chapter 6 trigonometry - de la salle high school · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5...

102
CHAPTER 6 Trigonometry Section 6.1 Angles and Their Measure . . . . . . . . . . . . . . . . . 532 Section 6.2 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 541 Section 6.3 Trigonometric Functions of Any Angle . . . . . . . . . . 552 Section 6.4 Graphs of Sine and Cosine Functions . . . . . . . . . . . 567 Section 6.5 Graphs of Other Trigonometric Functions . . . . . . . . . 579 Section 6.6 Inverse Trigonometric Functions . . . . . . . . . . . . . . 591 Section 6.7 Applications and Models . . . . . . . . . . . . . . . . . . 604 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

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Page 1: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

C H A P T E R 6Trigonometry

Section 6.1 Angles and Their Measure . . . . . . . . . . . . . . . . . 532

Section 6.2 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 541

Section 6.3 Trigonometric Functions of Any Angle . . . . . . . . . . 552

Section 6.4 Graphs of Sine and Cosine Functions . . . . . . . . . . . 567

Section 6.5 Graphs of Other Trigonometric Functions . . . . . . . . . 579

Section 6.6 Inverse Trigonometric Functions . . . . . . . . . . . . . . 591

Section 6.7 Applications and Models . . . . . . . . . . . . . . . . . . 604

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

Page 2: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

C H A P T E R 6Trigonometry

Section 6.1 Angles and Their Measure

532

You should know the following basic facts about angles, their measurement, and their applications.

■ Types of Angles:

(a) Acute: Measure between 0� and 90�.

(b) Right: Measure 90�.

(c) Obtuse: Measure between 90� and 180º.

(d) Straight: Measure 180�.

■ Two positive angles, and are complementary if They are supplementary if

■ Two angles in standard position that have the same terminal side are called coterminal angles.

■ To convert degrees to radians, use radians.

■ To convert radians to degrees, use 1 radian

■ one minute of 1�.

■ one second of 1�.

■ The length of a circular arc is where is measured in radians.

■ Speed

■ Angular speed � ��t � s�rt

� distance�time

�s � r�

� 1�60 of 1� � 1�36001� �

� 1�601� �

� �180����.1� � ��180

� � � 180�.� � � 90�.�

Vocabulary Check

1. Trigonometry 2. angle

3. coterminal 4. degree

5. acute; obtuse 6. complementary; supplementary

7. radian 8. linear

9. angular 10. A �12r 2�

1. The angle shown is approximately 210�.

2. The angle shown is approximately .120�

3. The angle shown is approximately 60�.

4. The angle shown is approximately 330�.

5. (a)

(b) Since 270� < 285� < 360�; 285� lies in Quadrant IV.

Since 90� < 130� < 180�; 130� lies in Quadrant II. 6. (a) Since lies in Quadrant I.

(b) Since lies inQuadrant III.

180� < 257� 30� < 270�; 257� 30�

0� < 8.3� < 90�; 8.3�

Page 3: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.1 Angles and Their Measure 533

7. (a) lies inQuadrant III.

(b) lies inQuadrant I.Since 360� < 336� < 270�; 336�

Since 180� < 132�50� < 90�; 132�50� 8. (a) Since lies inQuadrant II.

(b) Since lies in Quadrant IV.

90� < 3.4� < 0�; 3.4�

270� < 260� < 180�; 260�

9. (a)

x30°

y (b)

150°

x

y

10. (a)

−270°

x

y270� (b)

−120°

x

y120�

11. (a)

405°

x

y (b)

480°

x

y

12. (a)

−750°x

y750� (b)

−600°

x

y600�

13. (a) Coterminal angles for 45

(b) Coterminal angles for 36

36� 360� � 396�

36� � 360� � 324�

45� 360� � 315�

45� � 360� � 405�

� 14. (a)

(b)

420� � 360� � 60�

420� � 720� � 300�

120� 360� � 240�

120� � 360� � 480� 15. (a) Coterminal angles for 300

(b) Coterminal angles for 740

20� 360� � 340�

740° 2�360°� � 20°

300� 360� � 60�

300� � 360� � 660�

Page 4: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

534 Chapter 6 Trigonometry

16. (a)

(b)

230� 360� � 130�

230� � 360� � 590�

520� � 360� � 160�

520� � 720� � 200� 17. (a)

(b) 128�30� � 128� �3060�� � 128.5�

54�45� � 54� � �4560�� � 54.75�

18. (a)

(b)

� 2� � 0.2� � 2.2�

2�12� � 2� � �1260��

� 245.167�

� 245� � 0.167�

245�10� � 245� � �1060�� 19. (a)

(b) 330�25� � �330 �25

3600�� � 330.007�

85�18� 30� � �85 �1860 �

303600�� � 85.308�

20. (a)

(b)

� 408.272�

� �408� � 0.2667� � 0.0056��

408�16�20� � �408� � �1660�� � � 20

3600�� � � 135� 0.01� � 135.01�

135�36� � 135� � 363600�� 21. (a)

(b) 145.8� � �145� � 0.8�60��� � 145�48�

240.6� � 240� � 0.6�60�� � 240�36�

22. (a)

(b)

� 0�27�

� 0� � 27�

0.45� � 0� � �0.45��60��

� 345� 7� 12�

� �345� � 7� � 0.2�60� ��

345.12� � �345� � �0.12��60� �� 23. (a)

(b)

� 3�34�48�

� [3�34� � 0.8(60)�]

� [3�34.8�]

3.58� � [3� � 0.58(60)�]

2.5� � 2� � 0.5(60)� � 2�30�

24. (a)

(b)

� 0�47�11.4�

� 0� � 47� � 11.4�

� 0� � 47� � �0.19��60��0.7865 � 0� � �0.7865��60��

� 0�21�18�

� �0� � 21� � 18��

� �0� � 21� � �0.3��60���

0.355� � �0� � �0.355��60��� 25. (a) Complement:

Supplement:

(b) Complement: Not possible. is greater than

Supplement: 180� 126� � 54�

90�.126�

180� 24� � 156�

90� 24� � 66�

26. (a) Complement:

Supplement:

(b) Complement: Not possible.

Supplement: 180� 166� � 14�

�166� > 90��

180� 87� � 93�

90� 87� � 3� 27. The angle shown is approximately 2 radians.

28. The angle shown is approximately 5.5 radians.

29. The angle shown is approximately 3 radians.

Page 5: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.1 Angles and Their Measure 535

30. The angle shown is approximately radians.4

31. (a) Since lies in Quadrant I.

(b) Since lies in Quadrant III.7�

5� <

7�

5<

3�

2;

5 0 <

5<

2;

32. (a) Since lies in Quadrant IV.

(b) Since lies in Quadrant II.11�

9

3�

2<

11�

9< �;

12

2<

12< 0; 33. (a) Since lies in Quadrant IV.

(b) Since lies in Quadrant III.� < 2 < �

2; 2

2< 1 < 0; 1

34. (a) Since lies in Quadrant IV.3�

2< 6.02 < 2� ; 6.02 (b) Since lies in Quadrant II.2.25

2< 2.25 < �,

35. (a)

54π

x

y5�

4(b)

−23π

x

y

2�

3

36. (a) (b)

x

y

52π

5�

2

− 74π

y

x

7�

4

37. (a)

116π

x

y11�

6(b)

−3

x

y3

38. (a) 4 (b)

x

y7�

4

x

y

Page 6: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

536 Chapter 6 Trigonometry

39. (a) Coterminal angles for

(b) Coterminal angles for

5�

6 2� �

7�

6

5�

6� 2� �

17�

6

5�

6

6 2� �

11�

6

6� 2� �

13�

6

640. (a)

(b)

11�

6 2� �

23�

6

11�

6� 2� �

6

7�

6 2� �

5�

6

7�

6� 2� �

19�

641. (a)

(b)

2�

15 2� �

32�

15

2�

15� 2� �

28�

15

Coterminal angles for 2�

15

9�

4� 2� �

4

9�

4� 4� �

7�

4

Coterminal angles for 9�

4

42. (a)

(b)

8�

45 2� �

82�

45

8�

45� 2� �

98�

45

8�

9 2� �

10�

9

8�

9� 2� �

26�

943. (a) Complement:

Supplement:

(b) Complement:

Supplement: � 11�

12�

12

Not possible; 11�

12 is greater than

2

� �

12�

11�

12

2

12�

5�

12

44. (a) Complement:

Supplement:

(b) Complement: none

Supplement: � 3�

4�

4

�3�

4>

2�

� �

3�

2�

3

2

3�

645. (a)

(b) 150� � 150� �

180� �5�

6

30� � 30� �

180� ��

6

46. (a)

(b) 120� � 120�� �

180�� �2�

3

315� � 315�� �

180�� �7�

447. (a)

(b) 240� � 240� �

180� � 4�

3

20� � 20� �

180� � �

9

48. (a)

(b) 144� � 144�� �

180�� �4�

5

270� � 270�� �

180�� � 3�

249. (a)

(b)7�

6�

7�

6 �180� ��

� 210�

3�

2�

3�

2 �180� ��

� 270�

50. (a)

(b)�

9�

9�180�

� � � 20�

7�

12�

7�

12�180�

� � � 105� 51. (a)

(b) 11�

30�

11�

30 �180

� ��� 66�

7�

3�

7�

3 �180

� ��� 420� 52. (a)

(b)34�

15�

34�

15 �180�

� � � 408�

11�

6�

11�

6 �180�

� � � 330�

53. 115� � 115� �

180� � 2.007 radians 54. 87.4� � 87.4�� �

180�� � 1.525 radians

Page 7: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.1 Angles and Their Measure 537

55. 216.35� � 216.35� �

180� � 3.776 radians 56. radians48.27� � 48.27�� �

180�� � 0.842

57. 0.83� � 0.83� �

180� � 0.014 radian 58. 0.54� � 0.54�� �

180�� � 0.009 radians

59.�

7�

7 �180

� ��� 25.714� 60.

5�

11�

5�

11�180�

� � � 81.818� 61.15�

8�

15�

8 �180

� ��� 337.500�

62. 4.8� � 4.8��180�

� � � 864.000� 63. 2 � 2�180

� ��� 114.592� 64. 0.57 � 0.57�180�

� � � 32.659�

65.

� �65 radians

6 � 5�

s � r� 66.

� �2910 radians

29 � 10�

s � r� 67.

� �327 radians

32 � 7�

s � r� 68.

Because the angle represented is clock-wise, this angle is radian.

45

� �6075 �

45 radian

60 � 75�

s � r�

69.

radian� �6

27 �

2

9

6 � 27�

s � r� 70. feet, feet

� �s

r�

8

14�

4

7 radian

s � 8r � 14 71.

3� �25

14.5�

50

29 radians

25 � 14.5�

3s � r� 72. kilometers,

kilometers

� �s

r�

160

80� 2 radians

s � 160

r � 80

73.

� 47.12 inches

s � 15�180�� �

180� � 15� inches

s � r�, � in radians

75.

3s � 3(1) � 3 meters

3s � r�, � in radians

74. feet,

s � r� � 9��

3� � 3� feet � 9.42 feet

� � 60� ��

3r � 9

76. centimeters,

s � r� � 20��

4� � 5� centimeters � 15.71 centimeters

� ��

4r � 20

77.

square inches

square inches � 8.38

A �12

�4�2��

3� �8�

3

A �12

r 2� 78.

� 56.5 mm2 � 18� mm2

A �12

r 2� �12

�12�2��

4�

r � 12 mm, � ��

4

79.

square feetA �12

�2.5�2�225�� �

180� � 12.27

A �12

r 2� 80.

square miles� 5.6 �21.56

12�

A �12

�1.4�2�330�

180���

r � 1.4 miles, � � 330�

Page 8: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

538 Chapter 6 Trigonometry

83. � �s

r�

450

6378� 0.071 radian � 4.04°

81. radian

miless � r� � 4000�0.14782� � 591

15�50� 32�47�39� � 8.46972� � 0.14782� � 41�

82. radian

miles

miless � r� � 4000�0.17154� � 686

r � 4000

37�18� 37�47�36� � 9�49�42� � 9.82833� � 0.17154� � 47�

84. kilometers

radian

The difference in latitude is about 3.59�.

0.062716�180� � � 3.59�

� �sr

�400

6378� 0.062716

r � 6378

85. � �s

r�

2.5

6�

25

60�

5

12 radian 86. � �

s

r�

24

5� 4.8 radians � 4.8�180�

� � � 275�

88. (a) 2-inch diameter pulley

1700 rpm

Since the belt moves 10681.4

On the 4-inch diameter pulley:

This pulley is turning at

(b)5340.7

2�� 850 rpm

5340.7 radians�minute.

� �10681.4

2� 5340.7

s � 10681.4 � 2 � �

r � 2

inches�minute.r � 1,

� 10681.4 radians�minute

� 1700 � 2� radians�minute

89. (a) Angular speed

(b)

� 164.5 feet per second

� 314123

� feet per minute

Linear speed ��7.25

2 in.�� 1 ft

12 in.��5200��2�� feet

1 minute

� 10,400� radians per minute

��5200��2�� radians

1 minute

90. (a)

� 25.13274 radians�minute

� 8�

4 rpm � 4�2�� radians�minute (b)

� 628.32 ft�minute

Linear speed � 25�25.13274� ft�minute

r�

t� 200� ft�minute

r � 25 ft

87. (a)

The circumference of the tire is feet.

The number of revolutions per minute isrev/minute.r � 5720�2.5� � 728.3

C � 2.5�

� 5720 feet per minute

65 miles per hour � 65�5280��60 (b) The angular speed is

Angular speed �4576 radians

1 minute� 4576 radians�minute

� �5720

2.5�(2�) � 4576 radians

��t.

Page 9: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.1 Angles and Their Measure 539

91. (a)

Interval:

(b)

Interval: �2400�, 6000�� centimeters per minute

�6��200��2�� ≤ Linear speed ≤ �6��500��2�� centimeters per minute

�400�, 1000�� radians per minute

�200��2�� ≤ Angular speed ≤ �500��2�� radians per minute

92.

� 1445 in.2

A �12�

125180�� � �392 142� � 460.069�

r � 14

R � 25 � 14 � 39 25125°

14r

A �12

��R2 r2� 93.

� 1496.62 square meters

� 476.39� square meters

�12

�35�2�140��� �

180��140°

35

A �12

r 2�

94. (a) Arc length of larger sprocket in feet:

Therefore, the chain moves feet as does the smaller rear sprocket.

Thus, the angle of the smaller sprocket is

and the arc length of the tire in feet is:

(b) Since the arc length of the tire is feet and the cyclist is pedaling at a rate of one revolution per second, we have:

(c) Distance

(d) The functions are both linear.

�7�

7920 t miles

� �14�

3 feet�second�� 1 mile

5280 feet��t seconds�

� Rate � Time

�7�

7920 n miles

Distance � �14�

3

feetrevolutions��

1 mile5280 feet��n revolutions�

14�

3

14� feet

3 seconds

3600 seconds

1 hour

1 mile

5280 feet� 10 miles per hour

Speed �s

t�

14�

3

1 sec �

14�

3 feet per second

s � �4���14

12� �14�

3 feet

s � �r

�r � 2 inches �2

12 feet�� �

s

r�

2�

3 ft

2

12 ft

� 4�

2�

3

s �1

3�2�� �

2�

3 feet

s � r�

Page 10: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

540 Chapter 6 Trigonometry

95. False. A measurement of radians corresponds to twocomplete revolutions from the initial to the terminal sideof an angle.

4�

97. False. The terminal side of lies on the negative x-axis.1260�

99. The speed increases, since the linear speed is proportional to the radius.

101. Since the length is given by , if the central angle is fixed while the radius increases,then increases in proportion to .rs

r�s � r�sarc

96. True. If and are coterminal angles, thenor where is an

integer. The difference between isor if expressed in

radians.� � n�2��� � n�360��,

� and n� � � n�2��,� � � n�360��

98. (a) An angle is in standard position if its vertex is at the origin and its initial side is on the positive x-axis.

(b) A negative angle is generated by a clockwise rotation.

(c) Two angles with the same initial and terminal sides are coterminal.

(d) An obtuse angle measures between 90° and 180°.

100. so one radian is much

larger than one degree.

1 radian � �180

� ��� 57.3�,

102.

Area of sector � ��r2�� �

2�� �12

r2�

Area of sector

�r2 ��

2�

Area of sectorArea of circle

�Measure of central angle of sectorMeasure of central angle of circle

θr

Area of circle � �r2

103.4

42�

4

42�22

�42

8�

2

2

105.236

� 236

� 212

�22

�22

� 2

107. 22 � 62 � 4 � 36 � 40 � 4 � 10 � 210

109.

� 144 � 2 � 122

182 62 � 324 36 � 288

104.23

�23

�33

�23

3

106.55

210�

52 5

10�

52

12

�5

22�22

�52

4

108.

� 36 � 13 � 613

182 � 122 � 324 � 144 � 468

110.

� 16 � 13 � 413

172 92 � 289 81 � 208

111. Horizontal shift two unitsto the right

432−2

3

2

1

−1

−2

−3

x

yy = x5

y = (x − 2)5

112.

Vertical shift four unitsdownward

4

2

−2

−6

32−2−3 1

y = x5

y = x5 − 4

x

yf (x) � x5 4

Page 11: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.2 Right Triangle Trigonometry 541

113. Reflection in the x-axis and a vertical shift twounits upward

321−2−3

6

5

4

3

1

−1

−2

−3

x

y

y = x5

y = 2 − x5

114.Reflection in the x-axis and a horizontal shift three units to the left

3

2

1

−1

−2

−3

21−3−4−5x

y

y = x5

y = −(x + 3)5f �x� � ��x � 3�5

■ You should know the right triangle definition of trigonometric functions.

(a) (b) (c)

(d) (e) (f)

■ You should know the following identities.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

( j) (k)

■ You should know that two acute angles are complementary if and that the cofunctions ofcomplementary angles are equal.

■ You should know the trigonometric function values of 30 , 45 , and 60 , or be able to construct triangles from which you can determine them.

���

� � � � 90�,� and �

1 � cot2 � � csc2 �1 � tan2 � � sec2 �

sin2 � � cos2 � � 1cot � �cos �

sin �tan � �

sin �

cos �

cot � �1

tan �tan � �

1

cot �sec � �

1

cos �

cos � �1

sec �csc � �

1

sin �sin � �

1

csc �

cot � �adj

oppsec � �

hyp

adjcsc � �

hyp

opp

Adjacent side

Opp

osite

sid

e

Hypote

nuse

θ

tan � �opp

adjcos � �

adj

hypsin � �

opp

hyp

Section 6.2 Right Triangle Trigonometry

Vocabulary Check

1. (i) (e) 2. opposite; adjacent; hypotenuse

(ii) (f) 3. elevation; depression

(iii) (d)

(iv) (b)

(v) (a)

(vi) (c)oppositeadjacent

� tan �

oppositehypotenuse

� sin �

adjacenthypotenuse

� cos �

hypotenuseopposite

� csc �

adjacentopposite

� cot �

hypotenuseadjacent

� sec �

Page 12: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

542 Chapter 6 Trigonometry

1.

tan � �opp

adj�

6

8�

3

4

cos � �adj

hyp�

8

10�

4

5

sin � �opp

hyp�

6

10�

3

5

θ

6

8

hyp � �62 � 82 � �36 � 64 � �100 � 10

cot � �adj

opp�

8

6�

4

3

sec � �hyp

adj�

10

8�

5

4

csc � �hyp

opp�

10

6�

5

3

3.

tan � �opp

adj�

9

40

cos � �adj

hyp�

40

41

sin � �opp

hyp�

9

41θ9

41

adj � �412 � 92 � �1681 � 81 � �1600 � 40

cot � �adj

opp�

40

9

sec � �hyp

adj�

41

40

csc � �hyp

opp�

41

9

5.

tan � �opp

adj�

1

2�2�

�2

4

cos � �adj

hyp�

2�2

3

sin � �opp

hyp�

1

3θ 3

1

adj � �32 � 12 � �8 � 2�2

cot � �adj

opp� 2�2

sec � �hyp

adj�

3

2�2�

3�2

4

csc � �hyp

opp� 3

2.

sin csc

cos sec

tan cot � �adj

opp�

12

5� �

opp

adj�

5

12

� �hyp

adj�

13

12� �

adj

hyp�

12

13

� �hyp

opp�

13

5� �

opp

hyp�

5

135

13

adj � �132 � 52 � �169 � 25 � 12

4.

sin csc

cos sec

tan cot � �adj

opp�

4

4� 1� �

opp

adj�

4

4� 1

� �hyp

adj�

4�2

4� �2� �

adj

hyp�

4

4�2�

1�2

��2

2

� �hyp

opp�

4�2

4� �2� �

opp

hyp�

4

4�2�

1�2

��2

24

4

θ

hyp � �42 � 42 � �32 � 4�2

tan � �opp

adj5

2

4�2�

1

2�2�

�2

4

cos � �adj

hyp�

4�2

6�

2�2

3

sin � �opp

hyp�

2

6�

1

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.

Page 13: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.2 Right Triangle Trigonometry 543

6.

cot � �adj

opp�

15

8tan � �

opp

adj�

8

15

sec � �hyp

adj�

17

15cos � �

adj

hyp�

15

17

csc � �hyp

opp�

17

8sin � �

opp

hyp�

8

17

hyp � �152 � 82 � �289 � 17

15

8

θ

cot � �adj

opp�

7.5

4�

15

8tan � �

opp

adj�

4

7.5�

8

15

sec � �hyp

adj�

�17�2�7.5

�17

15cos � �

adj

hyp�

7.5

�17�2��

15

17

csc � �hyp

opp�

�17�2�4

�17

8sin � �

opp

hyp�

4

�17�2��

8

17

hyp � �7.52 � 42 �17

2

7.5

The function values are the same because the triangles are similar, and corresponding sides are proportional.

