chapter 6 trigonometry - de la salle high school · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5...
TRANSCRIPT
C H A P T E R 6Trigonometry
Section 6.1 Angles and Their Measure . . . . . . . . . . . . . . . . . 532
Section 6.2 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 541
Section 6.3 Trigonometric Functions of Any Angle . . . . . . . . . . 552
Section 6.4 Graphs of Sine and Cosine Functions . . . . . . . . . . . 567
Section 6.5 Graphs of Other Trigonometric Functions . . . . . . . . . 579
Section 6.6 Inverse Trigonometric Functions . . . . . . . . . . . . . . 591
Section 6.7 Applications and Models . . . . . . . . . . . . . . . . . . 604
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
C H A P T E R 6Trigonometry
Section 6.1 Angles and Their Measure
532
You should know the following basic facts about angles, their measurement, and their applications.
■ Types of Angles:
(a) Acute: Measure between 0� and 90�.
(b) Right: Measure 90�.
(c) Obtuse: Measure between 90� and 180º.
(d) Straight: Measure 180�.
■ Two positive angles, and are complementary if They are supplementary if
■ Two angles in standard position that have the same terminal side are called coterminal angles.
■ To convert degrees to radians, use radians.
■ To convert radians to degrees, use 1 radian
■ one minute of 1�.
■ one second of 1�.
■ The length of a circular arc is where is measured in radians.
■ Speed
■ Angular speed � ��t � s�rt
� distance�time
�s � r�
� 1�60 of 1� � 1�36001� �
� 1�601� �
� �180����.1� � ��180
� � � 180�.� � � 90�.�
Vocabulary Check
1. Trigonometry 2. angle
3. coterminal 4. degree
5. acute; obtuse 6. complementary; supplementary
7. radian 8. linear
9. angular 10. A �12r 2�
1. The angle shown is approximately 210�.
2. The angle shown is approximately .120�
3. The angle shown is approximately 60�.
4. The angle shown is approximately 330�.
5. (a)
(b) Since 270� < 285� < 360�; 285� lies in Quadrant IV.
Since 90� < 130� < 180�; 130� lies in Quadrant II. 6. (a) Since lies in Quadrant I.
(b) Since lies inQuadrant III.
180� < 257� 30� < 270�; 257� 30�
0� < 8.3� < 90�; 8.3�
Section 6.1 Angles and Their Measure 533
7. (a) lies inQuadrant III.
(b) lies inQuadrant I.Since 360� < 336� < 270�; 336�
Since 180� < 132�50� < 90�; 132�50� 8. (a) Since lies inQuadrant II.
(b) Since lies in Quadrant IV.
90� < 3.4� < 0�; 3.4�
270� < 260� < 180�; 260�
9. (a)
x30°
y (b)
150°
x
y
10. (a)
−270°
x
y270� (b)
−120°
x
y120�
11. (a)
405°
x
y (b)
480°
x
y
12. (a)
−750°x
y750� (b)
−600°
x
y600�
13. (a) Coterminal angles for 45
(b) Coterminal angles for 36
36� 360� � 396�
36� � 360� � 324�
�
45� 360� � 315�
45� � 360� � 405�
� 14. (a)
(b)
420� � 360� � 60�
420� � 720� � 300�
120� 360� � 240�
120� � 360� � 480� 15. (a) Coterminal angles for 300
(b) Coterminal angles for 740
20� 360� � 340�
740° 2�360°� � 20°
�
300� 360� � 60�
300� � 360� � 660�
�
534 Chapter 6 Trigonometry
16. (a)
(b)
230� 360� � 130�
230� � 360� � 590�
520� � 360� � 160�
520� � 720� � 200� 17. (a)
(b) 128�30� � 128� �3060�� � 128.5�
54�45� � 54� � �4560�� � 54.75�
18. (a)
(b)
� 2� � 0.2� � 2.2�
2�12� � 2� � �1260��
� 245.167�
� 245� � 0.167�
245�10� � 245� � �1060�� 19. (a)
(b) 330�25� � �330 �25
3600�� � 330.007�
85�18� 30� � �85 �1860 �
303600�� � 85.308�
20. (a)
(b)
� 408.272�
� �408� � 0.2667� � 0.0056��
408�16�20� � �408� � �1660�� � � 20
3600�� � � 135� 0.01� � 135.01�
135�36� � 135� � 363600�� 21. (a)
(b) 145.8� � �145� � 0.8�60��� � 145�48�
240.6� � 240� � 0.6�60�� � 240�36�
22. (a)
(b)
� 0�27�
� 0� � 27�
0.45� � 0� � �0.45��60��
� 345� 7� 12�
� �345� � 7� � 0.2�60� ��
345.12� � �345� � �0.12��60� �� 23. (a)
(b)
� 3�34�48�
� [3�34� � 0.8(60)�]
� [3�34.8�]
3.58� � [3� � 0.58(60)�]
2.5� � 2� � 0.5(60)� � 2�30�
24. (a)
(b)
� 0�47�11.4�
� 0� � 47� � 11.4�
� 0� � 47� � �0.19��60��0.7865 � 0� � �0.7865��60��
� 0�21�18�
� �0� � 21� � 18��
� �0� � 21� � �0.3��60���
0.355� � �0� � �0.355��60��� 25. (a) Complement:
Supplement:
(b) Complement: Not possible. is greater than
Supplement: 180� 126� � 54�
90�.126�
180� 24� � 156�
90� 24� � 66�
26. (a) Complement:
Supplement:
(b) Complement: Not possible.
Supplement: 180� 166� � 14�
�166� > 90��
180� 87� � 93�
90� 87� � 3� 27. The angle shown is approximately 2 radians.
28. The angle shown is approximately 5.5 radians.
29. The angle shown is approximately 3 radians.
Section 6.1 Angles and Their Measure 535
30. The angle shown is approximately radians.4
31. (a) Since lies in Quadrant I.
(b) Since lies in Quadrant III.7�
5� <
7�
5<
3�
2;
�
5 0 <
�
5<
�
2;
32. (a) Since lies in Quadrant IV.
(b) Since lies in Quadrant II.11�
9
3�
2<
11�
9< �;
�
12
�
2<
�
12< 0; 33. (a) Since lies in Quadrant IV.
(b) Since lies in Quadrant III.� < 2 < �
2; 2
�
2< 1 < 0; 1
34. (a) Since lies in Quadrant IV.3�
2< 6.02 < 2� ; 6.02 (b) Since lies in Quadrant II.2.25
�
2< 2.25 < �,
35. (a)
54π
x
y5�
4(b)
−23π
x
y
2�
3
36. (a) (b)
x
y
52π
5�
2
− 74π
y
x
7�
4
37. (a)
116π
x
y11�
6(b)
−3
x
y3
38. (a) 4 (b)
7π
x
y7�
4
x
y
536 Chapter 6 Trigonometry
39. (a) Coterminal angles for
(b) Coterminal angles for
5�
6 2� �
7�
6
5�
6� 2� �
17�
6
5�
6
�
6 2� �
11�
6
�
6� 2� �
13�
6
�
640. (a)
(b)
11�
6 2� �
23�
6
11�
6� 2� �
�
6
7�
6 2� �
5�
6
7�
6� 2� �
19�
641. (a)
(b)
2�
15 2� �
32�
15
2�
15� 2� �
28�
15
Coterminal angles for 2�
15
9�
4� 2� �
�
4
9�
4� 4� �
7�
4
Coterminal angles for 9�
4
42. (a)
(b)
8�
45 2� �
82�
45
8�
45� 2� �
98�
45
8�
9 2� �
10�
9
8�
9� 2� �
26�
943. (a) Complement:
Supplement:
(b) Complement:
Supplement: � 11�
12�
�
12
Not possible; 11�
12 is greater than
�
2
� �
12�
11�
12
�
2
�
12�
5�
12
44. (a) Complement:
Supplement:
(b) Complement: none
Supplement: � 3�
4�
�
4
�3�
4>
�
2�
� �
3�
2�
3
�
2
�
3�
�
645. (a)
(b) 150� � 150� �
180� �5�
6
30� � 30� �
180� ��
6
46. (a)
(b) 120� � 120�� �
180�� �2�
3
315� � 315�� �
180�� �7�
447. (a)
(b) 240� � 240� �
180� � 4�
3
20� � 20� �
180� � �
9
48. (a)
(b) 144� � 144�� �
180�� �4�
5
270� � 270�� �
180�� � 3�
249. (a)
(b)7�
6�
7�
6 �180� ��
� 210�
3�
2�
3�
2 �180� ��
� 270�
50. (a)
(b)�
9�
�
9�180�
� � � 20�
7�
12�
7�
12�180�
� � � 105� 51. (a)
(b) 11�
30�
11�
30 �180
� ��� 66�
7�
3�
7�
3 �180
� ��� 420� 52. (a)
(b)34�
15�
34�
15 �180�
� � � 408�
11�
6�
11�
6 �180�
� � � 330�
53. 115� � 115� �
180� � 2.007 radians 54. 87.4� � 87.4�� �
180�� � 1.525 radians
Section 6.1 Angles and Their Measure 537
55. 216.35� � 216.35� �
180� � 3.776 radians 56. radians48.27� � 48.27�� �
180�� � 0.842
57. 0.83� � 0.83� �
180� � 0.014 radian 58. 0.54� � 0.54�� �
180�� � 0.009 radians
59.�
7�
�
7 �180
� ��� 25.714� 60.
5�
11�
5�
11�180�
� � � 81.818� 61.15�
8�
15�
8 �180
� ��� 337.500�
62. 4.8� � 4.8��180�
� � � 864.000� 63. 2 � 2�180
� ��� 114.592� 64. 0.57 � 0.57�180�
� � � 32.659�
65.
� �65 radians
6 � 5�
s � r� 66.
� �2910 radians
29 � 10�
s � r� 67.
� �327 radians
32 � 7�
s � r� 68.
Because the angle represented is clock-wise, this angle is radian.
45
� �6075 �
45 radian
60 � 75�
s � r�
69.
radian� �6
27 �
2
9
6 � 27�
s � r� 70. feet, feet
� �s
r�
8
14�
4
7 radian
s � 8r � 14 71.
3� �25
14.5�
50
29 radians
25 � 14.5�
3s � r� 72. kilometers,
kilometers
� �s
r�
160
80� 2 radians
s � 160
r � 80
73.
� 47.12 inches
s � 15�180�� �
180� � 15� inches
s � r�, � in radians
75.
3s � 3(1) � 3 meters
3s � r�, � in radians
74. feet,
s � r� � 9��
3� � 3� feet � 9.42 feet
� � 60� ��
3r � 9
76. centimeters,
s � r� � 20��
4� � 5� centimeters � 15.71 centimeters
� ��
4r � 20
77.
square inches
square inches � 8.38
A �12
�4�2��
3� �8�
3
A �12
r 2� 78.
� 56.5 mm2 � 18� mm2
A �12
r 2� �12
�12�2��
4�
r � 12 mm, � ��
4
79.
square feetA �12
�2.5�2�225�� �
180� � 12.27
A �12
r 2� 80.
square miles� 5.6 �21.56
12�
A �12
�1.4�2�330�
180���
r � 1.4 miles, � � 330�
538 Chapter 6 Trigonometry
83. � �s
r�
450
6378� 0.071 radian � 4.04°
81. radian
miless � r� � 4000�0.14782� � 591
15�50� 32�47�39� � 8.46972� � 0.14782� � 41�
82. radian
miles
miless � r� � 4000�0.17154� � 686
r � 4000
37�18� 37�47�36� � 9�49�42� � 9.82833� � 0.17154� � 47�
84. kilometers
radian
The difference in latitude is about 3.59�.
0.062716�180� � � 3.59�
� �sr
�400
6378� 0.062716
r � 6378
85. � �s
r�
2.5
6�
25
60�
5
12 radian 86. � �
s
r�
24
5� 4.8 radians � 4.8�180�
� � � 275�
88. (a) 2-inch diameter pulley
1700 rpm
Since the belt moves 10681.4
On the 4-inch diameter pulley:
This pulley is turning at
(b)5340.7
2�� 850 rpm
5340.7 radians�minute.
� �10681.4
2� 5340.7
s � 10681.4 � 2 � �
r � 2
inches�minute.r � 1,
� 10681.4 radians�minute
� 1700 � 2� radians�minute
89. (a) Angular speed
(b)
� 164.5 feet per second
� 314123
� feet per minute
Linear speed ��7.25
2 in.�� 1 ft
12 in.��5200��2�� feet
1 minute
� 10,400� radians per minute
��5200��2�� radians
1 minute
90. (a)
� 25.13274 radians�minute
� 8�
4 rpm � 4�2�� radians�minute (b)
� 628.32 ft�minute
Linear speed � 25�25.13274� ft�minute
r�
t� 200� ft�minute
r � 25 ft
87. (a)
The circumference of the tire is feet.
The number of revolutions per minute isrev/minute.r � 5720�2.5� � 728.3
C � 2.5�
� 5720 feet per minute
65 miles per hour � 65�5280��60 (b) The angular speed is
Angular speed �4576 radians
1 minute� 4576 radians�minute
� �5720
2.5�(2�) � 4576 radians
��t.
Section 6.1 Angles and Their Measure 539
91. (a)
Interval:
(b)
Interval: �2400�, 6000�� centimeters per minute
�6��200��2�� ≤ Linear speed ≤ �6��500��2�� centimeters per minute
�400�, 1000�� radians per minute
�200��2�� ≤ Angular speed ≤ �500��2�� radians per minute
92.
� 1445 in.2
A �12�
125180�� � �392 142� � 460.069�
r � 14
R � 25 � 14 � 39 25125°
14r
A �12
��R2 r2� 93.
� 1496.62 square meters
� 476.39� square meters
�12
�35�2�140��� �
180��140°
35
A �12
r 2�
94. (a) Arc length of larger sprocket in feet:
Therefore, the chain moves feet as does the smaller rear sprocket.
Thus, the angle of the smaller sprocket is
and the arc length of the tire in feet is:
(b) Since the arc length of the tire is feet and the cyclist is pedaling at a rate of one revolution per second, we have:
(c) Distance
(d) The functions are both linear.
�7�
7920 t miles
� �14�
3 feet�second�� 1 mile
5280 feet��t seconds�
� Rate � Time
�7�
7920 n miles
Distance � �14�
3
feetrevolutions��
1 mile5280 feet��n revolutions�
14�
3
14� feet
3 seconds
3600 seconds
1 hour
1 mile
5280 feet� 10 miles per hour
Speed �s
t�
14�
3
1 sec �
14�
3 feet per second
s � �4���14
12� �14�
3 feet
s � �r
�r � 2 inches �2
12 feet�� �
s
r�
2�
3 ft
2
12 ft
� 4�
�
2�
3
s �1
3�2�� �
2�
3 feet
s � r�
540 Chapter 6 Trigonometry
95. False. A measurement of radians corresponds to twocomplete revolutions from the initial to the terminal sideof an angle.
4�
97. False. The terminal side of lies on the negative x-axis.1260�
99. The speed increases, since the linear speed is proportional to the radius.
101. Since the length is given by , if the central angle is fixed while the radius increases,then increases in proportion to .rs
r�s � r�sarc
96. True. If and are coterminal angles, thenor where is an
integer. The difference between isor if expressed in
radians.� � n�2��� � n�360��,
� and n� � � n�2��,� � � n�360��
�
98. (a) An angle is in standard position if its vertex is at the origin and its initial side is on the positive x-axis.
(b) A negative angle is generated by a clockwise rotation.
(c) Two angles with the same initial and terminal sides are coterminal.
(d) An obtuse angle measures between 90° and 180°.
100. so one radian is much
larger than one degree.
1 radian � �180
� ��� 57.3�,
102.
Area of sector � ��r2�� �
2�� �12
r2�
Area of sector
�r2 ��
2�
Area of sectorArea of circle
�Measure of central angle of sectorMeasure of central angle of circle
θr
Area of circle � �r2
103.4
42�
4
42�22
�42
8�
2
2
105.236
� 236
� 212
�22
�22
� 2
107. 22 � 62 � 4 � 36 � 40 � 4 � 10 � 210
109.
� 144 � 2 � 122
182 62 � 324 36 � 288
104.23
�23
�33
�23
3
106.55
210�
52 5
10�
52
12
�5
22�22
�52
4
108.
� 36 � 13 � 613
182 � 122 � 324 � 144 � 468
110.
� 16 � 13 � 413
172 92 � 289 81 � 208
111. Horizontal shift two unitsto the right
432−2
3
2
1
−1
−2
−3
x
yy = x5
y = (x − 2)5
112.
Vertical shift four unitsdownward
4
2
−2
−6
32−2−3 1
y = x5
y = x5 − 4
x
yf (x) � x5 4
Section 6.2 Right Triangle Trigonometry 541
113. Reflection in the x-axis and a vertical shift twounits upward
321−2−3
6
5
4
3
1
−1
−2
−3
x
y
y = x5
y = 2 − x5
114.Reflection in the x-axis and a horizontal shift three units to the left
3
2
1
−1
−2
−3
21−3−4−5x
y
y = x5
y = −(x + 3)5f �x� � ��x � 3�5
■ You should know the right triangle definition of trigonometric functions.
(a) (b) (c)
(d) (e) (f)
■ You should know the following identities.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
( j) (k)
■ You should know that two acute angles are complementary if and that the cofunctions ofcomplementary 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.
���
� � � � 90�,� and �
1 � cot2 � � csc2 �1 � tan2 � � sec2 �
sin2 � � cos2 � � 1cot � �cos �
sin �tan � �
sin �
cos �
cot � �1
tan �tan � �
1
cot �sec � �
1
cos �
cos � �1
sec �csc � �
1
sin �sin � �
1
csc �
cot � �adj
oppsec � �
hyp
adjcsc � �
hyp
opp
Adjacent side
Opp
osite
sid
e
Hypote
nuse
θ
tan � �opp
adjcos � �
adj
hypsin � �
opp
hyp
Section 6.2 Right Triangle Trigonometry
Vocabulary Check
1. (i) (e) 2. opposite; adjacent; hypotenuse
(ii) (f) 3. elevation; depression
(iii) (d)
(iv) (b)
(v) (a)
(vi) (c)oppositeadjacent
� tan �
oppositehypotenuse
� sin �
adjacenthypotenuse
� cos �
hypotenuseopposite
� csc �
adjacentopposite
� cot �
hypotenuseadjacent
� sec �
542 Chapter 6 Trigonometry
1.
tan � �opp
adj�
6
8�
3
4
cos � �adj
hyp�
8
10�
4
5
sin � �opp
hyp�
6
10�
3
5
θ
6
8
hyp � �62 � 82 � �36 � 64 � �100 � 10
cot � �adj
opp�
8
6�
4
3
sec � �hyp
adj�
10
8�
5
4
csc � �hyp
opp�
10
6�
5
3
3.
tan � �opp
adj�
9
40
cos � �adj
hyp�
40
41
sin � �opp
hyp�
9
41θ9
41
adj � �412 � 92 � �1681 � 81 � �1600 � 40
cot � �adj
opp�
40
9
sec � �hyp
adj�
41
40
csc � �hyp
opp�
41
9
5.
tan � �opp
adj�
1
2�2�
�2
4
cos � �adj
hyp�
2�2
3
sin � �opp
hyp�
1
3θ 3
1
adj � �32 � 12 � �8 � 2�2
cot � �adj
opp� 2�2
sec � �hyp
adj�
3
2�2�
3�2
4
csc � �hyp
opp� 3
2.
sin csc
cos sec
tan cot � �adj
opp�
12
5� �
opp
adj�
5
12
� �hyp
adj�
13
12� �
adj
hyp�
12
13
� �hyp
opp�
13
5� �
opp
hyp�
5
135
13
bθ
adj � �132 � 52 � �169 � 25 � 12
4.
sin csc
cos sec
tan cot � �adj
opp�
4
4� 1� �
opp
adj�
4
4� 1
� �hyp
adj�
4�2
4� �2� �
adj
hyp�
4
4�2�
1�2
��2
2
� �hyp
opp�
4�2
4� �2� �
opp
hyp�
4
4�2�
1�2
��2
24
4
θ
hyp � �42 � 42 � �32 � 4�2
tan � �opp
adj5
2
4�2�
1
2�2�
�2
4
cos � �adj
hyp�
4�2
6�
2�2
3
sin � �opp
hyp�
2
6�
1
3θ
62
adj � �62 � 22 � �32 � 4�2
cot � �adj
opp�
4�2
2� 2�2
sec � �hyp
adj�
6
4�2�
3
2�2�
3�2
4
csc � �hyp
opp�
6
2� 3
The function values are the same since the triangles are similar and the corresponding sides are proportional.
Section 6.2 Right Triangle Trigonometry 543
6.
cot � �adj
opp�
15
8tan � �
opp
adj�
8
15
sec � �hyp
adj�
17
15cos � �
adj
hyp�
15
17
csc � �hyp
opp�
17
8sin � �
opp
hyp�
8
17
hyp � �152 � 82 � �289 � 17
15
8
θ
cot � �adj
opp�
7.5
4�
15
8tan � �
opp
adj�
4
7.5�
8
15
sec � �hyp
adj�
�17�2�7.5
�17
15cos � �
adj
hyp�
7.5
�17�2��
15
17
csc � �hyp
opp�
�17�2�4
�17
8sin � �
opp
hyp�
4
�17�2��
8
17
hyp � �7.52 � 42 �17
2
7.5
4θ
The function values are the same because the triangles are similar, and corresponding sides are proportional.
7.
tan � �opp
adj�
3
4
cos � �adj
hyp�
4
5
sin � �opp
hyp�
3
5
θ
4
5
opp � �52 � 42 � 3
cot � �adj
opp�
4
3
sec � �hyp
adj�
5
4
csc � �hyp
opp�
5
3
tan � �opp
adj�
0.75
1�
3
4
cos � �adj
hyp�
1
1.25�
4
5
sin � �opp
hyp�
0.75
1.25�
3
5θ1
1.25
opp � �1.252 � 12 � 0.75
cot � �adj
opp�
1
0.75�
4
3
sec � �hyp
adj�
1.25
1�
5
4
csc � �hyp
opp�
1.25
0.75�
5
3
The function values are the same since the triangles are similar and the corresponding sides are proportional.
8.
cot � �adj
opp�
2
1� 2tan � �
opp
adj�
1
2
sec � �hyp
adj�
�5
2cos � �
adj
hyp�
2�5
�2�5
5
csc � �hyp
opp�
�5
1� �5sin � �
opp
hyp�
1�5
��5
5
hyp � �12 � 22 � �5
1
2
θ
cot � �adj
opp�
6
3� 2tan � �
opp
adj�
3
6�
1
2
sec � �hyp
adj�
3�5
6�
�5
2cos � �
adj
hyp�
6
3�5�
2�5
�2�5
5
csc � �hyp
opp�
3�5
3� �5sin � �
opp
hyp�
3
3�5�
1�5
��5
5
hyp � �32 � 62 � 3�5
3
6
θ
The function values are the same because the triangles are similar, and corresponding sides are proportional.
