set x 0
4
x
and
4
x
2
4
x
Key points: 0, 2,
7
4
34, 1, 32, 2, 94, 3, 3, 2
y
5
3
5
7
Key points: , 4 , , 0 ,
, 4 ,
,0 ,
,4
4
4
4
4
4
4
2
y
1
6
x
–3
–2
–1
1
2
−1
2
−π
−2
π
2π
x
−4
−6
50. y 3 5 cos
Period:
379
t
12
y
16
12
8
4
2
24
12
Amplitude: 5
Key points: 0, 2, 6, 3, 12, 8, 18, 3, 24, 2
t
− 12
4
−8
− 12
− 16
− 20
− 24
12
3
380
Chapter 4
51. y 2 Period:
Trigonometry
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,
Key points:
1
1
1
1
0, 2.1,
,2 ,
, 1.9 ,
,2 ,
, 2.1
120
60
40
30
2 , 3, , 5, 32, 3, 2, 1
y
1
−π
y
π
2π
2.2
−4
−5
−6
−7
1.8
x
− 0.1
0
0.1
0.2
53. y 3 cosx 3
y
Period: 2
4
2
Amplitude: 3
Shift: Set x 0
x 2
and
x π
−8
4
4
y
10
Period: 2
6
Amplitude: 4
Shift: Set x 0
4
x
Key points:
55. y and
x
4
4
2
4
x
2
− 2π − π
7
4
2π
3π
x
4 , 8, 4 , 4, 34, 0, 54, 4, 74, 8
y
4
3
Period: 4
2
2
Amplitude:
3
1
−1
x
x
0 and
2
2
4
2
4
x
Key points:
π
−4
2
x
2
1
cos ; a , b ,c
3
2
4
3
2
4
Shift:
x
x
Key points: , 0, , 3 , 0, 6, , 3 , , 0
2
2
54. y 4 cos x 2π
2
2
x
3
−2
−3
−4
9
2
5 2
7
9 2
2 , 3 , 2 , 0, 2 , 3 , 2 , 0, 2 , 3 π
4π
x
x
Section 4.5
Graphs of Sine and Cosine Functions
56. y 3 cos6x Period:
y
2 6
3
3
2
Amplitude: 3
Shift: Set 6x 0
x
Key points:
and
6
x
x
π
6x 2
6
6 , 3, 12 , 0, 0, 3, 12 , 0, 6 , 3
57. y 2 sin4x 58. y 4 sin
4
3x 3 2
59. y cos 2x 8
−6
6
1
2
3
− 12
12
−3
−4
60. y 3 cos
3
−8
x
2
2
2
61. y 0.1 sin
−1
x
10 62. y 1
sin 120 t
100
0.12
2
−6
0.02
6
− 20
− 0.03
− 0.12
−6
63. f x a cos x d
Amplitude:
20
1
3 1 2 ⇒ a 2
2
− 0.02
64. f x a cos x d
Amplitude:
1 3
2
2
Vertical shift one unit upward of
1 2 cos 0 d
gx 2 cos x ⇒ d 1
d 1 2 1
Thus, f x 2 cos x 1.
a 2, d 1
65. f x a cos x d
Amplitude:
1
8 0 4
2
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.
0.03
66. f x a cos x d
Amplitude:
2 4
1
2
Reflected in the x-axis: a 1
4 1 cos 0 d
d 3
a 1, d 3
381
382
Chapter 4
Trigonometry
68. y a sinbx c
67. y a sinbx c
Amplitude: a 3
Amplitude: 2 ⇒ a 2
Since the graph is reflected in the x-axis, we have
a 3.
Period: 4
2
1
4 ⇒ b b
2
2
⇒ b2
Period:
b
Phase shift: c 0
Phase shift: c 0
1
a 2, b , c 0
2
Thus, y 3 sin 2x.
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
c
4
.
4
71. y1 sin x
y2 2
4 ⇒ b
b
2
c
1 ⇒ c b
2
a 2, b ,c
2
2
72. y1 cos x
2
1
2
Phase shift:
−2
2
y2 1
2
−2␲
y1 y2 when x , In the interval 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
2
− 0.25
(c) Amplitude: 0.85; Period: 6
4
8
10
− 1.00
3
9
Key points: 0, 0, , 0.85 , 3, 0, , 0.85 , 6, 0
2
2
74. v 1.75 sin
t
2
(a)
Period (b)
1 cycle
4 seconds
2
4 seconds
2
2␲
60 seconds
15 cycles per minute
1 minute
(c)
v
3
2
1
t
1
−2
−3
3
5
7
Section 4.5
(b) f 2
1
seconds
880 440
(a) Period:
1
440 cycles per second
p
(b)
1
1
77. (a) a high low 83.5 29.6 26.95
2
2
(c)
383
5 t
3
76. P 100 20 cos
75. y 0.001 sin 880t
(a) Period:
Graphs of Sine and Cosine Functions
2
6
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
The model is a good fit.
(d) Tallahassee average maximum: 77.90
1
1
d high low 83.5 29.6 56.55
2
2
Ct 56.55 26.95 cos
(b)
12
0
Chicago average maximum: 56.55
The constant term, d, gives the average maximum
temperature.
6t 3.67
100
(e) The period for both models is
2
12 months.
6
This is as we expected since one full period is one
year.
0
12
(f) Chicago has the greater variability in temperature
throughout the year. The amplitude, a, determines this
1
variability since it is 2high temp low temp.
0
The model is a good fit.
78. (a) and (c)
(b) Vertical shift:
y
Amplitude:
Percent of moon’s
face illuminated
1.0
1
1
⇒ d
2
2
1
1
⇒ a
2
2
0.8
0.6
Period:
0.4
0.2
x
10
20
30
40
88768
7.4 (average length of interval in data)
5
2
47.4 29.6
b
Day of the year
Reasonably good fit
(d) Period is 29.6 days.
(e) March 12 ⇒ x 71. y 0.44 44%
The Naval observatory says that 50% of
the moon’s face will be illuminated on
March 12, 2007.
b
2
0.21
29.6
Horizontal shift: 0.213 7.4 C 0
C 0.92
y
1 1
sin0.21x 0.92
2 2
384
Chapter 4
Trigonometry
79. C 30.3 21.6 sin
Period (a)
2 t
365 10.9
80. (a) Period 2
365
2
365
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.