7.

tan � �opp

adj�

3

4

cos � �adj

hyp�

4

5

sin � �opp

hyp�

3

5

θ

4

5

opp � �52 � 42 � 3

cot � �adj

opp�

4

3

sec � �hyp

adj�

5

4

csc � �hyp

opp�

5

3

tan � �opp

adj�

0.75

1�

3

4

cos � �adj

hyp�

1

1.25�

4

5

sin � �opp

hyp�

0.75

1.25�

3

5θ1

1.25

opp � �1.252 � 12 � 0.75

cot � �adj

opp�

1

0.75�

4

3

sec � �hyp

adj�

1.25

1�

5

4

csc � �hyp

opp�

1.25

0.75�

5

3

The function values are the same since the triangles are similar and the corresponding sides are proportional.

8.

cot � �adj

opp�

2

1� 2tan � �

opp

adj�

1

2

sec � �hyp

adj�

�5

2cos � �

adj

hyp�

2�5

�2�5

5

csc � �hyp

opp�

�5

1� �5sin � �

opp

hyp�

1�5

��5

5

hyp � �12 � 22 � �5

1

2

θ

cot � �adj

opp�

6

3� 2tan � �

opp

adj�

3

6�

1

2

sec � �hyp

adj�

3�5

6�

�5

2cos � �

adj

hyp�

6

3�5�

2�5

�2�5

5

csc � �hyp

opp�

3�5

3� �5sin � �

opp

hyp�

3

3�5�

1�5

��5

5

hyp � �32 � 62 � 3�5

3

6

θ

The function values are the same because the triangles are similar, and corresponding sides are proportional.

Page 14: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

544 Chapter 6 Trigonometry

9. Given:

csc � �hyp

opp�

4

3

sec � �hyp

adj�

4�7

7

cot � �adj

opp�

�7

3

tan � �opp

adj�

3�7

7

cos � �adj

hyp�

�7

4

adj � �7

32 � �adj�2 � 42

θ

7

4 3

sin � �3

4�

opp

hyp

11. Given:

csc � �hyp

opp�

2�3

3

cot � �adj

opp�

�3

3

tan � �opp

adj� �3

cos � �adj

hyp�

1

2

sin � �opp

hyp�

�3

2

opp � �3

�opp�2 � 12 � 22

1

2 3

θ

sec � � 2 �2

1�

hyp

adj12.

sec � �hyp

adj�

�26

5

csc � �hyp

opp�

�26

1� �26

tan � �opp

adj�

1

5

cos � �adj

hyp�

5�26

�5�26

26

261

5

θsin � �opp

hyp�

1�26

��26

26

hyp � �52 � 12 � �26

13. Given:

csc � �hyp

opp�

�10

3

sec � �hyp

adj� �10

cot � �adj

opp�

1

3

cos � �adj

hyp�

�10

10

sin � �opp

hyp�

3�10

10

hyp � �10

32 � 12 � �hyp�2

3

1

10

θ

tan � � 3 �3

1�

opp

adj14.

cot � �adjopp

�1

�35�

�3535

csc � �hypopp

�6

�35�

6�3535

tan � �oppadj

��35

1� �35

cos � �adjhyp

�16

sin � �opphyp

��35

66

1

θ

35

opp � �62 � 12 � �35

10.

cot � �adj

opp�

5

2�6�

5�6

12

sec � �hyp

adj�

7

5

csc � �hyp

opp�

7

2�6�

7�6

12

tan � �opp

adj�

2�6

5

5

7 2 6

θ

sin � �opp

hyp�

2�6

7

opp � �72 � 52 � �24 � 2�6

Page 15: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.2 Right Triangle Trigonometry 545

15. Given:

sec � �hyp

adj�

�13

3

csc � �hyp

opp�

�13

2

tan � �opp

adj�

2

3

cos � �adj

hyp�

3�13

�3�13

13

sin � �opp

hyp�

2�13

�2�13

13

hyp � �13

22 � 32 � �hyp�22

3

13

θ

cot � �3

2�

adj

opp16.

cot � �adj

opp�

�273

4

sec � �hyp

adj�

17�273

�17�273

273

tan � �opp

adj�

4�273

�4�273

273

cos � �adj

hyp�

�273

17

417

273

θ

sin � �opp

hyp�

4

17

adj � �172 � 42 � �273

17.

sin 30� �opphyp

�12

30� � 30��

180�� �

6 radian

30°

60°12

3

18. degree radian value

cos

cos 45� �1�2

��22

�22

445�

21

1

45°

19.

tan

3�

oppadj

��31

� �3

3�

3�180�

� � 60�

1

23

π3

π6

20. degree radian value

sec

sec

4�

�21

� �2

�2

445�

21

1

π4

21.

� � 60� �

3 radian

cot � ��33

�1�3

�adjopp

1

3

30°

60°

22. degree radian value

csc

csc 45� ��21

�2

445�

21

1

45°

23.

cos

6�

adjhyp

��32

6�

6�180�

� � 30�

12

3

π3

π6

24. degree radian value

sin

sin

4�

1�2

��22

�22

445�

21

1

π4

25.

� � 45� � 45��

180�� �

4

cot � � 1 �11

�adjopp

21

1

45°

45°26. degree radian value

tan

tan 30� �1�3

1�3

630�

30°

60°12

3

Page 16: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

546 Chapter 6 Trigonometry

27.

(a)

(b)

(c)

(d) cot 60� �cos 60�

sin 60��

1�3

��3

3

cos 30� � sin 60� ��3

2

sin 30� � cos 60� �1

2

tan 60� �sin 60�

cos 60�� �3

sin 60� ��3

2, cos 60� �

1

228.

(a)

(b)

(c)

(d) cot 30� �1

tan 30��

3�3

�3�3

3� �3

cos 30� �sin 30�

tan 30��

1

2

�3

3

�3

2�3�

�3

2

cot 60� � tan�90� � 60�� � tan 30� ��3

3

csc 30� �1

sin 30�� 2

sin 30� �1

2, tan 30� �

�3

3

29.

(a)

(b)

(c)

(d) sec�90� � �� � csc � ��13

2

tan � �sin �

cos ��

2�13�13

3�13�13�

2

3

cos � �1

sec ��

3�13

�3�13

13

sin � �1

csc ��

2�13

�2�13

13

csc � ��13

2, sec � �

�13

330.

(a)

(b)

(c)

(d) sin � � tan � cos � � �2�6 ��1

5� �2�6

5

cot�90º � �� � tan � � 2�6

cot � �1

tan ��

1

2�6�

�6

12

cos � �1

sec ��

1

5

sec � � 5, tan � � 2�6

31.

(a)

(b)

(c)

(d) sin�90� � �� � cos � �1

3

cot � �cos �

sin ��

1

3

2�2

3

�1

2�2�

�2

4

sin � �2�2

3

sin2 � �8

9

sin2 � � �1

3�2

� 1

sin2 � � cos2 � � 1

sec � �1

cos �� 3

cos � �1

332.

(a)

(b)

(c)

(d)

��1 �1

25��26

25�

�26

5

��1 � �1

5�2

csc � � �1 � cot 2 �

tan�90º � �� � cot � �1

5

��2626

�1

�26

cos � �1

�1 � tan2 ��

1�1 � 52

cot � �1

tan ��

1

5

tan � � 5

33. tan � cot � � tan �� 1

tan �� � 1 34. cos � sec � � cos � 1

cos �� 1

35. tan � cos � � � sin �

cos �� cos � � sin � 36. cot � sin � �cos �

sin � sin � � cos �

Page 17: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.2 Right Triangle Trigonometry 547

37.

� sin2 �

� �sin2 � � cos2 �� � cos2 �

�1 � cos ���1 � cos �� � 1 � cos2 � 38. �1 � sin ���1 � sin �� � 1 � sin2 � � cos2 �

39.

� 1

� �1 � tan2 �� � tan2 �

�sec � � tan ���sec � � tan �� � sec2 � � tan2 � 40.

� 2 sin2 � � 1

� sin2 � � 1 � sin2 �

sin2 � � cos2 � � sin2 � � �1 � sin2 ��

41.

� csc � sec �

�1

sin �

1

cos �

�1

sin � cos �

sin �

cos ��

cos �

sin ��

sin2 � � cos2 �

sin � cos �42.

� 1 � cot2 � � csc2 �

� 1 � cot �

1

cot �

tan � � cot �

tan ��

tan �

tan ��

cot �

tan �

43. (a)

(b) cot 66.5� �1

tan 66.5�� 0.4348

tan 23.5� � 0.4348 44. (a)

(b) csc 16.35� �1

sin 16.35�� 3.5523

sin 16.35� � 0.2815

45. (a)

(b) sin 73�56� � sin�73 � 56

60��

� 0.9609

cos 16�18� � cos�16 �18

60��

� 0.9598 46. (a)

(b) csc 48�7� �1

sin �48 �7

60��� 1.3432

sec 42�12� � sec 42.2� �1

cos 42.2�� 1.3499

47. Make sure that your calculator is in radian mode.

(a)

(b) tan

16� 0.1989

cot

16�

1

tan

16

� 5.0273

48. (a)

(Note: 0.75 is in radians)

(b) cos 0.75 � 0.7317

sec 0.75 �1

cos 0.75� 1.3667

49. Make sure that your calculator is in radian mode.

(a)

(b) tan 1

2� 0.5463

csc 1 �1

sin 1� 1.1884

50. (a)

(b) cot�

2�

1

2� �1

tan�

2�

1

2�� 0.5463

sec�

2� 1� �

1

cos�

2� 1�

� 1.1884

51. (a)

(b) csc � � 2 ⇒ � � 30� �

6

sin � �1

2 ⇒ � � 30� �

652. (a)

(b) tan � � 1 ⇒ � � 45º �

4

cos � ��2

2 ⇒ � � 45� �

4

Page 18: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

548 Chapter 6 Trigonometry

53. (a)

(b) cot � � 1 ⇒ � � 45� �

4

sec � � 2 ⇒ � � 60� �

3

55. (a)

(b) sin � ��2

2 ⇒ � � 45� �

4

csc � �2�3

3 ⇒ � � 60� �

3

54. (a)

(b) cos � �1

2 ⇒ � � 60� �

3

tan � � �3 ⇒ � � 60� �

3

56. (a)

(b)

cos � �1�2

��2

2 ⇒ � � 45� �

4

sec � � �2

tan � �3�3

� �3 ⇒ � � 60� �

3

cot � ��3

3

57.

x � 30�3

1�3

�30

x

tan 30� �30

x

30

30

x

°

58.

� 9�3

y � 18 sin 60� � 18�3

2

sin 60� �y

18

59.

x �32�3

�32�3

3

�3x � 32

�3 �32

x

tan 60� �32

x

60

32

x

°

60.

r �20

sin 45��

20�2

2

� 20�2

sin 45� �20

r

61.

Height of the building:meters

Distance between friends:

meters � 323.34

cos 82� �45y

⇒ y �45

cos 82�

123 � 45 tan 82� � 443.2

x � 45 tan 82�

tan 82� �x

45

45 m

82°

xy

62. (a) (b)

(c)

h � 270 feet

2�135� � h

tan � �63

�h

135

h

3

6132

Not drawn to scale

63.

� � 30� �

6

sin � �15003000

�12

θ

1500 ft3000 ft

64.

w � 100 tan 54� � 137.6 feet

tan 54� �w

100

tan � �opp

adj

Page 19: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.2 Right Triangle Trigonometry 549

65. (a)

(b)

(c) Moving down the line: feet per second

Dropping vertically: feet per second1506

� 25

150�sin 23�

6� 63.98

y �150

tan 23�� 353.4 feet

tan 23� �150

y

x �150

sin 23�� 383.9 feet

1

5 ft23°

x

y

sin 23� �150

x

66. Let the height of the mountain.

Let the horizontal distance from where the angle of elevation is sighted to the point at that level directly below the mountain peak.

Then tan

Substitute into the expression for tan

The mountain is about 1.3 miles high.

1.2953 � h

13 tan 9� tan 3.5�

tan 9� � tan 3.5�� h

13 tan 9� tan 3.5� � h�tan 9� � tan 3.5�� h tan 3.5� � 13 tan 9� tan 3.5� � h tan 9�

tan 3.5� �h tan 9�

h � 13 tan 9�

tan 3.5� �h

htan 9�

� 13

3.5�.x �h

tan 9�

tan 9� �h

x ⇒ x �

h

tan 9�

3.5� �h

x � 13 and tan 9� �

hx.

9�x �

h � 67.

�x2, y2� � �28, 28�3�

x2 � �cos 60°��56� � �12��56� � 28

cos 60° �x2

56

y2 � sin 60°�56� � ��32 ��56� � 28�3

sin 60� �y2

56

60°

56

( , )x y2 2

�x1,y1� � �28�3, 28�

x1 � cos 30��56� ��32

�56� � 28�3

cos 30� �x1

56

y1 � �sin 30���56� � �12��56� � 28

sin 30� �y1

56

30°

56

( , )x y1 1

Page 20: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

(e)

550 Chapter 6 Trigonometry

69. (a)

(b)

(c)

(d) The side of the triangle labeled hwill become shorter.

h � 20 sin 85� � 19.9 meters

sin 85� �h

20

h20

85°

Angle, Height (in meters)

80 19.7

70 18.8

60 17.3

50 15.3

40 12.9

30 10.0

20 6.8

10 3.5�

70.

csc 20� �10

y� 2.92

sec 20� �10

x� 1.06

cot 20� �x

y� 2.75

tan 20� �y

x� 0.36

cos 20� �x

10� 0.94

sin 20� �y

10� 0.34

x � 9.397, y � 3.420

71.

True, csc x �1

sin x ⇒ sin 60� csc 60� � sin 60�� 1

sin 60�� � 1

sin 60� csc 60� � 1

72. True, because sec�90� � �� � csc �. 73.

False,�2

2�

�2

2� �2 � 1

sin 45� � sin 45� � 1

(f) The height of the balloondecreases.

20 h

θ

75.

False,

sin 2� � 0.0349

sin 60�

sin 30��

cos 30�

sin 30�� cot 30� � 1.7321

sin 60�

sin 30�� sin 2�74. True, because

cot 2 � � csc 2 � � �1.

cot 2� � csc 2 � � 1

1 � cot 2 � � csc 2 �

68.

d � 5 � 2x � 5 � 2�15 tan 3�� � 6.57 centimeters

x � 15 tan 3�

tan 3� �x

15

Page 21: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.2 Right Triangle Trigonometry 551

77. This is true because the corresponding sides of similar triangles are proportional.

76.

False.

tan 2�0.8� � �tan 0.8� 2 � 1.060

tan��0.8�2 � tan 0.64 � 0.745

tan��0.8� 2 � tan2�0.8�

78. Yes. Given can be found from the identity 1 � tan2 � � sec2 �.tan �, sec �

79. (a) (b) In the interval

(c) As and �

sin � → 1.� → 0, sin � → 0,

�0, 0.5, � > sin �.0.1 0.2 0.3 0.4 0.5

sin 0.0998 0.1987 0.2955 0.3894 0.4794�

80. (a)

(b) On sin is an increasing function.

(c) On cos is a decreasing function.

(d) As the angle increases the length of the side opposite the angle increases relative to the length of the hypotenuse and the length of the side adjacent to the angle decreases relative to the length of the hypotenuse. Thus the sine increases and the cosine decreases.

��0, 1.5,

��0, 1.5,

0 0.3 0.6 0.9 1.2 1.5

sin 0 0.2955 0.5646 0.7833 0.9320 0.9975

cos 1 0.9553 0.8253 0.6216 0.3624 0.0707�

81.

�x

x � 2, x � ±6

x2 � 6x

x2 � 4x � 12

x2 � 12x � 36

x2 � 36�

x�x � 6��x � 6��x � 2�

�x � 6��x � 6��x � 6��x � 6�

82.

�2t � 3

4 � t, t � ±

3

2, �4

��2t � 3��t � 4��3 � 2t��3 � 2t�

�2t � 3��2t � 3��t � 4��t � 4�

� ��2t � 3��t � 4�

�2t 2 � 5t � 12

9 � 4t 2 4t 2 � 12t � 9

t 2 � 16

2t 2 � 5t � 12

9 � 4t 2

t 2 � 16

4t 2 � 12t � 9

83.

�2x2 � 10x � 20

�x � 2��x � 2�2�

2�x2 � 5x � 10��x � 2��x � 2�2

�3�x2 � 4� � 2�x2 � 4x � 4� � x2 � 2x

�x � 2��x � 2�2

3

x � 2�

2

x � 2�

x

x2 � 4x � 4�

3�x � 2��x � 2� � 2�x � 2�2 � x�x � 2��x � 2��x � 2�2

84.�3

x�

1

4��12

x� 1�

12 � x

4x

12 � x

x

�12 � x

4x

x

12 � x�

1

4, x � 0, 12

Page 22: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

552 Chapter 6 Trigonometry

■ Know the Definitions of Trigonometric Functions of Any Angle.

If is in standard position, a point on the terminal side and then

■ You should know the signs of the trigonometric functions in each quadrant.

■ You should know the trigonometric function values of the quadrant angles

■ You should be able to find reference angles.

■ You should be able to evaluate trigonometric functions of any angle. (Use reference angles.)

■ You should know that the period of sine and cosine is

■ You should know which trigonometric functions are odd and even.

Even: cos x and sec x

Odd: sin x, tan x, cot x, csc x

2�.

0, �

2, �, and

3�

2.

cot � �x

y, y � 0tan � �

y

x, x � 0

sec � �r

x, x � 0cos � �

x

r

csc � �r

y, y � 0sin � �

y

r

r � �x2 � y2 � 0,�x, y��

Section 6.3 Trigonometric Functions of Any Angle

Vocabulary Check

1. reference 2. periodic

3. period 4. even; odd

1. (a)

r � �16 � 9 � 5

�x, y� � �4, 3�

tan � �y

x�

3

4

cos � �x

r�

4

5

sin � �y

r�

3

5

cot � �x

y�

4

3

sec � �r

x�

5

4

csc � �r

y�

5

3

(b)

r � �64 � 225 � 17

�x, y� � �8, �15�

tan � �y

x� �

15

8

cos � �x

r�

8

17

sin � �y

r� �

15

17

cot � �x

y� �

8

15

sec � �r

x�

17

8

csc � �r

y� �

17

15

Page 23: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.3 Trigonometric Functions of Any Angle 553

2. (a)

cot � �x

y�

�12

�5�

12

5tan � �

y

x�

�5

�12�

5

12

sec � �r

x� �

13

12cos � �

x

r� �

12

13

csc � �r

y� �

13

5sin � �

y

r� �

5

13

r � ���12�2 � ��5�2 � 13

x � �12, y � �5 (b)

cot � �x

y�

�1

1� �1tan � �

y

x�

1

�1� �1

sec � �r

x�

�2

�1� ��2cos � �

x

r�

�1�2

� ��2

2

csc � �r

y�

�2

1� �2sin � �

y

r�

1�2

��2

2

r � ���1�2 � 12 � �2

x � �1, y � 1

3. (a)

r � �3 � 1 � 2

�x, y� � ���3, �1� (b)

r � �16 � 1 � �17

�x, y� � ��4, 1�

tan � �y

x�

�3

3

cos � �x

r� �

�3

2

sin � �y

r� �

1

2

cot � �x

y� �3

sec � �r

x� �

2�3

3

csc � �r

y� �2

tan � �y

x� �

1

4

cos � �x

r� �

4�17

17

sin � �y

r�

�17

17

cot � �x

y� �4

sec � �r

x� �

�17

4

csc � �r

y� �17

4. (a)

cot � �x

y�

3

1� 3

sec � �r

x�

�10

3

csc � �r

y�

�10

1� �10

tan � �y

x�

1

3

cos � �x

r�

3�10

�3�10

10

sin � �y

r�

1�10

��10

10

r � �32 � 12 � �10

x � 3, y � 1 (b)

cot � �x

y�

4

�4� �1

sec � �r

x�

4�2

4� �2

csc � �r

y�

4�2

�4� ��2

tan � �y

x�

�4

4� �1

cos � �x

r�

4

4�2�

�2

2

sin � �y

r�

�4

4�2� �

�2

2

r � �42 � ��4�2 � 4�2

x � 4, y � �4

5.

r � �49 � 576 � 25

�x, y� � �7, 24�

tan � �y

x�

24

7

cos � �x

r�

7

25

sin � �y

r�

24

25

cot � �x

y�

7

24

sec � �r

x�

25

7

csc � �r

y�

25

24

6.

tan � �y

x�

15

8

cos � �x

r�

8

17

sin � �y

r�

15

17

r � �82 � 152 � 17

x � 8, y � 15

cot � �x

y�

8

15

sec � �r

x�

17

8

csc � �r

y�

17

15

Page 24: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

554 Chapter 6 Trigonometry

7.

r � �16 � 100 � 2�29

�x, y� � ��4, 10�

cot � �x

y� �

2

5

sec � �r

x� �

�29

2

csc � �r

y�

�29

5

tan � �y

x� �

5

2

cos � �x

r� �

2�29

29

sin � �y

r�

5�29

29

8.

cot � �x

y�

�5

�2�

5

2

sec � �r

x�

�29

�5� �

�29

5

csc � �r

y�

�29

�2� �

�29

2

tan � �y

x�

�2

�5�

2

5

cos � �x

r�

�5�29

� �5�29

29

sin � �y

r�

�2�29

� �2�29

29

r � ���5�2 � ��2�2 � �29

x � �5, y � �2

tan � �yx

� �6.83.5

cos � �xr

� �3.5

�58.49� �

3.5�58.4958.49

sin � �yr

�6.8

�58.49�

6.8�58.4958.49

9.

r � ���3.5�2 � �6.8�2 � �58.49

�x, y� � ��3.5, 6.8�

11.

sin � < 0 and cos � < 0 ⇒ � lies in Quadrant III.

cos � < 0 ⇒ � lies in Quadrant II or in Quadrant III.

sin � < 0 ⇒ � lies in Quadrant III or in Quadrant IV.

cot � �xy

� �3.56.8

sec � �rx

� ��58.49

3.5

csc � �ry

��58.49

6.8

10.

cot � �xy

72

�314

� �1431

tan � �yx

�314

72

� �3114

� �23

14

sec � �rx

�1157472

��1157

14cos � �

xr

72

�11574

�14�1157

1157

csc � �ry

�11574

�314

� ��1157

31sin � �

yr

�314

�11574

� �31�1157

1157

r ���72�

2

� ��314 �

2

��1157

4

x � 312

�72

, y � �734

� �314

12. lies in Quadrant I or in Quadrant II.

lies in Quadrant I or in Quadrant IV.

and lies in Quadrant I.cos � > 0 ⇒ �sin � > 0

cos � > 0 ⇒ �

sin � > 0 ⇒ �

13.

sin � > 0 and tan � < 0 ⇒ � lies in Quadrant II.

tan � < 0 ⇒ � lies in Quadrant II or in Quadrant IV.

sin � > 0 ⇒ � lies in Quadrant I or in Quadrant II. 14. lies in Quadrant I or in Quadrant IV.

lies in Quadrant II or in Quadrant IV.

and lies in Quadrant IV.cot � < 0 ⇒ �sec � > 0

cot � < 0 ⇒ �

sec � > 0 ⇒ �

Page 25: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.3 Trigonometric Functions of Any Angle 555

tan � �y

x� �

3

4

cos � �x

r� �

4

5

sin � �y

r�

3

5

cot � �x

y� �

4

3

sec � �r

x� �

5

4

csc � �r

y�

5

3

15.