544 Chapter 6 Trigonometry
9. Given:
csc � �hyp
opp�
4
3
sec � �hyp
adj�
4�7
7
cot � �adj
opp�
�7
3
tan � �opp
adj�
3�7
7
cos � �adj
hyp�
�7
4
adj � �7
32 � �adj�2 � 42
θ
7
4 3
sin � �3
4�
opp
hyp
11. Given:
csc � �hyp
opp�
2�3
3
cot � �adj
opp�
�3
3
tan � �opp
adj� �3
cos � �adj
hyp�
1
2
sin � �opp
hyp�
�3
2
opp � �3
�opp�2 � 12 � 22
1
2 3
θ
sec � � 2 �2
1�
hyp
adj12.
sec � �hyp
adj�
�26
5
csc � �hyp
opp�
�26
1� �26
tan � �opp
adj�
1
5
cos � �adj
hyp�
5�26
�5�26
26
261
5
θsin � �opp
hyp�
1�26
��26
26
hyp � �52 � 12 � �26
13. Given:
csc � �hyp
opp�
�10
3
sec � �hyp
adj� �10
cot � �adj
opp�
1
3
cos � �adj
hyp�
�10
10
sin � �opp
hyp�
3�10
10
hyp � �10
32 � 12 � �hyp�2
3
1
10
θ
tan � � 3 �3
1�
opp
adj14.
cot � �adjopp
�1
�35�
�3535
csc � �hypopp
�6
�35�
6�3535
tan � �oppadj
��35
1� �35
cos � �adjhyp
�16
sin � �opphyp
��35
66
1
θ
35
opp � �62 � 12 � �35
10.
cot � �adj
opp�
5
2�6�
5�6
12
sec � �hyp
adj�
7
5
csc � �hyp
opp�
7
2�6�
7�6
12
tan � �opp
adj�
2�6
5
5
7 2 6
θ
sin � �opp
hyp�
2�6
7
opp � �72 � 52 � �24 � 2�6
Section 6.2 Right Triangle Trigonometry 545
15. Given:
sec � �hyp
adj�
�13
3
csc � �hyp
opp�
�13
2
tan � �opp
adj�
2
3
cos � �adj
hyp�
3�13
�3�13
13
sin � �opp
hyp�
2�13
�2�13
13
hyp � �13
22 � 32 � �hyp�22
3
13
θ
cot � �3
2�
adj
opp16.
cot � �adj
opp�
�273
4
sec � �hyp
adj�
17�273
�17�273
273
tan � �opp
adj�
4�273
�4�273
273
cos � �adj
hyp�
�273
17
417
273
θ
sin � �opp
hyp�
4
17
adj � �172 � 42 � �273
17.
sin 30� �opphyp
�12
30� � 30��
180�� �
6 radian
30°
60°12
3
18. degree radian value
cos
cos 45� �1�2
��22
�22
445�
21
1
45°
19.
tan
3�
oppadj
��31
� �3
3�
3�180�
� � 60�
1
23
π3
π6
20. degree radian value
sec
sec
4�
�21
� �2
�2
445�
21
1
π4
21.
� � 60� �
3 radian
cot � ��33
�1�3
�adjopp
1
3
30°
60°
22. degree radian value
csc
csc 45� ��21
�2
445�
21
1
45°
23.
cos
6�
adjhyp
��32
6�
6�180�
� � 30�
12
3
π3
π6
24. degree radian value
sin
sin
4�
1�2
��22
�22
445�
21
1
π4
25.
� � 45� � 45��
180�� �
4
cot � � 1 �11
�adjopp
21
1
45°
45°26. degree radian value
tan
tan 30� �1�3
1�3
630�
30°
60°12
3
546 Chapter 6 Trigonometry
27.
(a)
(b)
(c)
(d) cot 60� �cos 60�
sin 60��
1�3
��3
3
cos 30� � sin 60� ��3
2
sin 30� � cos 60� �1
2
tan 60� �sin 60�
cos 60�� �3
sin 60� ��3
2, cos 60� �
1
228.
(a)
(b)
(c)
(d) cot 30� �1
tan 30��
3�3
�3�3
3� �3
cos 30� �sin 30�
tan 30��
1
2
�3
3
�3
2�3�
�3
2
cot 60� � tan�90� � 60�� � tan 30� ��3
3
csc 30� �1
sin 30�� 2
sin 30� �1
2, tan 30� �
�3
3
29.
(a)
(b)
(c)
(d) sec�90� � �� � csc � ��13
2
tan � �sin �
cos ��
2�13�13
3�13�13�
2
3
cos � �1
sec ��
3�13
�3�13
13
sin � �1
csc ��
2�13
�2�13
13
csc � ��13
2, sec � �
�13
330.
(a)
(b)
(c)
(d) sin � � tan � cos � � �2�6 ��1
5� �2�6
5
cot�90º � �� � tan � � 2�6
cot � �1
tan ��
1
2�6�
�6
12
cos � �1
sec ��
1
5
sec � � 5, tan � � 2�6
31.
(a)
(b)
(c)
(d) sin�90� � �� � cos � �1
3
cot � �cos �
sin ��
1
3
2�2
3
�1
2�2�
�2
4
sin � �2�2
3
sin2 � �8
9
sin2 � � �1
3�2
� 1
sin2 � � cos2 � � 1
sec � �1
cos �� 3
cos � �1
332.
(a)
(b)
(c)
(d)
��1 �1
25��26
25�
�26
5
��1 � �1
5�2
csc � � �1 � cot 2 �
tan�90º � �� � cot � �1
5
��2626
�1
�26
cos � �1
�1 � tan2 ��
1�1 � 52
cot � �1
tan ��
1
5
tan � � 5
33. tan � cot � � tan �� 1
tan �� � 1 34. cos � sec � � cos � 1
cos �� 1
35. tan � cos � � � sin �
cos �� cos � � sin � 36. cot � sin � �cos �
sin � sin � � cos �
Section 6.2 Right Triangle Trigonometry 547
37.
� sin2 �
� �sin2 � � cos2 �� � cos2 �
�1 � cos ���1 � cos �� � 1 � cos2 � 38. �1 � sin ���1 � sin �� � 1 � sin2 � � cos2 �
39.
� 1
� �1 � tan2 �� � tan2 �
�sec � � tan ���sec � � tan �� � sec2 � � tan2 � 40.
� 2 sin2 � � 1
� sin2 � � 1 � sin2 �
sin2 � � cos2 � � sin2 � � �1 � sin2 ��
41.
� csc � sec �
�1
sin �
1
cos �
�1
sin � cos �
sin �
cos ��
cos �
sin ��
sin2 � � cos2 �
sin � cos �42.
� 1 � cot2 � � csc2 �
� 1 � cot �
1
cot �
tan � � cot �
tan ��
tan �
tan ��
cot �
tan �
43. (a)
(b) cot 66.5� �1
tan 66.5�� 0.4348
tan 23.5� � 0.4348 44. (a)
(b) csc 16.35� �1
sin 16.35�� 3.5523
sin 16.35� � 0.2815
45. (a)
(b) sin 73�56� � sin�73 � 56
60��
� 0.9609
cos 16�18� � cos�16 �18
60��
� 0.9598 46. (a)
(b) csc 48�7� �1
sin �48 �7
60��� 1.3432
sec 42�12� � sec 42.2� �1
cos 42.2�� 1.3499
47. Make sure that your calculator is in radian mode.
(a)
(b) tan
16� 0.1989
cot
16�
1
tan
16
� 5.0273
48. (a)
(Note: 0.75 is in radians)
(b) cos 0.75 � 0.7317
sec 0.75 �1
cos 0.75� 1.3667
49. Make sure that your calculator is in radian mode.
(a)
(b) tan 1
2� 0.5463
csc 1 �1
sin 1� 1.1884
50. (a)
(b) cot�
2�
1
2� �1
tan�
2�
1
2�� 0.5463
sec�
2� 1� �
1
cos�
2� 1�
� 1.1884
51. (a)
(b) csc � � 2 ⇒ � � 30� �
6
sin � �1
2 ⇒ � � 30� �
652. (a)
(b) tan � � 1 ⇒ � � 45º �
4
cos � ��2
2 ⇒ � � 45� �
4
548 Chapter 6 Trigonometry
53. (a)
(b) cot � � 1 ⇒ � � 45� �
4
sec � � 2 ⇒ � � 60� �
3
55. (a)
(b) sin � ��2
2 ⇒ � � 45� �
4
csc � �2�3
3 ⇒ � � 60� �
3
54. (a)
(b) cos � �1
2 ⇒ � � 60� �
3
tan � � �3 ⇒ � � 60� �
3
56. (a)
(b)
cos � �1�2
��2
2 ⇒ � � 45� �
4
sec � � �2
tan � �3�3
� �3 ⇒ � � 60� �
3
cot � ��3
3
57.
x � 30�3
1�3
�30
x
tan 30� �30
x
30
30
x
°
58.
� 9�3
y � 18 sin 60� � 18�3
2
sin 60� �y
18
59.
x �32�3
�32�3
3
�3x � 32
�3 �32
x
tan 60� �32
x
60
32
x
°
60.
r �20
sin 45��
20�2
2
� 20�2
sin 45� �20
r
61.
Height of the building:meters
Distance between friends:
meters � 323.34
cos 82� �45y
⇒ y �45
cos 82�
123 � 45 tan 82� � 443.2
x � 45 tan 82�
tan 82� �x
45
45 m
82°
xy
62. (a) (b)
(c)
h � 270 feet
2�135� � h
tan � �63
�h
135
h
3
6132
Not drawn to scale
63.
� � 30� �
6
sin � �15003000
�12
θ
1500 ft3000 ft
64.
w � 100 tan 54� � 137.6 feet
tan 54� �w
100
tan � �opp
adj
Section 6.2 Right Triangle Trigonometry 549
65. (a)
(b)
(c) Moving down the line: feet per second
Dropping vertically: feet per second1506
� 25
150�sin 23�
6� 63.98
y �150
tan 23�� 353.4 feet
tan 23� �150
y
x �150
sin 23�� 383.9 feet
1
5 ft23°
x
y
sin 23� �150
x
66. Let the height of the mountain.
Let the horizontal distance from where the angle of elevation is sighted to the point at that level directly below the mountain peak.
Then tan
Substitute into the expression for tan
The mountain is about 1.3 miles high.
1.2953 � h
13 tan 9� tan 3.5�
tan 9� � tan 3.5�� h
13 tan 9� tan 3.5� � h�tan 9� � tan 3.5�� h tan 3.5� � 13 tan 9� tan 3.5� � h tan 9�
tan 3.5� �h tan 9�
h � 13 tan 9�
tan 3.5� �h
htan 9�
� 13
3.5�.x �h
tan 9�
tan 9� �h
x ⇒ x �
h
tan 9�
3.5� �h
x � 13 and tan 9� �
hx.
9�x �
h � 67.
�x2, y2� � �28, 28�3�
x2 � �cos 60°��56� � �12��56� � 28
cos 60° �x2
56
y2 � sin 60°�56� � ��32 ��56� � 28�3
sin 60� �y2
56
60°
56
( , )x y2 2
�x1,y1� � �28�3, 28�
x1 � cos 30��56� ��32
�56� � 28�3
cos 30� �x1
56
y1 � �sin 30���56� � �12��56� � 28
sin 30� �y1
56
30°
56
( , )x y1 1
(e)
550 Chapter 6 Trigonometry
69. (a)
(b)
(c)
(d) The side of the triangle labeled hwill become shorter.
h � 20 sin 85� � 19.9 meters
sin 85� �h
20
h20
85°
Angle, Height (in meters)
80 19.7
70 18.8
60 17.3
50 15.3
40 12.9
30 10.0
20 6.8
10 3.5�
�
�
�
�
�
�
�
�
70.
csc 20� �10
y� 2.92
sec 20� �10
x� 1.06
cot 20� �x
y� 2.75
tan 20� �y
x� 0.36
cos 20� �x
10� 0.94
sin 20� �y
10� 0.34
x � 9.397, y � 3.420
71.
True, csc x �1
sin x ⇒ sin 60� csc 60� � sin 60�� 1
sin 60�� � 1
sin 60� csc 60� � 1
72. True, because sec�90� � �� � csc �. 73.
False,�2
2�
�2
2� �2 � 1
sin 45� � sin 45� � 1
(f) The height of the balloondecreases.
20 h
θ
75.
False,
sin 2� � 0.0349
sin 60�
sin 30��
cos 30�
sin 30�� cot 30� � 1.7321
sin 60�
sin 30�� sin 2�74. True, because
cot 2 � � csc 2 � � �1.
cot 2� � csc 2 � � 1
1 � cot 2 � � csc 2 �
68.
d � 5 � 2x � 5 � 2�15 tan 3�� � 6.57 centimeters
x � 15 tan 3�
tan 3� �x
15
Section 6.2 Right Triangle Trigonometry 551
77. This is true because the corresponding sides of similar triangles are proportional.
76.
False.
tan 2�0.8� � �tan 0.8� 2 � 1.060
tan��0.8�2 � tan 0.64 � 0.745
tan��0.8� 2 � tan2�0.8�
78. Yes. Given can be found from the identity 1 � tan2 � � sec2 �.tan �, sec �
79. (a) (b) In the interval
(c) As and �
sin � → 1.� → 0, sin � → 0,
�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�
�
80. (a)
(b) On sin is an increasing function.
(c) On cos is a decreasing function.
(d) As the angle increases the length of the side opposite the angle increases relative to the length of the hypotenuse and the length of the side adjacent to the angle decreases relative to the length of the hypotenuse. Thus the sine increases and the cosine decreases.
��0, 1.5,
��0, 1.5,
0 0.3 0.6 0.9 1.2 1.5
sin 0 0.2955 0.5646 0.7833 0.9320 0.9975
cos 1 0.9553 0.8253 0.6216 0.3624 0.0707�
�
�
81.
�x
x � 2, x � ±6
x2 � 6x
x2 � 4x � 12
x2 � 12x � 36
x2 � 36�
x�x � 6��x � 6��x � 2�
�x � 6��x � 6��x � 6��x � 6�
82.
�2t � 3
4 � t, t � ±
3
2, �4
��2t � 3��t � 4��3 � 2t��3 � 2t�
�2t � 3��2t � 3��t � 4��t � 4�
� ��2t � 3��t � 4�
�2t 2 � 5t � 12
9 � 4t 2 4t 2 � 12t � 9
t 2 � 16
2t 2 � 5t � 12
9 � 4t 2
t 2 � 16
4t 2 � 12t � 9
83.
�2x2 � 10x � 20
�x � 2��x � 2�2�
2�x2 � 5x � 10��x � 2��x � 2�2
�3�x2 � 4� � 2�x2 � 4x � 4� � x2 � 2x
�x � 2��x � 2�2
3
x � 2�
2
x � 2�
x
x2 � 4x � 4�
3�x � 2��x � 2� � 2�x � 2�2 � x�x � 2��x � 2��x � 2�2
84.�3
x�
1
4��12
x� 1�
�
12 � x
4x
12 � x
x
�12 � x
4x
x
12 � x�
1
4, x � 0, 12
552 Chapter 6 Trigonometry
■ Know the Definitions of Trigonometric Functions of Any Angle.
If is in standard position, a point on the terminal side and then
■ You should know the signs of the trigonometric functions in each quadrant.
■ You should know the trigonometric function values of the quadrant angles
■ 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
■ You should know which trigonometric functions are odd and even.
Even: cos x and sec x
Odd: sin x, tan x, cot x, csc x
2�.
0, �
2, �, and
3�
2.
cot � �x
y, y � 0tan � �
y
x, x � 0
sec � �r
x, x � 0cos � �
x
r
csc � �r
y, y � 0sin � �
y
r
r � �x2 � y2 � 0,�x, y��
Section 6.3 Trigonometric Functions of Any Angle
Vocabulary Check
1. reference 2. periodic
3. period 4. even; odd
1. (a)
r � �16 � 9 � 5
�x, y� � �4, 3�
tan � �y
x�
3
4
cos � �x
r�
4
5
sin � �y
r�
3
5
cot � �x
y�
4
3
sec � �r
x�
5
4
csc � �r
y�
5
3
(b)
r � �64 � 225 � 17
�x, y� � �8, �15�
tan � �y
x� �
15
8
cos � �x
r�
8
17
sin � �y
r� �
15
17
cot � �x
y� �
8
15
sec � �r
x�
17
8
csc � �r
y� �
17
15
Section 6.3 Trigonometric Functions of Any Angle 553
2. (a)
cot � �x
y�
�12
�5�
12
5tan � �
y
x�
�5
�12�
5
12
sec � �r
x� �
13
12cos � �
x
r� �
12
13
csc � �r
y� �
13
5sin � �
y
r� �
5
13
r � ���12�2 � ��5�2 � 13
x � �12, y � �5 (b)
cot � �x
y�
�1
1� �1tan � �
y
x�
1
�1� �1
sec � �r
x�
�2
�1� ��2cos � �
x
r�
�1�2
� ��2
2
csc � �r
y�
�2
1� �2sin � �
y
r�
1�2
��2
2
r � ���1�2 � 12 � �2
x � �1, y � 1
3. (a)
r � �3 � 1 � 2
�x, y� � ���3, �1� (b)
r � �16 � 1 � �17
�x, y� � ��4, 1�
tan � �y
x�
�3
3
cos � �x
r� �
�3
2
sin � �y
r� �
1
2
cot � �x
y� �3
sec � �r
x� �
2�3
3
csc � �r
y� �2
tan � �y
x� �
1
4
cos � �x
r� �
4�17
17
sin � �y
r�
�17
17
cot � �x
y� �4
sec � �r
x� �
�17
4
csc � �r
y� �17
4. (a)
cot � �x
y�
3
1� 3
sec � �r
x�
�10
3
csc � �r
y�
�10
1� �10
tan � �y
x�
1
3
cos � �x
r�
3�10
�3�10
10
sin � �y
r�
1�10
��10
10
r � �32 � 12 � �10
x � 3, y � 1 (b)
cot � �x
y�
4
�4� �1
sec � �r
x�
4�2
4� �2
csc � �r
y�
4�2
�4� ��2
tan � �y
x�
�4
4� �1
cos � �x
r�
4
4�2�
�2
2
sin � �y
r�
�4
4�2� �
�2
2
r � �42 � ��4�2 � 4�2
x � 4, y � �4
5.
r � �49 � 576 � 25
�x, y� � �7, 24�
tan � �y
x�
24
7
cos � �x
r�
7
25
sin � �y
r�
24
25
cot � �x
y�
7
24
sec � �r
x�
25
7
csc � �r
y�
25
24
6.
tan � �y
x�
15
8
cos � �x
r�
8
17
sin � �y
r�
15
17
r � �82 � 152 � 17
x � 8, y � 15
cot � �x
y�
8
15
sec � �r
x�
17
8
csc � �r
y�
17
15
554 Chapter 6 Trigonometry
7.
r � �16 � 100 � 2�29
�x, y� � ��4, 10�
cot � �x
y� �
2
5
sec � �r
x� �
�29
2
csc � �r
y�
�29
5
tan � �y
x� �
5
2
cos � �x
r� �
2�29
29
sin � �y
r�
5�29
29
8.
cot � �x
y�
�5
�2�
5
2
sec � �r
x�
�29
�5� �
�29
5
csc � �r
y�
�29
�2� �
�29
2
tan � �y
x�
�2
�5�
2
5
cos � �x
r�
�5�29
� �5�29
29
sin � �y
r�
�2�29
� �2�29
29
r � ���5�2 � ��2�2 � �29
x � �5, y � �2
tan � �yx
� �6.83.5
cos � �xr
� �3.5
�58.49� �
3.5�58.4958.49
sin � �yr
�6.8
�58.49�
6.8�58.4958.49
9.
r � ���3.5�2 � �6.8�2 � �58.49
�x, y� � ��3.5, 6.8�
11.
sin � < 0 and cos � < 0 ⇒ � lies in Quadrant III.
cos � < 0 ⇒ � lies in Quadrant II or in Quadrant III.
sin � < 0 ⇒ � lies in Quadrant III or in Quadrant IV.
cot � �xy
� �3.56.8
sec � �rx
� ��58.49
3.5
csc � �ry
��58.49
6.8
10.
cot � �xy
�
72
�314
� �1431
tan � �yx
�
�314
72
� �3114
� �23
14
sec � �rx
�
�1157472
��1157
14cos � �
xr
�
72
�11574
�14�1157
1157
csc � �ry
�
�11574
�314
� ��1157
31sin � �
yr
�
�314
�11574
� �31�1157
1157
r ���72�
2
� ��314 �
2
��1157
4
x � 312
�72
, y � �734
� �314
12. lies in Quadrant I or in Quadrant II.
lies in Quadrant I or in Quadrant IV.
and lies in Quadrant I.cos � > 0 ⇒ �sin � > 0
cos � > 0 ⇒ �
sin � > 0 ⇒ �
13.
sin � > 0 and tan � < 0 ⇒ � lies in Quadrant II.
tan � < 0 ⇒ � lies in Quadrant II or in Quadrant IV.
sin � > 0 ⇒ � lies in Quadrant I or in Quadrant II. 14. lies in Quadrant I or in Quadrant IV.
lies in Quadrant II or in Quadrant IV.
and lies in Quadrant IV.cot � < 0 ⇒ �sec � > 0
cot � < 0 ⇒ �
sec � > 0 ⇒ �
Section 6.3 Trigonometric Functions of Any Angle 555
tan � �y
x� �
3
4
cos � �x
r� �
4
5
sin � �y
r�
3
5
cot � �x
y� �
4
3
sec � �r
x� �
5
4
csc � �r
y�
5
3
15.