The radius of the wheel is 50 feet.
(b)
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
(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
82. False. y 2 cos 2x has an amplitude that is half that
of y cos x. For y a cos bx, the amplitude is a .
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
85.
Since the graphs are the
same, the conjecture is that
y
2
sinx cos x f=g
1
− 3π
2
π
2
3π
2
.
2
x
−2
86. f x sin x, gx cos x x
2
0
1
0
1
0
sin x
cos x 2
2
y
2
Conjecture: sin x cos x f=g
1
3
2
2
0
1
0
0
1
0
− 3π
2
π
2
−2
3π
2
x
2
Section 4.5
87. (a)
Graphs of Sine and Cosine Functions
385
x3
x5
x7
3! 5! 7!
x4
x6
x2
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
0.5000
(c) sin 1 6
3!
5!
sin
(b) sin 1 1 0.5 (by calculator)
6
(e) cos 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
1
1
0.5417
2! 4!
cos
cos 1 0.5403 (by calculator)
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.
89. log10 x 2 log10x 212 12 log10x 2
90. log2x2x 3 log2 x2 log2x 3
2 log2 x log2x 3
91. ln
t3
ln t 3 lnt 1 3 ln t lnt 1
t1
z
92. ln
2
z
1
z
1
ln 2
ln z lnz2 1
1 2
z 1
2
93.
1
2 log10
x log10 y 12 log10xy
log10 xy
94. 2 log2 x log2xy log2 x2 log2xy
log2 x2(xy)
log2 x3y
95. ln 3x 4 ln y ln 3x ln y4
ln
3xy
4
1
1
ln z lnz2 1
2
2
386
96.
Chapter 4
Trigonometry
1
1
ln 2x 2 ln x 3 ln x ln 2x ln x2 ln x3
2
2
1
2x
ln 2 ln x3
2
x
2xx ln x
2x
lnx x
ln
3
2
3
2
lnx22x Section 4.6
■
■
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
5. x n
6. , 1 1, 7. 2
1. y sec 2x
Period:
2
2
Matches graph (e).
2. y tan
Period:
x
2
2
b
12
Asymptotes: x , x 4. 3. y 1
cot x
2
Period:
1
Matches graph (a).
Matches graph (c).
4. y csc x
Period: 2
Matches graph (d).
5. y 1
x
sec
2
2
Period:
2
2
4
b
2
Asymptotes: x 1, x 1
Matches graph (f).
6. y 2 sec
Period:
x
2
2
2
4
b
2
Asymptotes: x 1, x 1
Reflected in x-axis
Matches graph (b).
Section 4.6
7. y 1
tan x
3
Graphs of Other Trigonometric Functions
8. y y
1
tan x
4
y
Period: 3
2
Period: Two consecutive asymptotes:
1
Two consecutive asymptotes:
x
and x 2
2
−π
x
π
3
2
x ,x
2
2
4
0
4
x
4
0
4
y
1
3
0
1
3
y
1
4
0
1
4
Period:
4
2
1
Two consecutive asymptotes:
3x ⇒ x 2
6
3x Period:
3
−π
3
x
π
3
x
12
0
12
y
1
0
1
1
11. y sec x
2
y
1
−π
x
π
3
y
1
0
1
2
x
1
−3
1
2
5
6
y
2
1
2
Period:
2
−1
0
1
4
0
3
y
3
3
2
0
3
1
4
1
2
y
2
4
2
8
6
4
Two consecutive
asymptotes:
x 0, x x
2π
14. y 3 csc 4x
3
−2
1
4
−8
Two consecutive
asymptotes:
1
2
Two consecutive
asymptotes:
−4
1
sec x
4
y
1
1
6
3
1
4
x
y
y
x 0, x 1
x
2
2
Period:
x
3
13. y csc x
x
2
x ,x
2
2
x ,x
2
2
x
1
Period: 2
2
Two consecutive
asymptotes:
y
Two consecutive
asymptotes:
12. y 3
Period: 2
−3
1
1
x ,x
2
2
⇒ x
2
6
x
π
10. y 3 tan x
y
3
1
−π
x
9. y tan 3x
387
2
−π
4
4
−4
x
24
8
5
24
y
6
3
6
−2
π
4
x
388
Chapter 4
Trigonometry
15. y sec x 1
Period:
16. y 2 sec 4x 2
y
2
2
−3
−2
−1
x
1
−1
2
3
y
1
17. y csc
Period:
1
3
6
4
Two consecutive
asymptotes:
2
x ,x
8
8
1
1
x ,x
2
2
x
2 4
2
Period:
Two consecutive
asymptotes:
y
0
1
3
x
0
1
y
2
x
2
6
2
4
12
2
Two consecutive
asymptotes:
x
π
12
0
2
6
4
2
Two consecutive
asymptotes:
x
3
5
3
x
2
3
2
5
2
y
2
1
2
y
2
1
2
Period:
x
2
y
20. y 3 cot
3
2
12
2
Period:
1
Two consecutive
asymptotes:
− 2π
x
2π
x
0 ⇒ x0
2
x
y
2
1
1
sec 2x
2
Period:
x
y
1
3
2
0
1
y
6
2
2
4
2
Two consecutive
asymptotes:
x
−2
2
3
0
6
1
2
1
−π
x
1
2
1
3
2
y
3
0
3
1
22. y tan x
2
y
2
2
6
x
2
x 0, x 2
x
⇒ x 2
2
21. y x
π 2π
x 0, x 3
x 0, x 2
19. y cot
x
π
2
y
2
6
13
Period:
4
0
π
4
x
3
18. y csc
y
12
−π
4
y
3
Period: π
x
2
Two consecutive
asymptotes:
−π
x ,x
2
2
x
y
1
2
4
0
4
0
1
2
π
x
Section 4.6
23. y tan
x
4
y
24. y tanx 6
Period:
4
4
4
2
x
−4
Two consecutive asymptotes:
4
x
1
0
1
y
1
0
1
25. y csc x
Period: 2
4
Two consecutive
asymptotes:
2
1
− 3π
2
x
6
2
5
6
y
2
1
2
Two consecutive
asymptotes:
2
1
x ,x
2
2
y
29. y 4
−π
π
2π
3π
x
2
Two consecutive
asymptotes:
y
1
2
4
y
1
0
1
4
7
12
1
4
1
2
y
2
2
Two consecutive
asymptotes:
2
π
π
2
−1
3π
2
2π
3
4
x
−2
x
12
4
5
12
y
2
1
2
y
2
2
3
2
Two consecutive
asymptotes:
1
x
1
x
1
3
0
1
3
y
1
0
1
2
2
y
Period: 2π
3
x ,x
4
4
12
0
1
4
30. y 2 cot x y
Period: 2
x
4
x
π
1
1
x ,x
2
2
3
2
1
csc x 4
4
−1
1
x
Period:
3
0
2
3
28. y sec x 1
y
4
3
Two consecutive
asymptotes:
x 0, x Period: 2