� in Quadrant II ⇒ x � �4

sin � �yr

�35

⇒ x2 � 25 � 9 � 16 16.

in Quadrant III ⇒ y � �3�

cos � �x

r�

�4

5 ⇒ y � �3�

tan � �y

x�

3

4

cos � �x

r� �

4

5

sin � �y

r� �

3

5

cot � �4

3

sec � � �5

4

csc � � �5

3

17. is in Quadrant IV

tan � �yx

�� 15

8 ⇒ r � 17

y < 0 and x > 0.⇒sin � < 0 and tan � < 0 ⇒ �

tan � �y

x� �

15

8

cos � �x

r�

8

17

sin � �y

r� �

15

17

cot � �x

y� �

8

15

sec � �r

x�

17

8

csc � �r

y� �

17

15

tan � �y

x�

�15

8� �

15

8

cos � �x

r�

8

17

sin � �y

r�

�15

17� �

15

17

cot � � �8

15

sec � �17

8

csc � � �17

15

18.

tan � < 0 ⇒ y � �15

cos � �x

r�

8

17 ⇒ y � �15�

19.

x � 3, y � �1, r � �10

cos � > 0 ⇒ � is in Quadrant IV ⇒ x is positive;

cot � �x

y� �

3

1�

3

�1

tan � �y

x� �

1

3

cos � �x

r�

3�10

10

sin � �y

r� �

�10

10

cot � �x

y� �3

sec � �r

x�

�10

3

csc � �r

y� ��10

tan � �y

x� �

�15

15

cos � �x

r� �

�15

4

sin � �y

r�

1

4

cot � � ��15

sec � � �4�15

15

csc � � 4

20.

cot � < 0 ⇒ x � ��15

csc � �r

y�

4

1 ⇒ x � ��15�

21.

sin � > 0 ⇒ � is in Quadrant II ⇒ y � �3

sec � �r

x�

2

�1 ⇒ y2 � 4 � 1 � 3

tan � �y

x� ��3

cos � �x

r� �

1

2

sin � �y

r�

�3

2

cot � �x

y� �

�3

3

sec � �r

x� �2

csc � �r

y�

2�3

3

22.

tan � yr

��

r� 0

cos � �xr

� �rr

� �1

sin � � 0

y � 0, x � �r

sec � � �1 ⇒ � � � � 2�n

sin � � 0 ⇒ � � 0 � �n

is undefined.

is undefined.cot �

sec � rx

�r

�r� �1

csc �

23.

is undefined.

is undefined.cot �tan � � 0

sec � � �1cos � � �1

csc �sin � � 0

cot � is undefined, �

2≤ � ≤

3�

2 ⇒ y � 0 ⇒ � � �

Page 26: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

556 Chapter 6 Trigonometry

24. tan is undefined

� ≤ � ≤ 2� ⇒ � �3�

2, x � 0, y � �r

⇒ � � n� ��

2�

is undefined.tan �

cos � �x

r�

0

r� 0

sin � �y

r�

�r

r� �1

is undefined.

cot � �x

y�

0

y� 0

sec �

csc � �r

y� �1

25. use any point onQuadrant II. is one

such point.��1, 1�the line y � �x that lies in

To find a point on the terminal side of �,

tan � � �1

cos � � �1�2

� ��2

2

sin � �1�2

��2

2

x � �1, y � 1, r � �2

cot � � �1

sec � � ��2

csc � � �2

26. Let

Quadrant III

tan � �y

x�

��1

3�x

�x�

1

3

cos � �x

r�

�x

�10x

3

� �3�10

10

sin � �y

r�

��1

3�x

�10x

3

� ��10

10

r ��x2 �1

9x 2 �

�10x

3

��x, �1

3x�,

x > 0.

28. Let

Quadrant IV

cot � � �3

4tan � �

y

x�

��4

3� x

x� �

4

3

sec � �5

3cos � �

x

r�

x

5

3x

�3

5

csc � � �5

4sin � �

y

r�

��4

3�x

5

3x

� �4

5

r ��x2 �16

9x2 �

5

3x

�x, �4

3x�,

4x � 3y � 0 ⇒ y � �4

3x

x > 0.27. use any point onthe line Quadrant III. is onesuch point.

��1, �2�y � 2x that lies inTo find a point on the terminal side of �,

tan � ��2

�1� 2

cos � � �1�5

� ��5

5

sin � � �2�5

� �2�5

5

x � �1, y � �2, r � �5

cot � ��1

�2�

1

2

sec � ��5

�1� ��5

csc � ��5

�2� �

�5

2

cot � �x

y�

�x

��1

3� x

� 3

sec � �r

x�

�10x

3

�x� �

�10

3

csc � �r

y�

�10x

3

��1

3�x

� ��10

29.

sin � �y

r�

0

1� 0

�x, y� � ��1, 0�, r � 1 30.

csc 3�

2�

1

�1� �1

�x, y� � �0, �1�, r � 1 31.

sec undefined3�

2�

r

x�

1

0 ⇒

�x, y� � �0, �1�, r � 1

Page 27: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.3 Trigonometric Functions of Any Angle 557

32.

sec � �r

x�

1

�1� �1

�x, y� � ��1, 0�, r � 1 33.

sin �

2�

y

r�

1

1� 1

�x, y� � �0, 1�, r � 1 34.

undefined.cot � ��1

0

�x, y� � ��1, 0�, r � 1

35.

undefinedcsc � �r

y�

1

0 ⇒

�x, y� � ��1, 0�, r � 1 36.

cot �

2�

x

y�

0

1� 0

�x, y� � �0, 1� 37.

′θ

203°

x

y

�� � 203� � 180� � 23�

� � 203�

38.

′θ

309°

x

y

�� � 360� � 309� � 51�

� � 309� 39.

′θ

−245°

x

y

�� � 180� � 115� � 65�

360� � 245� � 115� �coterminal angle�� � �245�

40.

(coterminal angle)

�� � 215� � 180� � 35�

360� � 145� � 215�

′θ−145°

x

y � � �145� 41.

�� � � �2�

3�

3 ′θ

23π

x

y

� �2�

3

42.

�� � 2� �7�

4�

4

′θ

74π

x

y

� �7�

443.

�� � 3.5 � �

′θ

3.5

x

y

� � 3.5

Page 28: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

558 Chapter 6 Trigonometry

44. with .

�� � 2� �5�

3�

3

′θ

x

y

113π

5�

3� �

11�

3 is coterminal 45. Quadrant III

tan 225� � tan 45� � 1

cos 225� � �cos 45� � ��2

2

sin 225� � �sin 45� � ��2

2

�� � 45�,� � 225�,

46. in Quadrant IV.

sin

cos

tan 300� � �tan 60� � ��3

300� � cos 60� �1

2

300� � �sin 60� � ��3

2

� � 300�, �� � 360� � 300� � 60� 47. Quadrant I

tan 750� � tan 30� ��3

3

cos 750� � cos 30� ��3

2

sin 750� � sin 30� �1

2

� � 750�, �� � 30�,

48. in Quadrant IV.

sin

cos

tan��405�� � �tan 45� � �1

��405�� � cos 45� ��2

2

��405�� � �sin 45� � ��2

2

� � �405�, �� � 405� � 360� � 45� 49.

tan(�150�) � tan 30� ��33

cos(�150�) � �cos 30� � ��32

sin(�150�) � �sin 30� � �12

� � �150�, �� � 30�, Quadrant III

50. .

sin

cos

tan(�840�) � tan 60� � �3

(�840�) � �cos 60� � �1

2

(�840�) � �sin 60� � ��3

2

�� � 240� � 180� � 60� in Quadrant III.

� � �840� is coterminal with 240� 51. Quadrant III

tan 4�

3� tan

3� �3

cos 4�

3� �cos

3� �

1

2

sin 4�

3� �sin

3� �

�3

2

� �4�

3, �� �

3,

52. in Quadrant I.

sin

cos

tan �

4� 1

4�

�2

2

4�

�2

2

� ��

4, �� �

453. Quadrant IV

tan���

6� � �tan �

6� �

�3

3

cos���

6� � cos �

6�

�3

2

sin���

6� � �sin �

6� �

1

2

� � ��

6, �� �

6, 54. is coterminal with

sin

cos

tan is undefined.���

2� � tan 3�

2

���

2� � cos 3�

2� 0

���

2� � sin 3�

2� �1

3�

2.� � �

2

Page 29: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.3 Trigonometric Functions of Any Angle 559

55. Quadrant II

tan 11�

4� �tan

4� �1

cos 11�

4� �cos

4� �

�2

2

sin 11�

4� sin

4�

�2

2

� �11�

4, �� �

4, 56. is coterminal with

in Quadrant III.

sin

cos

tan 10�

3� tan

3� �3

10�

3� �cos

3� �

1

2

10�

3� �sin

3� �

�3

2

�� �4�

3� � �

3

4�

3.� �

10�

3

57.

tan��3�

2 � � tan �

2 which is undefined.

cos��3�

2 � � cos �

2� 0

sin��3�

2 � � sin �

2� 1

� � �3�

2, �� �

2 58. .

tan��25�

4 � � �tan��

4� � �1

cos��25�

4 � � cos��

4� ��22

sin��25�

4 � � �sin��

4� � ��22

�� � 2� �7�

4�

4 in Quadrant IV.

� � �25�

4 is coterminal with

7�

4

59.

in Quadrant IV.

cos � �4

5

cos � > 0

cos2 � �16

25

cos2 � � 1 �9

25

cos2 � � 1 � ��3

5�2

cos2 � � 1 � sin2 �

sin2 � � cos2 � � 1

sin � � �3

560.

sin � �1

csc ��

1�10

��10

10

csc � �1

sin �

�10 � csc �

csc � > 0 in Quadrant II.

10 � csc2 �

1 � ��3�2 � csc2 �

1 � cot2 � � csc2 �

cot � � �3 61.

in Quadrant III.

sec � � ��13

2

sec � < 0

sec2 � �13

4

sec2 � � 1 �9

4

sec2 � � 1 � �3

2�2

sec2 � � 1 � tan2 �

tan � �3

2

63.

sec � �158

�85

cos � �1

sec � ⇒ sec � �

1cos �

cos � �5862.

cot � � ��3

cot � < 0 in Quadrant IV.

cot2 � � 3

cot2 � � ��2�2 � 1

cot2 � � csc2 � � 1

1 � cot2 � � csc2 �

csc � � �2 64.

.

tan � ��65

4

tan � > 0 in Quadrant III

tan2 � �65

16

tan2 � � ��9

4�2

� 1

tan2 � � sec2 � � 1

1 � tan2 � � sec2 �

sec � � �9

4

Page 30: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

560 Chapter 6 Trigonometry

65. sin 10� � 0.1736 66. sec 225� �1

cos 225�� �1.4142 67. cos��110�� � �0.3420

68. csc��330�� �1

sin��330�� � 2.0000 69. tan 304� � �1.4826 70. cot 178� � �28.6363

71. sec 72� �1

cos 72�� 3.2361 72. tan��188�� � �0.1405 73. tan 4.5 � 4.6373

74. cot 1.35 �1

tan 1.35� 0.2245 75. cos��

9� � 0.9397 76. tan���

9� � �0.3640

77. sin��0.65� � �0.6052 78. sec 0.29 �1

cos 0.29� 1.0436

79. cot��11�

8 � �1

tan��11��8�� �0.4142 80. csc��

15�

14 � �1

sin��15��14� � 4.4940

81. (a) reference angle is 30 or is in

Quadrant I or Quadrant II.

Values in degrees: 30 , 150

Values in radian:

(b) reference angle is 30 or is

in Quadrant III or Quadrant IV.

Values in degrees: 210 , 330

Values in radians:7�

6,

11�

6

��

6 and ��sin � � �

1

2 ⇒

6,

5�

6

��

6 and ��sin � �

1

2 ⇒ 82. (a) cos reference angle is 45 or and is

in Quadrant I or IV.

Values in degrees: 45 , 315

Values in radians:

(b) cos reference angle is 45 or and

is in Quadrant II or III.

Values in degrees: 135 , 225

Values in radians:3�

4,

5�

4

��

��

4�� � �

�2

2 ⇒

4,

7�

4

��

��

4�� �

�2

2 ⇒

83. (a) reference angle is 60 or

is in Quadrant I or Quadrant II.

Values in degrees: 60 , 120

Values in radians:

(b) reference angle is 45 or is

in Quadrant II or Quadrant IV.

Values in degrees: 135 , 315

Values in radians:3�

4,

7�

4

��

4 and ��cot � � �1 ⇒

3,

2�

3

��

3 and ��csc � �

2�3

3 ⇒ 84. (a) sec reference angle is 60 or and is in

Quadrant I or IV.

Values in degrees: 60 , 300

Values in radians:

(b) sec reference angle is 60 or and is

in Quadrant II or III.

Values in degrees: 120 , 240

Values in radians:2�

3,

4�

3

��

��

3�� � �2 ⇒

3,

5�

3

��

��

3�� � 2 ⇒

Page 31: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.3 Trigonometric Functions of Any Angle 561

85. (a) reference angle is 45 or is in

Quadrant I or Quadrant III.

Values in degrees: 45 , 225

Values in radians:

(b) reference angle is 30 or

is in Quadrant II or Quadrant IV.

Values in degrees: 150 , 330

Values in radians:5�

6,

11�

6

��

6 and ��cot � � ��3 ⇒

4,

5�

4

��

4 and ��tan � � 1 ⇒ 86. (a) sin reference angle is 60 or and is

in Quadrant I or II.

Values in degrees: 60 , 120

Values in radians:

(b) sin reference angle is 60 or and

is in Quadrant III or IV.

Values in degrees: 240 , 300

Values in radians:4�

3,

5�

3

��

��

3�� � �

�3

2 ⇒

3,

2�

3

��

��

3�� �

�3

2 ⇒

87. corresponds to on the unit circle.

tan �

4� 1 since tan t �

y

x

cos �

4�

�2

2 since cos t � x

sin �

4�

�2

2 since sin t � y

t ��

4��2

2, �2

2 � 88.

sin since sin

cos since cos

tan since tan t �y

x.

3�

�3

2

1

2

� �3

t � x.�

3�

1

2

t � y.�

3�

�3

2

t ��

3, �x, y� � �1

2, �3

2 �

89. corresponds to on the unit circle.

tan 5�

6� �

�3

3 since tan t �

y

x

cos 5�

6� �

�3

2 since cos t � x

sin 5�

6�

1

2 since sin t � y

t �5�

6���3

2,

1

2� 90.

since sin

cos since cos

tan since tan t �y

x.

5�

4�

��2

2

��2

2

� 1

t � x.5�

4� �

�2

2

t � y.sin 5�

4� �

�2

2

t �5�

4, �x, y� � ��

�2

2, �

�2

2 �

91. corresponds to on the unit circle.

tan 4�

3� �3 since tan t �

y

x.

cos 4�

3� �

1

2 since cos t � x.

sin 4�

3� �

�3

2 since sin t � y.

t �4�

3��1

2, �

�3

2 � 92.

sin since sin

cos since cos

since tan t �y

x. � �

1

�3� �

�3

3

tan 11�

6�

�1

2

�3

2

t � x.11�

6�

�3

2

t � y.11�

6� �

1

2

t �11�

6, �x, y� � ��3

2, �

1

2�

Page 32: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

562 Chapter 6 Trigonometry

93. corresponds to on the unit circle.

tan 3�

2 is undefined since tan t �

y

x

cos 3�

2� �0 since cos t � x

sin 3�

2� �1 since sin t � y

t �3�

2�0, �1�

95. (a) (b) cos 2 � �0.4sin 5 � �1

97. (a)

(b)

t � 1.82 or 4.46

cos t � �0.25

t � 0.25 or 2.89

sin t � 0.25

94.

since sin

cos since cos

tan since tan t �y

x.� �

0

�1� 0

t � x.� � �1

t � y.sin � � 0

t � �, �x, y� � ��1, 0�

96. (a) (b) cos 2.5 � x � �0.8sin 0.75 � y � 0.7

98. (a)

(b)

t � 0.72 or t � 5.56

cos t � 0.75

t � 4.0 or t � 5.4

sin t � �0.75

99. (a) New York City:

Fairbanks:

(b)

(c) The periods are about the same for both models, approximately 12 months.

F � 36.6 sin�0.50t � 1.83� � 25.61

N � 22.1 sin�0.52t � 2.22� � 55.01

Month New York City Fairbanks

February 35

March 41 14

May 63 48

June 72 59

August 76 56

September 69 42

November 47 7��

��

��

��

��

��

�1��

100.

(a) snowboards

(b) snowboards

(c) snowboards

(d) snowboardst � 18; S � 23.1 � 0.442�18� � 4.3 cos ��18�

6� 26,756

t � 6; S � 23.1 � 0.442�6� � 4.3 cos ��6�

6� 21,452

t � 14; S � 23.1 � 0.442�14� � 4.3 cos ��14�

6� 31,438

t � 2; S � 23.1 � 0.442�2� � 4.3 cos ��2�

6� 26,134

S � 23.1 � 0.442t � 4.3 cos � t6

101.

(a)

(b)

(c) y�12� � 2 cos 3 � �1.98 centimeters

y�14� � 2 cos�3

2� � 0.14 centimeter

y�0� � 2 cos 0 � 2 centimeters

y�t� � 2 cos 6t

Page 33: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.3 Trigonometric Functions of Any Angle 563

102.

(a)

centimeters

(b)

centimeter

(c)

centimetersy�12� � 2e�1�2 cos�6 1

2� � �1.2

t �12

y�14� � 2e�1�4 cos�6 1

4� � 0.11

t �14

y�0� � 2e�0 cos 0 � 2

t � 0

y�t� � 2e�t cos 6t 103.

I�0.7� � 5e�1.4 sin 0.7 � 0.79

I � 5e�2t sin t

105. False. In each of the four quadrants, the sign of thesecant function and the cosine function will be the samesince they are reciprocals of each other.

106. False. The reference angle is always acute by definition.For in Quadrant II, For in

Quadrant III, For in Quadrant IV,

�� � 360� � �.

��� � � � 180�.

��� � 180� � �.�

107.

Therefore, is odd.h�t�

� �h�t�

� �f �t�g�t�

h��t� � f ��t�g��t�

h�t) � f �t�g�t� 108. As increases from x decreases from 12 cm to

0 cm, y increases from 0 cm to 12 cm,

increases from 0 to 1, decreases from 1 to 0,

and increases without bound (and is undefined

at � � 90�).

tan � �yx

cos � �x

12

sin � �y

12

0� to 90�,�

104. sin

(a)

�6

1�2� 12 miles

d �6

sin 30�

� � 30�

� �6

d ⇒ d �

6

sin �

(b)

milesd �6

sin 90º�

6

1� 6

� � 90� (c)

�6

�3�2� 6.9 miles

d �6

sin 120�

� � 120�

109. (a)

(b) Conjecture: sin � � sin�180� � ��

0 0.342 0.643 0.866 0.985

0 0.342 0.643 0.866 0.985sin�180� � ��

sin �

80�60�40�20�0��

110. (a)

(b) cos�3�

2� �� � �sin �

0 0.3 0.6 0.9 1.2 1.5

0

0 �0.9975�0.9320�0.7833�0.5646�0.2955�sin �

�0.9975�0.9320�0.7833�0.5646�0.2955cos�3�

2� ��

Page 34: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

564 Chapter 6 Trigonometry

111. If is obtuse, then The reference angle is and we have the following:�� � 180� � �90� < � < 180�.�

cot � � �cot ��tan � � �tan ��

sec � � �sec ��cos � � �cos ��

csc � � csc ��sin � � sin ��

112. Domain: All real numbers x

Range:

Period:

Zeros:

The function is odd.

n�

2�

�1, 1

−3

3

2��−

Domain: All real numbers x

Range:

Period:

Zeros:

The function is even.

n� ��

2

2�

�1, 1

−3

3

2��−

Domain: All real numbers x except

Range:

Period:

Zeros:

The function is odd.

n�

−3

3

2��−

��, �

x � n� ��

2Domain: All real numbers x except

Range:

Period:

Zeros: none

The function is odd.

2�

−3

3

2��−

��, �1 � 1, �

x � n�

Domain: All real numbers x except

Range:

Period:

Zeros: none

The function is even.