� in Quadrant II ⇒ x � �4
sin � �yr
�35
⇒ x2 � 25 � 9 � 16 16.
in Quadrant III ⇒ y � �3�
cos � �x
r�
�4
5 ⇒ y � �3�
tan � �y
x�
3
4
cos � �x
r� �
4
5
sin � �y
r� �
3
5
cot � �4
3
sec � � �5
4
csc � � �5
3
17. is in Quadrant IV
tan � �yx
�� 15
8 ⇒ r � 17
y < 0 and x > 0.⇒sin � < 0 and tan � < 0 ⇒ �
tan � �y
x� �
15
8
cos � �x
r�
8
17
sin � �y
r� �
15
17
cot � �x
y� �
8
15
sec � �r
x�
17
8
csc � �r
y� �
17
15
tan � �y
x�
�15
8� �
15
8
cos � �x
r�
8
17
sin � �y
r�
�15
17� �
15
17
cot � � �8
15
sec � �17
8
csc � � �17
15
18.
tan � < 0 ⇒ y � �15
cos � �x
r�
8
17 ⇒ y � �15�
19.
x � 3, y � �1, r � �10
cos � > 0 ⇒ � is in Quadrant IV ⇒ x is positive;
cot � �x
y� �
3
1�
3
�1
tan � �y
x� �
1
3
cos � �x
r�
3�10
10
sin � �y
r� �
�10
10
cot � �x
y� �3
sec � �r
x�
�10
3
csc � �r
y� ��10
tan � �y
x� �
�15
15
cos � �x
r� �
�15
4
sin � �y
r�
1
4
cot � � ��15
sec � � �4�15
15
csc � � 4
20.
cot � < 0 ⇒ x � ��15
csc � �r
y�
4
1 ⇒ x � ��15�
21.
sin � > 0 ⇒ � is in Quadrant II ⇒ y � �3
sec � �r
x�
2
�1 ⇒ y2 � 4 � 1 � 3
tan � �y
x� ��3
cos � �x
r� �
1
2
sin � �y
r�
�3
2
cot � �x
y� �
�3
3
sec � �r
x� �2
csc � �r
y�
2�3
3
22.
tan � yr
��
r� 0
cos � �xr
� �rr
� �1
sin � � 0
y � 0, x � �r
sec � � �1 ⇒ � � � � 2�n
sin � � 0 ⇒ � � 0 � �n
is undefined.
is undefined.cot �
sec � rx
�r
�r� �1
csc �
23.
is undefined.
is undefined.cot �tan � � 0
sec � � �1cos � � �1
csc �sin � � 0
cot � is undefined, �
2≤ � ≤
3�
2 ⇒ y � 0 ⇒ � � �
556 Chapter 6 Trigonometry
24. tan is undefined
� ≤ � ≤ 2� ⇒ � �3�
2, x � 0, y � �r
⇒ � � n� ��
2�
is undefined.tan �
cos � �x
r�
0
r� 0
sin � �y
r�
�r
r� �1
is undefined.
cot � �x
y�
0
y� 0
sec �
csc � �r
y� �1
25. use any point onQuadrant II. is one
such point.��1, 1�the line y � �x that lies in
To find a point on the terminal side of �,
tan � � �1
cos � � �1�2
� ��2
2
sin � �1�2
��2
2
x � �1, y � 1, r � �2
cot � � �1
sec � � ��2
csc � � �2
26. Let
Quadrant III
tan � �y
x�
��1
3�x
�x�
1
3
cos � �x
r�
�x
�10x
3
� �3�10
10
sin � �y
r�
��1
3�x
�10x
3
� ��10
10
r ��x2 �1
9x 2 �
�10x
3
��x, �1
3x�,
x > 0.
28. Let
Quadrant IV
cot � � �3
4tan � �
y
x�
��4
3� x
x� �
4
3
sec � �5
3cos � �
x
r�
x
5
3x
�3
5
csc � � �5
4sin � �
y
r�
��4
3�x
5
3x
� �4
5
r ��x2 �16
9x2 �
5
3x
�x, �4
3x�,
4x � 3y � 0 ⇒ y � �4
3x
x > 0.27. use any point onthe line Quadrant III. is onesuch point.
��1, �2�y � 2x that lies inTo find a point on the terminal side of �,
tan � ��2
�1� 2
cos � � �1�5
� ��5
5
sin � � �2�5
� �2�5
5
x � �1, y � �2, r � �5
cot � ��1
�2�
1
2
sec � ��5
�1� ��5
csc � ��5
�2� �
�5
2
cot � �x
y�
�x
��1
3� x
� 3
sec � �r
x�
�10x
3
�x� �
�10
3
csc � �r
y�
�10x
3
��1
3�x
� ��10
29.
sin � �y
r�
0
1� 0
�x, y� � ��1, 0�, r � 1 30.
csc 3�
2�
1
�1� �1
�x, y� � �0, �1�, r � 1 31.
sec undefined3�
2�
r
x�
1
0 ⇒
�x, y� � �0, �1�, r � 1
Section 6.3 Trigonometric Functions of Any Angle 557
32.
sec � �r
x�
1
�1� �1
�x, y� � ��1, 0�, r � 1 33.
sin �
2�
y
r�
1
1� 1
�x, y� � �0, 1�, r � 1 34.
undefined.cot � ��1
0
�x, y� � ��1, 0�, r � 1
35.
undefinedcsc � �r
y�
1
0 ⇒
�x, y� � ��1, 0�, r � 1 36.
cot �
2�
x
y�
0
1� 0
�x, y� � �0, 1� 37.
′θ
203°
x
y
�� � 203� � 180� � 23�
� � 203�
38.
′θ
309°
x
y
�� � 360� � 309� � 51�
� � 309� 39.
′θ
−245°
x
y
�� � 180� � 115� � 65�
360� � 245� � 115� �coterminal angle�� � �245�
40.
(coterminal angle)
�� � 215� � 180� � 35�
360� � 145� � 215�
′θ−145°
x
y � � �145� 41.
�� � � �2�
3�
�
3 ′θ
23π
x
y
� �2�
3
42.
�� � 2� �7�
4�
�
4
′θ
74π
x
y
� �7�
443.
�� � 3.5 � �
′θ
3.5
x
y
� � 3.5
558 Chapter 6 Trigonometry
44. with .
�� � 2� �5�
3�
�
3
′θ
x
y
113π
5�
3� �
11�
3 is coterminal 45. Quadrant III
tan 225� � tan 45� � 1
cos 225� � �cos 45� � ��2
2
sin 225� � �sin 45� � ��2
2
�� � 45�,� � 225�,
46. in Quadrant IV.
sin
cos
tan 300� � �tan 60� � ��3
300� � cos 60� �1
2
300� � �sin 60� � ��3
2
� � 300�, �� � 360� � 300� � 60� 47. Quadrant I
tan 750� � tan 30� ��3
3
cos 750� � cos 30� ��3
2
sin 750� � sin 30� �1
2
� � 750�, �� � 30�,
48. in Quadrant IV.
sin
cos
tan��405�� � �tan 45� � �1
��405�� � cos 45� ��2
2
��405�� � �sin 45� � ��2
2
� � �405�, �� � 405� � 360� � 45� 49.
tan(�150�) � tan 30� ��33
cos(�150�) � �cos 30� � ��32
sin(�150�) � �sin 30� � �12
� � �150�, �� � 30�, Quadrant III
50. .
sin
cos
tan(�840�) � tan 60� � �3
(�840�) � �cos 60� � �1
2
(�840�) � �sin 60� � ��3
2
�� � 240� � 180� � 60� in Quadrant III.
� � �840� is coterminal with 240� 51. Quadrant III
tan 4�
3� tan
�
3� �3
cos 4�
3� �cos
�
3� �
1
2
sin 4�
3� �sin
�
3� �
�3
2
� �4�
3, �� �
�
3,
52. in Quadrant I.
sin
cos
tan �
4� 1
�
4�
�2
2
�
4�
�2
2
� ��
4, �� �
�
453. Quadrant IV
tan���
6� � �tan �
6� �
�3
3
cos���
6� � cos �
6�
�3
2
sin���
6� � �sin �
6� �
1
2
� � ��
6, �� �
�
6, 54. is coterminal with
sin
cos
tan is undefined.���
2� � tan 3�
2
���
2� � cos 3�
2� 0
���
2� � sin 3�
2� �1
3�
2.� � �
�
2
Section 6.3 Trigonometric Functions of Any Angle 559
55. Quadrant II
tan 11�
4� �tan
�
4� �1
cos 11�
4� �cos
�
4� �
�2
2
sin 11�
4� sin
�
4�
�2
2
� �11�
4, �� �
�
4, 56. is coterminal with
in Quadrant III.
sin
cos
tan 10�
3� tan
�
3� �3
10�
3� �cos
�
3� �
1
2
10�
3� �sin
�
3� �
�3
2
�� �4�
3� � �
�
3
4�
3.� �
10�
3
57.
tan��3�
2 � � tan �
2 which is undefined.
cos��3�
2 � � cos �
2� 0
sin��3�
2 � � sin �
2� 1
� � �3�
2, �� �
�
2 58. .
tan��25�
4 � � �tan��
4� � �1
cos��25�
4 � � cos��
4� ��22
sin��25�
4 � � �sin��
4� � ��22
�� � 2� �7�
4�
�
4 in Quadrant IV.
� � �25�
4 is coterminal with
7�
4
59.
in Quadrant IV.
cos � �4
5
cos � > 0
cos2 � �16
25
cos2 � � 1 �9
25
cos2 � � 1 � ��3
5�2
cos2 � � 1 � sin2 �
sin2 � � cos2 � � 1
sin � � �3
560.
sin � �1
csc ��
1�10
��10
10
csc � �1
sin �
�10 � csc �
csc � > 0 in Quadrant II.
10 � csc2 �
1 � ��3�2 � csc2 �
1 � cot2 � � csc2 �
cot � � �3 61.
in Quadrant III.
sec � � ��13
2
sec � < 0
sec2 � �13
4
sec2 � � 1 �9
4
sec2 � � 1 � �3
2�2
sec2 � � 1 � tan2 �
tan � �3
2
63.
sec � �158
�85
cos � �1
sec � ⇒ sec � �
1cos �
cos � �5862.
cot � � ��3
cot � < 0 in Quadrant IV.
cot2 � � 3
cot2 � � ��2�2 � 1
cot2 � � csc2 � � 1
1 � cot2 � � csc2 �
csc � � �2 64.
.
tan � ��65
4
tan � > 0 in Quadrant III
tan2 � �65
16
tan2 � � ��9
4�2
� 1
tan2 � � sec2 � � 1
1 � tan2 � � sec2 �
sec � � �9
4
560 Chapter 6 Trigonometry
65. sin 10� � 0.1736 66. sec 225� �1
cos 225�� �1.4142 67. cos��110�� � �0.3420
68. csc��330�� �1
sin��330�� � 2.0000 69. tan 304� � �1.4826 70. cot 178� � �28.6363
71. sec 72� �1
cos 72�� 3.2361 72. tan��188�� � �0.1405 73. tan 4.5 � 4.6373
74. cot 1.35 �1
tan 1.35� 0.2245 75. cos��
9� � 0.9397 76. tan���
9� � �0.3640
77. sin��0.65� � �0.6052 78. sec 0.29 �1
cos 0.29� 1.0436
79. cot��11�
8 � �1
tan��11��8�� �0.4142 80. csc��
15�
14 � �1
sin��15��14� � 4.4940
81. (a) reference angle is 30 or is in
Quadrant I or Quadrant II.
Values in degrees: 30 , 150
Values in radian:
(b) reference angle is 30 or is
in Quadrant III or Quadrant IV.
Values in degrees: 210 , 330
Values in radians:7�
6,
11�
6
��
�
6 and ��sin � � �
1
2 ⇒
�
6,
5�
6
��
�
6 and ��sin � �
1
2 ⇒ 82. (a) cos reference angle is 45 or and is
in Quadrant I or IV.
Values in degrees: 45 , 315
Values in radians:
(b) cos reference angle is 45 or and
is in Quadrant II or III.
Values in degrees: 135 , 225
Values in radians:3�
4,
5�
4
��
��
4�� � �
�2
2 ⇒
�
4,
7�
4
��
��
4�� �
�2
2 ⇒
83. (a) reference angle is 60 or
is in Quadrant I or Quadrant II.
Values in degrees: 60 , 120
Values in radians:
(b) reference angle is 45 or is
in Quadrant II or Quadrant IV.
Values in degrees: 135 , 315
Values in radians:3�
4,
7�
4
��
�
4 and ��cot � � �1 ⇒
�
3,
2�
3
��
�
3 and ��csc � �
2�3
3 ⇒ 84. (a) sec reference angle is 60 or and is in
Quadrant I or IV.
Values in degrees: 60 , 300
Values in radians:
(b) sec reference angle is 60 or and is
in Quadrant II or III.
Values in degrees: 120 , 240
Values in radians:2�
3,
4�
3
��
��
3�� � �2 ⇒
�
3,
5�
3
��
��
3�� � 2 ⇒
Section 6.3 Trigonometric Functions of Any Angle 561
85. (a) reference angle is 45 or is in
Quadrant I or Quadrant III.
Values in degrees: 45 , 225
Values in radians:
(b) reference angle is 30 or
is in Quadrant II or Quadrant IV.
Values in degrees: 150 , 330
Values in radians:5�
6,
11�
6
��
�
6 and ��cot � � ��3 ⇒
�
4,
5�
4
��
�
4 and ��tan � � 1 ⇒ 86. (a) sin reference angle is 60 or and is
in Quadrant I or II.
Values in degrees: 60 , 120
Values in radians:
(b) sin reference angle is 60 or and
is in Quadrant III or IV.
Values in degrees: 240 , 300
Values in radians:4�
3,
5�
3
��
��
3�� � �
�3
2 ⇒
�
3,
2�
3
��
��
3�� �
�3
2 ⇒
87. corresponds to on the unit circle.
tan �
4� 1 since tan t �
y
x
cos �
4�
�2
2 since cos t � x
sin �
4�
�2
2 since sin t � y
t ��
4��2
2, �2
2 � 88.
sin since sin
cos since cos
tan since tan t �y
x.
�
3�
�3
2
1
2
� �3
t � x.�
3�
1
2
t � y.�
3�
�3
2
t ��
3, �x, y� � �1
2, �3
2 �
89. corresponds to on the unit circle.
tan 5�
6� �
�3
3 since tan t �
y
x
cos 5�
6� �
�3
2 since cos t � x
sin 5�
6�
1
2 since sin t � y
t �5�
6���3
2,
1
2� 90.
since sin
cos since cos
tan since tan t �y
x.
5�
4�
��2
2
��2
2
� 1
t � x.5�
4� �
�2
2
t � y.sin 5�
4� �
�2
2
t �5�
4, �x, y� � ��
�2
2, �
�2
2 �
91. corresponds to on the unit circle.
tan 4�
3� �3 since tan t �
y
x.
cos 4�
3� �
1
2 since cos t � x.
sin 4�
3� �
�3
2 since sin t � y.
t �4�
3��1
2, �
�3
2 � 92.
sin since sin
cos since cos
since tan t �y
x. � �
1
�3� �
�3
3
tan 11�
6�
�1
2
�3
2
t � x.11�
6�
�3
2
t � y.11�
6� �
1
2
t �11�
6, �x, y� � ��3
2, �
1
2�
562 Chapter 6 Trigonometry
93. corresponds to on the unit circle.
tan 3�
2 is undefined since tan t �
y
x
cos 3�
2� �0 since cos t � x
sin 3�
2� �1 since sin t � y
t �3�
2�0, �1�
95. (a) (b) cos 2 � �0.4sin 5 � �1
97. (a)
(b)
t � 1.82 or 4.46
cos t � �0.25
t � 0.25 or 2.89
sin t � 0.25
94.
since sin
cos since cos
tan since tan t �y
x.� �
0
�1� 0
t � x.� � �1
t � y.sin � � 0
t � �, �x, y� � ��1, 0�
96. (a) (b) cos 2.5 � x � �0.8sin 0.75 � y � 0.7
98. (a)
(b)
t � 0.72 or t � 5.56
cos t � 0.75
t � 4.0 or t � 5.4
sin t � �0.75
99. (a) New York City:
Fairbanks:
(b)
(c) The periods are about the same for both models, approximately 12 months.
F � 36.6 sin�0.50t � 1.83� � 25.61
N � 22.1 sin�0.52t � 2.22� � 55.01
Month New York City Fairbanks
February 35
March 41 14
May 63 48
June 72 59
August 76 56
September 69 42
November 47 7��
��
��
��
��
��
�1��
100.
(a) snowboards
(b) snowboards
(c) snowboards
(d) snowboardst � 18; S � 23.1 � 0.442�18� � 4.3 cos ��18�
6� 26,756
t � 6; S � 23.1 � 0.442�6� � 4.3 cos ��6�
6� 21,452
t � 14; S � 23.1 � 0.442�14� � 4.3 cos ��14�
6� 31,438
t � 2; S � 23.1 � 0.442�2� � 4.3 cos ��2�
6� 26,134
S � 23.1 � 0.442t � 4.3 cos � t6
101.
(a)
(b)
(c) y�12� � 2 cos 3 � �1.98 centimeters
y�14� � 2 cos�3
2� � 0.14 centimeter
y�0� � 2 cos 0 � 2 centimeters
y�t� � 2 cos 6t
Section 6.3 Trigonometric Functions of Any Angle 563
102.
(a)
centimeters
(b)
centimeter
(c)
centimetersy�12� � 2e�1�2 cos�6 1
2� � �1.2
t �12
y�14� � 2e�1�4 cos�6 1
4� � 0.11
t �14
y�0� � 2e�0 cos 0 � 2
t � 0
y�t� � 2e�t cos 6t 103.
I�0.7� � 5e�1.4 sin 0.7 � 0.79
I � 5e�2t sin t
105. False. In each of the four quadrants, the sign of thesecant function and the cosine function will be the samesince they are reciprocals of each other.
106. False. The reference angle is always acute by definition.For in Quadrant II, For in
Quadrant III, For in Quadrant IV,
�� � 360� � �.
��� � � � 180�.
��� � 180� � �.�
107.
Therefore, is odd.h�t�
� �h�t�
� �f �t�g�t�
h��t� � f ��t�g��t�
h�t) � f �t�g�t� 108. As increases from x decreases from 12 cm to
0 cm, y increases from 0 cm to 12 cm,
increases from 0 to 1, decreases from 1 to 0,
and increases without bound (and is undefined
at � � 90�).
tan � �yx
cos � �x
12
sin � �y
12
0� to 90�,�
104. sin
(a)
�6
1�2� 12 miles
d �6
sin 30�
� � 30�
� �6
d ⇒ d �
6
sin �
(b)
milesd �6
sin 90º�
6
1� 6
� � 90� (c)
�6
�3�2� 6.9 miles
d �6
sin 120�
� � 120�
109. (a)
(b) Conjecture: sin � � sin�180� � ��
0 0.342 0.643 0.866 0.985
0 0.342 0.643 0.866 0.985sin�180� � ��
sin �
80�60�40�20�0��
110. (a)
(b) cos�3�
2� �� � �sin �
0 0.3 0.6 0.9 1.2 1.5
0
0 �0.9975�0.9320�0.7833�0.5646�0.2955�sin �
�0.9975�0.9320�0.7833�0.5646�0.2955cos�3�
2� ��
�
564 Chapter 6 Trigonometry
111. If is obtuse, then The reference angle is and we have the following:�� � 180� � �90� < � < 180�.�
cot � � �cot ��tan � � �tan ��
sec � � �sec ��cos � � �cos ��
csc � � csc ��sin � � sin ��
112. Domain: All real numbers x
Range:
Period:
Zeros:
The function is odd.
n�
2�
�1, 1
−3
3
2��−
Domain: All real numbers x
Range:
Period:
Zeros:
The function is even.
n� ��
2
2�
�1, 1
−3
3
2��−
Domain: All real numbers x except
Range:
Period:
Zeros:
The function is odd.
n�
�
−3
3
2��−
��, �
x � n� ��
2Domain: All real numbers x except
Range:
Period:
Zeros: none
The function is odd.
2�
−3
3
2��−
��, �1 � 1, �
x � n�
Domain: All real numbers x except
Range:
Period:
Zeros: none
The function is even.
2�
−3
3
2��−
��, �1 � 1, �
x � n� ��
2
Domain: All real numbers x except
Range:
Period:
Zeros:
The function is odd.
The secant function is similar to the tangent function because they both have vertical asymptotes at
The cotangent function and the cosecant function both have vertical asymptotes at A maximum point on the sine curve corresponds to a relative minimum on the cosecant curve. The maximum points of sine and cosine are interchanged with the minimum points of cosecant and secant. The x-intercepts of the sine and cosine functions become vertical asymptotes of the cosecant and secant functions.
The graphs of sine and cosine may be translated left or right (respectively) to to coincide with each other.
��2
x � n�.
x � n� ��
2.
n� ��
2
�
−3
3
2��−
��, �
x � n�
Section 6.3 Trigonometric Functions of Any Angle 565
113.
Vertical asymptote:
Horizontal asymptote:
141210842−2
−8
−6
−4
2
4
6
8
x
y
y � 0
x � 4
f �x� �2
4 � x
x 0 1 2 3 5 6
y 1 2 �1�223
12
114.
Vertical asymptotes:
Horizontal asymptotes:
−4 −3 −1 31 2 4
−4
−3
−2
1
2
3
4
x
y
y � 0
x � 2, x � �2
g�x� � �1
x2 � 4
x 0
1.07 0.33 0.25�0.94�0.2g�x�
±1±1.75±2.25±3
115.
No asymptotes
The graph has a hole at �2, 12�.
y
x−3−6−9 3 6 9 12
−9
−12
3
6
9
12
h�x� �2x2 � 12x
x � 6�
2x�x � 6�x � 6
� 2x, x � 6
x 0 2 4 8
y 0 4 8 16�4
�2
117.
Domain: all real numbers x
Range:
y-intercept:
Horizontal asymptote: y � 0
�0, 12�y > 0 −2 −1 1 2 3 4
−1
2
3
4
5
10, 12 ))
x
yy � 2x�1
x 0 1 2 3
y 1 2 412
14
�1
116.
Vertical asymptote:
Horizontal asymptote:
y
x−6−8 2 4 6
−2
−4
−6
−8
2
4
6
8
y � 0
x � �3
� �undefined,
x � 5
x � 5
f �x� �x � 5
x2 � 2x � 15�
x � 5�x � 5��x � 3�
1x � 3
,
x 1 2
1 15
14
13
12�1f �x�
0�1�2�4
566 Chapter 6 Trigonometry
119.
Domain: all real numbers x
Range:
y-intercept:
Horizontal asymptote: y � 0
�0, 1�
y > 0−5 −4 −3 −2 −1 1 2 3
2
3
4
5
6
7
(0, 1)
x
yy � 3�x�2
x 0 2 4
y 9 3 1 19
13
�2�4
121.
Range: all real numbers
x-intercept:
Vertical asymptote: x � 1
�2, 0�
Domain: x � 1 > 0 ⇒ x > 1 1 2 3 4 5 6
−3
−2
−1
1
2
3
(2, 0)x
yy � ln�x � 1�
x 1.1 1.5 2 3 4
y 0 0.69 1.10�0.69�2.30
122.
Domain: all real numbers
Range: all real numbers
intercepts:
Vertical asymptote: x � 0
��1, 0�, �1, 0�x-
y
x−12 −9 −6 −3 3 6 9 12
6
9
12
(−1, 0) (1, 0)x
yy � ln x4
118.
Domain: all real numbers
Range:
intercept:
Horizontal asymptote:y � 2
�0, 5�y-
y > 2
x321−1−2−3−4−5
7
6
5
3
1
2
(0, 5)
yx
y � 3x�1 � 2
0 1
3 5 1173
199y
�1�2�3x
120.
Domain: all real numbers
Range:
intercept:
Horizontal asymptote: y � 0
�0, �3 � � �0, 1.73�y-
y > 0
x x321−1−2−3−4−5
7
6
5
3
4
1
2
y
0, 3( )
y � 3�x�1��2
0 1 2
1 3 3�3 � 5.20�3 � 1.73�33
� 0.57813
y
�1�2�3x
1 3 6
7.17 4.39 0 0 4.39 7.17�2.77�2.77y
12�
12�1�3�6x
Section 6.4 Graphs of Sine and Cosine Functions 567
123.