x
π
2
27. y 2 secx x
4
Period:
3
x 0, x Period: 26. y csc2x y
389
y
x ,x
2
2
x
⇒ x 2
4
2
x ⇒ x2
4
2
Graphs of Other Trigonometric Functions
x
Two consecutive
asymptotes:
− 3π
2
x ,x
2
2
x
y
2
4
3π
2
−2
0
4
0
2
x
390
Chapter 4
31. y tan
Trigonometry
x
3
32. y tan 2x
3
5
−5␲
4
− 3␲
4
5␲
3␲
4
−␲
2
␲
2
−3
−5
34. y sec x ⇒ y 1
cos x
35. y tan x −4
4
36. y 2
1
cot x 4
2
4 tan x 3
− 3␲
2
3␲
2
−2
1
3
−3
2
cos 4x
33. y 2 sec 4x 2
3
−3
− 3␲
2
3␲
2
−3
37. y csc4x y
38. y 2 sec(2x ) ⇒
3
1
sin4x −␲
2
y
␲
2
4
2
cos2x −␲
␲
−3
39. y 0.1 tan
4x 4 −4
40. y 0.6
−6
1
x 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
3
3
4
2 5
,
x , ,
3
3 3 3
y
2
y
2
3
2
1
1
π
2π
x
π
2π
x
−3
π
2
3π
2
x
Section 4.6
Graphs of Other Trigonometric Functions
46. sec x 2
45. sec x 2
44. cot x 1
x
7 3 5
, , ,
4
4 4 4
x±
2 4
,±
3
3
x
5 5
, , ,
3
3 3 3
y
y
391
y
3
2
− π2
− 3π
2
1
x
π
2
−2π
3π
2
−π
1
π
2π
x
−2π
−π
π
2π
x
−3
47. csc x 2
x
48. csc x y
7 5 3
, , ,
4
4 4 4
−π
2
2
1
cos x
π
2
2
1
x
3π
2
− 3π
2
y
−π
3π
2
y
tanx tan x
3
x
π
2
2
f x tan x
50.
4
f x secx
3
2
1
cosx
−1
3
2 4 5
x , , ,
3
3 3 3
2
− 3π
f x sec x y
3
1
49.
23
3
−π
π
2π
x
Thus, the function is odd
and the graph of y tan x
1
cos x
− 3π
2
− π2
π
2
−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
π
(b) f > g on the interval,
x
5
< x <
6
6
(c) As x → , f x 2 sin x → 0 and
gx 12 csc x → ± since g x is
the reciprocal of f x.
(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).
x
3π
2
392
Chapter 4
Trigonometry
54. y1 sin x sec x, y2 tan x
53. y1 sin x csc x and y2 1
2
4
−3
−2␲
3
2␲
−2
−4
sin x csc x sin x
sin x 1, sin x 0
1
sin x sec x sin x
The expressions are equivalent except when sin x 0
and y1 is undefined.
55. y1 The expressions are equivalent.
cos x
1
and y2 cot x sin x
tan x
cot x 1
sin x
tan x
cos x cos x
56. y1 sec 2 x 1, y2 tan2 x
3
1 tan2 x sec2 x
cos x
sin x
4
The expressions are equivalent.
tan2 x sec2 x 1
−2␲
2␲
− 3␲
2
3␲
2
The expressions are equivalent.
−1
−4
58. f x x sin x
57. f x x cos x
Matches graph (a) as x → 0, f x → 0.
As x → 0, f x → 0 and f x > 0.
Matches graph (d).
59. gx x sin x
60. gx x cos x
As x → 0, gx → 0 and gx is odd.
Matches graph (c) as x → 0, gx → 0.
Matches graph (b).
61. f x sin x cos x 2
3
gx 0
1
−3
−2
−1
2
y
4
gx 2 sin x
2
f x gx
The graph is the line y 0.
62. f x sin x cos x y
x
1
2
3
−1
sin x cos x −2
2
It appears that f x gx.
That is,
−π
2 sin x.
2
x
π
−4
−3
63. f x sin2 x
64. f x cos2
y
1
gx 1 cos 2x
2
x
2
y
3
3
1
gx 1 cos x
2
2
f x gx
−π
π
–1
x
2
It appears that f x gx.
That is,
x 1
1 cos x.
cos2
2
2
−6
x
−3
3
−1
6
Section 4.6
65. gx ex 2 sin x
2
Graphs of Other Trigonometric Functions
1
ex 2 ≤ gx ≤ ex 2
2
2
−8
8
The damping factor is y ex 2.
2
As x →
, gx →0.
−1
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
−3
As x →
−1
, 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 2
6
8␲
0
−6␲
6␲
0
−2
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␲
sin x
x
1
x
2
␲
−␲
−2
−1
d
−1
As x → 0, f x oscillates
between 1 and 1.
As x → 0, f (x) → 0.
75. tan x ␲
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
6
2
−2
−6
π
4
π
2
3π
4
π
x
60
Distance
Ground distance
10
40
20
−π
2
− 10
− 14
Angle of elevation
−π
4
0
π
4
Angle of camera
π
2
x
393
394
Chapter 4
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 4.6
Graphs of Other Trigonometric Functions
83. As x →
82. True.
y sec x 1
cos x
As x →
395
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
396
Chapter 4
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
14 e68
1.702 1029
2
96. lnx 4 5
1
2 lnx
x2 e3.2 1
4 5
lnx 4 10
x 4 e10
x e10 4
22,022.466
x ± e3.2 1 ± 4.851
97. log8 x log8x 1 13
98. log6 x log6x2 1 log664x
log8xx 1 13
log6x(x2 1 log664x
xx 1 813
x2 x 2
x2 x 2 0
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.
xx2 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 4.7
Section 4.7
■
■
Inverse Trigonometric Functions
397
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
398
Chapter 4
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
2
3
2
< y <
⇒ y0
2
2
2
⇒ sin y 3
3
⇒ tan y 33 for
< y <
⇒ y
2
2
6
2
−␲
2
Graph y2 tan1 x.
g
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
␲
2
g
Graph y3 x.