2�

−3

3

2��−

��, �1 � 1, �

x � n� ��

2

Domain: All real numbers x except

Range:

Period:

Zeros:

The function is odd.

The secant function is similar to the tangent function because they both have vertical asymptotes at

The cotangent function and the cosecant function both have vertical asymptotes at A maximum point on the sine curve corresponds to a relative minimum on the cosecant curve. The maximum points of sine and cosine are interchanged with the minimum points of cosecant and secant. The x-intercepts of the sine and cosine functions become vertical asymptotes of the cosecant and secant functions.

The graphs of sine and cosine may be translated left or right (respectively) to to coincide with each other.

��2

x � n�.

x � n� ��

2.

n� ��

2

−3

3

2��−

��, �

x � n�

Page 35: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.3 Trigonometric Functions of Any Angle 565

113.

Vertical asymptote:

Horizontal asymptote:

141210842−2

−8

−6

−4

2

4

6

8

x

y

y � 0

x � 4

f �x� �2

4 � x

x 0 1 2 3 5 6

y 1 2 �1�223

12

114.

Vertical asymptotes:

Horizontal asymptotes:

−4 −3 −1 31 2 4

−4

−3

−2

1

2

3

4

x

y

y � 0

x � 2, x � �2

g�x� � �1

x2 � 4

x 0

1.07 0.33 0.25�0.94�0.2g�x�

±1±1.75±2.25±3

115.

No asymptotes

The graph has a hole at �2, 12�.

y

x−3−6−9 3 6 9 12

−9

−12

3

6

9

12

h�x� �2x2 � 12x

x � 6�

2x�x � 6�x � 6

� 2x, x � 6

x 0 2 4 8

y 0 4 8 16�4

�2

117.

Domain: all real numbers x

Range:

y-intercept:

Horizontal asymptote: y � 0

�0, 12�y > 0 −2 −1 1 2 3 4

−1

2

3

4

5

10, 12 ))

x

yy � 2x�1

x 0 1 2 3

y 1 2 412

14

�1

116.

Vertical asymptote:

Horizontal asymptote:

y

x−6−8 2 4 6

−2

−4

−6

−8

2

4

6

8

y � 0

x � �3

� �undefined,

x � 5

x � 5

f �x� �x � 5

x2 � 2x � 15�

x � 5�x � 5��x � 3�

1x � 3

,

x 1 2

1 15

14

13

12�1f �x�

0�1�2�4

Page 36: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

566 Chapter 6 Trigonometry

119.

Domain: all real numbers x

Range:

y-intercept:

Horizontal asymptote: y � 0

�0, 1�

y > 0−5 −4 −3 −2 −1 1 2 3

2

3

4

5

6

7

(0, 1)

x

yy � 3�x�2

x 0 2 4

y 9 3 1 19

13

�2�4

121.

Range: all real numbers

x-intercept:

Vertical asymptote: x � 1

�2, 0�

Domain: x � 1 > 0 ⇒ x > 1 1 2 3 4 5 6

−3

−2

−1

1

2

3

(2, 0)x

yy � ln�x � 1�

x 1.1 1.5 2 3 4

y 0 0.69 1.10�0.69�2.30

122.

Domain: all real numbers

Range: all real numbers

intercepts:

Vertical asymptote: x � 0

��1, 0�, �1, 0�x-

y

x−12 −9 −6 −3 3 6 9 12

6

9

12

(−1, 0) (1, 0)x

yy � ln x4

118.

Domain: all real numbers

Range:

intercept:

Horizontal asymptote:y � 2

�0, 5�y-

y > 2

x321−1−2−3−4−5

7

6

5

3

1

2

(0, 5)

yx

y � 3x�1 � 2

0 1

3 5 1173

199y

�1�2�3x

120.

Domain: all real numbers

Range:

intercept:

Horizontal asymptote: y � 0

�0, �3 � � �0, 1.73�y-

y > 0

x x321−1−2−3−4−5

7

6

5

3

4

1

2

y

0, 3( )

y � 3�x�1��2

0 1 2

1 3 3�3 � 5.20�3 � 1.73�33

� 0.57813

y

�1�2�3x

1 3 6

7.17 4.39 0 0 4.39 7.17�2.77�2.77y

12�

12�1�3�6x

Page 37: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.4 Graphs of Sine and Cosine Functions 567

123.

Range: all real numbers

intercept:

intercept: �0, 0.301�y-

��1, 0�x-3

2

−1

−2

−3

321−1−3

(−1, 0)(0, 0.301)

x

y

Domain: x � 2 > 0 ⇒ x > �2

y � log10 �x � 2� 124.

Domain:

Range: all real numbers

intercept:

Vertical asymptote: x � 0

��13, 0�x-

, 0 13

−( (1−1−2−3

2

1

−1

−2

y

x

y

�3x > 0 ⇒ x < 0

y � log10��3x�

Section 6.4 Graphs of Sine and Cosine Functions

■ You should be able to graph

■ Amplitude:

■ Period:

■ Shift: Solve

■ Key increments: (period)1

4

bx � c � 0 and bx � c � 2�.

2�

b

�a�y � a sin�bx � c� and y � a cos�bx � c�. �Assume b > 0.�

Vocabulary Check

1. cycle 2. amplitude

3. 4. phase shift

5. vertical shift

2�

b

1.

Period:

Amplitude: �3� � 3

2�

2� �

y � 3 sin 2x 2.

Period:

Amplitude: �a� � 2

2�

b�

2�

3

y � 2 cos 3x 3.

Period:

Amplitude: �52� �5

2

2�

1

2

� 4�

y �5

2 cos

x

2

x 0 1 2

y 0 �0.602�0.477�0.301��0.301

�1�1.5 x

y 0 ��0.30�0.48�0.78�0.95

�16�

13�1�2�3

4.

Period:

Amplitude: �a� � ��3� � 3

2�

b�

2�13

� 6�

y � �3 sin x

35.

Period:

Amplitude: �12� �1

2

2�

3

� 6

y �1

2 sin

�x

36.

Period:

Amplitude: �a� �3

2

2�

b�

2�

2

� 4

y �3

2 cos

�x

2

Page 38: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

568 Chapter 6 Trigonometry

7.

Period:

Amplitude: ��2� � 2

2�

1� 2�

y � �2 sin x 8.

Period:

Amplitude: �a� � ��1� � 1

2�

b�

2�

2�3� 3�

y � �cos 2x

39.

Period:

Amplitude: �3� � 3

2�

10�

5

y � 3 sin 10x

10.

Period:

Amplitude: �a� �1

3

2�

b�

2�

8�

4

y �1

3 sin 8x 11.

Period:

Amplitude: �12� �1

2

2�

2�3� 3�

y �1

2 cos

2x

312.

Period:

Amplitude: �a� �5

2

2�

b�

2�

1�4� 8�

y �5

2 cos

x

4

13.

Amplitude: �14� �14

Period: 2�

2�� 1

y �14

sin 2�x 14.

Period:

Amplitude: �a� �2

3

2�

b�

2�

��10� 20

y �2

3 cos

�x

1015.

The graph of g is a horizontal shiftto the right units of the graph off �a phase shift�.

g�x� � sin�x � ��

f �x� � sin x

16.g is a horizontal shift of f unitsto the left.

�f�x� � cos x, g�x� � cos�x � �� 17.

The graph of g is a reflection inthe x-axis of the graph of f.

g�x� � �cos 2x

f �x� � cos 2x 18.

g is a reflection of f about the y-axis.

f �x� � sin 3x, g�x� � sin��3x�

19.

The period of f is twice that of g.

g�x� � cos 2x

f �x� � cos x 20.The period of g is one-third theperiod of f.

f�x� � sin x, g�x� � sin 3x 21.

The graph of g is a vertical shiftthree units upward of the graph of f.

f �x� � 3 � sin 2x

f �x� � sin 2x

22.g is a vertical shift of f two unitsdownward.

f�x� � cos 4x, g�x� � �2 � cos 4x 23. The graph of g has twice theamplitude as the graph of f. Theperiod is the same.

24. The period of g is one-third theperiod of f.

25. The graph of g is a horizontal shift units to the right ofthe graph of f.

� 26. Shift the graph of f two units upward to obtain the graphof g.

27.

Period:

Amplitude: 2

Symmetry: origin

Key points: Intercept Minimum Intercept Maximum Intercept

Since generate key points for the graph of by multiplyingthe coordinate of each key point of by �2.f �x�y-

g�x�g�x� � 4 sin x � ��2� f �x�,

�2�, 0��3�

2, 0���, 0���

2, �2��0, 0�

2�

b�

2�

1� 2�

x

−π π32 2

5

43

−5

f

g

yf �x� � �2 sin x

Page 39: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.4 Graphs of Sine and Cosine Functions 569

28.

Period:

Amplitude: 1

Symmetry: origin

Key points: Intercept Maximum Intercept Minimum Intercept

Since the graph of is the graph of but stretched horizontally by a factor of 3.

Generate key points for the graph of by multiplying the coordinate of each key point of by 3.f �x�x-g�x�

f �x�,g�x�g�x� � sin�x3� � f �x

3�,

�2�, 0��3�

2, �1���, 0���

2, 1��0, 0�

2�

b�

2�

1� 2�

− 2

2

π6

fg

x

yf �x� � sin x

29.

Period:

Amplitude: 1

Symmetry: axis

Key points: Maximum Intercept Minimum Intercept Maximum

Since the graph of is the graph of but translated upward by one unit.Generate key points for the graph of by adding 1 to the coordinate of each key point of f �x�.y-g�x�

f �x�,g�x�g�x� � 1 � cos�x� � f �x� � 1,

�2�, 1��3�

2, 0���, �1���

2, 0��0, 1�

y-

2�

b�

2�

1� 2�

−1

g

f

xππ 2

yf �x� � cos x

30.

Period:

Amplitude: 2

Symmetry: axis

Key points: Maximum Intercept Minimum Intercept Maximum

Since the graph of is the graph of but

i) shrunk horizontally by a factor of 2,

ii) shrunk vertically by a factor of and

iii) reflected about the axis.

Generate key points for the graph of by

i) dividing the coordinate of each key point of by 2, and

ii) dividing the coordinate of each key point of by �2.f �x�y-

f �x�x-

g�x�

x-

12,

f �x�,g�x�g�x� � �cos 4x � �12 f �2x�,

��, 2��3�

4, 0���

2, �2���

4, 0��0, 2�

y-

2�

b�

2�

2� �

−2

2

π

f

g

x

yf �x� � 2 cos 2x

Page 40: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

570 Chapter 6 Trigonometry

31.

Period:

Amplitude:

Symmetry: origin

Key points: Intercept Minimum Intercept Maximum Intercept

Since the graph of is the graph of but translated upward by three units.

Generate key points for the graph of by adding 3 to the coordinate of each key point of f �x�.y-g�x�

f �x�,g�x�g�x� � 3 �12

sin x2

� 3 � f �x�,

�4�, 0��3�, 12��2�, 0���, �

12��0, 0�

12

2�

b�

2�

1�2� 4�

−1

1

2

3

4

5

f

g

x−π π3

yf �x� � �12

sin x2

32.

Period:

Amplitude: 4

Symmetry: origin

Key points: Intercept Maximum Intercept Minimum Intercept

Since the graph of is the graph of but translated downward by three units.Generate key points for the graph of by subtracting 3 from the coordinate of each key point of f �x�.y-g�x�

f �x�,g�x�g�x� � 4 sin �x � 3 � f �x� � 3,

�2, 0��32

, �2��1, 0��12

, 2��0, 0�

2�

b�

2�

�� 2

−8

2

4f

g

1 x

yf �x� � 4 sin �x

33.

Period:

Amplitude: 2

Symmetry: axis

Key points: Maximum Intercept Minimum Intercept Maximum

Since the graph of is the graph of but with a phase shift (horizontal translation) of Generate key points for the graph of by shifting each key point of units to the left.�f �x�g�x���.

f �x�,g�x�g�x� � 2 cos�x � �� � f �x � ��,

�2�, 2��3�

2, 0���, �2���

2, 0��0, 2�

y-

2�

b�

2�

1� 2�

−3

3

f

g

xππ 2

yf �x� � 2 cos x

34.

Period:

Amplitude: 1

Symmetry: axis

Key points: Minimum Intercept Maximum Intercept Minimum

Since the graph of is the graph of but with a phase shift (horizontal translation) of Generate key points for the graph of by shifting each key point of units to the right.�f �x�g�x��.

f �x�,g�x�g�x� � �cos�x � �� � f �x � ��,

�2�, �1��3�

2, 0���, 1���

2, 0��0, �1�

y-

2�

b�

2�

1� 2�

−2

2

ππ 2

f

g

x

yf �x� � �cos x

Page 41: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.4 Graphs of Sine and Cosine Functions 571

35.

Period:

Amplitude:

Key points:

�3�

2, �3�, �2�, 0�

�0, 0�, ��

2, 3�, ��, 0�,

3

2�

y

xπ π32 2

−− π2

π32

−4

1

2

3

4

y � 3 sin x

37.

Period:

Amplitude:

Key points:

�3�

2, 0�, �2�,

1

3�

�0, 1

3�, ��

2, 0�, ��, �

1

3�,

1

3

2�

y

xπ π2

π2

1

−1

23

43

1323

43

y �1

3 cos x 38.

Period:

Amplitude:

Key points:

�3�

2, 0�, �2�, 4�

�0, 4�, ��

2, 0�, ��, �4�,

4

2�

y

xππ 2π−2 −π

−2

−4

4

y � 4 cos x

36.

Period:

Amplitude:

Key points:

�3�

2, �

1

4�, �2�, 0�

�0, 0�, ��

2,

1

4�, ��, 0�,

1

4

2�

y

xππ 2π−2 −π

−1

−2

1

2

y �1

4 sin x

39.

Period:

Amplitude: 1

Key points:

�3�, 0�, �4�, 1�

�0, 1�, ��, 0�, �2�, �1�,

4�

y

xπ4π2π−2

−1

−2

2

y � cos x

240.

Period:

Amplitude: 1

Key points:

�3�

8, �1�, ��

2, 0�

�0, 0�, ��

8, 1�, ��

4, 0�,

2

y

x

−2

1

2

π4

y � sin 4x

41.

Period:

Amplitude: 1

Key points:

�0, 1�, �14

, 0�, �12

, �1�, �34

, 0�

2�

2�� 1

1 2

−2

1

2

x

yy � cos 2�x 42.

Period:

Amplitude: 1

Key points:

�6, �1�, �8, 0��0, 0�, �2, 1�, �4, 0�,

2�

��4� 8

2

1

−2

62−2−6x

yy � sin �x

4

Page 42: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

572 Chapter 6 Trigonometry

45.

Period:

Amplitude: 1

Shift: Set

Key points: ��

4, 0�, �3�

4, 1�, �5�

4, 0�, �7�

4, �1�, �9�

4, 0�

x ��

4 x �

9�

4

x ��

4� 0 and x �

4� 2�

2�

−3

−2

1

2

3

ππ−x

yy � sin�x ��

4�; a � 1, b � 1, c ��

4

46.

Period:

Amplitude: 1

Shift: Set

Key points:

2

−2

−1

x

y

−π π32 2

��, 0�, �3�

2, 1�, �2�, 0�, �5�

2, �1�, �3�, 0�

x � � x � 3�

x � � � 0 and x � � � 2�

2�

y � sin�x � �� 47.

Period:

Amplitude: 3

Shift: Set

Key points:

−6

−4

2

4

6

− ππx

y

���, 3�, ���

2, 0�, �0, �3�, ��

2, 0�, ��, 3�

x � �� x � �

x � � � 0 and x � � � 2�

2�

y � 3 cos�x � ��

44.

Period:

Amplitude: 10

Key points:

1284−4−12

12

8

4

−12

x

y

�0, �10�, �3, 0�, �6, 10�, �9, 0�, �12, �10�

2�

��6� 12

y � �10 cos �x

643.

Period:

Amplitude: 1

Key points:

−1 2 3

−3

−2

2

3

x

y

�0, 0�, �3

4, �1�, �3

2, 0�, �9

4, 1�, �3, 0�

2�

2��3� 3

y � �sin 2�x

3; a � �1, b �

2�

3, c � 0

Page 43: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.4 Graphs of Sine and Cosine Functions 573

48.

Period:

Amplitude: 4

Shift: Set

Key points: �5�

4, 0�, �7�

4, 4���

4, 4�, ��

4, 0�, �3�

4, �4�,

x � ��

4 x �

7�

4

x ��

4� 0 and x �

4� 2�

2�

−6

−4

−2−

2

6

x

y

πππ 2

y � 4 cos�x ��

4�

49.

Period: 3

Amplitude: 1

Key points:

–3 –2 –1 1 2 3−1

1

2

4

5

x

y

�0, 2�, �34

, 1�, �32

, 2�, �94

, 3�, �3, 2�

y � 2 � sin 2�x

350.

Period:

Amplitude: 5

Key points:

−12 4 12

−24−20−16−12−8

48

1216

t

y

�0, 2�, �6, �3�, �12, �8�, �18, �3�, �24, 2�

2�

��12� 24

y � �3 � 5 cos � t

12

51.

Period:

Amplitude:

Vertical shift two units upward

Key points:

0.20.1−0.1 0

1.8

2.2

x

y

�0, 2.1�, � 1

120, 2�, � 1

60, 1.9�, � 1

40, 2�, � 1

30, 2.1�

1

10

2�

60��

1

30

y � 2 �1

10 cos 60�x 52.

Period:

Amplitude: 2

Key points:

−7

−6

−5

−4

1

−x

y

πππ 2

�0, �1�, ��

2, �3�, ��, �5�, �3�

2, �3�, �2�, �1�

2�

y � 2 cos x � 3

Page 44: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

574 Chapter 6 Trigonometry

55.

Period:

Amplitude:

Shift:

Key points: ��

2,

2

3�, �3�

2, 0�, �5�

2,

�2

3 �, �7�

2, 0�, �9�

2,

2

3�

x ��

2 x �

9�

2

x

2�

4� 0 and

x

2�

4� 2�

2

3

4�

−4

−3

−2

−1

1

2

3

4

π π4x

y

y �2

3 cos�x

2�

4�; a �2

3, b �

1

2, c �

4

56.

Period:

Amplitude: 3

Shift: Set

Key points: ���

6, �3�, ��

12, 0�, �0, 3�, � �

12, 0�, ��

6, �3�

x � ��

6 x �

6

6x � � � 0 and 6x � � � 2�

2�

6�

3

2

3

x

y

π

y � �3 cos�6x � ��

57.

−6 6

−4

4y � �2 sin�4x � �� 58.

−12

−8

12

8y � �4 sin�2

3x �

3�

53.

Period:

Amplitude: 3

Shift: Set

Key points: ���, 0�, ���

2, �3�, �0, �6�, ��

2, �3�, ��, 0�

x � �� x � �

x � � � 0 and x � � � 2�

2�

−8

2

4

ππ 2x

yy � 3 cos�x � �� � 3

54.

Period:

Amplitude: 4

Shift: Set

Key points: ���

4, 8�, ��

4, 4�, �3�

4, 0�, �5�

4, 4�, �7�

4, 8�

x � ��

4 x �

7�

4

x ��

4� 0 and x �

4� 2�

2�

−4

2

4

6

10

x

y

π3π2π−2 − ππ

y � 4 cos�x ��

4� � 4

Page 45: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.4 Graphs of Sine and Cosine Functions 575

63.

Amplitude:

Vertical shift one unit upward ofThus, f�x� � 2 cos x � 1.g�x� � 2 cos x ⇒ d � 1.

1

2�3 � ��1� � 2 ⇒ a � 2

f �x� � a cos x � d 64.

Amplitude:

a � 2, d � �1

d � 1 � 2 � �1

1 � 2 cos 0 � d

1 � ��3�2

� 2

f�x� � a cos x � d

65.

Amplitude:

Since is the graph of reflected in the x-axis and shifted vertically four units upward, we have

Thus, f �x� � �4 cos x � 4.a � �4 and d � 4.

g�x� � 4 cos xf �x�

1

2�8 � 0 � 4

f �x� � a cos x � d 66.

Amplitude:

Reflected in the x-axis:

a � �1, d � �3

d � �3

�4 � �1 cos 0 � d

a � �1

�2 � ��4�2

� 1

f�x� � a cos x � d

69.

Amplitude:

Period:

Phase shift:

Thus, y � 2 sin�x ��

4�.

�1����

4 � � c � 0 ⇒ c � ��

4

bx � c � 0 when x � ��

4

2� ⇒ b � 1

a � 2

y � a sin�bx � c� 70.

Amplitude:

Period: 2

Phase shift:

a � 2, b ��

2, c � �

2

c

b� �1 ⇒ c � �

2

2�

b� 4 ⇒ b �

2

2 ⇒ a � 2

y � a sin�bx � c�

67.

Amplitude: Since the graph is reflected in the x-axis, we have

Period:

Phase shift:

Thus, y � �3 sin 2x.

c � 0

2�

b� � ⇒ b � 2

a � �3.�a� � �3�

y � a sin�bx � c� 68.

Amplitude:

Period:

Phase shift:

a � 2, b �12

, c � 0

c � 0

2�

b� 4� ⇒ b �

1

2

4�

2 ⇒ a � 2

y � a sin�bx � c�

59.

−3

−1

3

3y � cos�2�x ��

2� � 1 60.

−6

−6

6

2y � 3 cos��x

2�

2� � 2

61.

−20

−0.12

20

0.12y � �0.1 sin��x

10� �� 62.

−0.03

−0.02

0.03

0.02y �1

100 sin 120 � t

Page 46: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

576 Chapter 6 Trigonometry

74.

(a) Period

(b)1 cycle

4 seconds�

60 seconds

1 minute� 15 cycles per minute

�2�

��2� 4 seconds

v � 1.75 sin � t

2

75.