Range: all real numbers
intercept:
intercept: �0, 0.301�y-
��1, 0�x-3
2
−1
−2
−3
321−1−3
(−1, 0)(0, 0.301)
x
y
Domain: x � 2 > 0 ⇒ x > �2
y � log10 �x � 2� 124.
Domain:
Range: all real numbers
intercept:
Vertical asymptote: x � 0
��13, 0�x-
, 0 13
−( (1−1−2−3
2
1
−1
−2
y
x
y
�3x > 0 ⇒ x < 0
y � log10��3x�
Section 6.4 Graphs of Sine and Cosine Functions
■ You should be able to graph
■ Amplitude:
■ Period:
■ Shift: Solve
■ Key increments: (period)1
4
bx � c � 0 and bx � c � 2�.
2�
b
�a�y � a sin�bx � c� and y � a cos�bx � c�. �Assume b > 0.�
Vocabulary Check
1. cycle 2. amplitude
3. 4. phase shift
5. vertical shift
2�
b
1.
Period:
Amplitude: �3� � 3
2�
2� �
y � 3 sin 2x 2.
Period:
Amplitude: �a� � 2
2�
b�
2�
3
y � 2 cos 3x 3.
Period:
Amplitude: �52� �5
2
2�
1
2
� 4�
y �5
2 cos
x
2
x 0 1 2
y 0 �0.602�0.477�0.301��0.301
�1�1.5 x
y 0 ��0.30�0.48�0.78�0.95
�16�
13�1�2�3
4.
Period:
Amplitude: �a� � ��3� � 3
2�
b�
2�13
� 6�
y � �3 sin x
35.
Period:
Amplitude: �12� �1
2
2�
�
3
� 6
y �1
2 sin
�x
36.
Period:
Amplitude: �a� �3
2
2�
b�
2�
�
2
� 4
y �3
2 cos
�x
2
568 Chapter 6 Trigonometry
7.
Period:
Amplitude: ��2� � 2
2�
1� 2�
y � �2 sin x 8.
Period:
Amplitude: �a� � ��1� � 1
2�
b�
2�
2�3� 3�
y � �cos 2x
39.
Period:
Amplitude: �3� � 3
2�
10�
�
5
y � 3 sin 10x
10.
Period:
Amplitude: �a� �1
3
2�
b�
2�
8�
�
4
y �1
3 sin 8x 11.
Period:
Amplitude: �12� �1
2
2�
2�3� 3�
y �1
2 cos
2x
312.
Period:
Amplitude: �a� �5
2
2�
b�
2�
1�4� 8�
y �5
2 cos
x
4
13.
Amplitude: �14� �14
Period: 2�
2�� 1
y �14
sin 2�x 14.
Period:
Amplitude: �a� �2
3
2�
b�
2�
��10� 20
y �2
3 cos
�x
1015.
The graph of g is a horizontal shiftto the right units of the graph off �a phase shift�.
�
g�x� � sin�x � ��
f �x� � sin x
16.g is a horizontal shift of f unitsto the left.
�f�x� � cos x, g�x� � cos�x � �� 17.
The graph of g is a reflection inthe x-axis of the graph of f.
g�x� � �cos 2x
f �x� � cos 2x 18.
g is a reflection of f about the y-axis.
f �x� � sin 3x, g�x� � sin��3x�
19.
The period of f is twice that of g.
g�x� � cos 2x
f �x� � cos x 20.The period of g is one-third theperiod of f.
f�x� � sin x, g�x� � sin 3x 21.
The graph of g is a vertical shiftthree units upward of the graph of f.
f �x� � 3 � sin 2x
f �x� � sin 2x
22.g is a vertical shift of f two unitsdownward.
f�x� � cos 4x, g�x� � �2 � cos 4x 23. The graph of g has twice theamplitude as the graph of f. Theperiod is the same.
24. The period of g is one-third theperiod of f.
25. The graph of g is a horizontal shift units to the right ofthe graph of f.
� 26. Shift the graph of f two units upward to obtain the graphof g.
27.
Period:
Amplitude: 2
Symmetry: origin
Key points: Intercept Minimum Intercept Maximum Intercept
Since generate key points for the graph of by multiplyingthe coordinate of each key point of by �2.f �x�y-
g�x�g�x� � 4 sin x � ��2� f �x�,
�2�, 0��3�
2, 0���, 0���
2, �2��0, 0�
2�
b�
2�
1� 2�
x
−π π32 2
5
43
−5
f
g
yf �x� � �2 sin x
Section 6.4 Graphs of Sine and Cosine Functions 569
28.
Period:
Amplitude: 1
Symmetry: origin
Key points: Intercept Maximum Intercept Minimum Intercept
Since the graph of is the graph of but stretched horizontally by a factor of 3.
Generate key points for the graph of by multiplying the coordinate of each key point of by 3.f �x�x-g�x�
f �x�,g�x�g�x� � sin�x3� � f �x
3�,
�2�, 0��3�
2, �1���, 0���
2, 1��0, 0�
2�
b�
2�
1� 2�
− 2
2
π6
fg
x
yf �x� � sin x
29.
Period:
Amplitude: 1
Symmetry: axis
Key points: Maximum Intercept Minimum Intercept Maximum
Since the graph of is the graph of but translated upward by one unit.Generate key points for the graph of by adding 1 to the coordinate of each key point of f �x�.y-g�x�
f �x�,g�x�g�x� � 1 � cos�x� � f �x� � 1,
�2�, 1��3�
2, 0���, �1���
2, 0��0, 1�
y-
2�
b�
2�
1� 2�
−1
g
f
xππ 2
yf �x� � cos x
30.
Period:
Amplitude: 2
Symmetry: axis
Key points: Maximum Intercept Minimum Intercept Maximum
Since the graph of is the graph of but
i) shrunk horizontally by a factor of 2,
ii) shrunk vertically by a factor of and
iii) reflected about the axis.
Generate key points for the graph of by
i) dividing the coordinate of each key point of by 2, and
ii) dividing the coordinate of each key point of by �2.f �x�y-
f �x�x-
g�x�
x-
12,
f �x�,g�x�g�x� � �cos 4x � �12 f �2x�,
��, 2��3�
4, 0���
2, �2���
4, 0��0, 2�
y-
2�
b�
2�
2� �
−2
2
π
f
g
x
yf �x� � 2 cos 2x
570 Chapter 6 Trigonometry
31.
Period:
Amplitude:
Symmetry: origin
Key points: Intercept Minimum Intercept Maximum Intercept
Since the graph of is the graph of but translated upward by three units.
Generate key points for the graph of by adding 3 to the coordinate of each key point of f �x�.y-g�x�
f �x�,g�x�g�x� � 3 �12
sin x2
� 3 � f �x�,
�4�, 0��3�, 12��2�, 0���, �
12��0, 0�
12
2�
b�
2�
1�2� 4�
−1
1
2
3
4
5
f
g
x−π π3
yf �x� � �12
sin x2
32.
Period:
Amplitude: 4
Symmetry: origin
Key points: Intercept Maximum Intercept Minimum Intercept
Since the graph of is the graph of but translated downward by three units.Generate key points for the graph of by subtracting 3 from the coordinate of each key point of f �x�.y-g�x�
f �x�,g�x�g�x� � 4 sin �x � 3 � f �x� � 3,
�2, 0��32
, �2��1, 0��12
, 2��0, 0�
2�
b�
2�
�� 2
−8
2
4f
g
1 x
yf �x� � 4 sin �x
33.
Period:
Amplitude: 2
Symmetry: axis
Key points: Maximum Intercept Minimum Intercept Maximum
Since the graph of is the graph of but with a phase shift (horizontal translation) of Generate key points for the graph of by shifting each key point of units to the left.�f �x�g�x���.
f �x�,g�x�g�x� � 2 cos�x � �� � f �x � ��,
�2�, 2��3�
2, 0���, �2���
2, 0��0, 2�
y-
2�
b�
2�
1� 2�
−3
3
f
g
xππ 2
yf �x� � 2 cos x
34.
Period:
Amplitude: 1
Symmetry: axis
Key points: Minimum Intercept Maximum Intercept Minimum
Since the graph of is the graph of but with a phase shift (horizontal translation) of Generate key points for the graph of by shifting each key point of units to the right.�f �x�g�x��.
f �x�,g�x�g�x� � �cos�x � �� � f �x � ��,
�2�, �1��3�
2, 0���, 1���
2, 0��0, �1�
y-
2�
b�
2�
1� 2�
−2
2
ππ 2
f
g
x
yf �x� � �cos x
Section 6.4 Graphs of Sine and Cosine Functions 571
35.
Period:
Amplitude:
Key points:
�3�
2, �3�, �2�, 0�
�0, 0�, ��
2, 3�, ��, 0�,
3
2�
y
xπ π32 2
−− π2
π32
−4
1
2
3
4
y � 3 sin x
37.
Period:
Amplitude:
Key points:
�3�
2, 0�, �2�,
1
3�
�0, 1
3�, ��
2, 0�, ��, �
1
3�,
1
3
2�
y
xπ π2
π2
1
−1
23
43
1323
43
−
−
−
y �1
3 cos x 38.
Period:
Amplitude:
Key points:
�3�
2, 0�, �2�, 4�
�0, 4�, ��
2, 0�, ��, �4�,
4
2�
y
xππ 2π−2 −π
−2
−4
4
y � 4 cos x
36.
Period:
Amplitude:
Key points:
�3�
2, �
1
4�, �2�, 0�
�0, 0�, ��
2,
1
4�, ��, 0�,
1
4
2�
y
xππ 2π−2 −π
−1
−2
1
2
y �1
4 sin x
39.
Period:
Amplitude: 1
Key points:
�3�, 0�, �4�, 1�
�0, 1�, ��, 0�, �2�, �1�,
4�
y
xπ4π2π−2
−1
−2
2
y � cos x
240.
Period:
Amplitude: 1
Key points:
�3�
8, �1�, ��
2, 0�
�0, 0�, ��
8, 1�, ��
4, 0�,
�
2
y
x
−2
1
2
π4
y � sin 4x
41.
Period:
Amplitude: 1
Key points:
�0, 1�, �14
, 0�, �12
, �1�, �34
, 0�
2�
2�� 1
1 2
−2
1
2
x
yy � cos 2�x 42.
Period:
Amplitude: 1
Key points:
�6, �1�, �8, 0��0, 0�, �2, 1�, �4, 0�,
2�
��4� 8
2
1
−2
62−2−6x
yy � sin �x
4
572 Chapter 6 Trigonometry
45.
Period:
Amplitude: 1
Shift: Set
Key points: ��
4, 0�, �3�
4, 1�, �5�
4, 0�, �7�
4, �1�, �9�
4, 0�
x ��
4 x �
9�
4
x ��
4� 0 and x �
�
4� 2�
2�
−3
−2
1
2
3
ππ−x
yy � sin�x ��
4�; a � 1, b � 1, c ��
4
46.
Period:
Amplitude: 1
Shift: Set
Key points:
2
−2
−1
x
y
−π π32 2
��, 0�, �3�
2, 1�, �2�, 0�, �5�
2, �1�, �3�, 0�
x � � x � 3�
x � � � 0 and x � � � 2�
2�
y � sin�x � �� 47.
Period:
Amplitude: 3
Shift: Set
Key points:
−6
−4
2
4
6
− ππx
y
���, 3�, ���
2, 0�, �0, �3�, ��
2, 0�, ��, 3�
x � �� x � �
x � � � 0 and x � � � 2�
2�
y � 3 cos�x � ��
44.
Period:
Amplitude: 10
Key points:
1284−4−12
12
8
4
−12
x
y
�0, �10�, �3, 0�, �6, 10�, �9, 0�, �12, �10�
2�
��6� 12
y � �10 cos �x
643.
Period:
Amplitude: 1
Key points:
−1 2 3
−3
−2
2
3
x
y
�0, 0�, �3
4, �1�, �3
2, 0�, �9
4, 1�, �3, 0�
2�
2��3� 3
y � �sin 2�x
3; a � �1, b �
2�
3, c � 0
Section 6.4 Graphs of Sine and Cosine Functions 573
48.
Period:
Amplitude: 4
Shift: Set
Key points: �5�
4, 0�, �7�
4, 4���
�
4, 4�, ��
4, 0�, �3�
4, �4�,
x � ��
4 x �
7�
4
x ��
4� 0 and x �
�
4� 2�
2�
−6
−4
−2−
2
6
x
y
πππ 2
y � 4 cos�x ��
4�
49.
Period: 3
Amplitude: 1
Key points:
–3 –2 –1 1 2 3−1
1
2
4
5
x
y
�0, 2�, �34
, 1�, �32
, 2�, �94
, 3�, �3, 2�
y � 2 � sin 2�x
350.
Period:
Amplitude: 5
Key points:
−12 4 12
−24−20−16−12−8
48
1216
t
y
�0, 2�, �6, �3�, �12, �8�, �18, �3�, �24, 2�
2�
��12� 24
y � �3 � 5 cos � t
12
51.
Period:
Amplitude:
Vertical shift two units upward
Key points:
0.20.1−0.1 0
1.8
2.2
x
y
�0, 2.1�, � 1
120, 2�, � 1
60, 1.9�, � 1
40, 2�, � 1
30, 2.1�
1
10
2�
60��
1
30
y � 2 �1
10 cos 60�x 52.
Period:
Amplitude: 2
Key points:
−7
−6
−5
−4
1
−x
y
πππ 2
�0, �1�, ��
2, �3�, ��, �5�, �3�
2, �3�, �2�, �1�
2�
y � 2 cos x � 3
574 Chapter 6 Trigonometry
55.
Period:
Amplitude:
Shift:
Key points: ��
2,
2
3�, �3�
2, 0�, �5�
2,
�2
3 �, �7�
2, 0�, �9�
2,
2
3�
x ��
2 x �
9�
2
x
2�
�
4� 0 and
x
2�
�
4� 2�
2
3
4�
−4
−3
−2
−1
1
2
3
4
π π4x
y
y �2
3 cos�x
2�
�
4�; a �2
3, b �
1
2, c �
�
4
56.
Period:
Amplitude: 3
Shift: Set
Key points: ���
6, �3�, ��
�
12, 0�, �0, 3�, � �
12, 0�, ��
6, �3�
x � ��
6 x �
�
6
6x � � � 0 and 6x � � � 2�
2�
6�
�
3
2
3
x
y
π
y � �3 cos�6x � ��
57.
−6 6
−4
4y � �2 sin�4x � �� 58.
−12
−8
12
8y � �4 sin�2
3x �
�
3�
53.
Period:
Amplitude: 3
Shift: Set
Key points: ���, 0�, ���
2, �3�, �0, �6�, ��
2, �3�, ��, 0�
x � �� x � �
x � � � 0 and x � � � 2�
2�
−8
2
4
ππ 2x
yy � 3 cos�x � �� � 3
54.
Period:
Amplitude: 4
Shift: Set
Key points: ���
4, 8�, ��
4, 4�, �3�
4, 0�, �5�
4, 4�, �7�
4, 8�
x � ��
4 x �
7�
4
x ��
4� 0 and x �
�
4� 2�
2�
−4
2
4
6
10
x
y
π3π2π−2 − ππ
y � 4 cos�x ��
4� � 4
Section 6.4 Graphs of Sine and Cosine Functions 575
63.
Amplitude:
Vertical shift one unit upward ofThus, f�x� � 2 cos x � 1.g�x� � 2 cos x ⇒ d � 1.
1
2�3 � ��1� � 2 ⇒ a � 2
f �x� � a cos x � d 64.
Amplitude:
a � 2, d � �1
d � 1 � 2 � �1
1 � 2 cos 0 � d
1 � ��3�2
� 2
f�x� � a cos x � d
65.
Amplitude:
Since is the graph of reflected in the x-axis and shifted vertically four units upward, we have
Thus, f �x� � �4 cos x � 4.a � �4 and d � 4.
g�x� � 4 cos xf �x�
1
2�8 � 0 � 4
f �x� � a cos x � d 66.
Amplitude:
Reflected in the x-axis:
a � �1, d � �3
d � �3
�4 � �1 cos 0 � d
a � �1
�2 � ��4�2
� 1
f�x� � a cos x � d
69.
Amplitude:
Period:
Phase shift:
Thus, y � 2 sin�x ��
4�.
�1����
4 � � c � 0 ⇒ c � ��
4
bx � c � 0 when x � ��
4
2� ⇒ b � 1
a � 2
y � a sin�bx � c� 70.
Amplitude:
Period: 2
Phase shift:
a � 2, b ��
2, c � �
�
2
c
b� �1 ⇒ c � �
�
2
2�
b� 4 ⇒ b �
�
2
2 ⇒ a � 2
y � a sin�bx � c�
67.
Amplitude: Since the graph is reflected in the x-axis, we have
Period:
Phase shift:
Thus, y � �3 sin 2x.
c � 0
2�
b� � ⇒ b � 2
a � �3.�a� � �3�
y � a sin�bx � c� 68.
Amplitude:
Period:
Phase shift:
a � 2, b �12
, c � 0
c � 0
2�
b� 4� ⇒ b �
1
2
4�
2 ⇒ a � 2
y � a sin�bx � c�
59.
−3
−1
3
3y � cos�2�x ��
2� � 1 60.
−6
−6
6
2y � 3 cos��x
2�
�
2� � 2
61.
−20
−0.12
20
0.12y � �0.1 sin��x
10� �� 62.
−0.03
−0.02
0.03
0.02y �1
100 sin 120 � t
576 Chapter 6 Trigonometry
74.
(a) Period
(b)1 cycle
4 seconds�
60 seconds
1 minute� 15 cycles per minute
�2�
��2� 4 seconds
v � 1.75 sin � t
2
75.
(a) Period:
(b) f �1
p� 440 cycles per second
2�
880��
1
440 seconds
y � 0.001 sin 880�t 76.
(a) Period:
(b)1 heartbeat
6�5 seconds�
60 seconds
1 minute� 50 heartbeats per minute
2�
�5���3�
6
5 seconds
P � 100 � 20 cos 5� t
3
77. (a)
—CONTINUED—
C�t� � 56.55 � 26.95 cos�� t6
� 3.67�
d �12
�high � low �12
�83.5 � 29.6 � 56.55
cb
� 7 ⇒ c � 7��
6� � 3.67
b �2�
p�
2�
12�
�
6
p � 2�high time � low time � 2�7 � 1 � 12
a �12
�high � low �12
�83.5 � 29.6 � 26.95
71.
In the interval
when x � �5�
6, �
�
6,
7�
6,
11�
6.sin x � �
1
2
��2�, 2�,
y2 � �1
2−2
2
2
�−2�
y1 � sin x72.
y1 � y2 when x � �, ��
y2 � �1
−2
2
2
�−2�
y1 � cos x
73.
(a) Time for one cycle
(b) Cycles per min cycles per min
(c) Amplitude: 0.85; Period: 6
Key points: �0, 0�, �3
2, 0.85�, �3, 0�, �9
2, �0.85�, �6, 0�
�60
6� 10
�2�
��3� 6 sec
t2 4 8 10
0.25
0.50
0.75
1.00
−0.25
−1.00
vy � 0.85 sin �t
3
(c)
1 3 5 7
−2
−3
1
2
3
t
v
(b)
The model is a good fit.
00
12
100
Section 6.4 Graphs of Sine and Cosine Functions 577
77. —CONTINUED—
(c)
The model is a good fit.
(d) Tallahassee average maximum:
Chicago average maximum:
The constant term, gives the average maximum temperature.
d,
56.55�
77.90�
00
12
100 (e) The period for both models is months.
This is as we expected since one full period is one year.
(f) Chicago has the greater variability in temperature through-out the year. The amplitude, a, determines this variabilitysince it is .1
2�high temp � low temp
2�
��6� 12
78. (a) and (c)
Reasonably good fit
(d) Period is 29.6 days.
(e) March 12
The Naval observatory says that 50% of the moon’sface will be illuminated on March 12, 2007.
y � 0.44 � 44% ⇒ x � 71.
y
x10 20 30 40
0.2
0.4
0.6
0.8
1.0
Perc
ent o
f m
oon’
sfa
ce il
lum
inat
ed
Day of the year
(b)
average length ofinterval in data
Horizontal shift:
y �12
�12
sin�0.21x � 0.92�
C � 0.92
0.21�3 � 7.4� � C � 0
b �2�
29.6� 0.21
2�
b� 4�7.4� � 29.6
��Period: 8 � 8 � 7 � 6 � 8
5� 7.4
Amplitude: 12
⇒ a �12
Vertical shift: 12
⇒ d �12
79.
(a) Period
Yes, this is what is expected because there are 365 days in a year.
(b) The average daily fuel consumption is given by theamount of the vertical shift (from 0) which is givenby the constant 30.3.
(c)
The consumption exceeds 40 gallons per day when124 < x < 252.
00
365
60
�2�
2�
365
� 365
C � 30.3 � 21.6 sin�2� t
365� 10.9� 80. (a) Period
The wheel takes 12 minutes to revolve once.
(b) Amplitude: 50 feet
The radius of the wheel is 50 feet.
(c)
00 20
110
�2�
��
6�� 12 minutes
81. False. The graph of is the graph of translated to the left by one period, and the graphs areindeed identical.
sin(x)sin(x � 2�) 82. False. has an amplitude that is half that
of For y � a cos bx, the amplitude is �a�.y � cos x.
y �12
cos 2x
578 Chapter 6 Trigonometry
85. Since the graphs are thesame, the conjecture is that
.sin�x� � cos�x ��
2�2
1
−2
f = g
xπ π32 2
− π32
y
86. f�x� � sin x, g�x� � �cos�x ��
2�
x 0
0 1 0 0
0 1 0 0�1�cos�x ��
2��1sin x
2�3�
2�
�
2
87. (a)
The graphs are nearly the same for
(b)
The graphs are nearly the same for ��
2< x <
�
2.
−2
−2 2� �
2
��
2< x <
�
2.
−2
−2 2� �
2 (c)
The graphs now agree over a wider range, �3�
4< x <
3�
4.
−2
−2 2� �
2
−2
2�−2�
2
cos x � 1 �x2
2!�
x4
4!�
x6
6!
sin x � x �x3
3!�
x5
5!�
x7
7!
88. (a)
(by calculator)
(c)
(by calculator)
(e)
(by calculator)cos 1 � 0.5403
cos 1 � 1 �1
2!�
1
4!� 0.5417
sin �
6� 0.5
sin �
6� 1 �
���6�3
3!�
���6�5
5!� 0.5000
sin 1
2� 0.4794
sin 1
2�
1
2�
�1�2�3
3!�
�1�2�5
5!� 0.4794 (b)
(by calculator)
(d)
(by calculator)
(f)
(by calculator)cos �
4� 0.7071
cos �
4� 1 �
���4�2
2!�
���4�2
4!� 0.7074
cos��0.5� � 0.8776
cos��0.5� � 1 ���0.5� 2
2!�
��0.5�4
4!� 0.8776
sin 1 � 0.8415
sin 1 � 1 �1
3!�
1
5!� 0.8417
The error in the approximation is not the same in each case. The error appears to increase as x moves farther away from 0.