−1
23. arctan3 tan13 1.25
for
≤ y ≤
⇒ y
2
2
4
Graph y1 tan x.
1.5
f
2
16. y arccos 1 ⇒ cos y 1 for 0 ≤ y ≤ ⇒ y 0
gx arcsin x
−1.5
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
10. y arctan3 ⇒ tan y 3 for
< y <
⇒ y
2
2
3
11. y arccos 2
8. y arcsin 25. arcsin 0.31 sin1 0.31 0.32
27. arccos0.41 cos10.41 1.99
26. arccos 0.26 1.31
28. arcsin0.125 0.13
29. arctan 0.92 tan1 0.92 0.74
30. arctan 2.8 1.23
31. arcsin4 sin10.75 0.85
32. arccos 3 1.91
33. arctan72 tan13.5 1.29
3
34. arctan 95
1.50
7
1
35. This is the graph of y arctan x. The coordinates are
3, 3 , 3 , 6 , and 1, 4 .
3
Section 4.7
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
399
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
400
Chapter 4
Trigonometry
5
52. Let u arccos
cos u 5
5
sin arccos
5
53. Let y arcsin
,
,0 < u <
5
5
,
2
sin u 5
sin y 2
5
25
5
5
,
13
5
12
, 0 < y < , and cos y .
13
2
13
y
.
2
13
u
5
y
1
x
12
54. Let u arctan tan u 5 ,
5
,
12
55. Let y arctan 34
3
tan y , < y < 0, and sec y .
5
2
5
5
, < u < 0,
12
2
csc arctan 5
12
3
csc u 5 .
13
y
5
x
y
−3
34
12
u
13
−5
3 ,
4 ,
56. Let u arcsin 3
57. Let y arccos 3 sin u , < u < 0,
4
2
3
tan arcsin 4
5
2 cos y ,
< y < , and sin y .
3 2
3
y
37
3
.
tan u 7
7
5
3
y
7
−2
u
4
2
−3
x
Section 4.7
5
58. Let u arctan ,
8
401
59. Let y arctan x,
5
tan u , 0 < u < ,
8
2
cot arctan
Inverse Trigonometric Functions
89
u
x
1
and cot y .
x
8
5
8
cot u .
8
5
x2 + 1
x
tan y x ,
1
5
x
60. Let u arctan x, tan u x ,
1
x
sinarctan x sin u .
x2 1
y
1
61. Let y arcsin2x,
sin y 2x 2x
,
1
1
2x
and cos y 1 4x2.
y
1 − 4x 2
x2 + 1
x
u
1
62. Let u arctan 3x,
tan u 3x 63. Let y arccos x,
3x
,
1
x
cos y x ,
1
9x 2 + 1
secarctan 3x sec u 9x2 1.
3x
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
3 ,
x
x
cos y ,
3
1
2x x2
.
3
and tan y 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
1
1
cot u x.
cot arctan
x
tan y x
2
and csc y u
x
x
2
,
x2 + 2
,
x
x2 2
x
.
y
2
402
Chapter 4
68. Let u arcsin
sin u Trigonometry
xh
,
r
xh
,
r
cos arcsin
r
xh
cos u r
r 2
x−h
x h
.
r
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
x
−2
x
Let u arccos .
2
2
x
tan u
2
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
arcsin y These are equal because:
f x tan arccos
tan y 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 4.7
74. If arccos
x2
u,
2
then cos u arccos
Inverse Trigonometric Functions
403
75. y 2 arccos x
Domain: 1 ≤ x ≤ 1
x2
,
2
Range: 0 ≤ y ≤ 2
x2
arctan
2
4x x 2
x2
This is the graph of f x arccos x with a factor of 2.
.
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
404
Chapter 4
82. f x arccos
Trigonometry
x
4
y
83. f x 2 arccos2x
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 graph 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 4.7
92. (a) tan s
750
Inverse Trigonometric Functions
93. arctan
arctan
(a)
s
750
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
95.
arctan
11
0.5743 32.9
17
1
(b) r 40 20
2
tan 20 ft
θ
41 ft
(a) tan h
h
r
20
h 20 tan 20 20
41
arctan
11
12.94 feet
17
(b) tan 26 41 26.0
20
h
50
h 50 tan 26 24.39 feet
96. (a) tan 6
x
97. (a) tan arctan
6
x
x
20
arctan
x
20
(b) x 7 miles
6
arctan 0.71 40.6
7
12
31.0
20
6
1.41 80.5
1
99. False.
98. False.
5
is not in the range of arcsinx.
6
arcsin
5
14.0
20
x 12: arctan
x 1 mile
arctan
(b) x 5: arctan
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 Since arctan 1 arcsin 1
.
arccos 1
2
arcsin 1
undefined.
, but
4
arccos 1
0
4
405
406
Chapter 4
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 ≤ .
2
2
The domain of y arcsec x is , 1 1, y
Range: 0 < x < π
2 2 , .
and the 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 4.7
Inverse Trigonometric Functions
407
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. 10142 10
10
0.051
142 196
408
Chapter 4
Trigonometry
111. 1.150 117.391
3
opp
4
hyp
sin 113.
112. 162 114. tan 2
4
3
θ
cos tan cot 4
3
37
7
3
csc 4
3
cos 5
6 hyp
opp2 25 36
opp 11
2
opp 11
cot sec sin 2
5
cot 1
2
2
θ
1
116. sec 3
opp2 52 62
tan 1
5
1
csc 5
2
4
47
7
7
sin cos sec 5
7
sec 115.
hyp 12 22 5
7
7
1
2.718 108
162
6
opp 32 12
8
θ
5
cos 1
3
sin 22
3
11
6
11
5
5
511
11
11
6
5
6
611
csc 11
11
3
22
θ
1
tan 22
sec 3
csc cot 3
22
32
4
2
1
4
22
Section 4.7
Inverse Trigonometric Functions
117. Let x the number of people presently in the group. Each person’s share is now 250,000x.
If two more join the group, each person’s share would then be 250,000x 2.