(a) Period:

(b) f �1

p� 440 cycles per second

2�

880��

1

440 seconds

y � 0.001 sin 880�t 76.

(a) Period:

(b)1 heartbeat

6�5 seconds�

60 seconds

1 minute� 50 heartbeats per minute

2�

�5���3�

6

5 seconds

P � 100 � 20 cos 5� t

3

77. (a)

—CONTINUED—

C�t� � 56.55 � 26.95 cos�� t6

� 3.67�

d �12

�high � low �12

�83.5 � 29.6 � 56.55

cb

� 7 ⇒ c � 7��

6� � 3.67

b �2�

p�

2�

12�

6

p � 2�high time � low time � 2�7 � 1 � 12

a �12

�high � low �12

�83.5 � 29.6 � 26.95

71.

In the interval

when x � �5�

6, �

6,

7�

6,

11�

6.sin x � �

1

2

��2�, 2�,

y2 � �1

2−2

2

2

�−2�

y1 � sin x72.

y1 � y2 when x � �, ��

y2 � �1

−2

2

2

�−2�

y1 � cos x

73.

(a) Time for one cycle

(b) Cycles per min cycles per min

(c) Amplitude: 0.85; Period: 6

Key points: �0, 0�, �3

2, 0.85�, �3, 0�, �9

2, �0.85�, �6, 0�

�60

6� 10

�2�

��3� 6 sec

t2 4 8 10

0.25

0.50

0.75

1.00

−0.25

−1.00

vy � 0.85 sin �t

3

(c)

1 3 5 7

−2

−3

1

2

3

t

v

(b)

The model is a good fit.

00

12

100

Page 47: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.4 Graphs of Sine and Cosine Functions 577

77. —CONTINUED—

(c)

The model is a good fit.

(d) Tallahassee average maximum:

Chicago average maximum:

The constant term, gives the average maximum temperature.

d,

56.55�

77.90�

00

12

100 (e) The period for both models is months.

This is as we expected since one full period is one year.

(f) Chicago has the greater variability in temperature through-out the year. The amplitude, a, determines this variabilitysince it is .1

2�high temp � low temp

2�

��6� 12

78. (a) and (c)

Reasonably good fit

(d) Period is 29.6 days.

(e) March 12

The Naval observatory says that 50% of the moon’sface will be illuminated on March 12, 2007.

y � 0.44 � 44% ⇒ x � 71.

y

x10 20 30 40

0.2

0.4

0.6

0.8

1.0

Perc

ent o

f m

oon’

sfa

ce il

lum

inat

ed

Day of the year

(b)

average length ofinterval in data

Horizontal shift:

y �12

�12

sin�0.21x � 0.92�

C � 0.92

0.21�3 � 7.4� � C � 0

b �2�

29.6� 0.21

2�

b� 4�7.4� � 29.6

��Period: 8 � 8 � 7 � 6 � 8

5� 7.4

Amplitude: 12

⇒ a �12

Vertical shift: 12

⇒ d �12

79.

(a) Period

Yes, this is what is expected because there are 365 days in a year.

(b) The average daily fuel consumption is given by theamount of the vertical shift (from 0) which is givenby the constant 30.3.

(c)

The consumption exceeds 40 gallons per day when124 < x < 252.

00

365

60

�2�

2�

365

� 365

C � 30.3 � 21.6 sin�2� t

365� 10.9� 80. (a) Period

The wheel takes 12 minutes to revolve once.

(b) Amplitude: 50 feet

The radius of the wheel is 50 feet.

(c)

00 20

110

�2�

��

6�� 12 minutes

81. False. The graph of is the graph of translated to the left by one period, and the graphs areindeed identical.

sin(x)sin(x � 2�) 82. False. has an amplitude that is half that

of For y � a cos bx, the amplitude is �a�.y � cos x.

y �12

cos 2x

Page 48: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

578 Chapter 6 Trigonometry

85. Since the graphs are thesame, the conjecture is that

.sin�x� � cos�x ��

2�2

1

−2

f = g

xπ π32 2

− π32

y

86. f�x� � sin x, g�x� � �cos�x ��

2�

x 0

0 1 0 0

0 1 0 0�1�cos�x ��

2��1sin x

2�3�

2�

2

87. (a)

The graphs are nearly the same for

(b)

The graphs are nearly the same for ��

2< x <

2.

−2

−2 2� �

2

��

2< x <

2.

−2

−2 2� �

2 (c)

The graphs now agree over a wider range, �3�

4< x <

3�

4.

−2

−2 2� �

2

−2

2�−2�

2

cos x � 1 �x2

2!�

x4

4!�

x6

6!

sin x � x �x3

3!�

x5

5!�

x7

7!

88. (a)

(by calculator)

(c)

(by calculator)

(e)

(by calculator)cos 1 � 0.5403

cos 1 � 1 �1

2!�

1

4!� 0.5417

sin �

6� 0.5

sin �

6� 1 �

���6�3

3!�

���6�5

5!� 0.5000

sin 1

2� 0.4794

sin 1

2�

1

2�

�1�2�3

3!�

�1�2�5

5!� 0.4794 (b)

(by calculator)

(d)

(by calculator)

(f)

(by calculator)cos �

4� 0.7071

cos �

4� 1 �

���4�2

2!�

���4�2

4!� 0.7074

cos��0.5� � 0.8776

cos��0.5� � 1 ���0.5� 2

2!�

��0.5�4

4!� 0.8776

sin 1 � 0.8415

sin 1 � 1 �1

3!�

1

5!� 0.8417

The error in the approximation is not the same in each case. The error appears to increase as x moves farther away from 0.

83. True.

Since and so is a reflection in the x-axis of y � sin�x ��

2�.cos x � sin�x ��

2�, y � �cos x � �sin�x ��

2�,

84. Answers will vary.

Conjecture: sin x � �cos�x ��

2�2

1

−2

f = g

x

y

π π32 2

π32

Page 49: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.5 Graphs of Other Trigonometric Functions 579

89. log10 �x � 2 � log10�x � 2�1�2 �1

2 log10�x � 2�

91. ln t3

t � 1� ln t3 � ln�t � 1� � 3 ln t � ln�t � 1�

93.

� log10 �xy

1

2�log10 x � log10 y� �

1

2log10�xy�

95.

� ln�3xy4�

ln 3x � 4 ln y � ln 3x � ln y4

90.

� 2 log2 x � log2�x � 3�

log2�x2�x � 3�� � log2 x2 � log2�x � 3�

92.

�1

2 ln z �

1

2 ln�z2 � 1�

ln� z

z2 � 1�

1

2 ln � z

z2 � 1� �1

2�ln z � ln�z2 � 1��

94.

� log2 x3y

� log2 x2(xy)

2 log2 x � log2�xy� � log2 x2 � log2�xy�

96.

� ln�x2�2x �

� ln�x3�2xx2 �

� ln �2xx2 � ln x3

�12 �ln

2xx2� � ln x3

12

�ln 2x � 2 ln x� � 3 ln x �12

�ln 2x � ln x2� � ln x3

Section 6.5 Graphs of Other Trigonometric Functions

■ You should be able to graph

■ When graphing or you should first graph orbecause

(a) The x-intercepts of sine and cosine are the vertical asymptotes of cosecant and secant.

(b) The maximums of sine and cosine are the local minimums of cosecant and secant.

(c) The minimums of sine and cosine are the local maximums of cosecant and secant.

■ You should be able to graph using a damping factor.

y � a sin�bx � c�y � a cos�bx � c�y � a csc�bx � c�y � a sec�bx � c�

y � a csc�bx � c�y � a sec�bx � c�

y � a cot�bx � c�y � a tan�bx � c�

Vocabulary Check

1. vertical 2. reciprocal

3. damping 4.

5. 6.

7. 2�

���, �1� � �1, ��x � n�

97. Answers will vary.

Page 50: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

580 Chapter 6 Trigonometry

1.

Period:

Matches graph (e).

2�

2� �

y � sec 2x 3.

Period:

Matches graph (a).

�� 1

y �1

2 cot � x2.

Period:

Asymptotes:

Matches graph (c).

x � ��, x � �

b�

1

2

� 2�

y � tan x

2

4.

Period:

Matches graph (d).

2�

y � �csc x 5.

Period:

Asymptotes:

Matches graph (f).

x � �1, x � 1

2�

b�

2�

2

� 4

y �1

2 sec

�x

26.

Period:

Asymptotes:

Reflected in x-axis

Matches graph (b).

x � �1, x � 1

2�

b�

2�

2

� 4

y � �2 sec �x

2

7.

Period:

Two consecutive asymptotes:

x � ��

2 and x �

2

1

2

3

xπ π

y

y �1

3 tan x

9.

Period:

Two consecutive asymptotes:

3x ��

2 ⇒ x �

6

3x � ��

2 ⇒ x � �

6

3

x

4

3

2

1

− π3

π3

yy � tan 3x

x 0

y 013

�13

4�

4

x 0

y 0 1�1

12�

12

8.

Period:

Two consecutiveasymptotes:

x � ��

2, x �

2

−3

1

2

3

ππ−x

yy �1

4 tan x

x 0

y 01

4�

1

4

4�

4

10.

Period:

Two consecutiveasymptotes:

x � �1

2, x �

1

2

�� 1

2

−8

−4

x

yy � �3 tan �x

x 0

y 3 0 �3

1

4�

1

4

Page 51: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.5 Graphs of Other Trigonometric Functions 581

13.

Period:

Two consecutiveasymptotes:

x � 0, x � 1

2�

�� 2

x

4

3

2

1

−3

−4

21−1−2

yy � csc �x

11.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

2�

1

2

3

y

− π

y � �1

2 sec x 12.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

2�

π

3

2x

yy �

1

4 sec x

14.

Period:

Two consecutiveasymptotes:

x � 0, x ��

4

2�

4 �

2

8

6

4

2

−2− π4

π4

x

yy � 3 csc 4x

15.

Two consecutiveasymptotes:

x � �12

, x �12

Period: 2�

�� 2

1−1−2−3 2 3−1

x

yy � sec �x � 1 16.

Period:

Two consecutiveasymptotes:

x � ��

8, x �

8

2�

4�

2

2

4

6

x

y

π4

π4

π2

y � �2 sec 4x � 2

17.

Period:

Two consecutiveasymptotes:

x � 0, x � 2�

2�

1�2� 4�

2

4

6

yy � csc x

2

0

�1�1

2�1y

3�

3x 0

12

1

412

y

3�

3x

2 1 2y

5

612

1

6x

6 3 6y

5�

24�

8�

24x

0

1 0 1y

1

3�

1

3x 0

0 �2�2y

12�

12x

2 1 2y

5�

3�

3x

18.

Period:

Two consecutiveasymptotes:

x � 0, x � 3�

2�

1�3� 6�

2

4

6

ππ 2x

yy � csc x

3

2 1 2y

5�

23�

2�

2x

Page 52: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

582 Chapter 6 Trigonometry

20.

Period:

Two consecutive asymptotes:

x � 0, x � 2

��2� 2

2

4

6

2−2x

yy � 3 cot �x

219.

Period:

Two consecutive asymptotes:

x

2� � ⇒ x � 2�

x

2� 0 ⇒ x � 0

1�2� 2�

1

2

3

x

y

π−2 2π

y � cot x

2

x

y 1 0 �1

3�

2�

2

21.

Period:2�

2� �

3

xπ π

y

y �1

2 sec 2x 22.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

�2

3

ππ−

y

x

y � �1

2 tan x

x 0

y 0 �1

2

1

2

4�

4

23.

Period:

Two consecutive asymptotes:

�x4

��

2 ⇒ x � 2

�x4

� ��

2 ⇒ x � �2

��4� 4

−4 4

2

4

6

x

y

y � tan �x

4

x 0 1

y 0 1�1

�1

1

3 0 �3y

3

2

1

2x

0

1 11

2y

6�

6x

24.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

�4

3

2

1

πx

yy � tan�x � ��

x 0

y 10�1

4�

4

25.

Period:

Two consecutive asymptotes:

x � 0, x � �

2�

1

2

3

4

x

− ππ322

yy � csc�� � x�

2 1 2y

5�

6�

2�

6x

26.

Period:

Two consecutive asymptotes:

x � 0, x ��

2

2�

2� �

y

x

−1

−2

π32

π2

2ππ

y � csc�2x � ��

�2�1�2y

5�

12�

4�

12x

Page 53: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.5 Graphs of Other Trigonometric Functions 583

27.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

2�

y

x

−1

1

2

3

4

π 2π−π 3π

y � 2 sec�x � �� 28.

Period:

Two consecutive asymptotes:

x � �12

, x �12

2�

�� 2

y

x1 2 3 4

1

2

3

y � �sec �x � 1

0

�4�2�4y

3�

3x

0

0 1�1y

1

3�

13

x

29.

Period:

Two consecutive asymptotes:

x � ��

4, x �

3�

4

2�

xπ2

1

2

yy �1

4 csc�x �

4� 30.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

−2

xπ32

− π32

yy � 2 cot�x ��

2�

x 0

y 2 0 �2

4�

4

12

1

412

y

7�

12�

4�

12x

31.

−5

5�−5�

5

y � tan x

333.

−4

�2

�2

4

y � �2 sec 4x ��2

cos 4x32.

−3

3

�4

3�4

3−

y � �tan 2x

34.

−3

−2

3

2

y � sec �x ⇒ y �1

cos��x� 35.

−3

− �2

3�2

3

3

y � tan�x ��

4� 36.

−3

�2

3�2

3−

3

�1

4 tan�x ��

2�

y �1

4 cot�x �

2�

37.

y ��1

sin�4x � ��

−3

− �2

�2

3y � �csc�4x � �� 38.

y �2

cos�2x � ��

−4

−� �

4 y � 2 sec(2x � �) ⇒

Page 54: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

584 Chapter 6 Trigonometry

39.

−6

−0.6

6

0.6y � 0.1 tan��x4

��

4�

41.

xπ π2

2

y

x � �7�

4, �

3�

4,

4,

5�

4

tan x � 1

40.

−6

−2

6

2

y �1

3 cos��x

2�

2� ⇒ y �

1

3 sec��x

2�

2�

42.

xπ π2

2

1

y

x � �5�

3, �

2�

3,

3,

4�

3

tan x � �3

44.

xπ2

π2

3π2

3π2

− −

2

3

−3

y

x � �7�

4, �

3�

4,

4,

5�

4

cot x � 1

43.

x

y

π2

3π2

1

2

3

−3

x � �4�

3, �

3,

2�

3,

5�

3

cot x � ��3

3

45.

xππ 2π π2

1

−−

y

x � ±2�

3, ±

4�

3

sec x � �2 46.

xππ π2π2

1

−−

y

x � �5�

3, �

3,

3,

5�

3

sec x � 2

47.

xπ 3ππ3π

2 2 2 2− −

1

2

3

−1

y

x � �7�

4, �

5�

4,

4,

3�

4

csc x � �2 48.

xπ 3ππ3π

2 2 2 2− −

1

2

3

y

x � �2�

3, �

3,

4�

3,

5�

3

csc x � �2�3

3

Page 55: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.5 Graphs of Other Trigonometric Functions 585

49.

Thus, is an even function and the graph hasaxis symmetry.y-

f �x� � sec x

� f �x�

�1

cos x

�1

cos��x�

f ��x� � sec��x�

y

x

3

4

ππ π2−

f �x� � sec x �1

cos x50.

Thus, the function is odd and the graph of is symmetric about the origin.

y � tan x

tan��x� � �tan x

x

2

3

−3

3π2

− π2− π

23π2

yf�x� � tan x

51.

(a)

(b) on the interval,

(c) As and

since is the reciprocal

of f �x�.

g �x�g�x� �1

2 csc x → ±�

f �x� � 2 sin x → 0x → �,

6< x <

5�

6f > g

1

−1

2

3

f

g

xπ ππ32 4

π4

y

g�x� �1

2 csc x

f �x� � 2 sin x 52.

(a)

(b) The interval in which

(c) The interval in which which is the same interval as part (b).

2f < 2g is ��1, 13�,f < g is ��1, 13�.

−3

g

f

3

1−1

f�x� � tan �x

2, g�x� �

1

2 sec

�x

2

53.

The expressions are equivalent except when and y1 is undefined.

sin x � 0

sin x csc x � sin x� 1

sin x� � 1, sin x � 0

−2

3−3

2

y1 � sin x csc x and y2 � 1 54.

The expressions are equivalent.

sin x sec x � sin x 1

cos x�

sin x

cos x� tan x

−4

−2 2

4

� �

y1 � sin x sec x, y2 � tan x

55.

The expressions are equivalent.

cot x �cos x

sin x

−4

2�−2�

4y1 �cos x

sin x and y2 � cot x �

1

tan x

Page 56: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

586 Chapter 6 Trigonometry

56.

The expressions are equivalent.

tan2 x � sec2 x � 1

1 � tan2 x � sec2 x

−1

3

�2

3�2

3

y1 � sec2 x � 1, y2 � tan2 x 57.

As

Matches graph (d).

x → 0, f �x� → 0 and f �x� > 0.

f �x� � x cos x

58.

Matches graph (a) as x → 0, f�x� → 0.

f�x� � x sin x

60.

Matches graph (c) as x → 0, g�x� → 0.

g�x� � x cos x

59.

As

Matches graph (b).

x → 0, g�x� → 0 and g�x� is odd.

g�x� � x sin x

61.

The graph is the line

−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

x

y

y � 0.f �x� � g�x�,

f �x� � sin x � cos�x ��

2�, g�x� � 0

62.

It appears that That is,

sin x � cos�x ��

2� � 2 sin x.

f�x� � g�x�.

−4

2

4

xππ−

y

g�x� � 2 sin x

f�x� � sin x � cos�x ��

2� 63.

–1

2

3

y

xππ−

f �x� � g�x�

f �x� � sin2 x, g�x� �1

2�1 � cos 2x�

64.

It appears that That is,

cos2 �x

2�

1

2�1 � cos �x�.

f�x� � g�x�.

g�x� �1

2�1 � cos �x�

−1

−3 3 6−6

2

3

x

yf�x� � cos2 �x

265.

The damping factor is

As x → �, g�x� →0.

y � e�x2�2.

�e�x2�2 ≤ g�x� ≤ e�x2�2

−1

8−8

1g�x� � e�x2�2 sin x

Page 57: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.5 Graphs of Other Trigonometric Functions 587

66.

Damping factor:

As x → �, f�x� → 0.

−3 6

−3

3

e�x

f�x� � e�x cos x 67.

Damping factor:

As x→�, f �x� → 0.

−9 9

−6

6

y � 2�x�4.

�2�x�4 ≤ f �x� ≤ 2�x�4

f �x� � 2�x�4 cos �x 68.

Damping factor:

As x → �, h �x� → 0.

−8

−1

8

1

2�x2�4

h�x� � 2�x2�4 sin x

69.

As x → 0, y → �.

0

−2

8�

6

y �6

x� cos x, x > 0 71.

As x → 0, g�x� → 1.

−1

6�−6�

2

g�x� �sin x

x70.

As x → 0, y → �.

0

−2

6�

6

y �4

x� sin 2x, x > 0

72.

As x → 0, f (x) → 0.

−1

−6� 6�

1

f (x) � 1 � cos x

x73.

As oscillates between and 1.�1

x → 0, f �x�

−2

−� �

2

f �x� � sin 1

x74.

As oscillates.x → 0, h(x)

−1

−� �

2

h(x) � x sin 1

x

75.

Gro

und

dist

ance

x

14

10

6

2

−2

−6

−10

−14Angle of elevation

d

π π π32 4

π4

d �7

tan x� 7 cot x

tan x �7

d76.

x

20

40

60

80

Angle of camera

Dis

tanc

e

d

0 π2

π4

π4

π2

− −

d �27

cos x� 27 sec x, �

2< x <

2

cos x �27

d

Page 58: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

588 Chapter 6 Trigonometry

77.

(a)

(b) As the predator population increases, the number of prey decreases. When the number of prey is small,the number of predators decreases.

(c) The period for both C and R is:

When the prey population is highest, the predator population is increasing most rapidly.When the prey population is lowest, the predator population is decreasing most rapidly.When the predator population is lowest, the prey population is increasing most rapidly.When the predator population is highest, the prey population is decreasing most rapidly.

In addition, weather, food sources for the prey, hunting, all affect the populations of both the predator and the prey.

p �2�

��12� 24 months

00 100

50,000

R

C

R � 25,000 � 15,000 cos � t12

C � 5000 � 2000 sin � t12

,

78.

Month (1 ↔ January)

Law

n m

ower

sal

es(i

n th

ousa

nds

of u

nits

) 150135120105907560453015

2 4 6 8 10 12t

SS � 74 � 3t � 40 cos �t6

79.

(a) Period of

Period of sin

Period of months

Period of monthsL�t� : 12

H�t� : 12

� t

6:

2�

��6� 12

cos � t

6:

2�

��6� 12

L�t� � 39.36 � 15.70 cos � t

6� 14.16 sin

� t

6

H�t� � 54.33 � 20.38 cos � t

6� 15.69 sin

� t

6

(b) From the graph, it appears that the greatest differencebetween high and low temperatures occurs in summer.The smallest difference occurs in winter.

(c) The highest high and low temperatures appear tooccur around the middle of July, roughly one monthafter the time when the sun is northernmost in the sky.

80. (a)

(b) The displacement is a damped sine wave.as t increases.y → 0

y �12

e�t�4 cos 4t

0 4�

−0.6

0.6 81. True. Since

for a given value of , the -coordinate of is thereciprocal of the -coordinate of sin x.y

csc xyx

y � csc x �1

sin x,

Page 59: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.5 Graphs of Other Trigonometric Functions 589

83. As from the left,

As from the right, f �x� � tan x → ��.x → �

2

f�x� � tan x → �.x → �

282. True.

If the reciprocal of is translated units to theleft, we have

y �1

sin�x ��

2��

1cos x

� sec x.

