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C H A P T E R 4Trigonometry
Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 335
Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . . 344
Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 350
Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 360
Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 373
Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 386
Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 397
Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 410
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
335
C H A P T E R 4Trigonometry
Section 4.1 Radian and Degree Measure
You should know the following basic facts about angles, their measurement, and their applications.
■ Types of Angles:
(a) Acute: Measure between 0� and 90�.
(b) Right: Measure 90�.
(c) Obtuse: Measure between 90� and 180 .
(d) Straight: Measure 180�.
■ 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.
■ Linear speed
■ Angular speed � ��t � s�rt
�arc length
time�
st
�s � r�
� 1�60 of 1� � 1�36001� �
� 1�601� �
� �180����.1� � ��180
� � � 180�.� � � 90�.�
�
Vocabulary Check
1. Trigonometry 2. angle
3. coterminal 4. radian
5. acute; obtuse 6. complementary; supplementary
7. degree 8. linear
9. angular 10. A �12r 2�
1.
The angle shown is approximately2 radians.
2.
The angle shown is approximately5.5 radians.
3.
The angle shown is approximatelyradians.3
4.
The angle shown is approximatelyradians.4
5.
The angle shown is approximately 1 radian.
6.
The angle shown is approximately6.5 radians.
336 Chapter 4 Trigonometry
7. (a) Since lies in Quadrant I.
(b) Since lies in Quadrant III.� <7�
5<
3�
2 ;
7�
5
0 <�
5<
�
2 ;
�
5
9. (a) Since lies in Quadrant IV.
(b) Since lies in Quadrant III.� < 2 < �
2; 2
�
2 <
�
12 < 0 ;
�
12
8. (a) Since lies in Quadrant III.
(b) Since lies in Quadrant III.� <9�
8<
3�
2;
9�
8
� <11�
8<
3�
2;
11�
8
10. (a) Since lies in Quadrant IV.
(b) Since lies in Quadrant II.11�
9
3�
2<
11�
9< �,
�
2< 1 < 0; 1
11. (a) Since lies in Quadrant III.
(b) Since lies in Quadrant II.�
2< 2.25 < � ; 2.25
� < 3.5 <3�
2 ; 3.5 12. (a) Since lies in Quadrant IV.
(b) Since 4.25 lies in Quadrant II.3�
2 < 4.25 < � ;
3�
2< 6.02 < 2� ; 6.02
13. (a)
(b)
− 23π
x
y
2�
3
54π
x
y
5�
414. (a)
(b)
x
y
52π
5�
2
− 74π
y
x
7�
4 15. (a)
(b)
−3
x
y
3
116π
x
y
11�
6
16. (a) 4
4
x
y (b)
7π
x
y7�
Section 4.1 Radian and Degree Measure 337
26.
The angle shown is approximately120�.
27.
The angle shown is approximately 60º.
28.
The angle shown is approximately330�.
17. (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
�
618. (a)
(b)
11�
6 2� �
23�
6
11�
6� 2� �
�
6
7�
6 2� �
5�
6
7�
6� 2� �
19�
619. (a) Coterminal angles for
(b) Coterminal angles for
�
12 2� �
23�
12
�
12� 2� �
25�
12
�
12
2�
3 2� �
4�
3
2�
3� 2� �
8�
3
2�
3
20. (a)
(b)
2�
15 2� �
32�
15
2�
15� 2� �
28�
15
9�
4� 4� �
7�
4
9�
4� 2� �
�
421. (a) Complement:
Supplement:
(b) Complement: Not possible, is greater than
Supplement: � 3�
4�
�
4
�
2.
3�
4
� �
3�
2�
3
�
2
�
3�
�
6
22. (a) Complement:
Supplement:
(b) Complement: Not possible, is greater than
Supplement: � 11�
12�
�
12
�
2.
11�
12
� �
12�
11�
12
�
2
�
12�
5�
1223. (a) Complement:
Supplement:
(b) Complement: Not possible, 2 is greater than
Supplement: � 2 � 1.14
�
2.
� 1 � 2.14
�
2 1 � 0.57
24. (a) Complement: Not possible, 3 is greater than
Supplement:
(b) Complement:
Supplement: � 1.5 � 1.64
�
2 1.5 � 0.07
� 3 � 0.14
�
2. 25.
The angle shown is approximately .210º
338 Chapter 4 Trigonometry
31. (a) Since lies in Quadrant II.
(b) Since lies in Quadrant IV.270� < 285� < 360�, 285�
90� < 130� < 180�, 130� 32. (a) Since lies in Quadrant I.
(b) Since lies inQuadrant III.
180� < 257� 30� < 270�, 257� 30�
0� < 8.3� < 90�, 8.3�
33. (a) Since lies in Quadrant III.
(b) Since lies inQuadrant I.
360� < 336� < 270�, 336�
180� < 132� 50� < 90�, 132� 50� 34. (a) Since lies inQuadrant II.
(b) Since lies in Quadrant IV.90� < 3.4� < 0�, 3.4�
270� < 260� < 180�, 260�
35. (a)
(b)
150°
x
y
150�
x30°
y
30� 36. (a)
(b)
−120°
x
y
120�
−270°
x
y
270�
37. (a)
(b)
480°
x
y
480�
405°
x
y
405�
29.
The angle shown is approximately .165�
30.
The angle shown is approximately 10�.
38. (a)
(b)
−600°
x
y
600�
−750°x
y
750�
Section 4.1 Radian and Degree Measure 339
54. (a)
(b)34�
15�
34�
15 �180�
� � � 408�
11�
6�
11�
6 �180�
� � � 330� 55.
� 2.007 radians
115� � 115� �
180� 56.
� 1.525 radians
87.4� � 87.4�� �
180��
57. 216.35� � 216.35� �
180� � 3.776 radians
59. 532� � 532� �
180� � 9.285 radians
58. radians48.27� � 48.27�� �
180�� � 0.842
60. 345� � 345�� �
180�� � 6.021 radians
39. (a) Coterminal angles for 45
45 � 360 � 405
45 360 � 315
(b) Coterminal angles for
36 � 360 � 324
36 360 � 396���
���
36�
���
���
� 40. (a)
(b)
420� � 360� � 60�
420� � 720� � 300�
120� 360� � 240�
120� � 360� � 480� 41. (a) Coterminal angles for
(b) Coterminal angles for
180� 360� � 540�
180� � 360� � 180�
180�
240� 360� � 120�
240� � 360� � 600�
240�
42. (a)
(b)
230� 360� � 130�
230� � 360� � 590�
420� � 360� � 60�
420� � 720� � 300� 43. (a) Complement: 90 18 � 72
Supplement:
(b) Complement: Not possible, 115 is greater than 90 .
Supplement: 1180� 115� � 65�
��
180� 18� � 162�
���
44. (a) Complement:
Supplement:
(b) Complement:
Supplement: 180� 64� � 116�
90� 64� � 26�
180� 3� � 177�
90� 3� � 87� 45. (a) Complement:
Supplement:
(b) Complement: Not possible, is greater than
Supplement: 180� 150� � 30�
90�.150�
180� 79� � 101�
90� 79� � 11�
46. (a) Complement: Not possible, is greater than
Supplement:
(b) Complement: Not possible, is greater than
Supplement: 180� 170� � 10�
90�.170�
180� 130� � 50�
90�.130� 47. (a)
(b) 150� � 150� �
180� �5�
6
30� � 30� �
180� ��
6
48. (a)
(b) 120� � 120�� �
180�� �2�
3
315� � 315�� �
180�� �7�
449. (a)
(b) 240� � 240� �
180� � 4�
3
20� � 20� �
180� � �
950. (a)
(b) 144� � 144�� �
180�� �4�
5
270� � 270�� �
180�� � 3�
2
51. (a)
(b)7�
6�
7�
6 �180
� �� � 210�
3�
2�
3�
2 �180
� �� � 270� 52. (a)
(b)�
9�
�
9�180�
� � � 20�
7�
12�
7�
12�180�
� � � 105� 53. (a)
(b) 11�
30�
11�
30 �180
� �� � 66�
7�
3�
7�
3 �180
� �� � 420�
340 Chapter 4 Trigonometry
61. radian0.83� � 0.83� �
180� � 0.014 62. 0.54� � 0.54�� �
180�� � 0.009 radians
63.�
7�
�
7�180
� �� � 25.714� 64.5�
11�
5�
11�180�
� � � 81.818� 65.15�
8�
15�
8 �180
� �� � 337.500�
66.13�
2�
13�
2 �180�
� � � 1170.000� 67.
� 756.000�
4.2� � 4.2��180
� �� 68. 4.8� � 4.8��180�
� � � 864.000�
69. 2 � 2�180
� ��� 114.592� 70. 0.57 � 0.57�180�
� � � 32.659�
71. (a)
(b) 128� 30� � 128� �3060�� � 128.5�
54� 45� � 54� � �4560�� � 54.75�
73. (a)
(b) 330� 25� � �330 �25
3600�� � 330.007�
85� 18� 30� � �85 �1860 �
303600�� � 85.308�
75. (a)
(b) 145.8� � �145� � 0.8�60��� � 145� 48�
240.6� � 240� � 0.6�60�� � 240� 36�
77. (a)
(b)
� 3� 34� 48�
� �3� � 34� � 0.8�60� ��
3.58� � �3� � �0.58��60� ��
2.5� � 2� 30�
72. (a)
(b) � 2� � 0.2� � 2.2�2�12� � 2� � �1260��
� 245.167�� 245� � 0.167�245�10� � 245� � �1060��
74. (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��
76. (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� ��
78. (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� ��
79.
� �65 radians
6 � 5�
s � r� 81.
� �327 � 4 4
7 radians
32 � 7�
s � r�80.
� �2910 radians
29 � 10 �
s � r�
82.
Because the angle represented isclockwise, this angle is radians.
45
� �60
75�
4
5 radians
60 � 75�
s � r� 83.
� �6
27�
2
9 radians
6 � 27�
s � r� 84. feet, feet
� �s
r�
8
14�
4
7 radians
s � 8r � 14
Section 4.1 Radian and Degree Measure 341
100. � �s
r�
24
5� 4.8 radians � 4.8�180�
� � � 275�
91.
� 8.38 square inches
A �12
�4�2��
3� �8�
3 square inches
A �12
r 2� 92.
� 56.55 mm2
� 18� mm2
A �12
r 2� �12
�12�2��
4�
r � 12 mm, � ��
493.
� 12.27 square feet
A �12
�2.5�2�225�� �
180�
A �12
r 2�
94.
square miles� 5.64�21.56
12�A �
12
�1.4�2�330�
180���
r � 1.4 miles, � � 330� 95.
miless � r� � 4000�0.14782� � 591.3
� 8.46972� � 0.14782 radian
� � 41� 15� 50� 32� 47� 39�
96.
s � r� � �4000��0.1715� � 686.2 miles
� 0.1715 radian
� � 47� 37� 18� 37� 47� 36� � 9� 49� 42�
r � 4000 miles97. � �
s
r�
450
6378� 0.071 radian � 4.04�
98. kilometers
The difference in latitude is about radians � 3.59�.0.062716
0.062716 � �
4006378
� �
400 � 6378�
s � r�
r � 3189 99. � �s
r�
2.5
6�
25
60�
5
12 radian
85.