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
rate time distance ⇒ t d
r
Time to go upstream Time to go downstream 4
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.
0.035
4
0.035
12
0.035
365
119. (a) A 15,000 1 (b) A 15,000 1 (c) A 15,000 1 (d) A 15,000e0.03510
410
\$21,253.63
1210
36510
\$21,275.17
\$21,285.66
\$21,286.01
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
742 458,504.31
632
2
er 4
409
410
Chapter 4
Section 4.8
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
A 90 B
sin B b
⇒ b c sin B
c
b = 10
3. Given: B 71, b 24
tan B b
b
24
⇒ a
8.26
a
tan B tan 71
sin B b
b
24
⇒ 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
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
a 71°
C
c = 15
15 sin 54 12.14
c
20°
C
54°
a
90 54 36
B
a
B
25
45.58
35
a
25
a
⇒ B arccos arccos
44.42
c
c
35
Section 4.8
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
411
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
C
A
b = 1.32
9. Given: A 12 15, c 430.5
B 90 12 15 77 45
sin 12 15 a
430.5
a 430.5 sin 12 15 91.34
cos 12 15 10. Given: B 65 12, a 14.2
A 90 B 90 65 12 24 48
cos B a
14.2
a
⇒ c
33.85
c
cos B cos 65 12
tan B b
⇒ b a tan B 14.2 tan 65 12 30.73
a
b
430.5
B
b 430.5 cos 12 15 420.70
a = 14.2
B
12°15′
b
C
11. tan 1
h 4 tan 52 2.56 inches
2
12. tan h
1
⇒ h b tan 12b
2
1
h 10 tan 18 1.62 meters
2
h
h
θ
θ
θ
θ
1
b
2
1
b
2
1
b
2
b
b
13. tan A
b
A
h
1
⇒ h b tan 12b
2
1
b
2
c
C
c = 430.5
a
65°12′
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
412
Chapter 4
15. tan 25 Trigonometry
50
x
16. tan 20 50
50
x
tan 25
x
25°
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 4.8
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
arctan
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
28.
θ
29. The plane has traveled 1.5600 900 miles.
100x
12x = y
4 miles = 21
,120 feet
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
413
414
Chapter 4
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 4.8
38. cot 55 d
⇒ d 7 kilometers
10
a
⇒ x a cot 57
x
a
tan 16 x 556
Distance between towns:
tan 16 D d 18.8 7 11.8 kilometers
cot 16 28°
55°
T1
55°
d
28°
P2
a
57°
16°
H
550
60
x
a
a cot 57 556
a cot 57 556
a
a cot 16 a cot 57 T2
415
P1
39. tan 57 D
cot 28 ⇒ D 18.8 kilometers
10
10 km
Applications and Models
D
55
⇒ a 3.23 miles
6
17,054 ft
40. tan 2.5 h
x
h
2.5°
17
h
x
tan 2.5
tan 9 x
x
9°
x − 17
41. L1: 3x 2y 5 ⇒ y 3
3
5
⇒ m1 x
2
2
2
L2: 3x y 1 ⇒ y x 1 ⇒ m2 1
Not drawn to scale
tan h
x 17
1 32
52
5
1 132
12
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 m2 m1
15 2
arctan
1 152
1 m2m1
9
52.1
arctan
7
44. tan 43. The diagonal of the base has a length of
a2 a2 2a. Now, we have
a
2a
arctan
1
2
a
1
2
θ
35.3.
a2
2
a
45. sin 36 2a
d
⇒ d 14.69
25
Length of side: 2d 29.4 inches
arctan 2 54.7º
a
2
d
θ
a
36°
25
416
Chapter 4
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
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.
t
3
54. Displacement at t 0 is 2 ⇒ d a cos t.
Amplitude: a 2
2
Period:
10 ⇒ 5
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.
4t
3 t 3 cos 3 .
d 2 cos
t
5
9
Section 4.8
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
417
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:
1
1
64
64
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
1
792 792
60. At t 0, buoy is at its high point ⇒ d a cos t.
Distance from high to low 2 a 3.5
7
a 4
Returns to high point every 10 seconds:
2264 528
Period:
d
2
10 ⇒ 5
7
t
cos
4
5
418
Chapter 4
61. y Trigonometry
1
cos 16t, t > 0
4
(a)
y
(b) Period:
1
(c)
π
8
π
4
3π
8
π
2
2 16
8
1
⇒ t
cos 16t 0 when 16t 4
2
32
t
−1
62. (a)
(b)
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
8 16 cos 60º
8 sin 60º
83.1
8
8 16 cos 70º
8 sin 70º
80.7
8 sin 2
16 16 cos 4 sin 641 cos sin (d)
100
0
90
0
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.
419
S
Average sales
(in millions of dollars)
Section 4.8
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 feet 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
1
70. Linear equation m 2 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
4
3
2
−2 −1
−1
13 23
12 14
x
1
2
3
4
5
y
y
3
1
34
1
14, 32 and 21, 13
2
1
4
3
−3
2
4
1
x
3
3
4
1
4
y x
3
3
−2
−1
x
2
−1
−2
−3
3
420
Chapter 4
Trigonometry
Review Exercises for Chapter 4
11
4
(a)
4. y
2
9
(a)
5. 4
3
(a)
y
y
11π
4
2π
9
x
x
x
−
(b) The angle lies in Quadrant II.
(c)
11
3
2 4
4
23
3
(a)
(a)
4
2
2 3
3
4
10
2 3
3
8. 280
7. 70
y
−
(c) Coterminal angles:
2
20
2 9
9
16
2
2 9
9
5
3
2 4
4
6. (b) The angle lies in Quadrant II.
(c) Coterminal angles:
4π
3
(a)
y
23π
3
y
280°
70°
x
x
x
(c) 23
8 3
3
17
23
2 3
3
(b) The angle lies in Quadrant I.