83. True.
Since and so is a reflection in the x-axis of y � sin�x ��
2�.cos x � sin�x ��
2�, y � �cos x � �sin�x ��
2�,
84. Answers will vary.
Conjecture: sin x � �cos�x ��
2�2
1
−2
f = g
x
y
π π32 2
π32
−
Section 6.5 Graphs of Other Trigonometric Functions 579
89. log10 �x � 2 � log10�x � 2�1�2 �1
2 log10�x � 2�
91. ln t3
t � 1� ln t3 � ln�t � 1� � 3 ln t � ln�t � 1�
93.
� log10 �xy
1
2�log10 x � log10 y� �
1
2log10�xy�
95.
� ln�3xy4�
ln 3x � 4 ln y � ln 3x � ln y4
90.
� 2 log2 x � log2�x � 3�
log2�x2�x � 3�� � log2 x2 � log2�x � 3�
92.
�1
2 ln z �
1
2 ln�z2 � 1�
ln� z
z2 � 1�
1
2 ln � z
z2 � 1� �1
2�ln z � ln�z2 � 1��
94.
� log2 x3y
� log2 x2(xy)
2 log2 x � log2�xy� � log2 x2 � log2�xy�
96.
� ln�x2�2x �
� ln�x3�2xx2 �
� ln �2xx2 � ln x3
�12 �ln
2xx2� � ln x3
12
�ln 2x � 2 ln x� � 3 ln x �12
�ln 2x � ln x2� � ln x3
Section 6.5 Graphs of Other Trigonometric Functions
■ You should be able to graph
■ When graphing or you should first graph orbecause
(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.
y � a sin�bx � c�y � a cos�bx � c�y � a csc�bx � c�y � a sec�bx � c�
y � a csc�bx � c�y � a sec�bx � c�
y � a cot�bx � c�y � a tan�bx � c�
Vocabulary Check
1. vertical 2. reciprocal
3. damping 4.
5. 6.
7. 2�
���, �1� � �1, ��x � n�
�
97. Answers will vary.
580 Chapter 6 Trigonometry
1.
Period:
Matches graph (e).
2�
2� �
y � sec 2x 3.
Period:
Matches graph (a).
�
�� 1
y �1
2 cot � x2.
Period:
Asymptotes:
Matches graph (c).
x � ��, x � �
�
b�
�
1
2
� 2�
y � tan x
2
4.
Period:
Matches graph (d).
2�
y � �csc x 5.
Period:
Asymptotes:
Matches graph (f).
x � �1, x � 1
2�
b�
2�
�
2
� 4
y �1
2 sec
�x
26.
Period:
Asymptotes:
Reflected in x-axis
Matches graph (b).
x � �1, x � 1
2�
b�
2�
�
2
� 4
y � �2 sec �x
2
7.
Period:
Two consecutive asymptotes:
x � ��
2 and x �
�
2
�
1
2
3
xπ π
y
−
y �1
3 tan x
9.
Period:
Two consecutive asymptotes:
3x ��
2 ⇒ x �
�
6
3x � ��
2 ⇒ x � �
�
6
�
3
x
4
3
2
1
− π3
π3
yy � tan 3x
x 0
y 013
�13
�
4�
�
4
x 0
y 0 1�1
�
12�
�
12
8.
Period:
Two consecutiveasymptotes:
x � ��
2, x �
�
2
�
−3
1
2
3
ππ−x
yy �1
4 tan x
x 0
y 01
4�
1
4
�
4�
�
4
10.
Period:
Two consecutiveasymptotes:
x � �1
2, x �
1
2
�
�� 1
2
−8
−4
x
yy � �3 tan �x
x 0
y 3 0 �3
1
4�
1
4
Section 6.5 Graphs of Other Trigonometric Functions 581
13.
Period:
Two consecutiveasymptotes:
x � 0, x � 1
2�
�� 2
x
4
3
2
1
−3
−4
21−1−2
yy � csc �x
11.
Period:
Two consecutive asymptotes:
x � ��
2, x �
�
2
2�
1
2
3
xπ
y
− π
y � �1
2 sec x 12.
Period:
Two consecutive asymptotes:
x � ��
2, x �
�
2
2�
π
3
2x
yy �
1
4 sec x
14.
Period:
Two consecutiveasymptotes:
x � 0, x ��
4
2�
4 �
�
2
8
6
4
2
−2− π4
π4
x
yy � 3 csc 4x
15.
Two consecutiveasymptotes:
x � �12
, x �12
Period: 2�
�� 2
1−1−2−3 2 3−1
x
yy � sec �x � 1 16.
Period:
Two consecutiveasymptotes:
x � ��
8, x �
�
8
2�
4�
�
2
2
4
6
x
y
π4
π4
π2
−
y � �2 sec 4x � 2
17.
Period:
Two consecutiveasymptotes:
x � 0, x � 2�
2�
1�2� 4�
2
4
6
xπ
yy � csc x
2
0
�1�1
2�1y
�
3�
�
3x 0
12
1
412
y
�
3�
�
3x
2 1 2y
5
612
1
6x
6 3 6y
5�
24�
8�
24x
0
1 0 1y
1
3�
1
3x 0
0 �2�2y
�
12�
�
12x
2 1 2y
5�
3�
�
3x
18.
Period:
Two consecutiveasymptotes:
x � 0, x � 3�
2�
1�3� 6�
2
4
6
ππ 2x
yy � csc x
3
2 1 2y
5�
23�
2�
2x
582 Chapter 6 Trigonometry
20.
Period:
Two consecutive asymptotes:
x � 0, x � 2
�
��2� 2
2
4
6
2−2x
yy � 3 cot �x
219.
Period:
Two consecutive asymptotes:
x
2� � ⇒ x � 2�
x
2� 0 ⇒ x � 0
�
1�2� 2�
1
2
3
x
y
π−2 2π
y � cot x
2
x
y 1 0 �1
3�
2�
�
2
21.
Period:2�
2� �
3
xπ π
y
−
y �1
2 sec 2x 22.
Period:
Two consecutive asymptotes:
x � ��
2, x �
�
2
�2
3
ππ−
y
x
y � �1
2 tan x
x 0
y 0 �1
2
1
2
�
4�
�
4
23.
Period:
Two consecutive asymptotes:
�x4
��
2 ⇒ x � 2
�x4
� ��
2 ⇒ x � �2
�
��4� 4
−4 4
2
4
6
x
y
y � tan �x
4
x 0 1
y 0 1�1
�1
1
3 0 �3y
3
2
1
2x
0
1 11
2y
�
6�
�
6x
24.
Period:
Two consecutive asymptotes:
x � ��
2, x �
�
2
�4
3
2
1
πx
yy � tan�x � ��
x 0
y 10�1
�
4�
�
4
25.
Period:
Two consecutive asymptotes:
x � 0, x � �
2�
1
2
3
4
x
− ππ322
yy � csc�� � x�
2 1 2y
5�
6�
2�
6x
26.
Period:
Two consecutive asymptotes:
x � 0, x ��
2
2�
2� �
y
x
−1
−2
π32
π2
2ππ
y � csc�2x � ��
�2�1�2y
5�
12�
4�
12x
Section 6.5 Graphs of Other Trigonometric Functions 583
27.
Period:
Two consecutive asymptotes:
x � ��
2, x �
�
2
2�
y
x
−1
1
2
3
4
π 2π−π 3π
y � 2 sec�x � �� 28.
Period:
Two consecutive asymptotes:
x � �12
, x �12
2�
�� 2
y
x1 2 3 4
1
2
3
y � �sec �x � 1
0
�4�2�4y
�
3�
�
3x
0
0 1�1y
1
3�
13
x
29.
Period:
Two consecutive asymptotes:
x � ��
4, x �
3�
4
2�
xπ2
1
2
yy �1
4 csc�x �
�
4� 30.
Period:
Two consecutive asymptotes:
x � ��
2, x �
�
2
�
−2
xπ32
− π32
yy � 2 cot�x ��
2�
x 0
y 2 0 �2
�
4�
�
4
12
1
412
y
7�
12�
4�
�
12x
31.
−5
5�−5�
5
y � tan x
333.
−4
�2
�2
−
4
y � �2 sec 4x ��2
cos 4x32.
−3
3
�4
3�4
3−
y � �tan 2x
34.
−3
−2
3
2
y � sec �x ⇒ y �1
cos��x� 35.
−3
− �2
3�2
3
3
y � tan�x ��
4� 36.
−3
�2
3�2
3−
3
�1
4 tan�x ��
2�
y �1
4 cot�x �
�
2�
37.
y ��1
sin�4x � ��
−3
− �2
�2
3y � �csc�4x � �� 38.
y �2
cos�2x � ��
−4
−� �
4 y � 2 sec(2x � �) ⇒
584 Chapter 6 Trigonometry
39.
−6
−0.6
6
0.6y � 0.1 tan��x4
��
4�
41.
xπ π2
2
y
x � �7�
4, �
3�
4,
�
4,
5�
4
tan x � 1
40.
−6
−2
6
2
y �1
3 cos��x
2�
�
2� ⇒ y �
1
3 sec��x
2�
�
2�
42.
xπ π2
2
1
y
x � �5�
3, �
2�
3,
�
3,
4�
3
tan x � �3
44.
xπ2
π2
3π2
3π2
− −
2
3
−3
y
x � �7�
4, �
3�
4,
�
4,
5�
4
cot x � 1
43.
x
y
π2
3π2
1
2
3
−3
x � �4�
3, �
�
3,
2�
3,
5�
3
cot x � ��3
3
45.
xππ 2π π2
1
−−
y
x � ±2�
3, ±
4�
3
sec x � �2 46.
xππ π2π2
1
−−
y
x � �5�
3, �
�
3,
�
3,
5�
3
sec x � 2
47.
xπ 3ππ3π
2 2 2 2− −
1
2
3
−1
y
x � �7�
4, �
5�
4,
�
4,
3�
4
csc x � �2 48.
xπ 3ππ3π
2 2 2 2− −
1
2
3
y
x � �2�
3, �
�
3,
4�
3,
5�
3
csc x � �2�3
3
Section 6.5 Graphs of Other Trigonometric Functions 585
49.
Thus, is an even function and the graph hasaxis symmetry.y-
f �x� � sec x
� f �x�
�1
cos x
�1
cos��x�
f ��x� � sec��x�
y
x
3
4
ππ π2−
f �x� � sec x �1
cos x50.
Thus, the function is odd and the graph of is symmetric about the origin.
y � tan x
tan��x� � �tan x
x
2
3
−3
3π2
− π2− π
23π2
yf�x� � tan x
51.
(a)
(b) on the interval,
(c) As and
since is the reciprocal
of f �x�.
g �x�g�x� �1
2 csc x → ±�
f �x� � 2 sin x → 0x → �,
�
6< x <
5�
6f > g
1
−1
2
3
f
g
xπ ππ32 4
π4
y
g�x� �1
2 csc x
f �x� � 2 sin x 52.
(a)
(b) The interval in which
(c) The interval in which which is the same interval as part (b).
2f < 2g is ��1, 13�,f < g is ��1, 13�.
−3
g
f
3
1−1
f�x� � tan �x
2, g�x� �
1
2 sec
�x
2
53.
The expressions are equivalent except when and y1 is undefined.
sin x � 0
sin x csc x � sin x� 1
sin x� � 1, sin x � 0
−2
3−3
2
y1 � sin x csc x and y2 � 1 54.
The expressions are equivalent.
sin x sec x � sin x 1
cos x�
sin x
cos x� tan x
−4
−2 2
4
� �
y1 � sin x sec x, y2 � tan x
55.
The expressions are equivalent.
cot x �cos x
sin x
−4
2�−2�
4y1 �cos x
sin x and y2 � cot x �
1
tan x
586 Chapter 6 Trigonometry
56.
The expressions are equivalent.
tan2 x � sec2 x � 1
1 � tan2 x � sec2 x
−1
−
3
�2
3�2
3
y1 � sec2 x � 1, y2 � tan2 x 57.
As
Matches graph (d).
x → 0, f �x� → 0 and f �x� > 0.
f �x� � x cos x
58.
Matches graph (a) as x → 0, f�x� → 0.
f�x� � x sin x
60.
Matches graph (c) as x → 0, g�x� → 0.
g�x� � x cos x
59.
As
Matches graph (b).
x → 0, g�x� → 0 and g�x� is odd.
g�x� � x sin x
61.
The graph is the line
−3 −2 −1 1 2 3
−3
−2
−1
1
2
3
x
y
y � 0.f �x� � g�x�,
f �x� � sin x � cos�x ��
2�, g�x� � 0
62.
It appears that That is,
sin x � cos�x ��
2� � 2 sin x.
f�x� � g�x�.
−4
2
4
xππ−
y
g�x� � 2 sin x
f�x� � sin x � cos�x ��
2� 63.
–1
2
3
y
xππ−
f �x� � g�x�
f �x� � sin2 x, g�x� �1
2�1 � cos 2x�
64.
It appears that That is,
cos2 �x
2�
1
2�1 � cos �x�.
f�x� � g�x�.
g�x� �1
2�1 � cos �x�
−1
−3 3 6−6
2
3
x
yf�x� � cos2 �x
265.
The damping factor is
As x → �, g�x� →0.
y � e�x2�2.
�e�x2�2 ≤ g�x� ≤ e�x2�2
−1
8−8
1g�x� � e�x2�2 sin x
Section 6.5 Graphs of Other Trigonometric Functions 587
66.
Damping factor:
As x → �, f�x� → 0.
−3 6
−3
3
e�x
f�x� � e�x cos x 67.
Damping factor:
As x→�, f �x� → 0.
−9 9
−6
6
y � 2�x�4.
�2�x�4 ≤ f �x� ≤ 2�x�4
f �x� � 2�x�4 cos �x 68.
Damping factor:
As x → �, h �x� → 0.
−8
−1
8
1
2�x2�4
h�x� � 2�x2�4 sin x
69.
As x → 0, y → �.
0
−2
8�
6
y �6
x� cos x, x > 0 71.
As x → 0, g�x� → 1.
−1
6�−6�
2
g�x� �sin x
x70.
As x → 0, y → �.
0
−2
6�
6
y �4
x� sin 2x, x > 0
72.
As x → 0, f (x) → 0.
−1
−6� 6�
1
f (x) � 1 � cos x
x73.
As oscillates between and 1.�1
x → 0, f �x�
−2
−� �
2
f �x� � sin 1
x74.
As oscillates.x → 0, h(x)
−1
−� �
2
h(x) � x sin 1
x
75.
Gro
und
dist
ance
x
14
10
6
2
−2
−6
−10
−14Angle of elevation
d
π π π32 4
π4
d �7
tan x� 7 cot x
tan x �7
d76.
x
20
40
60
80
Angle of camera
Dis
tanc
e
d
0 π2
π4
π4
π2
− −
d �27
cos x� 27 sec x, �
�
2< x <
�
2
cos x �27
d
588 Chapter 6 Trigonometry
77.
(a)
(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:
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.
p �2�
��12� 24 months
00 100
50,000
R
C
R � 25,000 � 15,000 cos � t12
C � 5000 � 2000 sin � t12
,
78.
Month (1 ↔ January)
Law
n m
ower
sal
es(i
n th
ousa
nds
of u
nits
) 150135120105907560453015
2 4 6 8 10 12t
SS � 74 � 3t � 40 cos �t6
79.
(a) Period of
Period of sin
Period of months
Period of monthsL�t� : 12
H�t� : 12
� t
6:
2�
��6� 12
cos � t
6:
2�
��6� 12
L�t� � 39.36 � 15.70 cos � t
6� 14.16 sin
� t
6
H�t� � 54.33 � 20.38 cos � t
6� 15.69 sin
� t
6
(b) From the graph, it appears that the greatest differencebetween high and low temperatures occurs in summer.The smallest difference occurs in winter.
(c) The highest high and low temperatures appear tooccur around the middle of July, roughly one monthafter the time when the sun is northernmost in the sky.
80. (a)
(b) The displacement is a damped sine wave.as t increases.y → 0
y �12
e�t�4 cos 4t
0 4�
−0.6
0.6 81. True. Since
for a given value of , the -coordinate of is thereciprocal of the -coordinate of sin x.y
csc xyx
y � csc x �1
sin x,
Section 6.5 Graphs of Other Trigonometric Functions 589
83. As from the left,
As from the right, f �x� � tan x → ��.x → �
2
f�x� � tan x → �.x → �
282. True.
If the reciprocal of is translated units to theleft, we have
y �1
sin�x ��
2��
1cos x
� sec x.
��2y � sin x
y � sec x �1
cos x
84. As from the left, .
As from the right, .f (x) � csc x → ��x → �
f (x) � csc x → �x → �
85.
(a)
The zero between 0 and 1 occurs at x 0.7391.
3
−2
−3
2
f �x� � x � cos x
(b)
This sequence appears to be approaching the zero of f : x 0.7391.
�
x9 � cos 0.7504 0.7314
x8 � cos 0.7221 0.7504
x7 � cos 0.7640 0.7221
x6 � cos 0.7014 0.7640
x5 � cos 0.7935 0.7014
x4 � cos 0.6543 0.7935
x3 � cos 0.8576 0.6543
x2 � cos 0.5403 0.8576
x1 � cos 1 0.5403
x0 � 1
xn � cos�xn�1�
86.
The graphs are nearly thesame for �1.1 < x < 1.1.
y � x �2x3
3!�
16x5
5!
−6
− �2
3�2
3
6y � tan x 87.
The graph appears to coincide on the interval�1.1 ≤ x ≤ 1.1.
y2 � 1 �x2
2!�
5x4
4!
−6
−
6
�2
3�2
3
y1 � sec x
88. (a)
—CONTINUED—
−3 3
−2
2
y2
−3 3
−2
2
y1
y2 �4��sin �x �
13
sin 3�x �15
sin 5�x�y1 �4��sin �x �
13
sin 3�x�
590 Chapter 6 Trigonometry
88. —CONTINUED—
(b)
(c) y4 �4��sin �x �
13
sin 3�x �15
sin 5�x �17
sin 7�x �19
sin 9�x�
−3 3
−2
2
y3 �4��sin �x �
13
sin 3�x �15
sin 5�x �17
sin 7�x�
89.
x �ln 54
2 1.994
2x � ln 54
e2x � 54 91.
x � �ln 2 �0.693
ln 2 � � x
2 � e�x
3 � 1 � e�x
300100
� 1 � e�x
300
1 � e�x � 100
93.
1.684 � 1031
x �2 � e73
3
3x � 2 � e73
3x � 2 � e73
ln�3x � 2� � 73
95.
x � ±�e3.2 � 1 ±4.851
x2 � e3.2 � 1
x2 � 1 � e3.2
ln�x2 � 1� � 3.2
97.
is extraneous (not in thedomain of ) so only isa solution.
x � 2log8 xx � �1
x � 2, �1
�x � 2��x � 1� � 0
x2 � x � 2 � 0
x2 � x � 2
x�x � 1� � 81�3
log8�x�x � 1�� �13
log8 x � log8�x � 1� �13
90.
x �ln 983 ln 8
0.735
3x � log8 98
83x � 98
92.
t �1
365 �log10 5
log10 1.00041096� 10.732
365t � log1.00041096 5
1.00041096365t � 5
1 �0.15365
1.00041096
�1 �0.15365 �
365t
� 5
94.
x �14 � e68
2 �1.702 � 1029
14 � e68 � 2x
14 � 2x � e68
ln(14 � 2x) � 68
96.
22,022.466
x � e10 � 4
x � 4 � e10
ln(x � 4) � 10
12
ln(x � 4) � 5
ln�x � 4 � 5 98.
Since is not in the domainof , the only solution isx � �65 8.062.
log6 x��65
x � ±�65
x2 � 1 � 64
x(x2 � 1) � 64x
log6�x(x2 � 1�� � log6(64x)
log6 x � log6(x2 � 1) � log6(64x)
Section 6.6 Inverse Trigonometric Functions 591
■ 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
y � arccos x ⇒ x � cos y
y � arctan x ⇒ x � tan y
■ You should know the inverse properties of the inverse trigonometric functions.
■ You should be able to use the triangle technique to convert trigonometric functions of inverse trigonometric functions into algebraic expressions.
tan�arctan x� � x and arctan�tan y� � y, ��
2< y <
�
2
cos�arccos x� � x and arccos�cos y� � y, 0 ≤ y ≤ �
sin�arcsin x� � x and arcsin�sin y� � y, ��
2 ≤ y ≤
�
2
��
2< x <
�
2�� < x < �
0 ≤ y ≤ ��1 ≤ x ≤ 1
��
2≤ y ≤
�
2�1 ≤ x ≤ 1
Vocabulary Check
Alternative Function Notation Domain Range
1.
2.
3. ��
2< y <
�
2�� < x < �y � tan�1 xy � arctan x
0 ≤ y ≤ ��1 ≤ x ≤ 1y � cos�1 xy � arccos x
��
2≤ y ≤
�
2�1 ≤ x ≤ 1y � sin�1 xy � arcsin x
1. ��
2≤ y ≤
�
2 ⇒ y �
�
6y � arcsin
1
2 ⇒ sin y �
1
2 for 2. y � arcsin 0 ⇒ sin y � 0 for �
�
2≤ y ≤
�
2 ⇒ y � 0
3. 0 ≤ y ≤ � ⇒ y ��
3y � arccos
1
2 ⇒ cos y �
1
2 for 4. y � arccos 0 ⇒ cos y � 0 for 0 ≤ y ≤ � ⇒ y �
�
2
5.
��
2< y <
�
2 ⇒ y �
�
6
y � arctan �3
3 ⇒ tan y �
�3
3 for 6.
��
2< y <
�
2 ⇒ y � �
�
4
y � arctan��1� ⇒ tan y � �1 for
Section 6.6 Inverse Trigonometric Functions
592 Chapter 6 Trigonometry
7.
0 ≤ y ≤ � ⇒ y �5�
6
y � arccos���3
2 � ⇒ cos y � ��3
2 for 8.
��
2≤ y ≤
�
2 ⇒ y � �
�
4
y � arcsin���2
2 � ⇒ sin y � ��2
2 for
9.
��
2< y <
�
2 ⇒ y � �
�
3
y � arctan���3� ⇒ tan y � ��3 for
11.
0 ≤ y ≤ � ⇒ y �2�
3
y � arccos��1
2� ⇒ cos y � �1
2 for
13.
��
2≤ y ≤
�
2 ⇒ y �
�
3
y � arcsin �3
2 ⇒ sin y �
�3
2 for
10.
��
2< y <
�
2 ⇒ y �
�
3
y � arctan��3 � ⇒ tan y � �3 for
12.