� �25
14.5�
50
29 radians
25 � 14.5�
s � r� 87. in radians
� 47.12 inches
s � 15�180�� �
180� � 15� inches
s � r�, �86. kilometers,kilometers
� �s
r�
160
80� 2 radians
s � 160r � 80
88. feet,
� 9.42 feet
s � r� � 9��
3� � 3� feet
� � 60� ��
3r � 9 89. in radians
s � 3�1� � 3 meters
s � r�, � 90. centimeters,
� 15.71 centimeters
s � r� � 20��
4� � 5� centimeters
� ��
4r � 20
101. (a) 65 miles per hour feet per minute
The circumference of the tire is feet.
The number of revolutions per minute is
revolutions per minuter �5720
2.5�� 728.3
C � 2.5�
�65�5280�
60� 5720 (b) The angular speed is
Angular speed radians per minute�4576 radians
1 minute� 4576
� �5720
2.5��2�� � 4576 radians
�
t.
342 Chapter 4 Trigonometry
102. Linear velocity for either pulley: inches per minute
(a) Angular speed of motor pulley: radians per minute
Angular speed of the saw arbor: radians per minute
(b) Revolutions per minute of the saw arbor: revolutions per minute1700�
2�� 850
� �v
r�
3400�
2� 1700�
� �v
r�
3400�
1� 3400�
1700�2�� � 3400�
103. (a)
(b)
� 9869.84 feet per minute
� 314123
� feet per minute
Linear speed ��7.25
2 in.�� 1 ft
12 in.��5200��2��
1 minute
� 32,672.56 radians per minute
� 10,400� radians per minute
Angular speed ��5200��2�� radians
1 minute104. (a)
(b)
� 628.32 feet per minute
Linear speed � 25�25.13274� feet per minute
r�
t� 200� feet per minute
r � 25 ft
� 25.13 radians per minute
� 8� radians per minute
4 rpm � 4�2�� radians per minute
105. (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
106.
A �12�
125180�� � �252 112� � 175� � 549.8 square inches
r � 25 14 � 11
R � 25 25125°
14r
A �12
��R2 r2� 107.
� 1496.62 square meters
� 476.39� square meters
�12
�35�2�140��� �
180��140°
35
A �12
r 2�
108. (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:
—CONTINUED—
14� feet
3 seconds
3600 seconds1 hour
1 mile
5280 feet� 10 miles per hour
Speed �st
��14���31 second
�14�
3 feet per second
s � �4���1412� �
14�
3 feet
s � �r
� �sr
��2���3 feet2�12 feet
� 4�
�r � 2 inches � 2�12 feet�.�2��3
s �13
�2�� �2�
3 feet
s � r�
Section 4.1 Radian and Degree Measure 343
115. The arc length is increasing. In order for the angle toremain constant as the radius r increases, the arc length s must increase in proportion to r, as can be seen fromthe formula s � r�.
�
116. The area of a circle is The circumference of a circle is
For a sector, Thus, for a sector.A ��r��r
2�
1
2�r 2C � s � r�.
Cr
2� A
C �2A
r
C � 2�A
r 2� r
C � 2�r.A � �r2 ⇒ � �A
r2.
108. —CONTINUED—
(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:
Distance
(c)
(d) The functions are both linear.
� �14�
3 feet per second�� 1 mile
5280 feet��t seconds� �7�
7920 t miles
Distance � Rate � Time
� �14�
3
feetrevolutions��
1 mile5280 feet��n revolutions� �
7�
7920n miles
�14���3
109. False. An angle measure of radians corresponds totwo complete revolutions from the initial to the terminalside of an angle.
4�
111. False. The terminal side of lies on the negative x-axis.1260�
113. Increases, since the linear speed is proportional to the radius.
110. True. If and are coterminal angles, thenwhere n is an integer. The difference
between is � � n�360�� � 2�n.� and � � � n�360��
�
112. (a) An angle is in standard position if its vertex is at the origin and its initial side is on the positive x-axis.
(b) A negative angle is generated by a clockwise rotation of the terminal side.
(c) Two angles in standard position with the same terminal sides are coterminal.
(d) An obtuse angle measures between 90° and 180°.
114. 1 radian
so one radian is much larger than one degree.
� �180
� ��� 57.3�,
117.4
42�
4
42�22
�42
8�
22
118.55
210�
52 5
10�
52
12
�5
22�22
�52
4
119. 22 � 62 � 4 � 36 � 40 � 4 � 10 � 210 120.
� 208 � 16 � 13 � 413
172 92 � 289 81
344 Chapter 4 Trigonometry
Section 4.2 Trigonometric Functions: The Unit Circle
■ You should know the definition of the trigonometric functions in terms of the unit circle. Let t be a real number and the point on the unit circle corresponding to t.
■ The cosine and secant functions are even.
■ The other four trigonometric functions are odd.
■ Be able to evaluate the trigonometric functions with a calculator.
cot��t� � �cot ttan��t� � �tan t
csc��t� � �csc tsin��t� � �sin t
sec��t� � sec tcos��t� � cos t
cot t �x
y, y � 0tan t �
y
x, x � 0
sec t �1
x, x � 0cos t � x
csc t �1
y, y � 0sin t � y
�x, y�
Vocabulary Check
1. unit circle 2. periodic
3. period 4. odd; even
121.
Graph of shifted to the right by two units
y � x5
432−2
3
2
1
−1
−2
−3
x
yy = x5
y = (x − 2)5
f �x� � �x � 2�5 122.
Vertical shift four unitsdownward
4
2
−2
−6
32−2−3 1
y = x5
y = x5 − 4
x
yf �x� � x5 � 4
123.
Graph of reflectedin x-axis and shiftedupward by two units
y � x5
321−2−3
6
5
4
3
1
−1
−2
−3
x
y
y = x5
y = 2 − x5
f �x� � 2 � x5 124.
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)5 f �x� � ��x � 3�5
Section 4.2 Trigonometric Functions: The Unit Circle 345
7. corresponds to ���3
2, �
1
2�.t �7�
68. t �
5�
4, �x, y� � ��
�22
, ��22 �
10. t �5�
3, �x, y� � �1
2, �
�32 �9. corresponds to ��1
2, �
�3
2 �.t �4�
3
11. corresponds to �0, �1�.t �3�
2
13. corresponds to
sin
cos
tan t �y
x� 1
t � x ��2
2
t � y ��2
2
��2
2, �2
2 �.t ��
4
15. corresponds to
sin
cos
tan t �y
x� �
1
�3� �
�3
3
t � x ��3
2
t � y � �1
2
��3
2, �
1
2�.t � ��
6
12. t � �, �x, y� � ��1, 0�
14.
tan �
3�
�3�21�2
� �3
cos �
3�
12
sin �
3�
�32
t ��
3, �x, y� � �1
2, �32 �
16.
tan���
4� ���2�2�2�2
� �1
cos���
4� ��22
sin���
4� � ��22
t � ��
4, �x, y� � ��2
2, �
�22 �
1.
sin csc
cos sec
tan cot � �x
y� �
8
15� �
y
x� �
15
8
� �1
x� �
17
8� � x � �
8
17
� �1
y�
17
15� � y �
15
17
x � �8
17, y �
15
17
3.
sin csc
cos sec
tan cot � �x
y� �
12
5� �
y
x� �
5
12
� �1
x�
13
12� � x �
12
13
� �1
y� �
13
5� � y � �
5
13
x �12
13, y � �
5
13
5. corresponds to ��2
2, �2
2 �.t ��
4
2.
cot � �xy
�125
tan � �y
x�
512
sec � �1x
�1312
cos � � x �1213
csc � �1y
�135
sin � � y �5
13
x �1213
, y �5
13
4.
cot � �xy
�43
tan � �y
x�
34
sec � �1x
� �54
cos � � x � �45
csc � �1y
� �53
sin � � y � �35
x � �45
, y � �35
6. t ��
3, �x y� � �1
2, �32 �
346 Chapter 4 Trigonometry
19. corresponds to
sin
cos
tan t �y
x� �
1
�3� �
�3
3
t � x ��3
2
t � y � �1
2
��3
2, �
1
2�.t �11�
620.
tan 5�
3�
��3�21�2
� ��3
cos 5�
3�
12
sin 5�
3� �
�32
t �5�
3, �x, y� � �1
2, �
�32 �
21. corresponds to
sin
cos
tan is undefined.t �y
x
t � x � 0
t � y � 1
�0, 1�.t � �3�
222.
tan��2�� �01
� 0
cos��2�� � 1
sin��2�� � 0
t � �2�, �x, y� � �1, 0�
23. corresponds to
sin csc
cos sec
tan cot t �xy � �1t �
y
x � �1
t �1
x� ��2t � x � �
�2
2
t �1
y� �2t � y �
�2
2
���2
2, �2
2 � .t �3�
424.
cot 5�
6�
1tan t
� ��3tan 5�
6�
1�2
��3�2� �
�33
sec 5�
6�
1cos t
� �2�3
3cos
5�
6� �
�32
csc 5�
6�
1sin t
� 2sin 5�
6�
12
t �5�
6, �x, y� � ��
�32
, 12�
25. corresponds to
sin csc
cos sec is undefined.
tan is undefined. cot t �xy � 0t �
y
x
t �1
xt � x � 0
t �1
y� �1t � y � �1
�0, �1�.t � ��
226.
is undefined.
is undefined. cot 3�
2�
0�1
� 0tan 3�
2
sec 3�
2cos
3�
2� 0
csc 3�
2�
1sin t
� �1sin 3�
2� �1
t �3�
2, �x, y� � �0, �1�
17. corresponds to
sin
cos
tan t �y
x� 1
t � x ��2
2
t � y ��2
2
��2
2, �2
2 �.t � �7�
418.
tan��4�
3 � ��3�2�1�2
� ��3
cos��4�
3 � � �12
sin��4�
3 � ��32
t � �4�
3, �x, y� � ��
12
, �32 �
Section 4.2 Trigonometric Functions: The Unit Circle 347
29. sin 5� � sin � � 0 31. cos 8�
3� cos
2�
3� �
1
2
33. cos��15�
2 � � cos �
2� 0
35. sin��9�
4 � � sin�7�
4 � � ��2
237.
(a)
(b) csc��t� � �csc t � �3
sin��t� � �sin t � �1
3
sin t �1
3
39.
(a)
(b) sec��t� �1
cos��t�� �5
cos t � cos��t� � �1
5
cos��t� � �1
5
30. cos 5� � cos � � �1
32. sin 9�
4� sin
�
4�
�22
34. sin 19�
6� sin
7�
6� �
12
36. cos��8�
3 � � cos 4�
3� �
12
38.
(a)
(b) csc t �1
sin t� �
83
sin t � �sin��t� � �38
sin��t� �38
40.
(a)
(b) sec��t� � sec t �1
cos t� �
43
cos��t� � cos t � �34
cos t � �34
41.
(a)
(b) sin�t � �� � �sin t � �4
5
sin�� � t� � sin t �4
5
sin t �4
543. sin
�
4� 0.7071
45. csc 1.3 �1
sin 1.3� 1.0378
42.