(c) Coterminal angles:
(c) 280 360 640
70 360 430
70 360 290
280 360 80
Review Exercises for Chapter 4
10. 405
9. 110
(a)
(a)
y
11. 480 480 y
180
8
3
− 405°
x
x
− 110°
(b) The angle lies in Quadrant III.
(c) 405 720 315
(c) Coterminal angles:
405 360 45
110 360 250
110 360 470
12. 127.5 2.225
180
13. 33 45 33.75 33.75 77 180 3.443
60
14. 196 77 196 16. 11
6
18. 5.7 180
15.
3
16
7
7
180
180
326.586
19. 138 138 23
180
30
s r 20
180
180
200.535
20. 60 2330 48.17 inches
60
180
s r 11 60
180
11
meters
3
s 11.52 meters
21. (a) Angular speed 3313 2 radians
22. linear speed angular speed radius
1 minute
(b) Linear speed 666 23 inches
1 minute
67.5 inches per second
212.1 inches per second
400 inches per minute
23. 120 1 5
1
6.5 2
24. A r 2 2
2 6
120 2
180
3
1
2
1
339.29 square inches
A r 2 182
2
2
3
25. t 12.05 miles per hour
2
1 3
corresponds to the point ,
.
3
2 2
A 55.31 square millimeters
26. t 2 2
3
, x, y ,
4
2
2
421
422
Chapter 4
Trigonometry
27. t 3 1
5
corresponds to the point , .
6
2 2
29. t 3
1
7
, .
corresponds to the point 6
2
2
28. t 30. t 4
1 3
, x, y ,
3
2 2
2 2
,
.
corresponds to the point
4
2
2
sin
1
7
y
6
2
csc
7 1
2
6
y
sin
2
y
4
2
csc
1
2
4
y
cos
3
7
x
6
2
sec
7 1
23
6
x
3
cos
2
x
4
2
sec
1
2
4
x
tan
3
7 y
1
6
x 3
3
cot
7 x
3
6
y
tan
y
1
4
x
cot
x
1
4
y
31. t 3
2
y
3
2
sin 2
1
23
3
y
3
2
1 3
.
corresponds to the point , 3
2
2
csc 32. t 2 corresponds to the point 1, 0.
sin 2 y 0
csc 2 1
is undefined.
y
2
1
x
cos 3
2
2
1
sec 2
3
x
cos 2 x 1
sec 2 1
1
x
2
y
tan 3
3
x
2
x 3
cot 3
y
3
tan 2 y
0
x
cot 2 x
is undefined.
y
33. sin
11
3 2
sin
4
4
2
36. cos 39. sec
13
1
5
cos
3
3
2
12
5 37. tan 33 75.3130
38. csc 10.5 1
1.1368
sin 10.5
9 0.3420
1
3.2361
12
cos
5
40. sin opp
4
441
hyp 41
41
17
5
1
sin 6
6
2
35. sin 41. opp 4, adj 5, hyp 42 52 41
sin 34. cos 4 cos 0 1
csc hyp 41
opp
4
5
541
cos hyp 41
41
41
hyp
5
opp 4
5
5
cot opp 4
43. adj 4, hyp 8, opp 82 42 48 43
sin opp 43 3
hyp
8
2
csc hyp
8
23
opp 43
3
cos 4 1
hyp 8 2
sec hyp
8
2
4
tan opp 43
3
4
cot 3
4
opp 43
3
hyp 62 62 62
sin 2
opp
6
hyp 62
2
csc hyp 62
2
opp
6
cos 2
6
hyp 62
2
sec hyp 62
2
6
tan opp 6
1
6
6
1
opp 6
Review Exercises for Chapter 4
44. opp 5, hyp 9
45. sin adj 92 52 214
1
3
(a) csc opp 5
sin hyp 9
1
3
sin (b) sin2 cos2 1
214
cos hyp
9
13
2
cos2 1
tan opp
5
514
28
214
csc hyp 9
opp 5
cos2 sec hyp
9
914
28
214
8
9
cos cot 214
opp
5
89
cos 22
3
46. tan 4
(a) cot (c) sec 1
3
32
cos 22
4
(d) tan 2
sin 13
1
cos 223 22
4
47. csc 4
1
1
tan 4
(a) sin (b) sec 1 tan2 1 16 17
(c) cos 1
9
cos2 1 (d) csc 1 cot2 (b) sin2 cos2 1
14
2
17
1
1
sec 17
17
1 161 1
1
csc 4
cos2 1
17
cos2 1 4
1
16
cos2 15
16
cos 1516
cos 15
4
1
4
415
cos 15
15
15
sin 14
1
(d) tan cos 154 15
15
(c) sec 48. csc 5
(a) sin 1
1
csc 5
(c) tan (b) cot csc2 1 25 1 26
49. tan 33 0.6494
50. csc 11 6
1
1
cot 26
12
(d) sec90 csc 5
1
5.2408
sin 11
51. sin 34.2 0.5621
423
424
Chapter 4
52. sec 79.3 Trigonometry
1
5.3860
cos 79.3
53. cot 15 14 1
tan15 14
60 54. cos 78 11 58 cos 78 3.6722
55. sin 1 10 x
3.5
0.2045
x 3.5 sin 1 10 0.07 kilometer or 71.3 meters
25
x
52°
x
Not drawn to scale
58. x, y 3, 4
57. x 12, y 16, r 144 256 400 20
sin 25
x
25
19.5 feet
x
tan 52
tan 52 56.