��2y � sin x

y � sec x �1

cos x

84. As from the left, .

As from the right, .f (x) � csc x → ��x → �

f (x) � csc x → �x → �

85.

(a)

The zero between 0 and 1 occurs at x 0.7391.

3

−2

−3

2

f �x� � x � cos x

(b)

This sequence appears to be approaching the zero of f : x 0.7391.

x9 � cos 0.7504 0.7314

x8 � cos 0.7221 0.7504

x7 � cos 0.7640 0.7221

x6 � cos 0.7014 0.7640

x5 � cos 0.7935 0.7014

x4 � cos 0.6543 0.7935

x3 � cos 0.8576 0.6543

x2 � cos 0.5403 0.8576

x1 � cos 1 0.5403

x0 � 1

xn � cos�xn�1�

86.

The graphs are nearly thesame for �1.1 < x < 1.1.

y � x �2x3

3!�

16x5

5!

−6

− �2

3�2

3

6y � tan x 87.

The graph appears to coincide on the interval�1.1 ≤ x ≤ 1.1.

y2 � 1 �x2

2!�

5x4

4!

−6

6

�2

3�2

3

y1 � sec x

88. (a)

—CONTINUED—

−3 3

−2

2

y2

−3 3

−2

2

y1

y2 �4��sin �x �

13

sin 3�x �15

sin 5�x�y1 �4��sin �x �

13

sin 3�x�

Page 60: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

590 Chapter 6 Trigonometry

88. —CONTINUED—

(b)

(c) y4 �4��sin �x �

13

sin 3�x �15

sin 5�x �17

sin 7�x �19

sin 9�x�

−3 3

−2

2

y3 �4��sin �x �

13

sin 3�x �15

sin 5�x �17

sin 7�x�

89.

x �ln 54

2 1.994

2x � ln 54

e2x � 54 91.

x � �ln 2 �0.693

ln 2 � � x

2 � e�x

3 � 1 � e�x

300100

� 1 � e�x

300

1 � e�x � 100

93.

1.684 � 1031

x �2 � e73

3

3x � 2 � e73

3x � 2 � e73

ln�3x � 2� � 73

95.

x � ±�e3.2 � 1 ±4.851

x2 � e3.2 � 1

x2 � 1 � e3.2

ln�x2 � 1� � 3.2

97.

is extraneous (not in thedomain of ) so only isa solution.

x � 2log8 xx � �1

x � 2, �1

�x � 2��x � 1� � 0

x2 � x � 2 � 0

x2 � x � 2

x�x � 1� � 81�3

log8�x�x � 1�� �13

log8 x � log8�x � 1� �13

90.

x �ln 983 ln 8

0.735

3x � log8 98

83x � 98

92.

t �1

365 �log10 5

log10 1.00041096� 10.732

365t � log1.00041096 5

1.00041096365t � 5

1 �0.15365

1.00041096

�1 �0.15365 �

365t

� 5

94.

x �14 � e68

2 �1.702 � 1029

14 � e68 � 2x

14 � 2x � e68

ln(14 � 2x) � 68

96.

22,022.466

x � e10 � 4

x � 4 � e10

ln(x � 4) � 10

12

ln(x � 4) � 5

ln�x � 4 � 5 98.

Since is not in the domainof , the only solution isx � �65 8.062.

log6 x��65

x � ±�65

x2 � 1 � 64

x(x2 � 1) � 64x

log6�x(x2 � 1�� � log6(64x)

log6 x � log6(x2 � 1) � log6(64x)

Page 61: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.6 Inverse Trigonometric Functions 591

■ You should know the definitions, domains, and ranges of y � arcsin x, y � arccos x, and y � arctan x.

Function Domain Range

y � arcsin x ⇒ x � sin y

y � arccos x ⇒ x � cos y

y � arctan x ⇒ x � tan y

■ You should know the inverse properties of the inverse trigonometric functions.

■ You should be able to use the triangle technique to convert trigonometric functions of inverse trigonometric functions into algebraic expressions.

tan�arctan x� � x and arctan�tan y� � y, ��

2< y <

2

cos�arccos x� � x and arccos�cos y� � y, 0 ≤ y ≤ �

sin�arcsin x� � x and arcsin�sin y� � y, ��

2 ≤ y ≤

2

��

2< x <

2�� < x < �

0 ≤ y ≤ ��1 ≤ x ≤ 1

��

2≤ y ≤

2�1 ≤ x ≤ 1

Vocabulary Check

Alternative Function Notation Domain Range

1.

2.

3. ��

2< y <

2�� < x < �y � tan�1 xy � arctan x

0 ≤ y ≤ ��1 ≤ x ≤ 1y � cos�1 xy � arccos x

��

2≤ y ≤

2�1 ≤ x ≤ 1y � sin�1 xy � arcsin x

1. ��

2≤ y ≤

2 ⇒ y �

6y � arcsin

1

2 ⇒ sin y �

1

2 for 2. y � arcsin 0 ⇒ sin y � 0 for �

2≤ y ≤

2 ⇒ y � 0

3. 0 ≤ y ≤ � ⇒ y ��

3y � arccos

1

2 ⇒ cos y �

1

2 for 4. y � arccos 0 ⇒ cos y � 0 for 0 ≤ y ≤ � ⇒ y �

2

5.

��

2< y <

2 ⇒ y �

6

y � arctan �3

3 ⇒ tan y �

�3

3 for 6.

��

2< y <

2 ⇒ y � �

4

y � arctan��1� ⇒ tan y � �1 for

Section 6.6 Inverse Trigonometric Functions

Page 62: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

592 Chapter 6 Trigonometry

7.

0 ≤ y ≤ � ⇒ y �5�

6

y � arccos���3

2 � ⇒ cos y � ��3

2 for 8.

��

2≤ y ≤

2 ⇒ y � �

4

y � arcsin���2

2 � ⇒ sin y � ��2

2 for

9.

��

2< y <

2 ⇒ y � �

3

y � arctan���3� ⇒ tan y � ��3 for

11.

0 ≤ y ≤ � ⇒ y �2�

3

y � arccos��1

2� ⇒ cos y � �1

2 for

13.

��

2≤ y ≤

2 ⇒ y �

3

y � arcsin �3

2 ⇒ sin y �

�3

2 for

10.

��

2< y <

2 ⇒ y �

3

y � arctan��3 � ⇒ tan y � �3 for

12.

��

2≤ y ≤

2 ⇒ y �

4

y � arcsin �2

2 ⇒ sin y �

�2

2 for

14.

��

2< y <

2 ⇒ y � �

6

y � arctan���3

3 � ⇒ tan y � ��3

3 for

15. y � arctan 0 ⇒ tan y � 0 for ��

2< y <

2 ⇒ y � 0 16. y � arccos 1 ⇒ cos y � 1 for 0 ≤ y ≤ � ⇒ y � 0

17.

y � x

g�x� � arcsin x

−1

1.5−1.5

fg

1 f�x� � sin x 18.

Graph

Graph

Graph y3 � x.

y2 � tan�1 x.

y1 � tan x.

−2

g

f

2

�2

�2

f �x� � tan x and g�x� � arctan x

19. arccos 0.28 � cos�1 0.28 � 1.29

21. arcsin��0.75� � sin�1��0.75� � �0.85

23. arctan��3� � tan�1��3� � �1.25 25. arcsin 0.31 � sin�1 0.31 � 0.32

27. arccos��0.41� � cos�1��0.41� � 1.99

29. arctan 0.92 � tan�1 0.92 � 0.74

31. arcsin�3

4� � sin�1�0.75� � 0.85 33. arctan�7

2� � tan�1�3.5� � 1.29

20. arcsin 0.45 � 0.47

22. arccos��0.7� � 2.35

24. arctan 15 � 1.50

28. arcsin��0.125� � �0.13

26. arccos 0.26 � 1.31

30. arctan 2.8 � 1.23

32. arccos��1

3� � 1.91

34. arctan��95

7 � � �1.50 35. This is the graph of The coordinates are

.���3

3,�

6�, and �1, �

4����3, ��

3�,

y � arctan x.

Page 63: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.6 Inverse Trigonometric Functions 593

36.

cos��

6� ��3

2

arccos��1

2� �2�

3

arccos��1� � � 37.

tan� � arctan x

4 θ4

x

tan � �x

4

38.

� � arccos 4

x

cos � �4

x39.

sin� � arcsin�x � 2

5 �θ

5x + 2

sin � �x � 2

5

40.

� � arctan�x � 1

10 �

tan � �x � 1

1041.

� � arccos�x � 32x �

θ

x + 3

2x

cos � �x � 3

2x

42.

x � 1

� � arctan 1

x � 1

tan � �x � 1

x2 � 1�

1

x � 143. sin�arcsin 0.3� � 0.3 44. tan�arctan 25� � 25

45. cos�arccos��0.1�� � �0.1 46. sin�arcsin��0.2�� � �0.2 47.

Note: 3 is not in the range ofthe arcsine function.

arcsin�sin 3�� � arcsin�0� � 0

48.

Note: is not in the range of the arccosine function.7�

2

arccos�cos 7�

2 � � arccos 0 ��

249. Let

and sin y �3

5.

tan y �3

4, 0 < y <

2

xy

53

4

y

y � arctan 3

4.

50. Let

sec�arcsin 4

5� � sec u �5

3

sin u �4

5, 0 < u <

2.

3

45

u

u � arcsin 4

5, 51. Let

and cos y �1�5

��5

5.

tan y � 2 �2

1, 0 < y <

2

x

y

5 2

1

yy � arctan 2.

Page 64: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

594 Chapter 6 Trigonometry

52. Let

1

25

u

sin�arccos �5

5 � � sin u �2

�5�

2�5

5

cos u ��5

5, 0 < u <

2.

u � arccos �5

5, 53. Let

and cos y �12

13.

sin y �5

13, 0 < y <

2

xy

5

12

13

yy � arcsin 5

13.

55. Let

and sec y ��34

5.

tan y � �3

5, �

2< y < 0

xy

34−3

5

y

y � arctan��3

5�.54. Let

13

12

−5u

cscarctan��5

12� � csc u � �13

5

tan u � �5

12, �

2 < u < 0.

u � arctan��5

12�,

56. Let

4−3

7u

tanarcsin��3

4� � tan u � �3�7

� �3�7

7

sin u � �3

4, �

2 < u < 0.

u � arcsin��3

4�, 57. Let

and sin y ��5

3.

cos y � �2

3,

2< y < �

xy

3

−2

5

y

y � arccos��2

3�.

Page 65: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.6 Inverse Trigonometric Functions 595

60. Let

x + 12

1

x

u

sin�arctan x� � sin u �x

�x2 � 1

tan u � x �x

1.

u � arctan x, 61. Let

and cos y � �1 � 4x2.

sin y � 2x �2x

1

1 − 4x2

2x

y

1

y � arcsin�2x�.

62. Let

sec�arctan 3x� � sec u � �9x2 � 1

tan u � 3x �3x

1.

1

3x9 + 1x2

u

u � arctan 3x, 63. Let

and sin y � �1 � x2.

cos y � x �x

1 1 − x2

xy

1

y � arccos x.

64. Let

x −1

2x x− 2

1

u

sec�arcsin�x � 1�� � sec u �1

�2x � x2

sin u � x � 1 �x � 1

1.

u � arcsin�x � 1�, 65. Let

and tan y ��9 � x2

x.

cos y �x

3 9 − x2

xy

3

y � arccos�x

3�.

66. Let

cot�arctan 1

x� � cot u � x

tan u �1

x. 1

ux

x + 12

u � arctan 1

x, 67. Let

and csc y ��x2 � 2

x.

tan y �x�2

x2 + 2

x

y

2

y � arctan x�2

.

59. Let

and cot y �1

x.

tan y � x �x

1

x2 + 1x

y

1

y � arctan x.58. Let

cot�arctan 5

8� � cot u �8

5

tan u �5

8, 0 < u <

2. 5

8

89

u

u � arctan 5

8,

Page 66: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

596 Chapter 6 Trigonometry

69.

They are equal. Let

and

The graph has horizontal asymptotes at y � ±1.

g�x� �2x

�1 � 4x2� f �x�

1 + 4x2

2x

y

1

sin y �2x

�1 � 4x2.

tan y � 2x �2x

1

y � arctan 2x. −3 3

−2

2

f �x� � sin�arctan 2x�, g�x� �2x

�1 � 4x2

70.

Asymptote:

These are equal because:

Let

Thus, f �x� � g�x�.

��4 � x2

x� g�x�

f �x� � tan�arccos x

2� � tan u

u � arccos x

2. 2

ux

4 − x 2

x � 0

g�x� ��4 � x2

x

−3

−2

3

2

f�x� � tan�arccos x

2� 71. Let

Thus,

x2 + 81

x

y

9

arcsin y ��9

�x2 � 81, x < 0.

arcsin y �9

�x2 � 81, x > 0;

tan y �9

x and sin y �

9�x2 � 81

, x > 0; �9

�x2 � 81, x < 0.

y � arctan 9

x.

72. If

then

arcsin �36 � x2

6� arccos

x

6

sin u ��36 � x2

6.

6

ux

36 − x 2

arcsin �36 � x2

6� u, 73. Let Then,

and

Thus,

(x − 1)2 + 9

3

y

x − 1

y � arcsin �x � 1��x2 � 2x � 10

.

sin y � �x � 1���x � 1�2 � 9

.

cos y �3

�x2 � 2x � 10�

3��x � 1�2 � 9

y � arccos 3

�x2 � 2x � 10.

68. Let

cos�arcsin x � h

r � � cos u ��r2 � �x � h�2

r

sin u �x � h

r. r

x h−

r x h− −( )22

u

u � arcsin x � h

r,

Page 67: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.6 Inverse Trigonometric Functions 597

75.

Domain:

Range:

This is the graph of with a factor of 2.

−2 −1 1 2

π

π2

x

y

f �x� � arccos x

0 ≤ y ≤ 2�

�1 ≤ x ≤ 1

y � 2 arccos x

76.

Domain:

Range:

This is the graph ofwith a

horizontal stretch of afactor of 2.

f �x� � arcsin x

��

2≤ y ≤

2

�2 ≤ x ≤ 2

1 2−2x

π

π

y

y � arcsin x

277.

Domain:

Range:

This is the graph of shifted

one unit to the right.g�x� � arcsin�x�

��

2≤ y ≤

2

0 ≤ x ≤ 2

−1 1 2 3

π

π

y

x

f �x� � arcsin�x � 1�

78.

Domain:

Range:

This is the graph ofshifted

two units to the left.y � arccos t

0 ≤ y ≤ �

�3 ≤ t ≤ �1

−4 −3 −2 −1t

π

yg�t� � arccos�t � 2� 79.

Domain: all real numbers

Range:

This is the graph ofwith a

horizontal stretch of a factor of 2.

g�x� � arctan�x�

��

2< y <

2−4 −2 2 4

π

π

y

x

f �x� � arctan 2x

80.

Domain: all real numbers

Range:

This is the graph ofshifted

upward units.��2

y � arctan x

0 < y ≤ �

−4 −2 2 4x

π

y

f�x� ��

2� arctan x 81.

Domain:

Range: all real numbers

−2 1 2

π

y

v

1 − v2

vy

1

�1 ≤ v ≤ 1, v � 0

h�v� � tan�arccos v� ��1 � v2

v

74. If

then

2

ux − 2

4x x− 2

arccos x � 2

2� arctan

�4x � x2

x � 2

cos u �x � 2

2.

arccos x � 2

2� u,

Page 68: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

598 Chapter 6 Trigonometry

82.

Domain:

Range: 0 ≤ y ≤ �

�4 ≤ x ≤ 4

−4 −2 2 4x

π

yf�x� � arccos

x

483.

0

2�

−1 1

f �x� � 2 arccos�2x�

84.

−0.5 0.5

2�

−2�

f�x� � � arcsin�4x� 85.

−2 4

�−

f �x� � arctan�2x � 3� 86.

−4 4

−2�

�2

f�x� � �3 � arctan��x�

87.

5

−2

−4

4

f �x� � � � arcsin�23� � 2.412 88.

5

−2

−4

4

f�x� ��

2� arccos�1

�� � 2.82

89.

The graphs are the same and implies that the identity is true.

� 3�2 sin�2t ��

4� � 3�2 sin�2t � arctan 1�

−6

−2

6

� 2�

f �t� � 3 cos 2t � 3 sin 2t � �32 � 32 sin�2t � arctan 3

3�

90.

The graph implies that

is true.

A cos �t � B sin �t ��A2 � B2 sin��t � arctan A

B�−6

6−6

6 � 5 sin�� t � arctan 4

3�

� �42 � 32 sin�� t � arctan 4

3�f�t� � 4 cos � t � 3 sin � t 91. (a)

(b)

s � 20: � � arcsin 5

20� 0.25

s � 40: � � arcsin 5

40� 0.13

sin� � arcsin 5

s

sin � �5

s

Page 69: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.6 Inverse Trigonometric Functions 599

92. (a)

(b) When

When

� � arctan 1200

750� 1.01 � 58.0.

s � 1200,

� � arctan 300

750� 0.38 � 21.8.

s � 300,

� � arctan s

750

tan � �s

75093.

(a)

(b) is maximum when feet.

(c) The graph has a horizontal asymptote at As xincreases, decreases.

� 0.

x � 2

0

−0.5

6

1.5

� arctan 3x

x2 � 4

94. (a)

(b)

feeth � 20 tan � � 20 �1117

� 12.94

tan � �hr

�h

20

r �12

�40� � 20

� � arctan 11

17� 0.5743 � 32.9

tan � �11

1795.

(a)

(b)

tanh � 50 tan 26 � 24.39 feet

tan 26 �h

50

tan� � arctan�20

41� � 26.0

tan � �20

41

20 ft

41 ft

θ

96. (a)

(b)

� � arctan 6

1� 1.41 � 80.5

x � 1 mile

� � arctan 6

7� 0.71 � 40.6

x � 7 miles

� � arctan 6

x

tan � �6

x97. (a)

(b)

x � 12: � � arctan 12

20� 31.0

x � 5: � � arctan 5

20� 14.0

tan� � arctan x

20

tan � �x

20

98. False.

is not in the range of

arcsin 12

��

6

arcsin�x�.5�

6

99. False.

is not in the range of the arctangent function.

arctan 1 ��

4

5�

4

100. False.

is defined for all real x, but and require

Also, for example,

Since , but undefined.arcsin 1arccos 1

���2

0�arctan 1 �

4

arctan 1 �arcsin 1arccos 1

.

�1 ≤ x ≤ 1.arccos xarcsin xarctan x

Page 70: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

600 Chapter 6 Trigonometry

101.

Domain:

Range: 0 < x < �

x−1−2 21

π2

y

π

�� < x < �

y � arccot x if and only if cot y � x.

103.

Domain:

Range: ��

2, 0� � �0,

2���, �1� � �1, ��

x−2 −1 21

y

π2

π2

y � arccsc x if and only if csc y � x.

102. if and only if where

and and The

domain of arcsec x is and the

range is

x−2 −1 1 2

y

π

π2

0, �

2� � ��

2, �.

���, �1� � �1, ��y �

2< y ≤ �.0 ≤ y <

2x ≤ �1 � x ≥ 1

sec y � xy � arcsec x

104. (a)

(b)

(c)

(d) y � arccsc 2 ⇒ csc y � 2 and ��

2≤ y < 0 � 0 < y ≤

2 ⇒ y �

6

y � arccot���3 � ⇒ cot y � ��3 and 0 < y < � ⇒ y �5�

6

y � arcsec 1 ⇒ sec y � 1 and 0 ≤ y <�

2�

2< y ≤ � ⇒ y � 0

y � arcsec �2 ⇒ sec y � �2 and 0 ≤ y <�

2�

2< y ≤ � ⇒ y �

4

105.

(a)

(b)

(c)

(d)

� 1.25 � ���

4� � 2.03

Area � arctan 3 � arctan��1�

a � �1, b � 3

� 1.25 � 0 � 1.25

Area � arctan 3 � arctan 0

a � 0, b � 3

��

4� ��

4� ��

2

Area � arctan 1 � arctan��1�

a � �1, b � 1

Area � arctan 1 � arctan 0 ��

4� 0 �

4

a � 0, b � 1

Area � arctan b � arctan a 106.

As x increases to infinity, g approaches but f has no maximum. Using the solve feature of the graphing utility, you find a � 87.54.

3�,

00

6

g

f

12

g�x� � 6 arctan x

f �x� � �x

Page 71: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.6 Inverse Trigonometric Functions 601

107.

(a)

(b) The graphs coincide with the graph of only for certain values of x.

over its entire domain, .

over the region , corresponding to the region where sin x is

one-to-one and thus has an inverse.

��

2≤ x ≤

2f �1 � f � x

�1 ≤ x ≤ 1f � f �1 � x

y � x

−2

−� �

2

−2

−� �

2

f �1 � f � arcsin�sin x�f � f �1 � sin�arcsin x�

f �x� � sin�x�, f �1�x� � arcsin�x�

108. (a) Let Then,

Therefore, arcsin arcsin x.

(c) Let

(e)

x1

y1

2y

1 − x 2

� arctan x

�1 � x2arcsin x � arcsin

x

1

x

1

y1

2y

� y1 � ��

2� y1� �

2

arctan x � arctan 1

x� y1 � y2

y2 ��

2� y1.

��x� � �

y � �arcsin x.

�y � arcsin x

sin��y� � x

�sin y � x

sin y � �x

y � arcsin��x�. (b) Let Then,

Thus,

(d) Let then andThus, which implies that and

are complementary angles and we have

arcsin x � arccos x ��

2.

� � ��

2

�sin � � cos cos � x.

sin � � x� � arcsin x and � arccos x,

arctan��x� � �arctan�x�.

y � �arctan x

�y � arctan x

arctan�tan��y�� � arctan x

tan��y� � x, ��

2< �y <

2

�tan y � x

tan y � �x, ��

2< y <

2

y � arctan��x�.