(a)
(b) cos�t � �� � �cos t � �45
cos�� � t� � �cos t � �45
cos t �45
44. tan �
3� 1.7321 46. cot 1 �
1tan 1
� 0.6421
47. cos��1.7� � �0.1288 49. csc 0.8 �1
sin 0.8� 1.3940
51. sec 22.8 �1
cos 22.8� �1.4486
53. (a)
(b) cos 2 � x � �0.4
sin 5 � y � �1
48. cos��2.5� � �0.8011
50. sec 1.8 �1
0051.8� �4.4014 52. sin��0.9� � �0.7833
54. (a)
(b) cos 2.5 � x � �0.8
sin 0.75 � y � 0.7
27. corresponds to
sin csc
cos sec
tan cot t �xy �
�3
3t �
y
x � �3
t �1
x� �2t � x � �
1
2
t �1
y� �
2�3
3t � y � �
�3
2
��1
2, �
�3
2 �.t �4�
328.
cot 7�
4�
1tan t
� �1 tan 7�
4�
��2�2�2�2
� �1
sec 7�
4�
1cos t
� �2 cos 7�
4�
�22
csc 7�
4�
1sin t
� ��2 sin 7�
4� �
�22
t �7�
4, �x, y� � ��2
2, �
�22 �
348 Chapter 4 Trigonometry
57.
(a)
(b) From the table feature of a graphing utility we see that seconds.
(c) As t increases, the displacement oscillates but decreases in amplitude.
y � 0 when t � 5
y�t� �14 e�t cos 6t
t 0 1
y 0.25 0.0138 0.0883�0.0249�0.1501
34
12
14
58.
(a)
(b)
(c) y�1
2� �1
4 cos 3 � �0.2475 feet
y�1
4� �1
4 cos
3
2� 0.0177 feet
y�0� �1
4 cos 0 � 0.2500 feet
y�t� �1
4 cos 6t 59. False. means the function is odd, not
that the sine of a negative angle is a negative number.
For example:
Even though the angle is negative, the sine value is positive.
sin��3�
2 � � �sin�3�
2 � � ���1� � 1.
sin��t� � �sin t
60. True. since the period of tan is �.tan a � tan�a � 6� � 61. (a) The points have y-axis symmetry.
(b) since they have the same y-value.
(c) since the x-values have theopposite signs.cos�� � t1� � �cos t1
sin t1 � sin �� � t1�
62.
So and are even.
So and are odd.
So and are odd.cot �tan �
cot���� �x
�y� �cot �
cot � �xy
tan���� ��yx
� �tan �
tan � �yx
csc �sin �
csc���� � �1y
� �csc �
x
y
θθ−
(x, y)
(x, −y)
csc � �1y
sin���� � �y � �sin �
sin � � y
cos �sec �
sec � �1x
� sec����
cos � � x � cos���� 63.
f �1�x� �23
x �23
�23
�x � 1�
23
x �23
� y
2x � 3y � 2
x �12
�3y � 2�
y �12
�3x � 2�
f �x� �12
�3x � 2�
55. (a)
(b)
t � 1.82 or 4.46
cos t � �0.25
t � 0.25 or 2.89
sin t � 0.25 56. (a)
(b)
t � 0.7 or t � 5.6
cos t � 0.75
t � 4.0 or t � 5.4
sin t � �0.75
Section 4.2 Trigonometric Functions: The Unit Circle 349
64.
f�1�x� � 3�4�x � 1�
y � 3�4�x � 1�
4�x � 1� � y3
x � 1 �14
y3
x �14
y3 � 1
y �14
x3 � 1
f �x� �14
x3 � 1 66.
f �1�x� �2�2x � 1�
x � 1
y �2�2x � 1�
x � 1
y�x � 1� � 4x � 2
xy � y � 4x � 2
xy � 4x � y � 2
x�y � 4� � y � 2
x �y � 2y � 4
y �x � 2x � 4
f �x� �x � 2x � 465.
f �1�x� � �x2 � 4, x ≥ 0
±�x2 � 4 � y
x2 � y2 � 4
x � �y2 � 4
y � �x2 � 4
f �x� � �x2 � 4, x ≥ 2
67.
Intercept:
Vertical asymptote:
Horizontal asymptote: y � 2
x � 3
�0, 0�
y
x−2−4−6 2 4 6 8 10
−2
−4
−6
−8
4
6
8
2
f �x� �2x
x � 3
x 0 1 2 4 5 6
y 0 8 5 4�4�112
�1
68.
Horizontal asymptote:
Vertical asymptote:
Intercept: �0, 0�
x � �3, x � 2
x � 0
y
x−4 4 6 8
−2
−4
−6
−8
2
4
6
8
2−2
f �x� �5x
x2 � x � 6�
5x�x � 3��x � 2�, x � �3, 2
x 0 1 3 5
y 02524
52
�54
52
�103
�54
�2�4�6
69.
Intercepts:
Vertical asymptote:
Horizontal asymptote:
Hole in the graph at �2, 78�
y �12
x � �2
��5, 0�, �0, 54�
y
x−1−5−6 1 2
−1
−2
−3
−4
1
2
3
4
−2
f �x� �x2 � 3x � 10
2x2 � 8�
�x � 5��x � 2�2�x � 2��x � 2� �
x � 52�x � 2�, x � 2
x 0 1 3
y 0 2 145
54
�1�14
�1�3�4�5
350 Chapter 4 Trigonometry
70.
Vertical asymptote:
Slant asymptote:
intercept:
intercept: ��5.86, 0�x-
�0, 18�y-
y �12
x �74
x �5 ± �89
4; x � �1.11, x � 3.61
y
x−4 46 82−2−6
−2
x �5 ± ���5�2 � �4��2���8�
2�2�
2x2 � 5x � 8 � 0
f �x� �x3 � 6x2 � x � 1
2x2 � 5x � 8�
12
x �74
�15�x � 4�
4�2x2 � 5x � 8�
Section 4.3 Right Triangle Trigonometry
■ 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 cofunctions of complemen-tary angles are equal.
■ You should know the trigonometric function values of 30 , 45 , and 60 , or be able to construct triangles from which youcan 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
tan � �opp
adjcos � �
adj
hypsin � �
opp
hyp
0 2 3 4 7
9 5 1�294
32
18�
15532�
175�
154y
�1�32�3�4x
Vocabulary Check
1. (i) (v) (ii) (iv) (iii) (vi)
(iv) (iii) (v) (i) (vi) (ii)
2. opposite; adjacent; hypotenuse
3. elevation; depression
oppositeadjacent
� tan �opposite
hypotenuse� sin �
adjacenthypotenuse
� cos �
hypotenuseopposite
� csc �adjacentopposite
� cot �hypotenuse
adjacent� sec �
Section 4.3 Right Triangle Trigonometry 351
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
tan � �opp
adj�
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.
1.
tan � �opp
adj�
6
8�
3
4
cos � �adj
hyp�
8
10�
4
5
sin � �opp
hyp�
6
10�
3
5
6
8
θ
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
2.
513
bθ
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
13
adj � �132 � 52 � �169 � 25 � 12
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
4.
4
4
θ
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
2
hyp � �42 � 42 � �32 � 4�2
352 Chapter 4 Trigonometry
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
θ4
5
opp � �52 � 42 � 3
cot � �adj
opp�
4
3
sec � �hyp
adj�
5
4
csc � �hyp
opp�
5
3
The function values are the same since the triangles are similar and the corresponding sides are proportional.
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
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 � �6
3� 2tan � �
3
6�
1
2
sec � �3�5
6�
�5
2cos � �
6
3�5�
2�5
�2�5
5
csc � �3�5
3� �5sin � �
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.
Section 4.3 Right Triangle Trigonometry 353
13. Given:
cot � �adj
opp�
1
3
sec � �hyp
adj� �10
csc � �hyp
opp�
�10
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. Given:
cot � �adjopp
�1
�35�
�3535
csc � �hypopp
�6
�35�
6�3535
tan � �oppadj
��35
1� �35
cos � �adjhyp
�16
sin � �opphyp
��35
6
opp � �62 � 12 � �356
1
θ
35
sec � �61
�hypadj
9. Given:
cot � �adj
opp�
�7
3
sec � �hyp
adj�
4�7
7
csc � �hyp
opp�
4
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:
cot � �adj
opp�
�3
3
csc � �hyp
opp�
2�3
3
tan � �opp
adj� �3
cos � �adj
hyp�
1
2
sin � �opp
hyp�
�3
2
opp � �3
1
2 3
θ
�opp�2 � 12 � 22
sec � � 2 �2
1�
hyp
adj
10. Given:
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
sin � �opp
hyp�
2�6
7
opp � �72 � 52 � �24 � 2�6
5
7 2 6
θ
cos � �57
�adjhyp
12. Given:
sec � �hyp
adj�
�26
5
csc � �hyp
opp�
�26
1� �26
tan � �opp
adj�
1
5
cos � �adj
hyp�
5�26
�5�26
26
sin � �opp
hyp�
1�26
��26
26
hyp � �52 � 12 � �26
261
5
θcot � �51
�adjopp
354 Chapter 4 Trigonometry
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. Given:
cot � �adjopp
��273
4
sec � �hyp
adj�
17�273
�17�273
273
tan � �opp
adj�
4�273
�4�273
273
cos � �adj
hyp�
�273
17
sin � �opp
hyp�
4
17
adj � �172 � 42 � �273
417
273
θ
csc � �174
�hypopp
17.
sin 30� �opphyp
�12
30� � 30�� �
180�� ��
6 radian
30°
60°12
3
18.
cos 45� �adjhyp
�1�2
��22
45� � 45�� �
180�� ��
4 radian
21
1
45°
19.
tan �
3�
oppadj
��31
� �3
�
3�
�
3�180�
� � � 60�
1
23
π3
π6
20.
sec �
4�
hypadj
��21
� �2
�
4�
�
4�180�
� � � 45�
21
1
π4
21.
� � 60� ��
3 radian
cot � ��33
�1�3
�adjopp
1
3
30°
60°
22.
� � 45� � 45�� �
180�� ��
4 radian
csc � � �2 �hypopp
21
1
45°
23.
cos �
6�
adjhyp
��32
�
6�
�
6�180�
� � � 30�
12
3
π3
π6
24.
sin �
4�
opphyp
�1�2
��22
� ��
4�
�
4�180�
� � � 45�
21
1
π4
25.
��
4 radian
� � 45� � 45�� �
180��
cot � � 1 �11
�adjopp
21
1
45°
45°26.
��
6 radian
� � 30� � 30�� �
180��
tan � ��33
�1�3
�oppadj
30°
60°12
3
Section 4.3 Right Triangle Trigonometry 355
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
tan ��
1
5
�1
�1 � 52�
1
�26�
�26
26
cos � �1
sec ��
1�1 � tan2 �
cot � �1
tan ��
1
5
tan � � 5
33. tan � cot � � tan �� 1
tan �� � 1 34. cos � sec � � cos �� 1
cos �� � 1
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
356 Chapter 4 Trigonometry
35. tan � cos � � � sin �
cos �� cos � � sin � 36. cot � sin � �cos �
sin � sin � � cos �
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 �
41.