m
3.5 k
1°10'
y 4
r
5
csc r 32 42 5
5
r
y 4
x 3
cos r
5
r
5
sec x 3
y 4
tan x 3
x 3
cot y 4
sin y
4
r
5
csc r
5
y
4
cos x 3
r
5
sec r
5
x 3
tan y
4
x
3
cot x
3
y
4
2
5
59. x , y 3
2
r
23 52
2
2
241
6
sin y
15241
52
15
r
241
2416
241
csc 2416
r
2241 241
y
52
30
15
cos x
4241
23
4
r
241
2416
241
sec 2416
241
r
x
23
4
tan y 52 15
x 23
4
cot x 23
4
y 52 15
60. x, y r
10 2
,
3
3
103 32
2
2
11
58 60 3600
226
3
sin y
26
23
r
26
2263
csc r
2263 26
y
23
cos x
103
526
r
26
2
26
3
sec r
2263 26
x
103
5
tan y
23
1
x 103 5
cot x 103
5
y
23
Review Exercises for Chapter 4
61. x 0.5, y 4.5
r 0.52 4.52 20.5 82
2
sin 4.5
y
982
r
82
822
csc 822
82
r
y
4.5
9
cos x
0.5
82
r
82
822
sec 822
r
82
x
0.5
tan 4.5
y
9
x 0.5
cot 1
x 0.5
y
4.5
9
62. x, y 0.3, 0.4
r 0.32 0.42 0.5
sin y 0.4 4
0.8
r
0.5 5
csc 0.5 5
r
1.25
y 0.4 4
cos x 0.3 3
0.6
r
0.5 5
sec r
0.5 5
1.67
x 0.3 3
tan y 0.4 4
1.33
x 0.3 3
cot x 0.3 3
0.75
y 0.4 4
63. x, 4x, x > 0
x x, y 4x
r x2 4x2 17 x
sin y
4x
417
r
17
17 x
csc 17 x
17
r
y
4x
4
cos 17
x
x
r
17
17 x
sec 17 x
r
17
x
x
tan y 4x
4
x
x
cot x
x
1
y 4x 4
64. x, y 2x, 3x, x > 0
r 2x2 3x2 13x
6
65. sec , tan < 0 ⇒ is in Quadrant IV.
5
r 6, x 5, y 36 25 11
sin y
313
3x
r
13
13x
sin 11
y
r
6
cos 213
x
2x
r
13
13x
cos x 5
r
6
tan y 3x
3
x 2x
2
tan 11
y
x
5
csc 13x
13
r
y
3x
3
csc r
611
y
11
sec 13x
13
r
x
2x
2
sec 6
5
cot x 2x 2
y 3x 3
cot 511
11
425
426
Chapter 4
Trigonometry
3
66. csc , cos < 0
2
3
67. sin , cos < 0 ⇒ is in Quadrant II.
8
y 3, r 8, x 55
sin 1
2
csc 3
cos 1 sin2 5
3
sin y 3
r
8
cos 55
x
r
8
tan sin 25
cos 5
tan 3
y
355
55
x
55
sec 1
35
cos 5
csc 8
3
sec 5
1
cot tan 2
cot 5
68. tan , cos < 0
4
69. cos 855
55
55
3
x 2
⇒ y2 21
r
5
sin > 0 ⇒ is in Quadrant II ⇒ y 21
sec 1 tan2 cos 8
55
1 2516 41
4
1
441
sec 41
sin 1 cos2 1 4116 5 4141
sin y 21
r
5
tan 21
y
x
2
csc 5
r
521
y 21
21
csc 41
1
sin 5
sec r
5
5
x 2
2
cot 1
4
tan 5
cot x
2
221
y
21
21
2
1
70. sin , cos > 0
4
2
71. 264
264 180 84
y
csc 1
2
sin cos 1 sin2 1
23
sec cos 3
tan 3
sin cos 3
cot 1
3
tan 1 14 3
264°
2
x
θ′
Review Exercises for Chapter 4
73.
72. 635 720 85
85
y
6
5
74. 17 18 3
3
3
6
4
2 5
5
6 635°
4 5
5
y
3
3
y
x
θ′
17π
3
θ′
x
−
3
3
2
75. sin
x
θ′
6π
5
76. sin
2
4
2
77. sin 3
7
sin 3
3
2
7
1
cos 3
3
2
7
tan 3
3
3
cos
1
3
2
cos
2
4
2
cos tan
3
3
tan
2
1
4
22
tan 5
2
sin 4
4
2
2
5
cos 4
4
2
5
tan
4
4
78. sin cos tan 79. sin 495 sin 45 2
22
2
1
2
cos150 2
tan150 3
2
3
12
3
32
1
81. sin240 sin 60 3
82. sin315 2
cos240 cos 60 1
2
cos315 tan240 tan 60 3
83. sin 4 0.7568
86. cot4.8 2
cos 495 cos 45 80. sin150 tan 495 tan 45 1
2
1
0.0878
tan4.8
tan315 84. tan 3 0.1425
12
5 87. sec
1
3.2361
12
cos
5
2
2
2
2
22
1
22
85. sin3.2 0.0584
88. tan
257 4.3813
427
428
Chapter 4
Trigonometry
89. y sin x
91. f x 5 sin
90. y cos x
Amplitude: 1
Amplitude: 1
Period: 2
Period: 2
Amplitude: 5
Period:
y
y
2
5
25
y
2
2
2x
5
6
1
4
x
π
2
− 3π
2
−π
2π
π
x
2
−1
6π
−2
x
−2
−2
−6
4x 92. f x 8 cos Shift the graph of y sin x
two units upward.
Amplitude: 8
Period:
94. y 4 cos x
93. y 2 sin x
2
8
14
Amplitude: 1
Period:
y
2
2
4
y
y
3
8
−3
2
−2
−1
x
1
2
3
−1
−2
4π
8π
−π
x
π
−1
2π
x
−3
−2
−4
−5
−6
−6
−8
95. gt 5
sint 2
96. gt 3 cost (a) a 2,
Amplitude: 3
5
2
Amplitude:
97. y a sin bx
Period: 2
Period: 2
2
1
⇒ b 528
b
264
y
y 2 sin528x
4
y
3
4
2
3
1
π
1
−1
(b) f π
t
1
1264
264 cycles per second.
t
−3
−2
−4
−3
−4
98. (a) St 18.09 1.41 sin
6t 4.60
(b) Period 2
26 12
6
12 months 1 year, so this is expected.
22
(c) Amplitude: 1.41
0
12
14
The amplitude represents the maximum change in the time
of sunset from the average time d 18.09.
Review Exercises for Chapter 4
100. f t tan t 99. f x tan x
y
4
429
101. f x cot x
y
y
4
4
3
3
2
2
1
1
3
2
1
x
π
−π
2
t
π
2
−π
x
π
−3
−4
102. gt 2 cot 2t
104. ht sec t 103. f x sec x
Graph y cos x first.
y
3
4
y
y
2
1
1
−π
π
t
t
π
−π
−1
π
x
−2
−3
−4
106. f t 3 csc 2t 105. f x csc x
Graph y sin x first.