109. �8.2�3.4 � 1279.284 110. 10(14)�2 �10

142�

10

196� 0.051

Page 72: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

602 Chapter 6 Trigonometry

111. �1.1�50 � 117.391 112. 16�2��1

162�� 2.718 10�8

113.

csc � �43

sec � �4�7

�4�7

7

cot � ��73

tan � �3�7

�3�7

7

cos � ��74

adj � �7

�adj�2 � 7

�adj�2 � 9 � 16

�adj�2 � �3�2 � �4�24 3

θ

sin � �34

�opphyp

114.

csc � �12�5

sec � � �5

cot � �12

sin � �2�5

cos � �1�5

hyp � �12 � 22 � �52

1

θ

tan � � 2

115.

csc � �6

�11�

6�1111

sec � �65

cot � �5

�11�

5�1111

tan � ��11

5

sin � ��11

6

opp � �11

�opp�2 � 11

�opp�2 � 25 � 36

�opp�2 � �5�2 � �6�2 6

5

θ

cos � �56

�adjhyp

116.

cot � �1

2�2�

�24

csc � �3

2�2�

3�24

sec � � 3

tan � � 2 �2

sin � �2�2

3

cos � �13

� 2�2

� �8

opp � �32 � 12

3

1

θ

sec � � 3

Page 73: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.6 Inverse Trigonometric Functions 603

117. Let the number of people presently in the group. Each person’s share is now

If two more join the group, each person’s share would then be

There were 8 people in the original group.

x � �10 is not possible.

x � �10 or x � 8

6250�x � 10��x � 8� � 0

6250�x2 � 2x � 80� � 0

6250x2 � 12500x � 500,000 � 0

250,000x � 250,000x � 500,000 � 6250x2 � 12500x

250,000x � 250,000�x � 2� � 6250x�x � 2�

250,000

x � 2�

250,000

x� 6250

Share per person with

two more people �

Original share

per person � 6250

250,000

x � 2.

250,000

x.x �

118. Rate downstream:

Rate upstream:

(Time to go upstream) (Time to go downstream)

The speed of the current is 3 miles per hour.

x � ±3

x2 � 9

315 � 324 � x2

1260 � 4�324 � x2�

630 � 35x � 630 � 35x � 4�324 � x2�

35�18 � x� � 35�18 � x� � 4�18 � x��18 � x�

35

18 � x�

35

18 � x� 4

� 4�

rate time � distance ⇒ t �d

r

18 � x

18 � x119. (a)

(b)

(c)

(d) A � 15,000e�0.035��10� � $21,286.01

A � 15,000�1 �0.035365 ��365��10�

� $21,285.66

A � 15,000�1 �0.035

12 ��12��10�� $21,275.17

A � 15,000�1 �0.035

4 ��4��10�� $21,253.63

120. Data:

To find:

Assume:

Then:

� 458,504.31

� 632,000 � �632742�

2

� 632,000 � �e�r�2�2

y � P0e�r�8 � P0e

�r�4 � e�r�4

e�r�2 �P0e�r�4

P0e�r�2 �632742

632,000 � P0e�r�4

742,000 � P0e�r�2

P � P0 � e�rt

�8, y�

�4, 632,000�

�2, 742,000�

Page 74: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.7 Applications and Models

604 Chapter 6 Trigonometry

■ You should be able to solve right triangles.

■ You should be able to solve right triangle applications.

■ You should be able to solve applications of simple harmonic motion.

Vocabulary Check

1. elevation; depression 2. bearing

3. harmonic motion

1. Given:

20°b = 10

a

AC

B

c

B � 90� � 20� � 70�

cos A �b

c ⇒ c �

b

cos A�

10

cos 20�� 10.64

tan A �a

b ⇒ a � b tan A � 10 tan 20� � 3.64

A � 20�, b � 10 2. Given:

cos B �a

c ⇒ a � c cos B � 15 cos 54� � 8.82

� 15 sin 54� � 12.14

sin B �b

c ⇒ b � c sin B

� 90� � 54� � 36�

A � 90� � B 54°

b

a

AC

B

c = 15

B � 54�, c � 15

3. Given:

71°

b = 24

a

AC

B

c

A � 90� � 71� � 19�

sin B �b

c ⇒ c �

b

sin B�

24

sin 71�� 25.38

tan B �b

a ⇒ a �

b

tan B�

24

tan 71�� 8.26

B � 71�, b � 24 4. Given:

8.4°

b

a = 40.5 AC

Bc

sin A �a

c ⇒ c �

a

sin A�

40.5

sin 8.4�� 277.24

�40.5

tan 8.4�� 274.27

tan A �a

b ⇒ b �

a

tan A

� 90� � 8.4� � 81.6�

B � 90� � A

A � 8.4�, a � 40.5

5. Given:

b = 10

a = 6

AC

B

c

B � 90� � 30.96� � 59.04�

tan A �a

b�

6

10 ⇒ A � arctan

3

5� 30.96º

� 2�34 � 11.66

c2 � a2 � b2 ⇒ c � �36 � 100

a � 6, b � 10 6. Given:

cos B �a

c ⇒ B � arccos

a

c� arccos

25

35� 44.42�

� arcsin 25

35� 45.58�

sin A �a

c ⇒ A � arcsin

a

c

� �600 � 24.49

� �352 � 252

b � �c2 � a2

b

a = 25 c = 35

AC

Ba � 25, c � 35

Page 75: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.7 Applications and Models 605

7. Given:

B � 90� � 72.08� � 17.92�

A � arccos 16

52� 72.08º

cos A �16

52

� �2448 � 12�17 � 49.48

a � �522 � 162

b = 16

c = 52a

AC

Bb � 16, c � 52 8. Given:

� 8.03�

� arcsin 1.32

9.45 a

b = 1.32

c = 9.45

AC

Bsin B �

b

c ⇒ B � arcsin

b

c

cos A �b

c ⇒ A � arccos

b

c � arccos

1.32

9.45� 81.97�

a � �c2 � b2 � �87.5601 � 9.36

b � 1.32, c � 9.45

9. Given:

b

c = 430.5a

AC

B

12°15′

b � 430.5 cos 12�15� � 420.70

cos 12�15� �b

430.5

a � 430.5 sin 12�15� � 91.34

sin 12�15� �a

430.5

B � 90� � 12�15� � 77�45�

A � 12�15�, c � 430.5 10. Given:

a = 14.2

b

c

AC

B

65°12′

tan B �b

a ⇒ b � a tan B � 14.2 tan 65�12� � 30.73

cos B �a

c ⇒ c �

a

cos B�

14.2

cos 65�12�� 33.85

A � 90� � B � 90� � 65�12� � 24�48�

B � 65�12�, a � 14.2

11.

12 b 1

2 b

b

h

θ θ

h �1

2�4� tan 52� � 2.56 inches

tan � �h

�1�2�b ⇒ h �

1

2b tan � 12.

12 b 1

2 b

b

h

θ θ

h �12

�10� tan 18� � 1.62 meters

tan � �h

�1�2�b ⇒ h �12

b tan �

13.

12 b 1

2 b

b

h

θ θ

h �12

�46� tan 41� � 19.99 inches

tan � �h

�1�2�b ⇒ h �12

b tan � 14.

12 b 1

2 b

b

h

θ θ

h �12

�11� tan 27� � 2.80 feet

tan � �h

�1�2�b ⇒ h �12

b tan �

Page 76: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

606 Chapter 6 Trigonometry

18.

� 81.2 feet

h � 125 tan 33

33°

h

125

tan 33� �h

125

19. (a)

(b) Let the height of the church and the height of thechurch and steeple . Then,

(c) feeth � 19.9

h � y � x � 50�tan 47�40� � tan 35��.

x � 50 tan 35� and y � 50 tan 47�40�

tan 35� �x

50 and tan 47�40� �

y

50

� y� x

50 ft

47° 40′

35°

h

x

y

20.

� 123.5 feet

h � 100 tan 51�

51°

h

100

tan 51� �h

100

21.

34°

4000x

� 2236.8 feet

x � 4000 sin 34º

sin 34� �x

400022.

θ

50 ft

75 ft

� � arctan 32

� 56.3�

tan � �7550

23. (a)

(b)

(c)

The angle of elevation of the sum is 35.8�.

� � arctan 121

2

1713

� 35.8�

tan � �121

2

1713

θ

17 ft

12 ft

1

1

3

2

24.

Angle of depression � � � 90� � 14.03� � 75.97�

� � 14.03�

� � arcsin� 400016,500�

sin � �4000

16,500

16,500 mi

4,000 mi

θα

Not drawn to scale

12,500 � 4000 � 16,500

17.

16 si 74 h � 19.7 feet

20 sin 80� � h

80°

h20 ft

16 sin 80� �h

20

15.

� 107.2 feet

x �50

tan 25�25°

50

x

tan 25� �50x

16.

� 1648.5 feet

x �600

tan 20

600

20°x

tan 20� �600

x

Page 77: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.7 Applications and Models 607

26. (a) Since the airplane speed is

after one minute its distance travelled is 16,500 feet.

18°

16500a

a � 16,500 sin 18� � 5099 ft

sin 18� �a

16,500

�275ft

sec��60sec

min� � 16,500ft

min,

(b)

275s 10,000feet

18°

� 117.7 seconds

s �10,000

275(sin 18�)

sin 18� �10,000

275s

27.

� 0.73 mile

x � 4 sin 10.5�

10.5°4 x sin 10.5� �

x

4

28.

Angle of grade:

Change in elevation:

� 2516.3 feet

� 21,120 sin�arctan 0.12�

y � 21,120 sin �

sin � �y

21,120

� � arctan 0.12 � 6.8�

tan � �12x

100x

100x12 =x y4 miles = 21,120 feet

θ 29. The plane has traveled

N

S

EW

90052°

38°a

b

cos 38� �b

900 ⇒ b � 709 miles east

sin 38� �a

900 ⇒ a � 554 miles north

1.5�600� � 900 miles.

30. (a) Reno is miles N of Miami.

Reno is miles W of Miami.

(b) The return heading is

N

S

EW

280°

10°

280�.

2472 cos 10� � 2434

N

S

EW

100°

80°10° Miami

Reno

2472 mi

2472 sin 10� � 429

25.

� � artan� 950

26,400� � 2.06�

tan � �950

26,400

5 miles � 5 miles�5280 feet1 mile � � 26,400 feet

5 miles

950feet θ

θ

Not drawn to scale1200 feet � 150 feet � 400 feet � 950 feet

Page 78: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

608 Chapter 6 Trigonometry

32.

(a) hours

(b) After 12 hours, the yacht will have traveled 240 nautical miles.

miles E

miles S

(c) Bearing from N is 178.6�.

240 cos 1.4� � 239.9

240 sin 1.4� � 5.9

t �42820

� 21.4

S

EW

N

Not drawn to scale

88.6°

1.4°

428

(b)

�d

50 ⇒ d � 68.82 meters

tan C �d

50 ⇒ tan 54�

C � � � � 54�

� 90� � � 22�

� � � � 32�

33.

(a)

Bearing from A to C: N 58º E

N

S

EW

C

B

d

A

θ α

φγ

β

β50

� � 90� � 32� � 58�

� � 32�, � 68�

34.

� 5.46 kilometers

d �30

cot 34� � cot 14�

d cot 34� � 30 � d cot 14�

cot 34� �30 � d cot 14�

d

�d

30 � d cot 14�

tan 34� �d

y�

d

30 � x

tan 14� �d

x ⇒ x � d cot 14� 35.

Bearing: N 56.3 W

N

S

EW

Port

Ship

45

30θ

tan � �45

30 ⇒ � � 56.3� 36.

N

S

EW

160

85

Plane

Airport

� 208.0� or 528� W

Bearing � 180� � arctan� 85160�

37.

Distance between ships: D � d � 1933.3 ft

tan 4� �350

D ⇒ D � 5005.23 ft

350

6.5°4°

dD

S1S2

Not drawn to scale

tan 6.5� �350

d ⇒ d � 3071.91 ft

31.

(a) nautical miles south

nautical miles west

(b)

Bearing: S W

Distance:

nautical miles from port� 130.9

d � �104.952 � 78.182

36.7�

tan � �20 � b

a�

78.18104.95

⇒ � � 36.7�

sin 29� �b

120 ⇒ b � 58.18

cos 29� �a

120 ⇒ a � 104.95

N

S

EW

120

20

29°

b

a

Page 79: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.7 Applications and Models 609

39.

� 17,054 ft

a cot 16� � a cot 57� �55

6 ⇒ a � 3.23 miles

cot 16� �a cot 57� �

556

a

tan 16� �a

a cot 57� �556

tan 16� �a

x � 556

57°16°

x55060

H

P1 P2

a

tan 57� �a

x ⇒ x � a cot 57�

41.

� � arctan 5 � 78.7�

tan � � � �1 �32

1 � ��1��32�� � ��5

2

�12� � 5

L2: 3x � y � 1 ⇒ y � �x � 1 ⇒ m2 � �1

L1: 3x � 2y � 5 ⇒ y � 3

2x �

5

2 ⇒ m1 �

3

240.

� 5410 feet

h �17

� 1tan 2.5�

�1

tan 9��� 1.025 miles

htan 2.5�

�h

tan 9�� 17

x �h

tan 9�� 17

tan 9� �h

x � 17

x �h

tan 2.5�

x − 17172.5° 9°

x

h

Not drawn to scale

tan 2.5� �hx

42.

� arctan�97� � 52.1�

� � arctan� m2 � m1

1 � m2m1� � arctan� 15 � 2

1 �15�2��

tan � � � m2 � m1

1 � m2m1�L2 � x � 5y � �4 ⇒ m2 �

1

5

L1 � 2x � y � 8 ⇒ m1 � 2 43. The diagonal of the base has a length ofNow, we have

� � 35.3�.

� � arctan 1

�2

a

2a

θ

tan � �a

�2a�

1�2

�a2 � a2 � �2a.

44.

� � arctan �2 � 54.7º

θa

a 2

tan � �a�2

a� �2 45.

Length of side:

36°25

d

2d � 29.4 inches

sin 36� �d

25 ⇒ d � 14.69

38.

Distance between towns:

dD

28°

28°

55°

55°10 km

T2T1

D � d � 18.8 � 7 � 11.8 kilometers

cot 28� �D

10 ⇒ D � 18.8 kilometers

cot 55 �d

10 ⇒ d � 7 kilometers

Page 80: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

610 Chapter 6 Trigonometry

46.

� 25 inches

Length of side � 2a � 2�12.5�

a � 25 sin 30� � 12.5

sin 30� �a

25

2530°

a 47.

y � 2b � 2��3r

2 � � �3r

b ��3r

2

b � r cos 30�

cos 30� �b

r

br

r

30°

2

x

y

48.

Distance � 2a � 9.06 centimeters

� 4.53

a � c sin 15� � 17.5 sin 15�

sin 15� �a

c

c �35

2� 17.5

c

15°a

49.

a �10

cos 35�� 12.2

cos 35� �10

a

b � 10 tan 35� � 7

tan 35� �b

10

35° 35°ba

10 10

10

10

10

10

50.

c � �10.82 � 7.22 � 13 feet

b �6

sin � 7.2 feet

sin �6

b

� 90 � 33.7 � 56.3�

f �21.6

2� 10.8 feet

a �18

cos �� 21.6 feet

6

6bca

f

θ φ9

36

cos � �18

a

� � arctan 2

3� 0.588 rad � 33.7�

tan � �12

18

51.

Use since

Thus, d � 4 sin�� t�.

2�

�� 2 ⇒ � � �

d � 0 when t � 0.d � a sin �t

d � 0 when t � 0, a � 4, Period � 2 52. Displacement at is

Amplitude:

Period:

d � 3 sin�� t

3 �

2�

�� 6 ⇒ � �

3

�a� � 3

0 ⇒ d � a sin � tt � 0

53.

Use since

Thus, d � 3 cos�4�

3t� � 3 cos�4�t

3 �.

2�

�� 1.5 ⇒ � �

4�

3

d � 3 when t � 0.d � a cos �t

d � 3 when t � 0, a � 3, Period � 1.5 54. Displacement at is

Amplitude:

Period:

d � 2 cos�� t

5 �

2�

�� 10 ⇒ � �

5

�a� � 2

2 ⇒ d � a cos � tt � 0

Page 81: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.7 Applications and Models 611

55.

(a) Maximum displacement amplitude

(b)

cycles per unit of time

(c)

(d) 8� t ��

2 ⇒ t �

1

16

d � 4 cos 40� � 4

� 4

Frequency ��

2��

8�

2�

� 4�

d � 4 cos 8�t 56.

(a) Maximum displacement:

(b) Frequency: cycles per unit of time

(c)

(d) Least positive value for t for which

t ��

2

1

20��

1

40

20� t ��

2

20� t � arccos 0

cos 20� t � 0

1

2 cos 20� t � 0

d � 0

t � 5 ⇒ d �12

cos 100� �12

2��

20�

2�� 10

�a� � �12� �1

2

d �1

2 cos 20 � t

57.

(a) Maximum displacement amplitude

(b)

cycles per unit of time

(c)

(d) 120�t � � ⇒ t �1

120

d �1

16 sin 600� � 0

� 60

Frequency ��

2��

120�

2�

�1

16�

d �1

16 sin 120�t 58.

(a) Maximum displacement:

(b) Frequency: cycles per unit of time

(c)

(d) Least positive value for t for which

t ��

792��

1

792

792� t � �

792� t � arcsin 0

sin 792� t � 0

1

64 sin 792� t � 0

d � 0

t � 5 ⇒ d �1

64 sin�3960�� � 0

2��

792�

2�� 396

�a� � � 1

64� �1

64

d �1

64 s in 792�t

59.

� � 2��264� � 528�

264 ��

2�

Frequency ��

2�

d � a sin �t

Page 82: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

612 Chapter 6 Trigonometry

60. At buoy is at its high point

Returns to high point every 10 seconds:

Period:

d �7

4 cos

� t

5

2�

�� 10 ⇒ � �

5

�a� �7

4

Distance from high to low � 2�a� � 3.5

⇒ d � a cos � t.t � 0, 61.

(a)

t

1

−1

y

π π38 8

π4

π2

y �1

4 cos 16t, t > 0

(b) Period:

(c)1

4cos 16t � 0 when 16t �

2 ⇒ t �

32

2�

16�

8

62. (a)

(c) L � L1 � L2 �2

sin ��

3

cos �

(b)

The minimum length of the elevator is 7.0 meters.

(d)

From the graph, it appears that the minimum length is 7.0 meters, which agrees with the estimate of part (b).

−12

−2� 2�

12

0.1 23.0

0.2 13.1

0.3 9.9

0.4 8.43

cos 0.4

2

sin 0.4

3

cos 0.3

2

sin 0.3

3

cos 0.2

2

sin 0.2

3

cos 0.1

2

sin 0.1

L1 � L2L2L1�

0.5 7.6

0.6 7.2

0.7 7.0

0.8 7.13

cos 0.8

2

sin 0.8

3

cos 0.7

2

sin 0.7

3

cos 0.6

2

sin 0.6

3

cos 0.5

2

sin 0.5

L1 � L2L2L1�

63. (a) and (b)

Base 1 Base 2 Altitude Area

8 22.1

8 42.5

8 59.7

8 72.7

8 80.5

8 83.1

8 80.7

The maximum occurs when and is approximately83.1 square feet.

� � 60�

8 sin 70º8 � 16 cos 70º

8 sin 60º8 � 16 cos 60º

8 sin 50º8 � 16 cos 50º

8 sin 40º8 � 16 cos 40º

8 sin 30º8 � 16 cos 30º

8 sin 20º8 � 16 cos 20º

8 sin 10º8 � 16 cos 10º

(c)

(d)

The maximum of 83.1 square feet occurs when

� ��

3� 60�.

0900

100

� 64�1 � cos ���sin ��

� �16 � 16 cos ���4 sin ��

A��� � 8 � �8 � 16 cos �� �8 sin �

2 �

Page 83: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Section 6.7 Applications and Models 613

66. False. One period is the time for one complete cycle ofthe motion.

68. Aeronautical bearings are always taken clockwise fromNorth (rather than the acute angle from a north-south line).

64. (a)

(c) Period:

This corresponds to the 12 months in a year. Since thesales of outerwear is seasonal this is reasonable.

2�

��6� 12

Month (1 January)↔

Ave

rage

sale

s(i

n m

illio

ns o

f do

llars

)

t

S

2 4 121086

3

6

9

12

15

(b)

Shift:

Note: Another model is

The model is a good fit.

(d) The amplitude represents the maximum displacement fromaverage sales of 8 million dollars. Sales are greatest inDecember (cold weather Christmas) and least in June.�

S � 8 � 6.3 sin��t6

��

2�.

S � 8 � 6.3 cos�� t

6 �S � d � a cos bt

d � 14.3 � 6.3 � 8

2�

b� 12 ⇒ b �

6

t2 4 121086

3

6

9

12

15

Ave

rage

sal

es(i

n m

illio

ns o

f do

llars

)

Month (1 ↔ January)

Sa �

1

2�14.3 � 1.7� � 6.3

65. False. Since the tower is not exactly vertical, a right triangle with sides 191 ft and d is not formed.

67. No. N 24 E means 24 east of north.��

69. passes through

y � 4x � 6

y � 2 � 4x � 4

y � 2 � 4�x � ��1��

y

x−1−2−3−4 1 2 3 4

−1

1

2

3

5

6

7

��1, 2�m � 4, 70. Linear equation through

y � �12x �

16

b �16

0 � �16 � b

0 � �12�1

3� � b

y

x−1−2−3 2 3

−1

−2

−3

1

2

3

y � �12x � b

�13, 0�.m � �

12

71. Passes through and

y � �45

x �225

y � 6 � �45

x �85

y � 6 � �45

x � ��2�

y

x−1−2 1 2 3 4 5

−1

1

2

3

4

6

7m �

2 � 63 � ��2� � �

45

�3, 2���2, 6� 72. Linear equation through and

y � �43

x �13

y �23

� �43�x �

14�

� �43

�1

�34

y

x−1−2−3 2 3

−1

−2

−3

1

2

3

m �13 � ��2

3��

12 �

14

��12

, 13�.�1

4, �

23�

Page 84: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Review Exercises for Chapter 6

614 Chapter 6 Trigonometry

1. � � 40� 2. � � 269�

3. (a)

(b) The angle lies in Quadrant I.