� csc � sec �
�1
sin �
1
cos �
�1
sin � cos �
sin �
cos ��
cos �
sin ��
sin2 � � cos2 �
sin � cos �
40.
� 2 sin2 � � 1
� sin2 � � 1 � sin2 �
sin2 � � cos2 � � sin2 � � �1 � sin2 ��
42.
� 1 � cot2 � � csc2 �
� 1 �cot �
� 1
cot ��
tan � � cot �
tan ��
tan �
tan ��
cot �
tan �
43. (a)
(b)
Note: cos 80� � sin�90� � 80�� � sin 10�
cos 80� � 0.1736
sin 10� � 0.1736 44. (a)
(b) cot 66.5� �1
tan 66.5�� 0.4348
tan 23.5� � 0.4348
45. (a)
(b) csc 16.35� �1
sin 16.35�� 3.5523
sin 16.35� � 0.2815 46. (a)
(b) sin 73� 56 � sin�73 � 56
60��
� 0.9609
cos 16� 18 � cos�16 �18
60��
� 0.9598
47. (a)
(b) csc 48� 7 �1
sin �48 �760��
� 1.3432
sec 42� 12 � sec 42.2� �1
cos 42.2�� 1.3499
49. (a)
(b) tan 11� 15 � tan 11.25� � 0.1989
cot 11� 15 �1
tan 11.25�� 5.0273
51. (a)
(b) tan 44� 28 16� � tan 44.4711� � 0.9817
csc 32� 40 3� �1
sin 32.6675�� 1.8527
48. (a)
(b)
� 1.0036
sec 4� 50 15� �1
cos 4� 50 15�
� 0.9964
cos 4� 50 15� � cos�4 �50
60�
15
3600��
50. (a)
(b) cos 56� 8 10� � cos�56 �8
60�
103600�
�� 0.5572
sec 56� 8 10� � sec�56 �8
60�
103600�
�� 1.7946
52. (a)
(b) cot�95
30 � 32��� 0.0699
sec�95
20 � 32��� 2.6695
Section 4.3 Right Triangle Trigonometry 357
62.
r �20
sin 45��
20
�2�2� 20�2
sin 45� �20
r
63.
Height of the building:
Distance between friends:
� 323.34 meters
cos 82� �45y
⇒ y �45
cos 82�
123 � 45 tan 82� � 443.2 meters
x � 45 tan 82�
45 m
82°
xy
tan 82� �x
45
53. (a)
(b) csc � � 2 ⇒ � � 30� ��
6
sin � �1
2 ⇒ � � 30� �
�
655. (a)
(b) cot � � 1 ⇒ � � 45� ��
4
sec � � 2 ⇒ � � 60� ��
354. (a)
(b) tan � � 1 ⇒ � � 45º ��
4
cos � ��2
2 ⇒ � � 45� �
�
4
59.
x � 30�3
1�3
�30
x
tan 30� �30
x
30°
30
x
60.
y � 18 sin 60� � 18��3
2 � � 9�3
sin 60� �y
18
61.
x �32�3
�32�3
3
�3x � 32
�3 �32
x
tan 60� �32
x
60°
32
x
64. (a) (b)
(c)
h � 270 feet
2�135� � h
tan � �63
�h
135
h
3
6132
Not drawn to scale
65.
� � 0� ��
6
sin � �15003000
�12
θ
1500 ft3000 ft
66.
w � 100 tan 54� � 137.6 feet
tan 54� �w
100
tan � �opp
adj
56. (a)
(b) cos � �1
2 ⇒ � � 60� �
�
3
tan � � �3 ⇒ � � 60� ��
3
57. (a)
(b) sin � ��2
2 ⇒ � � 45� �
�
4
csc � �2�3
3 ⇒ � � 60� �
�
3
58. (a)
tan � �3�3
� �3 ⇒ � � 60� ��
3
cot � ��3
3 (b)
cos � �1�2
��2
2 ⇒ � � 45� �
�
4
sec � � �2
358 Chapter 4 Trigonometry
67.
(a)
x �145
sin 23�� 371.1 feet
sin 23� �145
x
150 ft
5 ft23°
x
y
(b)
y �145
tan 23�� 341.6 feet
tan 23� �145
y(c) Moving down the line:
feet per second
Dropping vertically:
feet per second145
6� 24.17
145�sin 23�
6� 61.85
68. 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 � 69.
�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
70.
d � 5 � 2x � 5 � 2�15 tan 3�� � 6.57 centimeters
x � 15 tan 3�
tan 3� �x
15
Section 4.3 Right Triangle Trigonometry 359
72.
tan 20� �y
x� 0.36
cos 20� �x
10� 0.94
sin 20� �y
10� 0.34
x � 9.4, y � 3.4
csc 20� �10
y� 2.92
sec 20� �10
x� 1.06
cot 20� �x
y� 2.75
73. True,
csc x �1
sin x ⇒ sin 60� csc 60� � sin 60�� 1
sin 60�� � 1
75. False,�2
2�
�2
2� �2 � 1
77. False,
sin 2� � 0.0349
sin 60�
sin 30��
cos 30�
sin 30�� cot 30� � 1.7321;
74. True, because sec�90� � �� � csc �.sec 30� � csc 60�
76. True, because
cot2 � � csc2 � � �1.
cot2 � � csc2 � � 1
1 � cot2 � � csc2 �
cot2 10� � csc2 10� � �1
78. False,
tan2�5�� � �tan 5���tan 5�� � 0.0077
tan��5��2 � tan 25� � 0.4663
tan��5�� 2 � tan2�5��. 79. This is true because the corresponding sides of similar triangles are proportional.
80. Yes. Given can be found from the identity 1 � tan2 � � sec2 �.tan �, sec �
81. (a) (b) In the interval
(c) As approaches 0, approaches �.sin ��
�0, 0.5, � > sin �.0.1 0.2 0.3 0.4 0.5
0.0998 0.1987 0.2955 0.3894 0.4794sin �
�
71. (a)
(b)
(c)
(d) The side of the triangle labeled h will become shorter.
h � 20 sin 85� � 19.9 meters
sin 85� �h
20
h20
85°
(e)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�
�
�
�
�
�
�
�
�(f) The height of the balloon
decreases as decreases.
20 h
θ
�
82. (a)
—CONTINUED—
sin 0 0.3090 0.5878 0.8090 0.9511 1
cos 1 0.9511 0.8090 0.5878 0.3090 0�
�
90�72�54�36�18�0��
360 Chapter 4 Trigonometry
Section 4.4 Trigonometric Functions of Any Angle
■ 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 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��
82. —CONTINUED—
(b) increases from 0 to 1 as increases from to
(c) decreases from 1 to 0 as increases from to
(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.
90�.0��cos �
90�.0��sin �
83.
�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�
84.
�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
85.
�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
86.�3
x�
1
4��12
x� 1�
�
12 � x
4x
12 � x
x
�12 � x
4x�
x
12 � x�
1
4, x � 0, 12
Section 4.4 Trigonometric Functions of Any Angle 361
Vocabulary Check
1. 2. 3.
4. 5. 6.
7. reference
cot �xr
� cos �rx
tan � �yx
csc �sin � �yr
1. (a)
r � �16 � 9 � 5
�x, y� � �4, 3� (b)
r � �64 � 225 � 17
�x, y� � �8, �15�
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
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
2. (a)
cot � �x
y�
�12
�5�
12
5
sec � �r
x�
13
�12� �
13
12
csc � �r
y�
13
�5�
13
�5
tan � �y
x�
�5
�12�
5
12
cos � �x
r� �
12
13
sin � �y
r� �
5
13
r � ���12�2 � ��5�2 � 13
x � �12, y � �5 (b)
cot � �x
y�
�1
1� �1
sec � �r
x�
�2
�1� ��2
csc � �r
y�
�2
1� �2
tan � �y
x�
1
�1� �1
cos � �x
r�
�1�2
� ��2
2
sin � �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
362 Chapter 4 Trigonometry
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.
cot � �x
y�
7
24
sec � �r
x�
25
7
csc � �r
y�
25
24
tan � �y
x�
24
7
cos � �x
r�
7
25
sin � �y
r�
24
25
r � �49 � 576 � 25
�x, y� � �7, 24� 6.
cot � �x
y�
8
15
sec � �r
x�
17
8
csc � �r
y�
17
15
tan � �y
x�
15
8
cos � �x
r�
8
17
sin � �y
r�
15
17
r � �82 � 152 � 17
x � 8, y � 15 7.
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
r � �16 � 100 � 2�29
�x, y� � ��4, 10�
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 9.
cot � �xy
� �3568
� �0.5
sec � �rx
� ��5849
35� �2.2
csc � �ry
��5849
68� 1.1
tan � �yx
� �6835
� �1.9
cos � �xr
� �35�5849
5849� �0.5
sin � �yr
�68�5849
5849� 0.9
r � �12.25 � 46.24 ��5849
10
�x, y� � ��3.5, 6.8�
Section 4.4 Trigonometric Functions of Any Angle 363
18.
cot � � �8
15tan � �
y
x�
�15
8� �
15
8
sec � �17
8cos � �
x
r�
8
17
csc � � �17
15sin � �
y
r�
�15
17� �
15
17
tan � < 0 ⇒ y � �15
cos � �x
r�
8
17 ⇒ y � �15�
15.
tan � �y
x� �
3
4
cos � �x
r� �
4
5
sin � �y
r�
3
5
� in Quadrant II ⇒ x � �4
sin � �yr
�35
⇒ x2 � 25 � 9 � 16
10.
cot � �xy
�7�2
�31�4� �
1431
� �0.5tan � �yx
��31�4
7�2� �
3114
� �2.2
sec � �rx
��1157�4
7�2�
�115714
� 2.4cos � �xr
�7�2
�1157�4�
14�11571157
� 0.4
csc � �ry
��1157�4
�31�4� �
�115731
� �1.1sin � �yr
��31�4�1157�4
� �31�1157
1157� �0.9
r ���72�
2
� ��314 �
2
��1157
4
x � 312
�72
, y � �734
� �314
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.
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.