4
y
y
4
2
3
2
π
1
− 3π
2
π
2
t
x
−3
−4
107. f x x cos x
108. gx x 4 cos x
6
Graph y x and y x first.
As x → , f x → .
300
Damping factor: x 4
−9
9
As x → , f x → .
−2
−300
−6
21 arcsin 21 6
2
109. arcsin 110. arcsin1 112. arcsin0.213 0.21 radian
115. arccos
3
2
6
116. arccos
2
22 4
117. cos 11 430
Chapter 4
118. cos1
Trigonometry
23 6
123. f x 2 arcsin x 2 sin1x
124. y 3 arccos x
␲
3␲
−1.5
1.5
−1.5
−␲
125. f x arctan
1.5
0
2x tan 2x 126. f x arcsin 2x
1
␲
2
␲
2
−1.5
−4
1.5
4
−␲
2
−␲
2
127. cosarctan 34 45
128. Let u arccos 35.
Use a right triangle. Let
arctan 34 then tan and cos 45 .
tanarccos 35 tan u 43
5
3
4
5
3
θ
4
u
4
3
13
129. secarctan 12
5 5
130. Let u arcsin 12
13 .
Use a right triangle. Let arctan
13
then tan 12
5 and sec 5 .
12
5
13
12
cot arcsin 12
13
5
cot u 5
12
u
θ
13
−12
5
131. Let y arccos
cos y 2x . Then
132. secarcsinx 1
x
x
and tan y tan arccos
2
2
4 x 2
x
.
arcsinx 1 ⇒ ≤ ≤
2
2
sin x 1
cos 12 x 12 x2 x
2
sec 4 − x2
1
x2 x
1
y
x
θ
12 − (x − 1)2
x −1
Review Exercises for Chapter 4
133. tan 70
30
arctan
d1
⇒ d1 483
650
cos 25 d2
⇒ d2 734
810
d
cos 48 3 ⇒ d3 435
650
tan d4
⇒ d4 342
810
h
h 25 tan 21 9.6 feet
135. sin 48 sin 25 h
25
134. tan 21 70
66.8
30
431
21°
25
N
d1 d2 1217
48°
W
d3
A
d3 d4 93
B
25°
48° 65° 810
650
D
θ
d1
d2
E
d4
C
S
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.
136. Amplitude:
1.5
0.75 inches
2
Period: 3 seconds
137. False. The sine or cosine functions
are often useful for modeling
simple harmonic motion.
138. True. The inverse sine,
y arcsin x, is defined where
1 ≤ x ≤ 1 and
≤ y ≤ .
2
2
140. False. The range of arctan is
, , so arctan1 .
2 2
4
141. y 3 sin x
d a cos bt
a 0.75
b
2
3
d 0.75 cos
23 t
139. False. For each there
corresponds exactly one
value of y.
Amplitude: 3
Period: 2
Matches graph d
142. y 3 sin x matches graph (a).
143. y 2 sin x
Period: 2
Amplitude: 2
Amplitude: 3
Period: 2
144. y 2 sin
Period: 4
Amplitude: 2
Matches graph b
145. f sec is undefined at the zeros of g cos since sec x
matches graph (c).
2
1
.
cos 432
Chapter 4
146. (a)
Trigonometry
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
1
148. y Aekt cos bt 5 et10 cos 6t
147. 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
1
1
1
(a) A is changed from 5 to 3 : The displacement is
increased.
(b) k is changed from 10 to 3 : The friction damps the
oscillations more rapidly.
(c) b is changed from 6 to 9: The frequency of oscillation is increased.
1
149. A 2 r 2, s r
1
(a) A 2 r 20.8 0.4r 2, r > 0
(b) A 12 102 50, > 0
s r0.8 0.8r, r > 0
s 10, > 0
As r increases, the area function increases more rapidly.
30
A
4
A
s
s
3
0
0
0
6
0
Problem Solving for Chapter 4
1. (a) 8:57 6:45 2 hours 12 minutes 132 minutes
2. Gear 1:
132 11
revolutions
48
4
Gear 2:
24
26
Gear 3:
24
22
Gear 4:
40
5
32
2
Gear 5:
24
19
(b) s r 47.255.5 816.42 feet
3. (a) sin 39 d
(b) tan 39 x
3000
d
3000
4767 feet
sin 39
3000
x
3000
3705 feet
tan 39
(c)
tan 63 24
3
32
2
w 3705
3000
3000 tan 63 w 3705
w 3000 tan 63 3705 2183 feet
Problem Solving for Chapter 4
433
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
opp BC DE FG
sin A it does not matter which triangle is used to calculate sin A.
hyp
AB
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.
5. (a) hx cos2 x
6. Given: f is an even function and g is an odd function.
hx f x2
(a)
3
hx f x2
−2␲
f x2 since f is even
2␲
hx
−1
Thus, h is an even function.
h is even.
(b) hx hx gx2
(b)
sin2 x
hx gx2
3
gx2 since g is odd
−2␲
gx2
2␲
hx
−1
Thus, h is an even function.
h is even.
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.
8. P 100 20 cos
(a)
1
1
a max min 101 1 50
2
2
83t
130
1
1
d max min 101 1 51
2
2
0
b 8
5
70
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 Here we wanted the sine value to be 1 when t 0.
3
2
(b) Period 2
6 3
sec
83 8 4
This is the time between heartbeats.
(c) Amplitude: 20
The blood pressure ranges between 100 20 80
and 100 20 120.
(d) Pulse rate (e) Period 64 60 secmin
80 beatsmin
3
4 secbeat
60 15
sec
64 16
64
60
⇒ b
2b
60
32
2 15 434
Chapter 4
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
leap years
remaining
July days
August days
31
1
11
20 years
t 365 20
5
(b) Number of days since birth until September 1, 2006:
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)
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.
(b) The graphs intersect when x 5.35 32 0.65.
6
g
−␲
␲
f
−6
(b) The period of f x is 2.
The period of gx is .
(c) hx A cos x B 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 4
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
435
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
436
Chapter 4
Chapter 4
Trigonometry
Practice Test
1. Express 350° in radian measure.
2. Express 59 in degree measure.
3. Convert 135 14 12 to decimal form.
4. Convert 22.569 to D M S 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 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.
B
c
A
b
21. A 40, c 12
a
C
22. B 6.84, a 21.3
23. a 5, b 9
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
```