��
2≤ y ≤
�
2 ⇒ y �
�
4
y � arcsin �2
2 ⇒ sin y �
�2
2 for
14.
��
2< y <
�
2 ⇒ y � �
�
6
y � arctan���3
3 � ⇒ tan y � ��3
3 for
15. y � arctan 0 ⇒ tan y � 0 for ��
2< y <
�
2 ⇒ y � 0 16. y � arccos 1 ⇒ cos y � 1 for 0 ≤ y ≤ � ⇒ y � 0
17.
y � x
g�x� � arcsin x
−1
1.5−1.5
fg
1 f�x� � sin x 18.
Graph
Graph
Graph y3 � x.
y2 � tan�1 x.
y1 � tan x.
−2
g
f
2
�2
�2
−
f �x� � tan x and g�x� � arctan x
19. arccos 0.28 � cos�1 0.28 � 1.29
21. arcsin��0.75� � sin�1��0.75� � �0.85
23. arctan��3� � tan�1��3� � �1.25 25. arcsin 0.31 � sin�1 0.31 � 0.32
27. arccos��0.41� � cos�1��0.41� � 1.99
29. arctan 0.92 � tan�1 0.92 � 0.74
31. arcsin�3
4� � sin�1�0.75� � 0.85 33. arctan�7
2� � tan�1�3.5� � 1.29
20. arcsin 0.45 � 0.47
22. arccos��0.7� � 2.35
24. arctan 15 � 1.50
28. arcsin��0.125� � �0.13
26. arccos 0.26 � 1.31
30. arctan 2.8 � 1.23
32. arccos��1
3� � 1.91
34. arctan��95
7 � � �1.50 35. This is the graph of The coordinates are
.���3
3,�
�
6�, and �1, �
4����3, ��
3�,
y � arctan x.
Section 6.6 Inverse Trigonometric Functions 593
36.
cos��
6� ��3
2
arccos��1
2� �2�
3
arccos��1� � � 37.
tan� � arctan x
4 θ4
x
tan � �x
4
38.
� � arccos 4
x
cos � �4
x39.
sin� � arcsin�x � 2
5 �θ
5x + 2
sin � �x � 2
5
40.
� � arctan�x � 1
10 �
tan � �x � 1
1041.
� � arccos�x � 32x �
θ
x + 3
2x
cos � �x � 3
2x
42.
x � 1
� � arctan 1
x � 1
tan � �x � 1
x2 � 1�
1
x � 143. sin�arcsin 0.3� � 0.3 44. tan�arctan 25� � 25
45. cos�arccos��0.1�� � �0.1 46. sin�arcsin��0.2�� � �0.2 47.
Note: 3 is not in the range ofthe arcsine function.
�
arcsin�sin 3�� � arcsin�0� � 0
48.
Note: is not in the range of the arccosine function.7�
2
arccos�cos 7�
2 � � arccos 0 ��
249. Let
and sin y �3
5.
tan y �3
4, 0 < y <
�
2
xy
53
4
y
y � arctan 3
4.
50. Let
sec�arcsin 4
5� � sec u �5
3
sin u �4
5, 0 < u <
�
2.
3
45
u
u � arcsin 4
5, 51. Let
and cos y �1�5
��5
5.
tan y � 2 �2
1, 0 < y <
�
2
x
y
5 2
1
yy � arctan 2.
594 Chapter 6 Trigonometry
52. Let
1
25
u
sin�arccos �5
5 � � sin u �2
�5�
2�5
5
cos u ��5
5, 0 < u <
�
2.
u � arccos �5
5, 53. Let
and cos y �12
13.
sin y �5
13, 0 < y <
�
2
xy
5
12
13
yy � arcsin 5
13.
55. Let
and sec y ��34
5.
tan y � �3
5, �
�
2< y < 0
xy
34−3
5
y
y � arctan��3
5�.54. Let
13
12
−5u
cscarctan��5
12� � csc u � �13
5
tan u � �5
12, �
�
2 < u < 0.
u � arctan��5
12�,
56. Let
4−3
7u
tanarcsin��3
4� � tan u � �3�7
� �3�7
7
sin u � �3
4, �
�
2 < u < 0.
u � arcsin��3
4�, 57. Let
and sin y ��5
3.
cos y � �2
3,
�
2< y < �
xy
3
−2
5
y
y � arccos��2
3�.
Section 6.6 Inverse Trigonometric Functions 595
60. Let
x + 12
1
x
u
sin�arctan x� � sin u �x
�x2 � 1
tan u � x �x
1.
u � arctan x, 61. Let
and cos y � �1 � 4x2.
sin y � 2x �2x
1
1 − 4x2
2x
y
1
y � arcsin�2x�.
62. Let
sec�arctan 3x� � sec u � �9x2 � 1
tan u � 3x �3x
1.
1
3x9 + 1x2
u
u � arctan 3x, 63. Let
and sin y � �1 � x2.
cos y � x �x
1 1 − x2
xy
1
y � arccos x.
64. Let
x −1
2x x− 2
1
u
sec�arcsin�x � 1�� � sec u �1
�2x � x2
sin u � x � 1 �x � 1
1.
u � arcsin�x � 1�, 65. Let
and tan y ��9 � x2
x.
cos y �x
3 9 − x2
xy
3
y � arccos�x
3�.
66. Let
cot�arctan 1
x� � cot u � x
tan u �1
x. 1
ux
x + 12
u � arctan 1
x, 67. Let
and csc y ��x2 � 2
x.
tan y �x�2
x2 + 2
x
y
2
y � arctan x�2
.
59. Let
and cot y �1
x.
tan y � x �x
1
x2 + 1x
y
1
y � arctan x.58. Let
cot�arctan 5
8� � cot u �8
5
tan u �5
8, 0 < u <
�
2. 5
8
89
u
u � arctan 5
8,
596 Chapter 6 Trigonometry
69.
They are equal. Let
and
The graph has horizontal asymptotes at y � ±1.
g�x� �2x
�1 � 4x2� f �x�
1 + 4x2
2x
y
1
sin y �2x
�1 � 4x2.
tan y � 2x �2x
1
y � arctan 2x. −3 3
−2
2
f �x� � sin�arctan 2x�, g�x� �2x
�1 � 4x2
70.
Asymptote:
These are equal because:
Let
Thus, f �x� � g�x�.
��4 � x2
x� g�x�
f �x� � tan�arccos x
2� � tan u
u � arccos x
2. 2
ux
4 − x 2
x � 0
g�x� ��4 � x2
x
−3
−2
3
2
f�x� � tan�arccos x
2� 71. Let
Thus,
x2 + 81
x
y
9
arcsin y ��9
�x2 � 81, x < 0.
arcsin y �9
�x2 � 81, x > 0;
tan y �9
x and sin y �
9�x2 � 81
, x > 0; �9
�x2 � 81, x < 0.
y � arctan 9
x.
72. If
then
arcsin �36 � x2
6� arccos
x
6
sin u ��36 � x2
6.
6
ux
36 − x 2
arcsin �36 � x2
6� u, 73. Let Then,
and
Thus,
(x − 1)2 + 9
3
y
x − 1
y � arcsin �x � 1��x2 � 2x � 10
.
sin y � �x � 1���x � 1�2 � 9
.
cos y �3
�x2 � 2x � 10�
3��x � 1�2 � 9
y � arccos 3
�x2 � 2x � 10.
68. Let
cos�arcsin x � h
r � � cos u ��r2 � �x � h�2
r
sin u �x � h
r. r
x h−
r x h− −( )22
u
u � arcsin x � h
r,
Section 6.6 Inverse Trigonometric Functions 597
75.
Domain:
Range:
This is the graph of with a factor of 2.
−2 −1 1 2
π
π2
x
y
f �x� � arccos x
0 ≤ y ≤ 2�
�1 ≤ x ≤ 1
y � 2 arccos x
76.
Domain:
Range:
This is the graph ofwith a
horizontal stretch of afactor of 2.
f �x� � arcsin x
��
2≤ y ≤
�
2
�2 ≤ x ≤ 2
1 2−2x
π
π
y
−
y � arcsin x
277.
Domain:
Range:
This is the graph of shifted
one unit to the right.g�x� � arcsin�x�
��
2≤ y ≤
�
2
0 ≤ x ≤ 2
−1 1 2 3
π
π
y
x
−
f �x� � arcsin�x � 1�
78.
Domain:
Range:
This is the graph ofshifted
two units to the left.y � arccos t
0 ≤ y ≤ �
�3 ≤ t ≤ �1
−4 −3 −2 −1t
π
yg�t� � arccos�t � 2� 79.
Domain: all real numbers
Range:
This is the graph ofwith a
horizontal stretch of a factor of 2.
g�x� � arctan�x�
��
2< y <
�
2−4 −2 2 4
π
π
y
x
−
f �x� � arctan 2x
80.
Domain: all real numbers
Range:
This is the graph ofshifted
upward units.��2
y � arctan x
0 < y ≤ �
−4 −2 2 4x
π
y
f�x� ��
2� arctan x 81.
Domain:
Range: all real numbers
−2 1 2
π
y
v
1 − v2
vy
1
�1 ≤ v ≤ 1, v � 0
h�v� � tan�arccos v� ��1 � v2
v
74. If
then
2
ux − 2
4x x− 2
arccos x � 2
2� arctan
�4x � x2
x � 2
cos u �x � 2
2.
arccos x � 2
2� u,
598 Chapter 6 Trigonometry
82.
Domain:
Range: 0 ≤ y ≤ �
�4 ≤ x ≤ 4
−4 −2 2 4x
π
yf�x� � arccos
x
483.
0
2�
−1 1
f �x� � 2 arccos�2x�
84.
−0.5 0.5
2�
−2�
f�x� � � arcsin�4x� 85.
−2 4
�−
�
f �x� � arctan�2x � 3� 86.
−4 4
−2�
�2
f�x� � �3 � arctan��x�
87.
5
−2
−4
4
f �x� � � � arcsin�23� � 2.412 88.
5
−2
−4
4
f�x� ��
2� arccos�1
�� � 2.82
89.
The graphs are the same and implies that the identity is true.
� 3�2 sin�2t ��
4� � 3�2 sin�2t � arctan 1�
−6
−2
6
� 2�
f �t� � 3 cos 2t � 3 sin 2t � �32 � 32 sin�2t � arctan 3
3�
90.
The graph implies that
is true.
A cos �t � B sin �t ��A2 � B2 sin��t � arctan A
B�−6
6−6
6 � 5 sin�� t � arctan 4
3�
� �42 � 32 sin�� t � arctan 4
3�f�t� � 4 cos � t � 3 sin � t 91. (a)
(b)
s � 20: � � arcsin 5
20� 0.25
s � 40: � � arcsin 5
40� 0.13
sin� � arcsin 5
s
sin � �5
s
Section 6.6 Inverse Trigonometric Functions 599
92. (a)
(b) When
When
� � arctan 1200
750� 1.01 � 58.0.
s � 1200,
� � arctan 300
750� 0.38 � 21.8.
s � 300,
� � arctan s
750
tan � �s
75093.
(a)
(b) is maximum when feet.
(c) The graph has a horizontal asymptote at As xincreases, decreases.
� 0.
x � 2
0
−0.5
6
1.5
� arctan 3x
x2 � 4
94. (a)
(b)
feeth � 20 tan � � 20 �1117
� 12.94
tan � �hr
�h
20
r �12
�40� � 20
� � arctan 11
17� 0.5743 � 32.9
tan � �11
1795.
(a)
(b)
tanh � 50 tan 26 � 24.39 feet
tan 26 �h
50
tan� � arctan�20
41� � 26.0
tan � �20
41
20 ft
41 ft
θ
96. (a)
(b)
� � arctan 6
1� 1.41 � 80.5
x � 1 mile
� � arctan 6
7� 0.71 � 40.6
x � 7 miles
� � arctan 6
x
tan � �6
x97. (a)
(b)
x � 12: � � arctan 12
20� 31.0
x � 5: � � arctan 5
20� 14.0
tan� � arctan x
20
tan � �x
20
98. False.
is not in the range of
arcsin 12
��
6
arcsin�x�.5�
6
99. False.
is not in the range of the arctangent function.
arctan 1 ��
4
5�
4
100. False.
is defined for all real x, but and require
Also, for example,
Since , but undefined.arcsin 1arccos 1
���2
0�arctan 1 �
�
4
arctan 1 �arcsin 1arccos 1
.
�1 ≤ x ≤ 1.arccos xarcsin xarctan x
600 Chapter 6 Trigonometry
101.
Domain:
Range: 0 < x < �
x−1−2 21
π2
y
π
�� < x < �
y � arccot x if and only if cot y � x.
103.
Domain:
Range: ��
2, 0� � �0,
�
2���, �1� � �1, ��
x−2 −1 21
−
y
π2
π2
y � arccsc x if and only if csc y � x.
102. if and only if where
and and The
domain of arcsec x is and the
range is
x−2 −1 1 2
y
π
π2
0, �
2� � ��
2, �.
���, �1� � �1, ��y �
�
2< y ≤ �.0 ≤ y <
�
2x ≤ �1 � x ≥ 1
sec y � xy � arcsec x
104. (a)
(b)
(c)
(d) y � arccsc 2 ⇒ csc y � 2 and ��
2≤ y < 0 � 0 < y ≤
�
2 ⇒ y �
�
6
y � arccot���3 � ⇒ cot y � ��3 and 0 < y < � ⇒ y �5�
6
y � arcsec 1 ⇒ sec y � 1 and 0 ≤ y <�
2�
�
2< y ≤ � ⇒ y � 0
y � arcsec �2 ⇒ sec y � �2 and 0 ≤ y <�
2�
�
2< y ≤ � ⇒ y �
�
4
105.
(a)
(b)
(c)
(d)
� 1.25 � ���
4� � 2.03
Area � arctan 3 � arctan��1�
a � �1, b � 3
� 1.25 � 0 � 1.25
Area � arctan 3 � arctan 0
a � 0, b � 3
��
4� ��
�
4� ��
2
Area � arctan 1 � arctan��1�
a � �1, b � 1
Area � arctan 1 � arctan 0 ��
4� 0 �
�
4
a � 0, b � 1
Area � arctan b � arctan a 106.
As x increases to infinity, g approaches but f has no maximum. Using the solve feature of the graphing utility, you find a � 87.54.
3�,
00
6
g
f
12
g�x� � 6 arctan x
f �x� � �x
Section 6.6 Inverse Trigonometric Functions 601
107.
(a)
(b) The graphs coincide with the graph of only for certain values of x.
over its entire domain, .
over the region , corresponding to the region where sin x is
one-to-one and thus has an inverse.
��
2≤ x ≤
�
2f �1 � f � x
�1 ≤ x ≤ 1f � f �1 � x
y � x
−2
−� �
2
−2
−� �
2
f �1 � f � arcsin�sin x�f � f �1 � sin�arcsin x�
f �x� � sin�x�, f �1�x� � arcsin�x�
108. (a) Let Then,
Therefore, arcsin arcsin x.
(c) Let
(e)
x1
y1
2y
1 − x 2
� arctan x
�1 � x2arcsin x � arcsin
x
1
x
1
y1
2y
� y1 � ��
2� y1� �
�
2
arctan x � arctan 1
x� y1 � y2
y2 ��
2� y1.
��x� � �
y � �arcsin x.
�y � arcsin x
sin��y� � x
�sin y � x
sin y � �x
y � arcsin��x�. (b) Let Then,
Thus,
(d) Let then andThus, which implies that and
are complementary angles and we have
arcsin x � arccos x ��
2.
� � ��
2
�sin � � cos cos � x.
sin � � x� � arcsin x and � arccos x,
arctan��x� � �arctan�x�.
y � �arctan x
�y � arctan x
arctan�tan��y�� � arctan x
tan��y� � x, ��
2< �y <
�
2
�tan y � x
tan y � �x, ��
2< y <
�
2
y � arctan��x�.
109. �8.2�3.4 � 1279.284 110. 10(14)�2 �10
142�
10
196� 0.051
602 Chapter 6 Trigonometry
111. �1.1�50 � 117.391 112. 16�2��1
162�� 2.718 10�8
113.
csc � �43
sec � �4�7
�4�7
7
cot � ��73
tan � �3�7
�3�7
7
cos � ��74
adj � �7
�adj�2 � 7
�adj�2 � 9 � 16
�adj�2 � �3�2 � �4�24 3
θ
sin � �34
�opphyp
114.
csc � �12�5
sec � � �5
cot � �12
sin � �2�5
cos � �1�5
hyp � �12 � 22 � �52
1
θ
tan � � 2
115.
csc � �6
�11�
6�1111
sec � �65
cot � �5
�11�
5�1111
tan � ��11
5
sin � ��11
6
opp � �11
�opp�2 � 11
�opp�2 � 25 � 36
�opp�2 � �5�2 � �6�2 6
5
θ
cos � �56
�adjhyp
116.
cot � �1
2�2�
�24
csc � �3
2�2�
3�24
sec � � 3
tan � � 2 �2
sin � �2�2
3
cos � �13
� 2�2
� �8
opp � �32 � 12
3
1
θ
sec � � 3
Section 6.6 Inverse Trigonometric Functions 603
117. Let the number of people presently in the group. Each person’s share is now
If two more join the group, each person’s share would then be
There were 8 people in the original group.
x � �10 is not possible.
x � �10 or x � 8
6250�x � 10��x � 8� � 0
6250�x2 � 2x � 80� � 0
6250x2 � 12500x � 500,000 � 0
250,000x � 250,000x � 500,000 � 6250x2 � 12500x
250,000x � 250,000�x � 2� � 6250x�x � 2�
250,000
x � 2�
250,000
x� 6250
Share per person with
two more people �
Original share
per person � 6250
250,000
x � 2.
250,000
x.x �
118. Rate downstream:
Rate upstream:
(Time to go upstream) (Time to go downstream)
The speed of the current is 3 miles per hour.
x � ±3
x2 � 9
315 � 324 � x2
1260 � 4�324 � x2�
630 � 35x � 630 � 35x � 4�324 � x2�
35�18 � x� � 35�18 � x� � 4�18 � x��18 � x�
35
18 � x�
35
18 � x� 4
� 4�
rate time � distance ⇒ t �d
r
18 � x
18 � x119. (a)
(b)
(c)
(d) A � 15,000e�0.035��10� � $21,286.01
A � 15,000�1 �0.035365 ��365��10�
� $21,285.66
A � 15,000�1 �0.035
12 ��12��10�� $21,275.17
A � 15,000�1 �0.035
4 ��4��10�� $21,253.63
120. Data:
To find:
Assume:
Then:
� 458,504.31
� 632,000 � �632742�
2
� 632,000 � �e�r�2�2
y � P0e�r�8 � P0e
�r�4 � e�r�4
e�r�2 �P0e�r�4
P0e�r�2 �632742
632,000 � P0e�r�4
742,000 � P0e�r�2
P � P0 � e�rt
�8, y�
�4, 632,000�
�2, 742,000�
Section 6.7 Applications and Models
604 Chapter 6 Trigonometry
■ 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
1. Given:
20°b = 10
a
AC
B
c
B � 90� � 20� � 70�
cos A �b
c ⇒ c �
b
cos A�
10
cos 20�� 10.64
tan A �a
b ⇒ a � b tan A � 10 tan 20� � 3.64
A � 20�, b � 10 2. Given:
cos B �a
c ⇒ a � c cos B � 15 cos 54� � 8.82
� 15 sin 54� � 12.14
sin B �b
c ⇒ b � c sin B
� 90� � 54� � 36�
A � 90� � B 54°
b
a
AC
B
c = 15
B � 54�, c � 15
3. Given:
71°
b = 24
a
AC
B
c
A � 90� � 71� � 19�
sin B �b
c ⇒ c �
b
sin B�
24
sin 71�� 25.38
tan B �b
a ⇒ a �
b
tan B�
24
tan 71�� 8.26
B � 71�, b � 24 4. Given:
8.4°
b
a = 40.5 AC
Bc
sin A �a
c ⇒ c �
a
sin A�
40.5
sin 8.4�� 277.24
�40.5
tan 8.4�� 274.27
tan A �a
b ⇒ b �
a
tan A
� 90� � 8.4� � 81.6�
B � 90� � A
A � 8.4�, a � 40.5
5. Given:
b = 10
a = 6
AC
B
c
B � 90� � 30.96� � 59.04�
tan A �a
b�
6
10 ⇒ A � arctan
3
5� 30.96º
� 2�34 � 11.66
c2 � a2 � b2 ⇒ c � �36 � 100
a � 6, b � 10 6. Given:
cos B �a
c ⇒ B � arccos
a
c� arccos
25
35� 44.42�
� arcsin 25
35� 45.58�
sin A �a
c ⇒ A � arcsin
a
c
� �600 � 24.49
� �352 � 252
b � �c2 � a2
b
a = 25 c = 35
AC
Ba � 25, c � 35
Section 6.7 Applications and Models 605
7. Given:
B � 90� � 72.08� � 17.92�
A � arccos 16
52� 72.08º
cos A �16
52
� �2448 � 12�17 � 49.48
a � �522 � 162
b = 16
c = 52a
AC
Bb � 16, c � 52 8. Given:
� 8.03�
� arcsin 1.32
9.45 a
b = 1.32
c = 9.45
AC
Bsin B �
b
c ⇒ B � arcsin
b
c
cos A �b
c ⇒ A � arccos
b
c � arccos
1.32
9.45� 81.97�
a � �c2 � b2 � �87.5601 � 9.36
b � 1.32, c � 9.45
9. Given:
b
c = 430.5a
AC
B
12°15′
b � 430.5 cos 12�15� � 420.70
cos 12�15� �b
430.5
a � 430.5 sin 12�15� � 91.34
sin 12�15� �a
430.5
B � 90� � 12�15� � 77�45�
A � 12�15�, c � 430.5 10. Given:
a = 14.2
b
c
AC
B
65°12′
tan B �b
a ⇒ b � a tan B � 14.2 tan 65�12� � 30.73
cos B �a
c ⇒ c �
a
cos B�
14.2
cos 65�12�� 33.85
A � 90� � B � 90� � 65�12� � 24�48�
B � 65�12�, a � 14.2
11.
12 b 1
2 b
b
h
θ θ
h �1
2�4� tan 52� � 2.56 inches
tan � �h
�1�2�b ⇒ h �
1
2b tan � 12.
12 b 1
2 b
b
h
θ θ
h �12
�10� tan 18� � 1.62 meters
tan � �h
�1�2�b ⇒ h �12
b tan �
13.
12 b 1
2 b
b
h
θ θ
h �12
�46� tan 41� � 19.99 inches
tan � �h
�1�2�b ⇒ h �12
b tan � 14.
12 b 1
2 b
b
h
θ θ
h �12
�11� tan 27� � 2.80 feet
tan � �h
�1�2�b ⇒ h �12
b tan �
606 Chapter 6 Trigonometry
18.