12. and
and
Quadrant I
x
r> 0
y
r> 0
cos � > 0sin � > 0
14. and
and
Quadrant IV
x
y< 0
r
x> 0
cot � < 0sec � > 0
cot � �x
y� �
4
3
sec � �r
x� �
5
4
csc � �r
y�
5
3
16.
in Quadrant III
tan � �y
x�
3
4
cos � �x
r� �
4
5
sin � �y
r� �
3
5
⇒ y � �3�
cos � �x
r�
�4
5 ⇒ y2 � 25 � 16 � 9
cot � �4
3
sec � � �5
4
csc � � �5
3
17.
is in Quadrant IV
tan � �y
x� �
15
8
cos � �x
r�
8
17
sin � �y
r� �
15
17
x � 8, y � �15, r � 17
y < 0 and x > 0.⇒sin � < 0 and tan � < 0 ⇒ �
tan � �yx
��15
8
cot � �x
y� �
8
15
sec � �r
x�
17
8
csc � �r
y� �
17
15
364 Chapter 4 Trigonometry
19.
cot � �x
y� �3tan � �
y
x� �
1
3
sec � �r
x�
�10
3cos � �
x
r�
3�10
10
csc � �r
y� ��10sin � �
y
r� �
�10
10
x � 3, y � �1, r � �10
cos � > 0 ⇒ � is in Quadrant IV ⇒ x is positive;
cot � �x
y� �
3
1�
3
�120.
cot � � ��15tan � �y
x� �
�15
15
sec � � �4�15
15cos � �
x
r� �
�15
4
csc � � 4sin � �y
r�
1
4
cot � < 0 ⇒ x � ��15
csc � �r
y�
4
1 ⇒ x � ��15�
21.
cot � �x
y� �
�3
3tan � �
y
x� ��3
sec � �r
x� �2cos � �
x
r� �
1
2
csc � �r
y�
2�3
3sin � �
y
r�
�3
2
sin � > 0 ⇒ � is in Quadrant II ⇒ y � �3
sec � �r
x�
2
�1 ⇒ y2 � 4 � 1 � 3 22.
is undefined
is undefinedcot � �xy
tan � �yx
� 0
sec � �rx
� �1cos � �xr
��rr
� �1
csc � �ry
sin � � 0
y � 0, x � �r
sec � � �1 ⇒ � � � � 2�n
sin � � 0 ⇒ � � 0 � �n
23.
cot � is undefinedtan � � 0
sec � � �1cos � � �1
csc � is undefinedsin � � 0
cot � is undefined, �
2≤ � ≤
3�
2 ⇒ y � 0 ⇒ � � � 24. tan is undefined
is undefined.
is undefined. cot � �x
y�
0
y� 0tan � �
y
x
sec � �r
xcos � �
x
r�
0
r� 0
csc � �r
y� �1sin � �
y
r�
�r
r� �1
� ≤ � ≤ 2� ⇒ � �3�
2, x � 0, y � �r
⇒ � � n� ��
2�
25. To find a point on the terminal side of use any point onthe line that lies in Quadrant II. is onesuch point.
cot � � �1
sec � � ��2
csc � � �2
tan � � �1
cos � � �1�2
� ��2
2
sin � �1�2
��2
2
x � �1, y � 1, r � �2
��1, 1�y � �x� 26. Let
Quadrant III
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
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.
Section 4.4 Trigonometric Functions of Any Angle 365
27. use any point onthe line Quadrant III. is onesuch point.
cot � ��1
�2�
1
2
sec � ��5
�1� ��5
csc � ��5
�2� �
�5
2
tan � ��2
�1� 2
cos � � �1�5
� ��5
5
sin � � �2�5
� �2�5
5
x � �1, y � �2, r � �5
��1, �2�y � 2x that lies inTo find a point on the terminal side of �, 28. Let
Quadrant IV
tan � � �3
4tan � �
y
x�
��4�3� x
x� �
4
3
sec � �5
3cos � �
x
r�
x
�5�3�x�
3
5
csc � � �5
4sin � �
y
r�
��4�3�x
�5�3� x� �
4
5
r ��x2 �16
9x2 �
5
3x
�x, �4
3x�,
4x � 3y � 0 ⇒ y � �4
3x
x > 0.
29.
sin � �yr
� 0
�x, y� � ��1, 0�, r � 1 30.
since corresponds to �0, �1�.3�
2
csc 3�
2�
r
y�
1
�1� �1 31.
sec undefined3�
2�
r
x�
1
0 ⇒
�x, y� � �0, �1�, r � 1
32.
since corresponds to ��1, 0�.3�
2
sec � �r
x�
1
�1� �1 33.
sin �
2�
yr
� 1
�x, y� � �0, 1�, r � 1 34. (undefined)
since corresponds to ��1, 0�.�
cot � �x
y�
�1
0
35.
undefinedcsc � �r
y�
1
0 ⇒
�x, y� � ��1, 0�, r � 1 36.
since corresponds to �0, 1�.�
2
cot �
2�
x
y�
0
1� 0
37.
′θ
203°
x
y
� � 203� � 180� � 23�
� � 203� 38.
′θ
309°
x
y
� � 360� � 309� � 51�
� � 309�
45. Quadrant III
tan 225� � tan 45� � 1
cos 225� � �cos 45� � ��2
2
sin 225� � �sin 45� � ��2
2
� � 225�, � � 360� � 225� � 45�,
47. is coterminal with
Quadrant I
tan 750� � tan 30� ��3
3
cos 750� � cos 30� ��3
2
sin 750� � sin 30� �1
2
� � 30�,
30�.� � 750�
46. Quadrant IV
sin
cos
tan 300� � �tan 60� � ��3
300� � cos 60� �1
2
300� � �sin 60� � ��3
2
� � 300�, � � 360� � 300� � 60�,
48. is coterminal with
Quadrant IV
sin
cos
tan��405�� � �tan 45� � �1
��405�� � cos 45� ��2
2
��405�� � �sin 45� � ��2
2
� � 360� � 315� � 45�,
315�.� � �405�
366 Chapter 4 Trigonometry
39.
′θ
−245°
x
y
� � 180� � 115� � 65�
360� � 245� � 115� �coterminal angle�� � �245� 40. is coterminal with
′θ−145°
x
y
� � 215� � 180� � 35�
215�.� � �145�
41.
� � � �2�
3�
�
3′θ
23π
x
y � �2�
342.
� � 2� �7�
4�
�
4
′θ
74π
x
y� �7�
4
43.
� � 3.5 � �
′θ
3.5
x
y � � 3.5 44. is coterminal
with
� � 2� �5�
3�
�
3
5�
3.
′θ
x
y
113π
� �11�
3
Section 4.4 Trigonometric Functions of Any Angle 367
49. is coterminal with
Quadrant III
tan��150�� � tan 30� ��33
cos��150�� � �cos 30� � ��32
sin��150�� � �sin 30� � �12
� � 210� � 180� � 30�,
210�.� � �150� 50. .
sin
cos
tan��840�� � tan 60� � �3
��840�� � �cos 60� � �1
2
��840�� � �sin 60� � ��3
2
� � 240� � 180� � 60�, 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
� ��
3,� �
4�
3, 52. Quadrant I
sin
cos
tan �
4� 1
�
4�
�2
2
�
4�
�2
2
� ��
4, � �
�
4, 53. 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.� � �
�
255. is coterminal with
Quadrant II
tan 11�
4� �tan
�
4� �1
cos 11�
4� �cos
�
4� �
�2
2
sin 11�
4� sin
�
4�
�2
2
� � � �34
� ��
4,
34
�.� �11�
456. is coterminal with
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. is coterminal with
tan��3�
2 � � tan �
2 which is undefined.
cos��3�
2 � � cos �
2� 0
sin��3�
2 � � sin �
2� 1
� ��
2.
�
2,� � �
3�
258. .
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
74. cot 1.35 �1
tan 1.35� 0.2245
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
368 Chapter 4 Trigonometry
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. cot
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 �
� � �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
62.
cot � � ��3
cot � < 0 in Quadrant IV.
cot2 � � 3
cot2 � � ��2�2 � 1
cot2 � � csc2 � � 1
1 � cot2 � � csc2 �
csc � � �2 63.
sec � �1
5�8�
85
cos � �1
sec � ⇒ sec � �
1cos �
cos � �58
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
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� �1
tan 178�� �28.6363
71. sec 72� �1
cos 72�� 3.2361 72. tan��188�� � �0.1405 73. tan 4.5 � 4.6373
75. tan �
9� 0.3640 76. tan��
�
9� � �0.3640
Section 4.4 Trigonometric Functions of Any Angle 369
81. (a) reference angle is 30 or is in Quadrant I or Quadrant II.
Values in degrees: 30 , 150
Values in radians:
(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 and is in Quadrant I or Quadrant II.
Values in degrees: 60 , 120
Values in radians:
(b) reference angle is 45 or and is in Quadrant II or Quadrant IV.
Values in degrees: 135 , 315
Values in radians:3�
4,
7�
4
��
��
4�cot � � �1 ⇒
�
3,
2�
3
��
��
3�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 ⇒ 85. (a) reference angle is 45 or and is in
Quadrant I or Quadrant III.
Values in degrees: 45 , 225
Values in radians:
(b) reference angle is 30 or and
is in Quadrant II or Quadrant IV.
Values in degrees: 150 , 330
Values in radians:5�
6,
11�
6
��
��
6�cot � � ��3 ⇒
�
4,
5�
4
��
��
4�tan � � 1 ⇒
370 Chapter 4 Trigonometry
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. (a) New York City:
Fairbanks:
(b)
(c) The periods are about the same for both models,approximately 12 months.
F � 36.641 sin�0.502t � 1.831� � 25.610
N � 22.099 sin�0.522t � 2.219� � 55.008
88.
(a) For February 2006,
(b) For February 2007,
(c) For June 2006,
(d) For June 2007,
S � 23.1 � 0.442�18� � 4.3 cos ��18�
6� 26,756 units
t � 18.
S � 23.1 � 0.442�6� � 4.3 cos ��6�
6� 21,452 units
t � 6.
S � 23.1 � 0.442�14� � 4.3 cos ��14�
6� 31,438 units
t � 14.
S � 23.1 � 0.442�2� � 4.3 cos ��2�
6� 26,134 units
t � 2.
S � 23.1 � 0.442t � 4.3 cos �t
689.
(a)
(b)
(c) y�1
2� � 2 cos 3 � �1.98 centimeters
y�1
4� � 2 cos�3
2� � 0.14 centimeter
y�0� � 2 cos 0 � 2 centimeters
y�t� � 2 cos 6t
Month New York City Fairbanks
February
March
May
June
August
September
November 6.5�46.8�
41.7�68.6�
55.6�75.5�
59.5�72.5�
48.6�63.4�
13.9�41.6�
�1.4�34.6�
90.
(a)
centimeters
(b)
centimeters
(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 91.
ampere I�0.7� � 5e�1.4 sin 0.7 � 0.79
I � 5e�2t sin t
Section 4.4 Trigonometric Functions of Any Angle 371
92. sin
(a)
miles
(b)
miles
(c)
milesd �6
sin 120�� 6.9
� � 120�
d �6
sin 90º�
6
1� 6
� � 90�
d �6
sin 30��
6
1�2� 12
� � 30�
� �6
d ⇒ d �
6
sin �93. False. In each of the four quadrants, the sign of the secant
function and the cosine function will be the same sincethey are reciprocals of each other.
94. False. For example, if and but is not the reference angle. The reference angle would be For in Quadrant II, For in Quadrant III,For in Quadrant IV, � � 360� � �.�
� � � � 180�.�� � 180� � �.�45�.360�n � � � 135�0 ≤ 135 ≤ 360,� � 225�,n � 1
95. As increases from to x decreases from 12 cm to 0 cm and y increases from 0 cm to 12 cm.
Therefore, increases from 0 to 1 and decreases from 1 to 0. Thus,
increases without bound, and when the tangent is undefined.� � 90�tan � �yx
cos � �x
12sin � �
y12
90�,0��
96. Determine the trigonometric function of the reference angle and prefix the appropriate sign.
97.
x-intercepts:
y-intercept:
No asymptotes
Domain: All real numbers x
�0, �4�
−2−6−8 2 4 6 8−2
−4
−8
2
4
6
8
y
x(1, 0)(−4, 0)
(0, −4)
��4, 0�, �1, 0�
y � x2 � 3x � 4 � �x � 4��x � 1� 98.
x-intercepts:
y-intercepts:
No asymptotes
Domain: All real numbers x
�0, 0�
−1−2−3 1 2 3 4 5−1
−2
−3
−4
1
2
y
x(0, 0) , 05
2( (
�0, 0�, �52, 0�
y � 2x2 � 5x � x�2x � 5�
99.
x-intercept:
y-intercept:
No asymptotes
Domain: All real numbers x
�0, 8�
��2, 0�
−6 −4−8 2 4 6 8
−4
10
12
y
x
(0, 8)
(−2, 0)
f �x� � x3 � 8 100.
x-intercepts:
y-intercepts:
No asymptotes
Domain: All real numbers x
�0, �3�
−2−3−4 2 3 4
−3
−4
1
2
3
4
y
x(−1, 0)
(0, −3)
(1, 0)
��1, 0�, �1, 0�
� �x2 � 3��x � 1��x � 1�
g�x� � x4 � 2x2 � 3��x2 � 3��x2 � 1�
372 Chapter 4 Trigonometry
101.
x-intercept:
y-intercept:
Vertical asymptote:
Horizontal asymptote:
Domain: All real numbers except x � �2
y � 0
x � �2
�0, �74�
�7, 0�−8 −2 2 4 6 8
2
4
y
x
74
0, ( (−
(7, 0)
f �x� �x � 7
x2 � 4x � 4�
x � 7�x � 2�2
102.
x-intercepts:
To find the y-intercept, let
y-intercept:
Vertical asymptote:
To find the slant asymptote, use long division:
Slant asymptote:
Domain: All real numbers except
−8−12 4
−8
−16
−24
8
y
x(−1, 0)
(1, 0)
0, 15
−( (
x � �5
y � x � 5
x2 � 1x � 5
� x � 5 �24
x � 5
x � �5
�0, �15�
x � 0: 02 � 10 � 5
� �15
��1, 0�, �1, 0�,
h�x� �x2 � 1x � 5
��x � 1��x � 1�
x � 5103.
y-intercept:
Horizontal asymptote:
Domain: All real numbers
−2 −1 1 2 3 4−1
2
3
4
5
10, 12 ))
x
y
x
y � 0
�0, 12�
y � 2x�1
x 0 1 2 3
y 1 2 412
14
�1
104.
This is an exponential function (always positive) translated two units upward. There are no x-intercepts.
To find the y-intercept, let
y-intercepts:
The horizontal asymptote is the horizontal asymptote oftranslated two units upward.
Horizontal asymptote:
Domain: All real numbers x
y � 2
y � 3x�1
�0, 5�
y � 30�1 � 2 � 3 � 2 � 5
x � 0:
x321−1−2−3−4−5
7
6
5
3
1
2
(0, 5)
yy � 3x�1 � 2
Section 4.5 Graphs of Sine and Cosine Functions 373
Section 4.5 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
105.
Domain: All real numbers except
x-intercepts:
Vertical asymptote: x � 0
�±1, 0�
x � 0
−12 −9 −6 −3 3 6 9 12
6
9
12
(−1, 0) (1, 0)x
yy � ln x4
106.
To find the x-intercept, let
x-intercepts:
To find the y-intercept, let
y-intercepts:
The vertical asymptote is the horizontal asymptote of translated two units to the left.
Vertical asymptote:
Domain: All real numbers such that x > �2x
x � �2
y � log10 x
�0, 0.301�
y � log10�x � 2� � log10 2 � 0.301
x � 0:
��1, 0�
0 � log10�x � 2� ⇒ 10� � x � 2 ⇒ x � �1
y � 0: 3
2
−1
−2
−3
321−1−3
(−1, 0)x
yy � log10�x � 2�
374 Chapter 4 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
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
4.
Period:
Amplitude: �a� � ��3� � 3
2�
b�
2�
1�3� 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
Section 4.5 Graphs of Sine and Cosine Functions 375
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
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
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.
376 Chapter 4 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
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
Section 4.5 Graphs of Sine and Cosine Functions 377
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
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
378 Chapter 4 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
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
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
Section 4.5 Graphs of Sine and Cosine Functions 379
48.
Period:
Amplitude: 4
Shift: Set
Key points:
−6
−4
−2−
2
6
x
y
πππ 2
�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�
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
3
50.
Period:
Amplitude: 5
Key points: �0, 2�, �6, �3�, �12, �8�, �18, �3�, �24, 2�
2�
��12� 24
−12 4 12
−24−20−16−12−8
48
1216
t
yy � �3 � 5 cos � t
12
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 � ��
380 Chapter 4 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
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
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
Section 4.5 Graphs of Sine and Cosine Functions 381
63.
Amplitude:
Vertical shift one unit upward of
Thus, 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 axis:
a � �1, d � �3
d � �3
�4 � �1 cos 0 � d
a � �1x-
�2 � ��4�2
� 1
f�x� � a cos x � d
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
4
y � �2 sin�4x � �� 58.
−12
−8
12
8
y � �4 sin�2
3x �
�
3� 59.
−3
−1
3
3
y � cos�2�x ��
2� � 1
60.
−6
−6
6
2
y � 3 cos��x
2�
�
2� � 2 61.
−20
−0.12
20
0.12
y � �0.1 sin��x
10� �� 62.
−0.03
−0.02
0.03
0.02
y �1
100 sin 120 � t
382 Chapter 4 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
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 x 72.
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
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�
Section 4.5 Graphs of Sine and Cosine Functions 383
78. (a) and (c)
Reasonably good fit
(d) Period is 29.6 days.
(e) March 12
The Naval observatory says that 50% ofthe moon’s face will be illuminated onMarch 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)
y �12
�12
sin�0.21x � 0.92�
C � 0.92
Horizontal shift: 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 � 85
� 7.4 (average length of interval in data)
Amplitude: 12
⇒ a �12
Vertical shift: 12
⇒ d �12
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)
(b)
The model is a good fit.
00
12
100
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 (c)
The model is a good fit.
(d) Tallahassee average maximum:
Chicago average maximum:
The constant term, gives the average maximum temperature.
(e) The period for both models is months.
This is as we expected since one full period is oneyear.
(f ) Chicago has the greater variability in temperaturethroughout the year. The amplitude, a, determines thisvariability since it is .1
2�high temp � low temp
2�
��6� 12
d,
56.55�
77.90�
00
12
100
384 Chapter 4 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
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
−
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
Section 4.5 Graphs of Sine and Cosine Functions 385
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.
89. log10 x � 2 � log10�x � 2�1�2 �12 log10�x � 2�
91. ln t3
t � 1� ln t3 � ln�t � 1� � 3 ln t � ln�t � 1�
93.
� log10 xy
12 �log10 x � log10 y� �
12 log10�xy�
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�
95.
� ln�3xy4�
ln 3x � 4 ln y � ln 3x � ln y4
386 Chapter 4 Trigonometry
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 4.6 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�
97. Answers will vary.
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
Vocabulary Check
1. vertical 2. reciprocal 3. damping 4.
5. 6. 7. 2����, �1� � �1, ��x � n�
�
Section 4.6 Graphs of Other Trigonometric Functions 387
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 consecutive asymptotes:
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
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
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
388 Chapter 4 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
1
3 0 �3y
3
2
1
2x
0
1 11
2y
�
6�
�
6x
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 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
Section 4.6 Graphs of Other Trigonometric Functions 389
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
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
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
390 Chapter 4 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 43.
x
y
π2
3π2
1
2
3
−3
x � �4�
3, �
�
3,
2�
3,
5�
3
cot x � ��3
3
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 � �) ⇒
Section 4.6 Graphs of Other Trigonometric Functions 391
44.
xπ2
π2
3π2
3π2
− −
2
3
−3
y
x � �7�
4, �
3�
4,
�
4,
5�
4
cot x � 1 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.
x � �7�
4, �
5�
4,
�
4,
3�
4
xπ 3ππ3π
2 2 2 2− −
1
2
3
−1
ycsc x � �2 48.
x � �2�
3, �
�
3,
4�
3,
5�
3
xπ 3ππ3π
2 2 2 2− −
1
2
3
y csc x � �2�3
3
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 x 50.
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
y f�x� � tan x
51.
(a)
(b) on the interval,
(c) As andsince is
the reciprocal of f �x�.g �x�g�x� �
12 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
392 Chapter 4 Trigonometry
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 x59.
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 y � 0.
f �x� � g�x�
g�x� � 0
−3 −2 −1 1 2 3
−3
−2
−1
1
2
3
x
yf �x� � sin x � cos�x ��
2� 62.
It appears that That is,
sin x � cos�x ��
2� � 2 sin x.
f�x� � g�x�.
g�x� � 2 sin x
−4
2
4
xππ−
y
f�x� � sin x � cos�x ��
2�
63.
f �x� � g�x�
g�x� �1
2�1 � cos 2x�
–1
2
3
y
xππ−
f �x� � sin2 x 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
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.
−4
2�−2�
4cot x �cos x
sin x
y1 �cos x
sin x and y2 � cot x �
1
tan x56.
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
Section 4.6 Graphs of Other Trigonometric Functions 393
65.
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
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
394 Chapter 4 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 4.6 Graphs of Other Trigonometric Functions 395
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�
396 Chapter 4 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 4.7 Inverse Trigonometric Functions 397
■ 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 4.7 Inverse Trigonometric Functions
398 Chapter 4 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�34� � 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��13� � 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 4.7 Inverse Trigonometric Functions 399
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,
400 Chapter 4 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
xy
5
12
13
y
cos y �12
13.sin y �
5
13, 0 < y <
�
2,
y � arcsin 5
13,
55. Let
and
xy
34−3
5
y
sec y ��34
5.tan y � �
3
5, �
�
2< y < 0,
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
xy
3
−2
5
y
sin y ��5
3.cos y � �
2
3,
�
2< y < �,
y � arccos��2
3�,
Section 4.7 Inverse Trigonometric Functions 401
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,
402 Chapter 4 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 4.7 Inverse Trigonometric Functions 403
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.��2y � 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,
404 Chapter 4 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 graph 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 4.7 Inverse Trigonometric Functions 405
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 x increases, 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)
h � 50 tan 26 � 24.39 feet
tan 26 �h
50
� � 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
� � 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
406 Chapter 4 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 4.7 Inverse Trigonometric Functions 407
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
408 Chapter 4 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 4.7 Inverse Trigonometric Functions 409
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:
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
�Time to go upstream� � �Time to go downstream� � 4
rate time � distance ⇒ t �d
r
18 � x
18 � x
119. (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 4.8 Applications and Models
410 Chapter 4 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 4.8 Applications and Models 411
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 �
412 Chapter 4 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 4.8 Applications and Models 413
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.
x � 4 sin 10.5� � 0.73 mile
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.
� � arctan� 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
414 Chapter 4 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 4.8 Applications and Models 415
39.