� 81.2 feet
h � 125 tan 33
33°
h
125
tan 33� �h
125
19. (a)
(b) Let the height of the church and the height of thechurch and steeple . Then,
(c) feeth � 19.9
h � y � x � 50�tan 47�40� � tan 35��.
x � 50 tan 35� and y � 50 tan 47�40�
tan 35� �x
50 and tan 47�40� �
y
50
� y� x
50 ft
47° 40′
35°
h
x
y
20.
� 123.5 feet
h � 100 tan 51�
51°
h
100
tan 51� �h
100
21.
34°
4000x
� 2236.8 feet
x � 4000 sin 34º
sin 34� �x
400022.
θ
50 ft
75 ft
� � arctan 32
� 56.3�
tan � �7550
23. (a)
(b)
(c)
The angle of elevation of the sum is 35.8�.
� � arctan 121
2
1713
� 35.8�
tan � �121
2
1713
θ
17 ft
12 ft
1
1
3
2
24.
Angle of depression � � � 90� � 14.03� � 75.97�
� � 14.03�
� � arcsin� 400016,500�
sin � �4000
16,500
16,500 mi
4,000 mi
θα
Not drawn to scale
12,500 � 4000 � 16,500
17.
16 si 74 h � 19.7 feet
20 sin 80� � h
80°
h20 ft
16 sin 80� �h
20
15.
� 107.2 feet
x �50
tan 25�25°
50
x
tan 25� �50x
16.
� 1648.5 feet
x �600
tan 20
600
20°x
tan 20� �600
x
Section 6.7 Applications and Models 607
26. (a) Since the airplane speed is
after one minute its distance travelled is 16,500 feet.
18°
16500a
a � 16,500 sin 18� � 5099 ft
sin 18� �a
16,500
�275ft
sec��60sec
min� � 16,500ft
min,
(b)
275s 10,000feet
18°
� 117.7 seconds
s �10,000
275(sin 18�)
sin 18� �10,000
275s
27.
� 0.73 mile
x � 4 sin 10.5�
10.5°4 x sin 10.5� �
x
4
28.
Angle of grade:
Change in elevation:
� 2516.3 feet
� 21,120 sin�arctan 0.12�
y � 21,120 sin �
sin � �y
21,120
� � arctan 0.12 � 6.8�
tan � �12x
100x
100x12 =x y4 miles = 21,120 feet
θ 29. The plane has traveled
N
S
EW
90052°
38°a
b
cos 38� �b
900 ⇒ b � 709 miles east
sin 38� �a
900 ⇒ a � 554 miles north
1.5�600� � 900 miles.
30. (a) Reno is miles N of Miami.
Reno is miles W of Miami.
(b) The return heading is
N
S
EW
280°
10°
280�.
2472 cos 10� � 2434
N
S
EW
100°
80°10° Miami
Reno
2472 mi
2472 sin 10� � 429
25.
� � artan� 950
26,400� � 2.06�
tan � �950
26,400
5 miles � 5 miles�5280 feet1 mile � � 26,400 feet
5 miles
950feet θ
θ
Not drawn to scale1200 feet � 150 feet � 400 feet � 950 feet
608 Chapter 6 Trigonometry
32.
(a) hours
(b) After 12 hours, the yacht will have traveled 240 nautical miles.
miles E
miles S
(c) Bearing from N is 178.6�.
240 cos 1.4� � 239.9
240 sin 1.4� � 5.9
t �42820
� 21.4
S
EW
N
Not drawn to scale
88.6°
1.4°
428
(b)
�d
50 ⇒ d � 68.82 meters
tan C �d
50 ⇒ tan 54�
C � � � � 54�
� 90� � � 22�
� � � � 32�
33.
(a)
Bearing from A to C: N 58º E
N
S
EW
C
B
d
A
θ α
φγ
β
β50
� � 90� � 32� � 58�
� � 32�, � 68�
34.
� 5.46 kilometers
d �30
cot 34� � cot 14�
d cot 34� � 30 � d cot 14�
cot 34� �30 � d cot 14�
d
�d
30 � d cot 14�
tan 34� �d
y�
d
30 � x
tan 14� �d
x ⇒ x � d cot 14� 35.
Bearing: N 56.3 W
N
S
EW
Port
Ship
45
30θ
�
tan � �45
30 ⇒ � � 56.3� 36.
N
S
EW
160
85
Plane
Airport
� 208.0� or 528� W
Bearing � 180� � arctan� 85160�
37.
Distance between ships: D � d � 1933.3 ft
tan 4� �350
D ⇒ D � 5005.23 ft
350
6.5°4°
dD
S1S2
Not drawn to scale
tan 6.5� �350
d ⇒ d � 3071.91 ft
31.
(a) nautical miles south
nautical miles west
(b)
Bearing: S W
Distance:
nautical miles from port� 130.9
d � �104.952 � 78.182
36.7�
tan � �20 � b
a�
78.18104.95
⇒ � � 36.7�
sin 29� �b
120 ⇒ b � 58.18
cos 29� �a
120 ⇒ a � 104.95
N
S
EW
120
20
29°
b
a
Section 6.7 Applications and Models 609
39.
� 17,054 ft
a cot 16� � a cot 57� �55
6 ⇒ a � 3.23 miles
cot 16� �a cot 57� �
556
a
tan 16� �a
a cot 57� �556
tan 16� �a
x � 556
57°16°
x55060
H
P1 P2
a
tan 57� �a
x ⇒ x � a cot 57�
41.
� � arctan 5 � 78.7�
tan � � � �1 �32
1 � ��1��32�� � ��5
2
�12� � 5
L2: 3x � y � 1 ⇒ y � �x � 1 ⇒ m2 � �1
L1: 3x � 2y � 5 ⇒ y � 3
2x �
5
2 ⇒ m1 �
3
240.
� 5410 feet
h �17
� 1tan 2.5�
�1
tan 9��� 1.025 miles
htan 2.5�
�h
tan 9�� 17
x �h
tan 9�� 17
tan 9� �h
x � 17
x �h
tan 2.5�
x − 17172.5° 9°
x
h
Not drawn to scale
tan 2.5� �hx
42.
� arctan�97� � 52.1�
� � arctan� m2 � m1
1 � m2m1� � arctan� 15 � 2
1 �15�2��
tan � � � m2 � m1
1 � m2m1�L2 � x � 5y � �4 ⇒ m2 �
1
5
L1 � 2x � y � 8 ⇒ m1 � 2 43. The diagonal of the base has a length ofNow, we have
� � 35.3�.
� � arctan 1
�2
a
2a
θ
tan � �a
�2a�
1�2
�a2 � a2 � �2a.
44.
� � arctan �2 � 54.7º
θa
a 2
tan � �a�2
a� �2 45.
Length of side:
36°25
d
2d � 29.4 inches
sin 36� �d
25 ⇒ d � 14.69
38.
Distance between towns:
dD
28°
28°
55°
55°10 km
T2T1
D � d � 18.8 � 7 � 11.8 kilometers
cot 28� �D
10 ⇒ D � 18.8 kilometers
cot 55 �d
10 ⇒ d � 7 kilometers
610 Chapter 6 Trigonometry
46.
� 25 inches
Length of side � 2a � 2�12.5�
a � 25 sin 30� � 12.5
sin 30� �a
25
2530°
a 47.
y � 2b � 2��3r
2 � � �3r
b ��3r
2
b � r cos 30�
cos 30� �b
r
br
r
30°
2
x
y
48.
Distance � 2a � 9.06 centimeters
� 4.53
a � c sin 15� � 17.5 sin 15�
sin 15� �a
c
c �35
2� 17.5
c
15°a
49.
a �10
cos 35�� 12.2
cos 35� �10
a
b � 10 tan 35� � 7
tan 35� �b
10
35° 35°ba
10 10
10
10
10
10
50.
c � �10.82 � 7.22 � 13 feet
b �6
sin � 7.2 feet
sin �6
b
� 90 � 33.7 � 56.3�
f �21.6
2� 10.8 feet
a �18
cos �� 21.6 feet
6
6bca
f
θ φ9
36
cos � �18
a
� � arctan 2
3� 0.588 rad � 33.7�
tan � �12
18
51.
Use since
Thus, d � 4 sin�� t�.
2�
�� 2 ⇒ � � �
d � 0 when t � 0.d � a sin �t
d � 0 when t � 0, a � 4, Period � 2 52. Displacement at is
Amplitude:
Period:
d � 3 sin�� t
3 �
2�
�� 6 ⇒ � �
�
3
�a� � 3
0 ⇒ d � a sin � tt � 0
53.
Use since
Thus, d � 3 cos�4�
3t� � 3 cos�4�t
3 �.
2�
�� 1.5 ⇒ � �
4�
3
d � 3 when t � 0.d � a cos �t
d � 3 when t � 0, a � 3, Period � 1.5 54. Displacement at is
Amplitude:
Period:
d � 2 cos�� t
5 �
2�
�� 10 ⇒ � �
�
5
�a� � 2
2 ⇒ d � a cos � tt � 0
Section 6.7 Applications and Models 611
55.
(a) Maximum displacement amplitude
(b)
cycles per unit of time
(c)
(d) 8� t ��
2 ⇒ t �
1
16
d � 4 cos 40� � 4
� 4
Frequency ��
2��
8�
2�
� 4�
d � 4 cos 8�t 56.
(a) Maximum displacement:
(b) Frequency: cycles per unit of time
(c)
(d) Least positive value for t for which
t ��
2
1
20��
1
40
20� t ��
2
20� t � arccos 0
cos 20� t � 0
1
2 cos 20� t � 0
d � 0
t � 5 ⇒ d �12
cos 100� �12
�
2��
20�
2�� 10
�a� � �12� �1
2
d �1
2 cos 20 � t
57.
(a) Maximum displacement amplitude
(b)
cycles per unit of time
(c)
(d) 120�t � � ⇒ t �1
120
d �1
16 sin 600� � 0
� 60
Frequency ��
2��
120�
2�
�1
16�
d �1
16 sin 120�t 58.
(a) Maximum displacement:
(b) Frequency: cycles per unit of time
(c)
(d) Least positive value for t for which
t ��
792��
1
792
792� t � �
792� t � arcsin 0
sin 792� t � 0
1
64 sin 792� t � 0
d � 0
t � 5 ⇒ d �1
64 sin�3960�� � 0
�
2��
792�
2�� 396
�a� � � 1
64� �1
64
d �1
64 s in 792�t
59.
� � 2��264� � 528�
264 ��
2�
Frequency ��
2�
d � a sin �t
612 Chapter 6 Trigonometry
60. At buoy is at its high point
Returns to high point every 10 seconds:
Period:
d �7
4 cos
� t
5
2�
�� 10 ⇒ � �
�
5
�a� �7
4
Distance from high to low � 2�a� � 3.5
⇒ d � a cos � t.t � 0, 61.
(a)
t
1
−1
y
π π38 8
π4
π2
y �1
4 cos 16t, t > 0
(b) Period:
(c)1
4cos 16t � 0 when 16t �
�
2 ⇒ t �
�
32
2�
16�
�
8
62. (a)
(c) L � L1 � L2 �2
sin ��
3
cos �
(b)
The minimum length of the elevator is 7.0 meters.
(d)
From the graph, it appears that the minimum length is 7.0 meters, which agrees with the estimate of part (b).
−12
−2� 2�
12
0.1 23.0
0.2 13.1
0.3 9.9
0.4 8.43
cos 0.4
2
sin 0.4
3
cos 0.3
2
sin 0.3
3
cos 0.2
2
sin 0.2
3
cos 0.1
2
sin 0.1
L1 � L2L2L1�
0.5 7.6
0.6 7.2
0.7 7.0
0.8 7.13
cos 0.8
2
sin 0.8
3
cos 0.7
2
sin 0.7
3
cos 0.6
2
sin 0.6
3
cos 0.5
2
sin 0.5
L1 � L2L2L1�
63. (a) and (b)
Base 1 Base 2 Altitude Area
8 22.1
8 42.5
8 59.7
8 72.7
8 80.5
8 83.1
8 80.7
The maximum occurs when and is approximately83.1 square feet.
� � 60�
8 sin 70º8 � 16 cos 70º
8 sin 60º8 � 16 cos 60º
8 sin 50º8 � 16 cos 50º
8 sin 40º8 � 16 cos 40º
8 sin 30º8 � 16 cos 30º
8 sin 20º8 � 16 cos 20º
8 sin 10º8 � 16 cos 10º
(c)
(d)
The maximum of 83.1 square feet occurs when
� ��
3� 60�.
0900
100
� 64�1 � cos ���sin ��
� �16 � 16 cos ���4 sin ��
A��� � 8 � �8 � 16 cos �� �8 sin �
2 �
Section 6.7 Applications and Models 613
66. False. One period is the time for one complete cycle ofthe motion.
68. Aeronautical bearings are always taken clockwise fromNorth (rather than the acute angle from a north-south line).
64. (a)
(c) Period:
This corresponds to the 12 months in a year. Since thesales of outerwear is seasonal this is reasonable.
2�
��6� 12
Month (1 January)↔
Ave
rage
sale
s(i
n m
illio
ns o
f do
llars
)
t
S
2 4 121086
3
6
9
12
15
(b)
Shift:
Note: Another model is
The model is a good fit.
(d) The amplitude represents the maximum displacement fromaverage sales of 8 million dollars. Sales are greatest inDecember (cold weather Christmas) and least in June.�
S � 8 � 6.3 sin��t6
��
2�.
S � 8 � 6.3 cos�� t
6 �S � d � a cos bt
d � 14.3 � 6.3 � 8
2�
b� 12 ⇒ b �
�
6
t2 4 121086
3
6
9
12
15
Ave
rage
sal
es(i
n m
illio
ns o
f do
llars
)
Month (1 ↔ January)
Sa �
1
2�14.3 � 1.7� � 6.3
65. False. Since the tower is not exactly vertical, a right triangle with sides 191 ft and d is not formed.
67. No. N 24 E means 24 east of north.��
69. passes through
y � 4x � 6
y � 2 � 4x � 4
y � 2 � 4�x � ��1��
y
x−1−2−3−4 1 2 3 4
−1
1
2
3
5
6
7
��1, 2�m � 4, 70. Linear equation through
y � �12x �
16
b �16
0 � �16 � b
0 � �12�1
3� � b
y
x−1−2−3 2 3
−1
−2
−3
1
2
3
y � �12x � b
�13, 0�.m � �
12
71. Passes through and
y � �45
x �225
y � 6 � �45
x �85
y � 6 � �45
x � ��2�
y
x−1−2 1 2 3 4 5
−1
1
2
3
4
6
7m �
2 � 63 � ��2� � �
45
�3, 2���2, 6� 72. Linear equation through and
y � �43
x �13
y �23
� �43�x �
14�
� �43
�1
�34
y
x−1−2−3 2 3
−1
−2
−3
1
2
3
m �13 � ��2
3��
12 �
14
��12
, 13�.�1
4, �
23�
Review Exercises for Chapter 6
614 Chapter 6 Trigonometry
1. � � 40� 2. � � 269�
3. (a)
(b) The angle lies in Quadrant I.
(c) Coterminal angles:70� � 360� � �290�70� � 360� � 430�
70°
y
x
� � 70�
5. (a)
(b) The angle lies in Quadrant III.
(c) Coterminal angles:�110� � 360� � �470��110� � 360� � 250�
−110°
x
y
� � �110�
4. (a)
(b) The angle lies in Quadrant IV.
(c) Coterminal angles:280° � 360° � �80°280° � 360° � 640°
x
280°
y
� � 280�
6. (a)
(b) The angle lies in Quadrant IV.
(c) Coterminal angles:�405° � 360° � �45°�405° � 720° � 315°
x
−405°
y
� � �405�
7. (a)
(b) The angle lies in Quadrant II.
(c) Coterminal angles:
3�
4� 2� � �
5�
4
11�
4� 2� �
3�
4
114π
x
y
� �11�
4 8. (a)
(b) The angle lies in Quadrant I.
(c) Coterminal angles:
2�
9� 2� � �
16�
9
2�
9� 2� �
20�
9
x
y
29π
� �2�
9
Review Exercises for Chapter 6 615
9. (a)
(b) The angle lies in Quadrant II.
(c) Coterminal angles:
�4�
3� 2� � �
10�
3
�4�
3� 2� �
2�
3
−43π
x
y
� � �4�
310. (a)
(b) The angle lies in Quadrant I.
(c) Coterminal angles:
�23�
3� 6� � �
5�
3
�23�
3� 8� �
�
3
x
−
y
233π
� � �23�
3
11. 480� � 480� �� rad
180��
8�
3 rad � 8.378 rad
13. �16.5� � �16.5� �� rad
180�� �0.288 rad
12. 120� � 120� �� rad
180�� 2.094 rad
14. �127.5� � �127.5� �� rad
180�� �2.225 rad
17. 84�15� � 84.25� � 84.25� �� rad
180�� 1.470 rad
15. �33�45� � �33.75� � �33.75� �� rad
180�� �
3�
16 rad � �0.589 rad
16. �98�25� � �98.416� �� rad
180�� �1.718 rad
18. 196�77� � 197.283 �� rad
180�� 3.443 rad 19.
5� rad
7�
5� rad
7�
180�
� rad� 128.571�
20.7�
5�
7�
5�
180�
� rad� 252� 21. �
3�
5 rad � �
3� rad
5�
180�
� rad� �108�
22. �11�
6� �
11�
6�
180�
� rad� �330� 23. �3.5 rad � �3.5 rad �
180�
� rad� �200.535�
24. �8.3 � �8.3 �180�
� rad� �475.555� 25. 1.75 rad �
1.75
1 rad �
180�
� rad� 100.268�
27. radians
inchess � r� � 20�23�
30 � � 48.17
138� �138�
180�
23�
3026. 5.7 � 5.7 �
180�
� rad� 326.586�
28.
ms � 11.52
s � r� � 11 � � 60180�� �
113
� rad
616 Chapter 6 Trigonometry
30.
� 212.1 inchessecond
Linear speed � �272
inches��5� rad
second� 31. radians
square inchesA �12
r 2� �12
�18�2�2�
3 � � 339.29
120� �120�
180�
2�
3
32.
A � 55.31 mm2
A �12
�r 2 �12�
5�
6 �65 2 33.
cot � �adjopp
�54
tan � �oppadj
�45
cos � �adjhyp
�5
�41�
5�4141
sec � �hypadj
��41
5
sin � �opphyp
�4
�41�
4�4141
csc � �hypopp
��41
4
opp � 4, adj � 5, hyp � �42 � 52 � �41
34.
cot � �adjopp
� 1
sec � �hypadj
�6�2
6� �2
csc � �hypopp
�6�2
6� �2
tan � �oppadj
� 1
cos � �adjhyp
�6
6�2�
�22
sin � �opphyp
�6
6�2�
�22
opp � 6, adj � 6, hyp � 6�2 35.
cot � �adjopp
�4
4�3�
�33
sec � �hypadj
�84
� 2
csc � �hypopp
�8
4�3�
2�33
tan � �oppadj
�4�3
4� �3
cos � �adjhyp
�48
�12
sin � �opphyp
�4�3
8�
�32
adj � 4, hyp � 8, opp � �82 � 42 � �48 � 4�3
36.
cot � �adjopp
�2�14
5
sec � �hypadj
�9
2�14�
9�1428
csc � �hypopp
�95
tan � �oppadj
�5
2�14�
5�1428
cos � �adjhyp
�2�14
9
sin � �opphyp
�59
adj � �92 � 52 � 2�14
opp � 5 hyp � 9
29. (a)
� 66 23� radians per minute
Angular speed ��331
3��2�� radians1 minute (b)
� 400� inches per minute
Linear speed �6�66 2
3�� inches
1 minute
Review Exercises for Chapter 6 617
38.
(a)
(b)
(c)
(d)
csc � ��17
4
csc2 � � �14�
2
� 1 �1716
cot2 � � 1 � csc2 �
cos � �1
sec ��
1�17
��1717
sec � � �17
sec2 � � 1 � 16 � 17
1 � tan2 � � sec2 �
cot � �1
tan ��
14
tan � � 4
40.
(a)
(b)
(c)
(d) sec�90 � �� � csc � � 5
tan � �1
cot ��
1
2�6�
�612
cot � � 2�6
cot2 � � csc2 � � 1 � 52 � 1 � 24
cot2 � � 1 � csc2 �
sin � �1
csc ��
15
csc � � 5
41. tan 33� � 0.6494 42. csc 11� �1
sin 11�� 5.2408 43. sin 34.2� � 0.5621
37.
(a)
(b)
(c)
(d) tan � �sin �cos �
�
13
2�23
�1
2�2�
�24
sec � �1
cos ��
3
2�2�
3�24
cos � �2�2
3
cos � ��89
cos2 � �89
cos2 � � 1 �19
�13�
2
� cos2 � � 1
sin2 � � cos2 � � 1
csc � �1
sin �� 3
sin � �13
39.
(a)
(b)
(c)
(d) tan � �sin �cos �
�
14
�154
�1
�15�
�1515
sec � �1
cos ��
4�15
�4�15
15
cos � ��15
4
cos � ��1516
cos2 � �1516
cos2 � � 1 �1
16
�14�
2
� cos2 � � 1
sin2 � � cos2 � � 1
sin � �1
csc ��
14
csc � � 4
44. sec 79.3� �1
cos 79.3�� 5.3860 45. cot 15�14� � cot 15.2333� �
1tan 15.2333�
� 3.6722
618 Chapter 6 Trigonometry
50.
x �25
tan 52�� 19.5 feet
tan 52� �25x
52°
x
25
51. x � 12, y � 16, r � �144 � 256 � �400 � 20
tan � �y
x�
4
3
cos � �x
r�
3
5
sin � �y
r�
4
5
cot � �x
y�
3
4
sec � �r
x�
5
3
csc � �r
y�
5
4
52.
tan � �yx
� �43
cos � �xr
�35
sin � �yr
� �45
x � 3, y � �4, r � �32 � (�4)2 � 5
cot � �xy
� �34
sec � �rx
�53
csc � �ry
� �54
53.
cot � �x
y�
2
3
5
2
�4
15
sec � �r
x�
�241
6
2
3
��241
4
csc � �r
y�
�241
6
5
2
�2�241
30�
�241
15
tan � �y
x�
5
2
2
3
�15
4
cos � �x
r�
2
3
�241
6
�4
�241�
4�241
241
sin � �y
r�
5
2
�241
6
�15
�241�
15�241
241
r ���2
3�2
� �5
2�2
��241
6
x �2
3, y �
5
254.
cot � �xy
�
�103
�23
� 5
sec � �rx
�
2�263
�103
� ��26
5
csc � �ry
�
2�263
�23
� ��26
tan � �yx
�
�23
�103
�15
cos � �xr
�
�103
2�263
� �5�26
26
sin � �yr
�
�23
2�263
� ��2626
r ����103 �
2
� ��23�
2
�2�26
3
x � �103
, y � �23
49.
� 71.3 meters
x � 3.5 sin 1�10��1000�1 10'°
x
Not drawn to scale
3.5 km sin 1�10� �x
3.5
46.