� 17,054 ft
a cot 16� � a cot 57� �55
6 ⇒ a � 3.23 miles
cot 16� �a cot 57� � �55�6�
a
tan 16� �a
a cot 57� � �55�6�
tan 16� �a
x � �55�6�
57°16°
x55060
H
P1 P2
a
tan 57� �a
x ⇒ x � a cot 57�
41.
� � arctan 5 � 78.7�
tan � � � �1 � �3�2�1 � ��1��3�2�� � ��5�2
�1�2� � 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� �1�5� � 2
1 � �1�5��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
416 Chapter 4 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 � t.t � 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 � t.t � 0
Section 4.8 Applications and Models 417
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 sin 792�t
59.
� � 2��264� � 528�
264 ��
2�
Frequency ��
2�
d � a sin �t 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,
418 Chapter 4 Trigonometry
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 4.8 Applications and Models 419
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 feet 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
�3�4
y
x−1−2−3 2 3
−1
−2
−3
1
2
3
m ��1�3� � ��2�3���1�2� � �1�4�
��12
, 13��1
4, �
23�
Review Exercises for Chapter 4
420 Chapter 4 Trigonometry
1. � � 0.5 radian 2. � � 4.5 radians
3.
(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�
44.
(a)
(b) Quadrant I
(c)
2�
9� 2� � �
16�
9
2�
9� 2� �
20�
9
x
y
29π
� �2�
95.
(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�
3
6.
(a)
(b) Quadrant I
(c)
�23�
3� 2� � �
17�
3
�23�
3� 8� �
�
3
x
−
y
233π
� � �23�
37.
(a)
(b) The angle lies in Quadrant I.
(c) Coterminal angles:
70� � 360� � �290�
70� � 360� � 430�
70°
y
x
� � 70� 8.
(a)
(b) Quadrant IV
(c)
280� � 360� � �80�
280� � 360� � 640�
x
280°
y
� � 280�
Review Exercises for Chapter 4 421
9.
(a)
(b) The angle lies in Quadrant III.
(c) Coterminal angles:
�110� � 360� � �470�
�110� � 360� � 250�
−110°
x
y
� � �110� 10.
(a)
(b) Quadrant IV
(c)
�405� � 360� � �45�
�405� � 720� � 315�
x
−405°
y
� � �405� 11.
� 8.378 radians
�8�
3 radians
480� � 480� �� rad
180�
12. �127.5� ��
180�� �2.225 13.
� �3�
16 radian � �0.589 radian
�33� 45� � �33.75� � �33.75� �� rad
180�
14. 196� 77� � �196 �7760�
��
�
180�� 3.443 15.
5� rad
7�
5� rad
7�
180�
� rad� 128.571�
16. �11�
6 �
180�
�� �330.000� 17. �3.5 rad � �3.5 rad �
180�
� rad� �200.535�
18. 5.7 �180�
�� 326.586� 19. radians
inchess � r� � 20�23�
30 � � 48.17
138� �138�
180�
23�
30 20.
meterss � 11.52
�113
� meters
s � r� � 11 � � 60180��
60� �60�
180 radians
21. (a)
(b)
� 400� inches per minute
Linear speed �6�66 23�� inches
1 minute
� 66 23� radians per minute
Angular speed ��331
3��2�� radians1 minute
22.
� 12.05 miles per hour
� 212.1 inches per second
� 67.5� inches per second
� �5� rad�s� � �13.5 inches�
�linear speed� � �angular speed� � �radius�
23. radians
square inchesA �12
r 2� �12
�18�2�2�
3 � � 339.29
120� �120�
180�
2�
324.
A � 55.31 square millimeters
A �12
�r 2 �12�
5�
6 �6.5 2
26. t �3�
4, �x, y� � ��
�22
, �22 �25. corresponds to the point .��
12
, �32 �t �
2�
3
422 Chapter 4 Trigonometry
27. corresponds to the point ���32
, 12�.t �
5�
628. t � �
4�
3, �x, y� � ��
12
, �32 �
29. corresponds to the point
cot 7�
6�
xy
� �3tan 7�
6�
y
x�
1�3
��3
3
sec 7�
6�
1x
� �2�3
3cos
7�
6� x � �
�3
2
csc 7�
6�
1y
� �2sin 7�
6� y � �
1
2
���3
2, �
1
2�.t �7�
630. corresponds to the point
cot �
4�
xy
� 1tan �
4�
yx
� 1
sec �
4�
1x
� �2cos �
4� x �
�22
csc �
4�
1y
� �2sin �
4� y �
�22
��22
, �22 �.t �
�
4
31. corresponds to the point
cot��2�
3 � �xy
��33
tan��2�
3 � �y
x� �3
sec��2�
3 � �1x
� �2cos��2�
3 � � x � �1
2
csc��2�
3 � �1y
� �2�3
3sin��2�
3 � � y � ��3
2
��1
2, �
�3
2 �.t � �2�
332. corresponds to the point
is undefined.
is undefined.cot 2� �xy
tan 2� �yx
� 0
sec 2� �1x
� 1cos 2� � x � 1
csc 2� �1y
sin 2� � y � 0
�1, 0�.t � 2�
33. sin 11�
4� sin
3�
4�
�22
34. cos 4� � cos 0 � 1 35. sin��17�
6 � � sin��5�
6 � � �12
36. cos��13�
3 � � cos�5�
3 � �12 37. tan 33 � �75.3130 38. csc 10.5 �
1sin 10.5
� �1.1368
39. sec�12�
5 � �1
cos�12�
5 �� 3.2361 40. sin��
�
9� � �0.3420
41.
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 42.
cot � �adjopp
�66
� 1tan � �oppadj
�66
� 1
sec � �hypadj
�6�2
6� �2cos � �
adjhyp
�6
6�2�
�22
csc � �hypopp
�6�2
6� �2sin � �
opphyp
�6
6�2�
�22
hyp � �62 � 62 � 6�2
adj � 6, opp � 6
43.
cot � �adjopp
�4
4�3�
�33tan � �
oppadj
�4�3
4� �3
cos � �adjhyp
�48
�12
sec � �hypadj
�84
� 2
sin � �opphyp
�4�3
8�
�32
csc � �hypopp
�8
4�3�
2�33
adj � 4, hyp � 8, opp � �82 � 42 � �48 � 4�3
Review Exercises for Chapter 4 423
44.
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 45.
(a)
(b)
(c)
(d) tan � �sin �cos �
�1�3
�2�2��3�
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
46.
(a)
(b)
(c)
(d) csc � � �1 � cot2 � ��1 �1
16�
�174
cos � �1
sec ��
1�17
��1717
sec � � �1 � tan2 � � �1 � 16 � �17
cot � �1
tan ��
14
tan � � 4 47.
(a)
(b)
(c)
(d) tan � �sin �cos �
�1�4
�15�4�
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
48.
(a)
(b) cot � � �csc2 � � 1 � �25 � 1 � 2�6
sin � �1
csc ��
15
csc � � 5
(c)
(d) sec�90� � �� � csc � � 5
tan � �1
cot ��
12�6
��612
49. tan 33� � 0.6494 51. sin 34.2� � 0.562150. csc 11� �1
sin 11�� 5.2408
424 Chapter 4 Trigonometry
55.
kilometer or 71.3 meters
1 10'°x
Not drawn to scale
3.5 km
x � 3.5 sin 1� 10� � 0.07
sin 1� 10� �x
3.556.
x �25
tan 52�� 19.5 feet
tan 52� �25x
52°
x
25
57.
cot � �xy
�34
tan � �yx
�43
sec � �rx
�53
cos � �xr
�35
csc � �ry
�54
sin � �yr
�45
x � 12, y � 16, r � �144 � 256 � �400 � 20 58.
cot � �xy
� �34
tan � �yx
� �43
sec � �rx
�53
cos � �xr
�35
csc � �ry
� �54
sin � �yr
� �45
r � �32 � ��4�2 � 5
�x, y� � �3, �4�
59.
cot � �xy
�2�35�2
�4
15tan � �
yx
�5�22�3
�154
sec � �rx
��241�6
2�3�
�2414
cos � �xr
�2�3
�241�6�
4�241
�4�241
241
csc � �ry
��241�6
5�2�
2�24130
��241
15sin � �
yr
�5�2
�241�6�
15�241
�15�241
241
r ���23�
2
� �52�
2
��241
6
x �23
, y �52
52. sec 79.3� �1
cos 79.3�� 5.3860 53.
� 3.6722
cot 15� 14� �1
tan�15 �1460� 54.
� 0.2045
cos 78� 11� 58 � cos�78 �1160
�58
3600��
60.
cot � �xy
��10�3�2�3
� 5tan � �y
x�
�2�3�10�3
�15
sec � �rx
��2�26��3
�10�3� �
�265
cos � �xr
��10�3
�2�26��3� �
5�2626
csc � �ry
��2�26��3
�2�3� ��26sin � �
y
r�
�2�3
�2�26��3� �
�2626
r ����103 �
2
� ��23�
2
�2�26
3
�x, y� � ��103
, �23�
Review Exercises for Chapter 4 425
61.
cot � �xy
��0.54.5
� �19
tan � �yx
�4.5
�0.5� �9
sec � �rx
��82�2�0.5
� ��82cos � �xr
��0.5�82�2
���82
82
csc � �ry
��82�2
4.5�
�829
sin � �yr
�4.5
�82�2�
9�8282
r � ���0.5�2 � �4.5�2 � �20.5 ��82
2
x � �0.5, y � 4.5
62.
cot � �xy
�0.30.4
�34
� 0.75tan � �y
x�
0.40.3
�43
� 1.33
sec � �rx
�0.50.3
�53
� 1.67cos � �xr
�0.30.5
�35
� 0.6
csc � �ry
�0.50.4
�54
� 1.25sin � �y
r�
0.40.5
�45
� 0.8
r � ��0.3�2 � �0.4�2 � 0.5
�x, y� � �0.3, 0.4�
64.
cot � �x�
y��
�2x�3x
�23
sec � �rx�
��13x�2x
� ��13
2
csc � �ry�
��13x�3x
� ��13
3
tan � �y�
x��
�3x�2x
�32
cos � �x�
r�
�2x�13x
� �2�13
13
sin � �y�
r�
�3x�13x
� �3�13
13
r � ���2x�2 � ��3x�2 � �13x
�x�, y� � � ��2x, �3x�, x > 0 65. is in Quadrant IV.
cot � � �5�11
11
sec � �6
5
csc � �r
y� �
6�11
11
tan � �y
x� �
�11
5
cos � �x
r�
5
6
sin � �y
r� �
�11
6
r � 6, x � 5, y � ��36 � 25 � ��11
sec � �6
5, tan � < 0 ⇒ �
63.
cot � �x�
y��
x4x
�14
tan � �y�
x��
4xx
� 4
sec � �rx�
��17x
x� �17cos � �
x�
r�
x�17x
��1717
csc � �ry�
��17x
4x�
�174
sin � �y�
r�
4x�17x
�4�17
17
r � �x2 � �4x�2 � �17x
x� � x, y� � 4x
�x, 4x�, x > 0
426 Chapter 4 Trigonometry
66.
is in Quadrant II.
cot � �1
tan �� �
�52
sec � �1
cos �� �
3�55
tan � �sin �cos �
� �2�5
5
cos � � ��1 � sin2 � � ��53
sin � �1
csc ��
23
�
csc � �32
, cos � < 0 67. is in Quadrant II.
cot � � ��55
3
sec � � �8
�55� �
8�55
55
csc � �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 ⇒ �
68.
is in Quadrant III.
cot � �1
tan ��
45
csc � �1
sin �� �
�415
sin � � ��1 � cos2 � � ��1 �1641
� �5�41
41
cos � �1
sec �� �
4�4141
sec � � ��1 � tan2 � � ��1 � �2516� � �
�414
�
tan � �54
, cos � < 0 69.
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 � �xr
��25
⇒ y2 � 21
70.
is in Quadrant IV.
cot � �1
tan �� ��3
tan � �sin �cos �
� ��33
sec � �1
cos ��
2�33
cos � � �1 � sin2 � ��1 � �14� �
�32
csc � �1
sin �� �2
�
sin � � �24
� �12
, cos � > 0 71.
′θ
264°
x
y
�� � 264� � 180� � 84�
� � 264�
Review Exercises for Chapter 4 427
72.
′θ
635°
x
y
�� � 85�
� � 635� � 720� � 85� 73.
′θx
y
65π−
�� � � �4�
5�
�
5
�6�
5� 2� �
4�
5
� � �6�
574.
′θ
x
y
173π
�� ��
3
� 6� ��
3
� �17�
3�
18�
3�
�
3
75.
tan �
3� �3
cos �
3�
12
sin �
3�
�32
76.
tan �
4�
2
�2�2� 1
cos �
4�
�2
2
sin �
4�
�2
277.
tan��7�
3 � � �tan �
3� ��3
cos��7�
3 � � cos �
3�
12
sin��7�
3 � � �sin �
3� �
�32
78.
�2
��2�2� �1
tan�� 5�
4 � � �tan �
4
cos�� 5�
4 � � �cos �
4� �
�22
sin�� 5�
4 � � sin �
4�
�22
79.
tan 495� � �tan 45� � �1
cos 495� � �cos 45� � ��2
2
sin 495� � sin 45� ��2
280.
tan��150�� ��1�2
��3�2�
�33
cos��150�� � ��32
sin��150�� � �12
81.
tan��240�� � �tan 60� � ��3
cos��240�� � �cos 60� � �1
2
sin��240�� � sin 60� ��3
282.
tan�315�� ���2�2�2�2
� �1
cos�315�� ��22
sin�315�� � ��22
83. sin 4 � �0.7568 84. tan 3 � �0.1425 85. sin��3.2� � 0.0584
86. cot��4.8� �1
tan��4.8� � 0.0878 87. sec�12�
5 � �1
cos�12�
5 �� 3.2361 88. tan��25�
7 � � 4.3813
428 Chapter 4 Trigonometry
89.
2
1
−2
x
− ππ322
y
Period: 2�
Amplitude: 1
y � sin x 91.
Amplitude: 5
Period:
−6
−2
2
4
6
xπ6
y
2�
2�5� 5�
f �x� � 5 sin 2x
590.
Amplitude: 1
Period:
x
2
−1
−2
2πππ−
y
2�
y � cos x
92.
Amplitude: 8
Period:
−8
−6
−4
8
x
y
π8π4
2�
1�4� 8�
f �x� � 8 cos��x4� 93.
Shift the graph of two units upward.
4
3
2
−1
−2
xπππ 2
y
−
y � sin x
y � 2 � sin x 94.
Amplitude: 1
Period:
321−2−3x
−1
−2
−3
−5
−6
−1
y
2�
�� 2
y � �4 � cos �x
96.
Amplitude: 3
Period:
−4
−3
1
2
3
4
tπ
y
2�
g�t� � 3 cos�t � � � 97.
(a)
(b)
� 264 cycles per second.
f �1
1�264
y � 2 sin�528�x�
2�
b�
1264
⇒ b � 528�
a � 2,
y � a sin bx
98. (a)
014
12
22
S�t� � 18.09 � 1.41 sin��t6
� 4.60�
95.
Amplitude:
Period:
−4
−3
−2
−1
1
3
4
tπ
y
2�
5
2
g�t� �5
2 sin�t � ��
(b)
so this is expected.
(c) Amplitude: 1.41
The amplitude represents the maximum change in the time of sunset from the average time .�d � 18.09�
12 months � 1 year,
Period �2�
��6� �2��6� � 12
Review Exercises for Chapter 4 429
105.
Graph first.
4
3
2
1
−3
−4
x
− ππ322
y
y � sin x
f �x� � csc x 106.
tπ
2
y
f �t� � 3 csc�2t ��
4�
107.
and first.
As x →, f �x� →.
y � �xGraph y � x−9
−6
9
6f �x� � x cos x 108.
Damping factor:
As x →, f �x� →.
x 4
−2
−300
300
� 2�
g�x� � x 4 cos x
109. arcsin��12� � �arcsin
12
� ��
6111. arcsin 0.4 � 0.41 radian
113. sin�1��0.44� � �0.46 radian
115. arccos �32
��
6 117. cos�1��1� � �
110. arcsin��1� � ��
2
112. radianarcsin�0.213� � 0.21 114. radianssin�1�0.89� � 1.10
116. arccos��22 � �
�
4
99.
4
3
2
1
πx
y
f �x� � tan x 100.
t
1
2
3
y
π2
π2
−
f �t� � tan�t ��
4� 101.
x
4
3
2
1
−3
−4
ππ−
y
f �x� � cot x
102.
t
3
2
1
y
ππ−
g�t� � 2 cot 2t 103.
Graph first.
ππ−−1
−2
−3
−4
x
y
y � cos x
f �x� � sec x 104.
tπ
1
y
h�t� � sec�t ��
4�
430 Chapter 4 Trigonometry
119. arccos 0.324 � 1.24 radians118. cos�1��32 � �
�
6120. radiansarccos��0.888� � 2.66
122. radianstan�1�8.2� � 1.45121. tan�1� � 1.5� � �0.98 radian
127.
Use a right triangle. Letthen
and cos � � 45.tan � � 34 � � arctan 34
θ
5
4
3
cos�arctan 34� �45
126.
−1.5 1.5
�2
−�2
f �x� � �arcsin 2x125.
−4 4
�2
−�2
f �x� � arctan�x2� � tan�1�x
2�
123.
−1.5 1.5
�
−�
f �x� � 2 arcsin x � 2 sin�1�x� 124.
−1.5 1.5
�3
0
y � 3 arccos x
128. Let
tan�arccos 35� � tan u �43
4
u
3
5
u � arccos 35.
129.
Use a right triangle. Let then and sec � �
135 .tan � �
125
� � arctan 125
θ
5
1213
sec�arctan 125 � �
135 130. Let
cot �arcsin��1213� � cot u � �
512
u
−1213
5u � arcsin��1213�.
131. Let Then
and
2
yx
4 − x 2
tan y � tan�arccos�x2�� �
�4 � x2
x.cos y �
x2
y � arccos�x2�. 132.
x −11
θ
12 − (x − 1)2
sec � �1
�x�2 � x�
cos � � �12 � �x � 1�2 � �x�2 � x�
sin � � x � 1
� � arcsin�x � 1� ⇒ ��
2≤ � ≤
�
2
sec�arcsin�x � 1��
Review Exercises for Chapter 4 431
133.
� � arctan�7030� � 66.8�
tan � �7030
134.
h � 25 tan 21� � 9.6 feeth
25
21°
tan 21� �h
25
135.
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
136. Amplitude:
Period: 3 seconds
d � 0.75 cos�2� t3 �
b �2�
3
a � 0.75
d � a cos bt
1.52
� 0.75 inches 137. False. The sine or cosine functionsare often useful for modeling simple harmonic motion.
138. True. The inverse sine,is defined whereand
��
2≤ y ≤
�
2.
�1 ≤ x ≤ 1y � arcsin x,
139. False. For each there corresponds exactly one value of .y
� 140. False. The range of arctan is
so arctan��1� � ��
4.��
�
2,
�
2�,
141.
Matches graph �d�
Period: 2�
Amplitude: 3
y � 3 sin x
142. matches graph (a).
Period:
Amplitude: 3
2�
y � �3 sin x 143.
Matches graph �b�
Period: 2
Amplitude: 2
y � 2 sin �x 144. matches graph (c).
Period:
Amplitude: 2
4�
y � 2 sin x2
145. is undefined at the zeros of since sec � �1
cos � .g��� � cos �f ��� � sec �
432 Chapter 4 Trigonometry
Problem Solving for Chapter 4
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)
(b)
x �3000
tan 39�� 3705 feet
tan 39� �3000
x
d �3000
sin 39�� 4767 feet
sin 39� �3000
d(c)
w � 3000 tan 63� � 3705 � 2183 feet
3000 tan 63� � w � 3705
tan 63� �w � 3705
3000
146. (a) (b) tan�� ��
2� � �cot �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��
147. The ranges for the other four trigonometric functions are not bounded. For the rangeis For the range is���, �1� � �1, ��.
y � sec x and y � csc x,���, �� .y � tan x and y � cot x,
148.
(a) A is changed from The displacement isincreased.
(b) k is changed from The friction damps theoscillations more rapidly.
(c) b is changed from 6 to 9: The frequency of oscilla-tion is increased.
110 to 13 :
15 to 13 :
y � Ae�kt cos bt �15e�t�10 cos 6t
149.
(a)
As r increases, the area function increases more rapidly.
00 6
4
A s
s � r�0.8� � 0.8r, r > 0
A �12 r 2�0.8� � 0.4r 2, r > 0
A �12 r 2�, s � r�
150. Answers will vary.
(b)
03
30
A s
0
s � 10�, � > 0
A �12 �10�2� � 50�, � > 0
Problem Solving for Chapter 4 433
5. (a)
h is even.
(b)
h is even.
−1
−2
3
2� �
h�x� � sin2 x
−1
−2
3
2� �
h�x� � cos2 x
7. If we alter the model so that we can useeither 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 sinevalue 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)
(b)
This is the time between heartbeats.
(c) Amplitude: 20
The blood pressure ranges between and
(d) Pulse rate
(e) Period
64 �60
2��b ⇒ b �
6460
� 2� �3215
�
�6064
�1516
sec
�60 sec�min34 sec�beat
� 80 beats�min
100 � 20 � 120.100 � 20 � 80
Period �2�
8��3�
68
�34
sec
070
5
130
P � 100 � 20 cos�8�
3t�
6. Given: f is an even function and g is an odd function.
(a)
since f is even
Thus, h is an even function.
(b)
since g is odd
Thus, h is an even function.
Conjecture: The square of either an even function or anodd function is an even function.
� h�x�
� �g�x��2
� ��g�x��2
h��x� � �g��x��2
h�x� � �g�x��2
� h�x�
� � f �x��2
h��x� � � f ��x��2
h�x� � � f �x��2
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
434 Chapter 4 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 � B 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 4 435
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
Chapter 4 Practice Test
436 Chapter 4 Trigonometry
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 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°
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