� 0.2045
cos 78�11� 58�� � cos 78.1994 47. cos� �
18� � 0.9848 48. tan 5�
6� �0.5774
Review Exercises for Chapter 6 619
55.
cot � �xy
��1�29�2
� �19
sec � �rx
��82�2�1�2
� ��82
csc � �ry
��82�2
9�2�
�829
tan � �yx
�9�2
�1�2� �9
cos � �xr
��1�2�82�2
� �1
�82� �
�8282
sin � �yr
�9�2
�82�2�
9�82
�9�82
82
r ����12�
2
� �92�
2
��82
2
x � �0.5 � �12
, y � 4.5 �92
56.
tan � �yx
�0.40.3
�43
cos � �xr
�0.30.5
�35
sin � �yr
�0.40.5
�45
r � �(0.3)2 � (0.4)2 � 0.5
x � 0.3, y � 0.4
cot � �xy
�0.30.4
�34
sec � �rx
�0.50.3
�53
csc � �ry
� �0.50.4
�54
57.
cot � �x
y�
x
4x�
1
4
sec � �r
x�
�17x
x� �17
csc � �r
y�
�17x
4x�
�17
4
tan � �y
x�
4x
x� 4
cos � �x
r�
x�17x
��17
17
sin � �y
r�
4x�17x
�4�17
17
r � �x2 � �4x�2 � �17x
x � x, y � 4x
�x, 4x �, x > 0 58.
cot � �xy
��2x�3x
�23
sec � �rx
��13x�2x
� ��13
2
csc � �ry
��13x�3x
� ��13
3
tan � �yx
��3x�2x
�32
cos � �xr
��2x�13x
� �2�13
13
sin � �yr
��3x�13x
� �3�13
13
r � �(�2x)2 � (�3x )2 � �13x
x � �2x, y � �3x, x > 0
59. is in Quadrant IV.
cot � �x
y� �
5�11
11tan � �
y
x� �
�11
5
sec � �r
x�
6
5cos � �
x
r�
5
6
csc � �r
y� �
6�11
11sin � �
y
r� �
�11
6
r � 6, x � 5, y � ��36 � 25 � ��11
sec � �6
5, tan � < 0 ⇒ � 60.
cot � �xy
���5
2
sec � �rx
�3
��5� �
3�55
csc � �32
tan � �yx
�2
��5� �
2�55
cos � �xr
� ��53
sin � �yr
�23
r � 3, y � 2, x � ��32 � 22 � ��5
⇒ � is in Quadrant II. csc � �32
, cos � < 0
620 Chapter 6 Trigonometry
61.
cot � �xy
��4�5
�45
sec � �rx
��41�4
� ��41
4
csc � �ry
��41�5
� ��41
5
tan � �54
cos � �xr
��4�41
� �4�41
41
sin � �yr
��5�41
� �5�41
41
r � �(�4)2 � (�5)2 � �41
x � �4, y � �5
tan � �54
, cos � < 0 ⇒ � is in Quadrant III. 62. is in Quadrant II.
cot � �x
y� �
�55
3
sec � �r
x� �
8�55
� �8�55
55
csc � �r
y�
8
3
tan � �y
x� �
3�55
� �3�55
55
cos � �x
r� �
�55
8
sin � �y
r�
3
8
y � 3, r � 8, x � ��55
sin � �3
8, cos � < 0 ⇒ �
63.
cot � �x
y� �
5
12tan � �
yx
� �125
sec � �r
x� �
13
5cos � �
x
r� �
5
13
csc � �r
y�
13
12sin � �
y
r�
12
13
sin � > 0 ⇒ � is in Quadrant II ⇒ y � 12, x � �5
tan � �y
x� �
12
5 ⇒ r � 13 64.
cot � �x
y�
�2�21
� �2�21
21
sec � �r
x�
5
�2� �
5
2
csc � �r
y�
5�21
�5�21
21
tan � �y
x� �
�21
2
sin � �y
r�
�21
5
sin � > 0 ⇒ � is in Quadrant II ⇒ y � �21
cos � �x
r�
�2
5 ⇒ y2 � 21
65.
�� � 264� � 180� � 84�
′θ
264°
x
y� � 264� 66.
�� � 85�
′θ
635°
x
y� � 635� � 720� � 85�
67.
�� � � �4�
5�
�
5
�6�
5� 2� �
4�
5 ′θx
y
65π−
� � �6�
568.
�� ��
3
� 6� ��
3
′θ
x
y
173π
� �17�
3�
18�
3�
�
3
Review Exercises for Chapter 6 621
69.
tan �
3� �3
cos �
3�
12
sin �
3�
�32
70.
tan �
4�
�2
2
�2
2� 1
cos �
4�
�2
2
sin �
4�
�2
271.
tan 5�
6� �tan
�
6� �
�33
cos 5�
6� �cos
�
6� �
�32
sin 5�
6� sin
�
6�
12
72.
tan 5�
3� �
�3
2
1
2� ��3
cos 5�
3� cos�2� �
5�
3 � � cos �
3�
1
2
sin 5�
3� �sin�2� �
5�
3 � � �sin �
3� �
�3
273.
tan��7�
3 � � �tan �
3� ��3
cos��7�
3 � � cos �
3�
12
sin��7�
3 � � �sin �
3� �
�32
74. is coterminal with
tan��5�
4 � ��2
2 ��
�2
2 � � �1
cos��5�
4 � � cos 3�
4� �cos�� �
3�
4 � � �cos �
4� �
�2
2
sin��5�
4 � � sin 3�
4� sin�� �
3�
4 � � sin �
4�
�2
2
3�
4.�
5�
4
75.
tan 495� � �tan 45� � �1
cos 495� � �cos 45� � ��22
sin 495� � sin 45� ��22
76.
tan 120� ��32
��12� � ��3
cos 120� � �cos�180� � 120�� � �cos 60� � �12
sin 120� � sin�180� � 120�� � sin 60� ��32
77.
tan��150�� � tan 30� ��33
cos��150�� � �cos 30� � ��32
sin��150�� � �sin 30� � �12
78. is coterminal with
tan��420�� � ��32
12
� ��3
cos��420�� � cos 300� � cos�360� � 300�� � cos 60� �12
sin��420�� � sin 300� � �sin�360� � 300�� � �sin 60� � ��32
300�.�420�
79. sin 4� � 0.0698 80. tan 231� � 1.2349 81. sec 2.8 �1
cos 2.8� �1.0613
622 Chapter 6 Trigonometry
82. cos 5.5 � 0.7087 83. sin��17�
15 � � 0.4067
85.
tan 2�
3�
yx
��3�2�1�2
� ��3
cos 2�
3� x � �
12
sin 2�
3� y �
�32
t �2�
3, �x, y� � ��
12
, �32 � 86.
tan�7�
4 � � �1
sin�7�
4 � � ��22
x
y
2, ( (22
2−
cos�7�
4 � ��22
84. tan��25�
7 � � 4.3813
87.
tan 7�
6�
yx
��1�2
��3�2�
1�3
��33
cos 7�
6� x � �
�32
sin 7�
6� y � �
12
t �7�
6, �x, y� � ��
�32
, �12� 88.
tan�3�
4 � � �1
sin�3�
4 � ��22
x
y2 , ( (2
22
−cos�3�
4 � � ��22
89.
Period: 2�
Amplitude: 1 2
1
−2
x
− ππ322
yy � sin x 90.
Amplitude: 1
Period: 2�
x
2
−1
−2
2πππ−
yy � cos x
91.
Amplitude: 3
Period:2�
2�� 1
1 2 3
1
2
3
x
yy � 3 cos 2�x 92.
Amplitude: 2
Period:2�
�� 2
−3
−2
−1
1
3
x1
yy � �2 sin �x
93.
Amplitude: 5
Period:2�
25
� 5�
−6
−2
2
4
6
xπ6
yf �x� � 5 sin
2x
594.
Amplitude: 8
Period:2�
14
� 8�
−8
−6
−4
8
x
y
π8π4
f (x) � 8 cos��x4�
Review Exercises for Chapter 6 623
95.
Amplitude: 1
Period:
Shift the graph of
two units upward.
y � sin x
2�
4
3
2
−1
−2
xπππ 2
y
−
y � 2 � sin x 96.
Amplitude: 1
Period: 2321−2−3
x
−1
−2
−3
−5
−6
−1
yy � �4 � cos �x
97.
Amplitude:
Period: 2�
52
−4
−3
−2
−1
1
3
4
tπ
yg�t� �52 sin�t � �� 98.
Amplitude: 3
Period: 2�
−4
−3
1
2
3
4
tπ
yg(t) � 3 cos(t � �)
99.
(a)
(b) f �11
264
� 264 cycles per second
y � 2 sin�528�x�
a � 2, 2�
b�
1264
⇒ b � 528�
y � a sin bx100. (a)
(b) Period:
12 months 1 year, so this is expected.
(c) Amplitude: 1.41
The amplitude represents the maximum change intime from the average time of sunset.�d � 18.09�
�
2�
���6� � �2��6� � 12
014
12
22
s�t� � 18.09 � 1.41 sin��t6
� 4.60�
101.
4
3
2
1
πx
y
f �x� � tan x 102.
Graph of shifted to theright by
t
1
2
3
y
π2
π2
−
��4.tan�t�
f (t) � tan�t ��
4� 103.
x
4
3
2
1
−3
−4
ππ−
y
f �x� � cot x
116. sin�1(0.89) � 1.10
118. arccos��22 � �
�
4120. cos�1��3
2 � ��
6
122. radiansarccos(�0.888) � 2.66
115. sin�1� � 0.44� � � 0.46 radian 117. arccos �32
��
6
119. cos�1��1� � �
121. arccos 0.324 � 1.24 radians 123. tan�1��1.5� � �0.98 radian
112. arcsin(�1) � ��
2113. arcsin 0.4 � 0.41 radian 114. arcsin(0.213) � 0.21
624 Chapter 6 Trigonometry
104.
Period:
t
3
2
1
y
ππ−
�
2
g(t) � 2 cot 2t 105.
Graph first.
ππ−−1
−2
−3
−4
x
y
y � cos x
f �x� � sec x
107.
Graph first.
4
3
2
1
−3
−4
x
− ππ322
y
y � sin x
f �x� � csc x
106.
Graph of sec t shifted to the right
by .
tπ
1
y
�
4
h(t) � sec�t ��
4�
109.
Graph
As
−9
−6
9
6
x →�, f �x� →�.
y3 � �x
y2 � x
y1 � x cos�x�
f �x� � x cos x108.
tπ
2
y
f (t) � 3 csc�2t ��
4�
110.
Graph .y � ±ex first
−1
−150
7
300g(x) � ex cos x 111. arcsin ��12� � �arcsin
12
� ��
6
124. radianstan�1(8.2) � 1.45
Review Exercises for Chapter 6 625
128.
−1.5 1.5
�2
−�2
f �x� � �arcsin 2x
127.
−4 4
�2
−�2
f �x� � arctan�x2� � tan�1�x
2�
129.
Then
and cos � � 45.tan � �34
θ4
53
Let � � arctan 34.
cos�arctan 34� �45. Use a right triangle.
130.
Use a right triangle. Let
Then
and
tan�arccos 35� � tan u �43.
cos u �35
u � arccos 35.
u
45
3
tan�arccos 35� �43 131.
Use a right triangle. Let
Then
and sec � �135 .tan � �
125
� � arctan 125 .
θ
5
1213
sec�arctan 125 � �
135
132.
Use a right triangle. Let Then
and
cot�arcsin��1213� � cot u � �
512.
sin u � �1213
u � arcsin��1213�. u
−12
5
13
�arcsin��1213� � �
512
126.
−1.5 1.5
�3
0
y � 3 arccos x125.
−1.5 1.5
�
−�
f �x� � 2 arcsin x � 2 sin�1�x�
133. Let
Then and
��4 � x2
x.
tan y � tan�arccos�x2��
cos y �x2 2
yx
4 − x 2
y � arccos�x2�. 134.
sec � �1
�x�2 � x�
� �x�2 � x�
cos � � �12 � �x � 1�2
x −11
θ
12 − (x − 1)2
sin � � x � 1
��
2≤ � ≤
�
2� � arcsin�x � 1� ⇒
sec�arcsin�x � 1��
141. False. The sine or cosine functions are often useful for modeling simple harmonic motion.
142. True. The inverse sine, , cannot be defined asa function over any interval that is greater than the
interval defined as ��
2≤ y ≤
�
2.
y � arcsin x
143. False. For each there corresponds exactly one value of .y
� 144. False.
is not in the range of the arctan function.3�
4
arctan(�1) � ��
4
626 Chapter 6 Trigonometry
135.
� � arctan�7030� � 66.8�
tan � �7030
136.
ftr �1808
sin 25.2�� 4246.33
r
25.2°
1808 ft
139.
b � 42.43 nautical miles east6045°a
bN
S
EW
sin 45� �b
60 ⇒
a � 42.43 nautical miles north
cos 45� �a
60 ⇒ 140. High point at time
Period
d � 0.75 cos�2�t3 �
� 3 seconds �2�
b ⇒ b �
2�
3
a �12
�1.5� � 0.75 inches
d � a cos btt � 0 ⇒
138.
The distance is 1221 miles and the bearing is 85.6 .�
sec 4.4� �D
1217 ⇒ D � 1217 sec 4.4� � 1221
tan � �93
1217 ⇒ � � 4.4�
sin 25� �d4
810 ⇒ d4 � 342
cos 48� �d3
650 ⇒ d3 � 435
cos 25� �d2
810 ⇒ d2 � 734
sin 48� �d1
650 ⇒ d1 � 483
48°
48°
65°25°
d3d4
d d1 2
810650
D
B
CA θ
N
S
W E
d1 � d2 � 1217
d3 � d4 � 93
137.
h � 25 tan 21� � 9.6 feet
tan 21� �h
25
21°
h
25 feet
Review Exercises for Chapter 6 627
145.
Matches graph �d�.
Period: 2�
Amplitude: 3
y � 3 sin x 147.
Matches graph �b�.
Period: 2
Amplitude: 2
y � 2 sin �x146.
Amplitude:
Period:
graph is reflected in the x-axis
Matches graph (a).
a < 0 ⇒
2�
3
y � �3 sin x
148.
Amplitude:
Period:
Matches graph (c).
4�
2
y � 2 sin x2
149. is undefined at the zeros of
since sec � �1
cos � .
g��� � cos �f ��� � sec �
150. (a)
(b) tan�� ��
2� � �cot �
151. The ranges for the other four trigonometric functions arenot bounded. For the range is
For the range is���, �1 � �1, ��.
y � sec x and y � csc x,���, ��.y � tan x and y � cot x,
152.
(a) If is changed from to the amplitude of eachoscillation is increased.
(b) If k is changed from to the oscillations aredamped more quickly.
(c) If b is changed from 6 to 9, the frequency of theoscillations increases.
13,1
10
13,1
5A
y � Ae�kt cos bt �15 e�t�10 cos 6t
0.1 0.4 0.7 1.0 1.3
�0.2776�0.6421�1.1872�2.3652�9.9666�cot �
�0.2776�0.6421�1.1872�2.3652�9.9666tan�� ��
2��
153. (a)
12θ
x
� 72�tan � � ��
� 72 tan � � 72�
�1
2�12��12 tan �� �
1
2�122����
� �1
2bh� � �1
2r 2��
Area � Area of triangle � Area of sector
x � 12 tan �
tan � �x
12(b)
As .
The area increases without bound as � approaches �
2.
� ⇒ �2
, � ⇒ �
�2
00
800
628 Chapter 6 Trigonometry
154. (a)
The area function increases more rapidly than thearc length function because it is a function of thesquare of the radius, while the arc length function isa function of the radius.
00 6
4
A s
s � r� � 0.8r, r > 0
A �12
r 2� �12 r 2�0.8� � 0.4r 2, r > 0 (b)
03
30
A s
0
s � r� � 10�, � > 0
A �12
r 2� �12�10 2�� � 50�, � > 0
155. Answers will vary.
Problem Solving for Chapter 6
1. (a)
revolutions
radians or
(b) s � r� � 47.25�5.5�� � 816.42 feet
990�� � �114 ��2�� �
11�
2
13248
�114
8:57 � 6:45 � 2 hours 12 minutes � 132 minutes 2. Gear 1:
Gear 2:
Gear 3:
Gear 4:
Gear 5:2419
�360�� � 454.737� � 7.94 radians
4032
�360�� � 450� �5�
2 radians
2422
�360�� � 392.727� � 6.85 radians
2426
�360�� � 332.308� � 5.80 radians
2432
�360�� � 270� �3�
2 radians
3. (a)
d �3000
sin 39�� 4767 feet
sin 39� �3000
d(b)
x �3000
tan 39�� 3705 feet
tan 39� �3000
x(c)
w � 3000 tan 63� � 3705 � 2183 feet
3000 tan 63� � w � 3705
tan 63� �w � 3705
3000
4. (a) are all similar triangles since they all have the same angles. is part of all three triangles and Thus,
(b) Since the triangles are similar, the ratios of corresponding sides are equal.
(c) Since the ratios: it does not matter which triangle is used to calculate sin A.
Any triangle similar to these three triangles could be used to find sin A. The value of sin A would not change.
(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.
opphyp
�BCAB
�DEAD
�FGAF
� sin A
BCAB
�DEAD
�FGAF
�B � �D � �F.�C � �E � �G � 90�.�A�ABC, �ADE, and �AFG
Problem Solving for Chapter 6 629
5. (a)
h is even.
−1
−2
3
2� �
h�x� � cos2 x
6. Given: f is an even function and g is an odd function.
(a)
since f is even
Thus, h is an even function.
� h�x�
� � f �x��2
h��x� � � f ��x��2
h�x� � � f �x��2
7. If we alter the model so that we can use either a sine or a cosine model.
For the cosine model we have:
For the sine model we have:
Notice that we needed the horizontal shift so that the sine value was one when .
Another model would be:
Here we wanted the sine value to be 1 when t � 0.
h � 51 � 50 sin�8� t �3�
2 �t � 0
h � 51 � 50 sin�8� t ��
2�h � 51 � 50 cos�8� t�
b � 8�
d �12
�max � min� �12
�101 � 1� � 51
a �12
�max � min� �12
�101 � 1� � 50
h � 1 when t � 0,
8.
(a)
(c) Amplitude: 20
The blood pressure ranges between and
(e) Period
64 �60
�2��b� ⇒ b �6460
� 2� �3215
�
�6064
�1516
sec
100 � 20 � 120.100 � 20 � 80
070
5
130
P � 100 � 20 cos�8�
3t�
(b)
This is the time between heartbeats.
(d) Pulse rate �60 sec�min34 sec�beat
� 80 beats�min
Period �2�
�8�
3 ��
68
�34
sec
(b)
h is even.
−1
−2
3
2� �
h�x� � sin2 x
(b)
since g is odd
Thus, h is an even function.
� h�x�
� �g�x��2
� ��g�x��2
h��x� � �g��x��2
h�x� � �g�x��2
Conjecture: The square of either an even function or an odd function is an even function.
630 Chapter 6 Trigonometry
9. Physical (23 days):
Emotional (28 days):
Intellectual (33 days):
(a)
(b) Number of days since birth until September 1, 2006:
5 11 31 1
20 years leap years remaining August days day inJuly days September
All three drop early in the month, then peak toward the middle of the month, and drop againtoward the latter part of the month.
(c) For September 22, 2006, use
I � 0.945
E � 0.901
P � 0.631
t � 7369.
−2
7349 7379
2
P
I
E
t � 7348
���
����t � 365 � 20
−2
7300 7380
2
P IE
I � sin 2� t33
, t ≥ 0
E � sin 2� t28
, t ≥ 0
P � sin 2� t23
, t ≥ 0
10.
(a)
(b) The period of
The period of
(c) is periodic since the sineand cosine functions are periodic.h�x� � A cos x � sin �x
g�x� is �.
f �x� is 2�.
−6
−
6
� �
g
f
g�x� � 2 cos 2x � 3 sin 4x
f �x� � 2 cos 2x � 3 sin 3x 11. (a) Both graphs have a period of 2 and intersect when They should also intersect when
and
(b) The graphs intersect when
(c) Since and the graphs will intersect again at these values. Therefore f �13.35� � g��4.65�.
�4.65 � 5.35 � 5�2�13.35 � 5.35 � 4�2�
x � 5.35 � 3�2� � �0.65.
x � 5.35 � 2 � 7.35.x � 5.35 � 2 � 3.35x � 5.35.
Problem Solving for Chapter 6 631
12. (a) is true since this is a two period horizontal shift.
(b) is not true.
is a horizontal translation of
is a doubling of the period of
(c) is not true.
is a horizontal
translation of by half a period.
For example, sin�12
�� � 2��� � sin�12
��.
f �12
t�f �1
2 �t � c�� � f �1
2t �
12
c�
f �12
�t � c�� � f �12
t�
f �t�.f �12
t�
f �t�.f �t �12
c�
f �t �12
c� � f �12
t�
f �t � 2c� � f �t� 13.
(a)
(b)
(c) feet
(d) As you more closer to the rock, decreases, whichcauses y to decrease, which in turn causes d todecrease.
�1
d � y � x � 3.46 � 1.71 � 1.75
tan �1 �y2
⇒ y � 2 tan 60� � 3.46 feet
tan �2 �x2
⇒ x � 2 tan 40.52� � 1.71 feet
�2 � 40.52�
sin �2 �sin �1
1.333� sin 60�
1.333� 0.6497
sin �1
sin �2� 1.333
2 ft
xy
d
θθ
1
2
14.
(a)
The graphs are nearly the same for �1 < x < 1.
−2
2
�2
−�2
arctan x � x �x3
3�
x5
5�
x7
7
(b)
The accuracy of the approximation improved slightly byadding the next term �x9�9�.
−2
2
�2
−�2
632 Chapter 6 Trigonometry
Chapter 6 Practice Test
1. Express 350° in radian measure. 2. Express in degree measure.�5���9
3. Convert to decimal form.135�14�12� 4. Convert form.�22.569� to D�M�S�
5. If use the trigonometric identities to find tan .�cos � �23, 6. Find given .sin � � 0.9063�
7. Solve for in the figure below.
20°
35
x
x 8. Find the magnitude of the reference angle for .� � �6���5
9. Evaluate .csc 3.92 10. Find .sec � given that � lies in Quadrant III and tan � � 6
11. Graph y � 3 sin x
2. 12. Graph y � �2 cos�x � ��.
13. Graph y � tan 2x. 14. Graph .y � �csc�x ��
4�
15. Graph using a graphing calculator.y � 2x � sin x, 16. Graph using a graphing calculator.y � 3x cos x,
17. Evaluate .arcsin 1 18. Evaluate arctan��3�.
19. Evaluate sin�arccos 4
�35�. 20. Write an algebraic expression for .cos�arcsin x
4�
For Exercises 21–23, solve the right triangle.
A C
B
ca
b
21. A � 40�, c � 12 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 , how far out is the ship?5°