(c) Coterminal angles:70� � 360� � �290�70� � 360� � 430�

70°

y

x

� � 70�

5. (a)

(b) The angle lies in Quadrant III.

(c) Coterminal angles:�110� � 360� � �470��110� � 360� � 250�

−110°

x

y

� � �110�

4. (a)

(b) The angle lies in Quadrant IV.

(c) Coterminal angles:280° � 360° � �80°280° � 360° � 640°

x

280°

y

� � 280�

6. (a)

(b) The angle lies in Quadrant IV.

(c) Coterminal angles:�405° � 360° � �45°�405° � 720° � 315°

x

−405°

y

� � �405�

7. (a)

(b) The angle lies in Quadrant II.

(c) Coterminal angles:

3�

4� 2� � �

5�

4

11�

4� 2� �

3�

4

114π

x

y

� �11�

4 8. (a)

(b) The angle lies in Quadrant I.

(c) Coterminal angles:

2�

9� 2� � �

16�

9

2�

9� 2� �

20�

9

x

y

29π

� �2�

9

Page 85: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Review Exercises for Chapter 6 615

9. (a)

(b) The angle lies in Quadrant II.

(c) Coterminal angles:

�4�

3� 2� � �

10�

3

�4�

3� 2� �

2�

3

−43π

x

y

� � �4�

310. (a)

(b) The angle lies in Quadrant I.

(c) Coterminal angles:

�23�

3� 6� � �

5�

3

�23�

3� 8� �

3

x

y

233π

� � �23�

3

11. 480� � 480� �� rad

180��

8�

3 rad � 8.378 rad

13. �16.5� � �16.5� �� rad

180�� �0.288 rad

12. 120� � 120� �� rad

180�� 2.094 rad

14. �127.5� � �127.5� �� rad

180�� �2.225 rad

17. 84�15� � 84.25� � 84.25� �� rad

180�� 1.470 rad

15. �33�45� � �33.75� � �33.75� �� rad

180�� �

3�

16 rad � �0.589 rad

16. �98�25� � �98.416� �� rad

180�� �1.718 rad

18. 196�77� � 197.283 �� rad

180�� 3.443 rad 19.

5� rad

7�

5� rad

7�

180�

� rad� 128.571�

20.7�

5�

7�

5�

180�

� rad� 252� 21. �

3�

5 rad � �

3� rad

5�

180�

� rad� �108�

22. �11�

6� �

11�

6�

180�

� rad� �330� 23. �3.5 rad � �3.5 rad �

180�

� rad� �200.535�

24. �8.3 � �8.3 �180�

� rad� �475.555� 25. 1.75 rad �

1.75

1 rad �

180�

� rad� 100.268�

27. radians

inchess � r� � 20�23�

30 � � 48.17

138� �138�

180�

23�

3026. 5.7 � 5.7 �

180�

� rad� 326.586�

28.

ms � 11.52

s � r� � 11 � � 60180�� �

113

� rad

Page 86: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

616 Chapter 6 Trigonometry

30.

� 212.1 inchessecond

Linear speed � �272

inches��5� rad

second� 31. radians

square inchesA �12

r 2� �12

�18�2�2�

3 � � 339.29

120� �120�

180�

2�

3

32.

A � 55.31 mm2

A �12

�r 2 �12�

5�

6 �65 2 33.

cot � �adjopp

�54

tan � �oppadj

�45

cos � �adjhyp

�5

�41�

5�4141

sec � �hypadj

��41

5

sin � �opphyp

�4

�41�

4�4141

csc � �hypopp

��41

4

opp � 4, adj � 5, hyp � �42 � 52 � �41

34.

cot � �adjopp

� 1

sec � �hypadj

�6�2

6� �2

csc � �hypopp

�6�2

6� �2

tan � �oppadj

� 1

cos � �adjhyp

�6

6�2�

�22

sin � �opphyp

�6

6�2�

�22

opp � 6, adj � 6, hyp � 6�2 35.

cot � �adjopp

�4

4�3�

�33

sec � �hypadj

�84

� 2

csc � �hypopp

�8

4�3�

2�33

tan � �oppadj

�4�3

4� �3

cos � �adjhyp

�48

�12

sin � �opphyp

�4�3

8�

�32

adj � 4, hyp � 8, opp � �82 � 42 � �48 � 4�3

36.

cot � �adjopp

�2�14

5

sec � �hypadj

�9

2�14�

9�1428

csc � �hypopp

�95

tan � �oppadj

�5

2�14�

5�1428

cos � �adjhyp

�2�14

9

sin � �opphyp

�59

adj � �92 � 52 � 2�14

opp � 5 hyp � 9

29. (a)

� 66 23� radians per minute

Angular speed ��331

3��2�� radians1 minute (b)

� 400� inches per minute

Linear speed �6�66 2

3�� inches

1 minute

Page 87: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Review Exercises for Chapter 6 617

38.

(a)

(b)

(c)

(d)

csc � ��17

4

csc2 � � �14�

2

� 1 �1716

cot2 � � 1 � csc2 �

cos � �1

sec ��

1�17

��1717

sec � � �17

sec2 � � 1 � 16 � 17

1 � tan2 � � sec2 �

cot � �1

tan ��

14

tan � � 4

40.

(a)

(b)

(c)

(d) sec�90 � �� � csc � � 5

tan � �1

cot ��

1

2�6�

�612

cot � � 2�6

cot2 � � csc2 � � 1 � 52 � 1 � 24

cot2 � � 1 � csc2 �

sin � �1

csc ��

15

csc � � 5

41. tan 33� � 0.6494 42. csc 11� �1

sin 11�� 5.2408 43. sin 34.2� � 0.5621

37.

(a)

(b)

(c)

(d) tan � �sin �cos �

13

2�23

�1

2�2�

�24

sec � �1

cos ��

3

2�2�

3�24

cos � �2�2

3

cos � ��89

cos2 � �89

cos2 � � 1 �19

�13�

2

� cos2 � � 1

sin2 � � cos2 � � 1

csc � �1

sin �� 3

sin � �13

39.

(a)

(b)

(c)

(d) tan � �sin �cos �

14

�154

�1

�15�

�1515

sec � �1

cos ��

4�15

�4�15

15

cos � ��15

4

cos � ��1516

cos2 � �1516

cos2 � � 1 �1

16

�14�

2

� cos2 � � 1

sin2 � � cos2 � � 1

sin � �1

csc ��

14

csc � � 4

44. sec 79.3� �1

cos 79.3�� 5.3860 45. cot 15�14� � cot 15.2333� �

1tan 15.2333�

� 3.6722

Page 88: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

618 Chapter 6 Trigonometry

50.

x �25

tan 52�� 19.5 feet

tan 52� �25x

52°

x

25

51. x � 12, y � 16, r � �144 � 256 � �400 � 20

tan � �y

x�

4

3

cos � �x

r�

3

5

sin � �y

r�

4

5

cot � �x

y�

3

4

sec � �r

x�

5

3

csc � �r

y�

5

4

52.

tan � �yx

� �43

cos � �xr

�35

sin � �yr

� �45

x � 3, y � �4, r � �32 � (�4)2 � 5

cot � �xy

� �34

sec � �rx

�53

csc � �ry

� �54

53.

cot � �x

y�

2

3

5

2

�4

15

sec � �r

x�

�241

6

2

3

��241

4

csc � �r

y�

�241

6

5

2

�2�241

30�

�241

15

tan � �y

x�

5

2

2

3

�15

4

cos � �x

r�

2

3

�241

6

�4

�241�

4�241

241

sin � �y

r�

5

2

�241

6

�15

�241�

15�241

241

r ���2

3�2

� �5

2�2

��241

6

x �2

3, y �

5

254.

cot � �xy

�103

�23

� 5

sec � �rx

2�263

�103

� ��26

5

csc � �ry

2�263

�23

� ��26

tan � �yx

�23

�103

�15

cos � �xr

�103

2�263

� �5�26

26

sin � �yr

�23

2�263

� ��2626

r ����103 �

2

� ��23�

2

�2�26

3

x � �103

, y � �23

49.

� 71.3 meters

x � 3.5 sin 1�10��1000�1 10'°

x

Not drawn to scale

3.5 km sin 1�10� �x

3.5

46.

� 0.2045

cos 78�11� 58�� � cos 78.1994 47. cos� �

18� � 0.9848 48. tan 5�

6� �0.5774

Page 89: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Review Exercises for Chapter 6 619

55.

cot � �xy

��1�29�2

� �19

sec � �rx

��82�2�1�2

� ��82

csc � �ry

��82�2

9�2�

�829

tan � �yx

�9�2

�1�2� �9

cos � �xr

��1�2�82�2

� �1

�82� �

�8282

sin � �yr

�9�2

�82�2�

9�82

�9�82

82

r ����12�

2

� �92�

2

��82

2

x � �0.5 � �12

, y � 4.5 �92

56.

tan � �yx

�0.40.3

�43

cos � �xr

�0.30.5

�35

sin � �yr

�0.40.5

�45

r � �(0.3)2 � (0.4)2 � 0.5

x � 0.3, y � 0.4

cot � �xy

�0.30.4

�34

sec � �rx

�0.50.3

�53

csc � �ry

� �0.50.4

�54

57.

cot � �x

y�

x

4x�

1

4

sec � �r

x�

�17x

x� �17

csc � �r

y�

�17x

4x�

�17

4

tan � �y

x�

4x

x� 4

cos � �x

r�

x�17x

��17

17

sin � �y

r�

4x�17x

�4�17

17

r � �x2 � �4x�2 � �17x

x � x, y � 4x

�x, 4x �, x > 0 58.

cot � �xy

��2x�3x

�23

sec � �rx

��13x�2x

� ��13

2

csc � �ry

��13x�3x

� ��13

3

tan � �yx

��3x�2x

�32

cos � �xr

��2x�13x

� �2�13

13

sin � �yr

��3x�13x

� �3�13

13

r � �(�2x)2 � (�3x )2 � �13x

x � �2x, y � �3x, x > 0

59. is in Quadrant IV.

cot � �x

y� �

5�11

11tan � �

y

x� �

�11

5

sec � �r

x�

6

5cos � �

x

r�

5

6

csc � �r

y� �

6�11

11sin � �

y

r� �

�11

6

r � 6, x � 5, y � ��36 � 25 � ��11

sec � �6

5, tan � < 0 ⇒ � 60.

cot � �xy

���5

2

sec � �rx

�3

��5� �

3�55

csc � �32

tan � �yx

�2

��5� �

2�55

cos � �xr

� ��53

sin � �yr

�23

r � 3, y � 2, x � ��32 � 22 � ��5

⇒ � is in Quadrant II. csc � �32

, cos � < 0

Page 90: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

620 Chapter 6 Trigonometry

61.

cot � �xy

��4�5

�45

sec � �rx

��41�4

� ��41

4

csc � �ry

��41�5

� ��41

5

tan � �54

cos � �xr

��4�41

� �4�41

41

sin � �yr

��5�41

� �5�41

41

r � �(�4)2 � (�5)2 � �41

x � �4, y � �5

tan � �54

, cos � < 0 ⇒ � is in Quadrant III. 62. is in Quadrant II.

cot � �x

y� �

�55

3

sec � �r

x� �

8�55

� �8�55

55

csc � �r

y�

8

3

tan � �y

x� �

3�55

� �3�55

55

cos � �x

r� �

�55

8

sin � �y

r�

3

8

y � 3, r � 8, x � ��55

sin � �3

8, cos � < 0 ⇒ �

63.

cot � �x

y� �

5

12tan � �

yx

� �125

sec � �r

x� �

13

5cos � �

x

r� �

5

13

csc � �r

y�

13

12sin � �

y

r�

12

13

sin � > 0 ⇒ � is in Quadrant II ⇒ y � 12, x � �5

tan � �y

x� �

12

5 ⇒ r � 13 64.

cot � �x

y�

�2�21

� �2�21

21

sec � �r

x�

5

�2� �

5

2

csc � �r

y�

5�21

�5�21

21

tan � �y

x� �

�21

2

sin � �y

r�

�21

5

sin � > 0 ⇒ � is in Quadrant II ⇒ y � �21

cos � �x

r�

�2

5 ⇒ y2 � 21

65.

�� � 264� � 180� � 84�

′θ

264°

x

y� � 264� 66.

�� � 85�

′θ

635°

x

y� � 635� � 720� � 85�

67.

�� � � �4�

5�

5

�6�

5� 2� �

4�

5 ′θx

y

65π−

� � �6�

568.

�� ��

3

� 6� ��

3

′θ

x

y

173π

� �17�

3�

18�

3�

3

Page 91: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Review Exercises for Chapter 6 621

69.

tan �

3� �3

cos �

3�

12

sin �

3�

�32

70.

tan �

4�

�2

2

�2

2� 1

cos �

4�

�2

2

sin �

4�

�2

271.

tan 5�

6� �tan

6� �

�33

cos 5�

6� �cos

6� �

�32

sin 5�

6� sin

6�

12

72.

tan 5�

3� �

�3

2

1

2� ��3

cos 5�

3� cos�2� �

5�

3 � � cos �

3�

1

2

sin 5�

3� �sin�2� �

5�

3 � � �sin �

3� �

�3

273.

tan��7�

3 � � �tan �

3� ��3

cos��7�

3 � � cos �

3�

12

sin��7�

3 � � �sin �

3� �

�32

74. is coterminal with

tan��5�

4 � ��2

2 ��

�2

2 � � �1

cos��5�

4 � � cos 3�

4� �cos�� �

3�

4 � � �cos �

4� �

�2

2

sin��5�

4 � � sin 3�

4� sin�� �

3�

4 � � sin �

4�

�2

2

3�

4.�

5�

4

75.

tan 495� � �tan 45� � �1

cos 495� � �cos 45� � ��22

sin 495� � sin 45� ��22

76.

tan 120� ��32

��12� � ��3

cos 120� � �cos�180� � 120�� � �cos 60� � �12

sin 120� � sin�180� � 120�� � sin 60� ��32

77.

tan��150�� � tan 30� ��33

cos��150�� � �cos 30� � ��32

sin��150�� � �sin 30� � �12

78. is coterminal with

tan��420�� � ��32

12

� ��3

cos��420�� � cos 300� � cos�360� � 300�� � cos 60� �12

sin��420�� � sin 300� � �sin�360� � 300�� � �sin 60� � ��32

300�.�420�

79. sin 4� � 0.0698 80. tan 231� � 1.2349 81. sec 2.8 �1

cos 2.8� �1.0613

Page 92: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

622 Chapter 6 Trigonometry

82. cos 5.5 � 0.7087 83. sin��17�

15 � � 0.4067

85.

tan 2�

3�

yx

��3�2�1�2

� ��3

cos 2�

3� x � �

12

sin 2�

3� y �

�32

t �2�

3, �x, y� � ��

12

, �32 � 86.

tan�7�

4 � � �1

sin�7�

4 � � ��22

x

y

2, ( (22

2−

cos�7�

4 � ��22

84. tan��25�

7 � � 4.3813

87.

tan 7�

6�

yx

��1�2

��3�2�

1�3

��33

cos 7�

6� x � �

�32

sin 7�

6� y � �

12

t �7�

6, �x, y� � ��

�32

, �12� 88.

tan�3�

4 � � �1

sin�3�

4 � ��22

x

y2 , ( (2

22

−cos�3�

4 � � ��22

89.

Period: 2�

Amplitude: 1 2

1

−2

x

− ππ322

yy � sin x 90.

Amplitude: 1

Period: 2�

x

2

−1

−2

2πππ−

yy � cos x

91.

Amplitude: 3

Period:2�

2�� 1

1 2 3

1

2

3

x

yy � 3 cos 2�x 92.

Amplitude: 2

Period:2�

�� 2

−3

−2

−1

1

3

x1

yy � �2 sin �x

93.

Amplitude: 5

Period:2�

25

� 5�

−6

−2

2

4

6

xπ6

yf �x� � 5 sin

2x

594.

Amplitude: 8

Period:2�

14

� 8�

−8

−6

−4

8

x

y

π8π4

f (x) � 8 cos��x4�

Page 93: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Review Exercises for Chapter 6 623

95.

Amplitude: 1

Period:

Shift the graph of

two units upward.

y � sin x

2�

4

3

2

−1

−2

xπππ 2

y

y � 2 � sin x 96.

Amplitude: 1

Period: 2321−2−3

x

−1

−2

−3

−5

−6

−1

yy � �4 � cos �x

97.

Amplitude:

Period: 2�

52

−4

−3

−2

−1

1

3

4

yg�t� �52 sin�t � �� 98.

Amplitude: 3

Period: 2�

−4

−3

1

2

3

4

yg(t) � 3 cos(t � �)

99.

(a)

(b) f �11

264

� 264 cycles per second

y � 2 sin�528�x�

a � 2, 2�

b�

1264

⇒ b � 528�

y � a sin bx100. (a)

(b) Period:

12 months 1 year, so this is expected.

(c) Amplitude: 1.41

The amplitude represents the maximum change intime from the average time of sunset.�d � 18.09�

2�

���6� � �2��6� � 12

014

12

22

s�t� � 18.09 � 1.41 sin��t6

� 4.60�

101.

4

3

2

1

πx

y

f �x� � tan x 102.

Graph of shifted to theright by

t

1

2

3

y

π2

π2

��4.tan�t�

f (t) � tan�t ��

4� 103.

x

4

3

2

1

−3

−4

ππ−

y

f �x� � cot x

Page 94: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

116. sin�1(0.89) � 1.10

118. arccos��22 � �

4120. cos�1��3

2 � ��

6

122. radiansarccos(�0.888) � 2.66

115. sin�1� � 0.44� � � 0.46 radian 117. arccos �32

��

6

119. cos�1��1� � �

121. arccos 0.324 � 1.24 radians 123. tan�1��1.5� � �0.98 radian

112. arcsin(�1) � ��

2113. arcsin 0.4 � 0.41 radian 114. arcsin(0.213) � 0.21

624 Chapter 6 Trigonometry

104.

Period:

t

3

2

1

y

ππ−

2

g(t) � 2 cot 2t 105.

Graph first.

ππ−−1

−2

−3

−4

x

y

y � cos x

f �x� � sec x

107.

Graph first.

4

3

2

1

−3

−4

x

− ππ322

y

y � sin x

f �x� � csc x

106.

Graph of sec t shifted to the right

by .

1

y

4

h(t) � sec�t ��

4�

109.

Graph

As

−9

−6

9

6

x →�, f �x� →�.

y3 � �x

y2 � x

y1 � x cos�x�

f �x� � x cos x108.

2

y

f (t) � 3 csc�2t ��

4�

110.

Graph .y � ±ex first

−1

−150

7

300g(x) � ex cos x 111. arcsin ��12� � �arcsin

12

� ��

6

124. radianstan�1(8.2) � 1.45

Page 95: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Review Exercises for Chapter 6 625

128.

−1.5 1.5

�2

−�2

f �x� � �arcsin 2x

127.

−4 4

�2

−�2

f �x� � arctan�x2� � tan�1�x

2�

129.

Then

and cos � � 45.tan � �34

θ4

53

Let � � arctan 34.

cos�arctan 34� �45. Use a right triangle.

130.

Use a right triangle. Let

Then

and

tan�arccos 35� � tan u �43.

cos u �35

u � arccos 35.

u

45

3

tan�arccos 35� �43 131.

Use a right triangle. Let

Then

and sec � �135 .tan � �

125

� � arctan 125 .

θ

5

1213

sec�arctan 125 � �

135

132.

Use a right triangle. Let Then

and

cot�arcsin��1213� � cot u � �

512.

sin u � �1213

u � arcsin��1213�. u

−12

5

13

�arcsin��1213� � �

512

126.

−1.5 1.5

�3

0

y � 3 arccos x125.

−1.5 1.5

−�

f �x� � 2 arcsin x � 2 sin�1�x�

133. Let

Then and

��4 � x2

x.

tan y � tan�arccos�x2��

cos y �x2 2

yx

4 − x 2

y � arccos�x2�. 134.

sec � �1

�x�2 � x�

� �x�2 � x�

cos � � �12 � �x � 1�2

x −11

θ

12 − (x − 1)2

sin � � x � 1

��

2≤ � ≤

2� � arcsin�x � 1� ⇒

sec�arcsin�x � 1��

Page 96: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

141. False. The sine or cosine functions are often useful for modeling simple harmonic motion.

142. True. The inverse sine, , cannot be defined asa function over any interval that is greater than the

interval defined as ��

2≤ y ≤

2.

y � arcsin x

143. False. For each there corresponds exactly one value of .y

� 144. False.

is not in the range of the arctan function.3�

4

arctan(�1) � ��

4

626 Chapter 6 Trigonometry

135.

� � arctan�7030� � 66.8�

tan � �7030

136.

ftr �1808

sin 25.2�� 4246.33

r

25.2°

1808 ft

139.

b � 42.43 nautical miles east6045°a

bN

S

EW

sin 45� �b

60 ⇒

a � 42.43 nautical miles north

cos 45� �a

60 ⇒ 140. High point at time

Period

d � 0.75 cos�2�t3 �

� 3 seconds �2�

b ⇒ b �

2�

3

a �12

�1.5� � 0.75 inches

d � a cos btt � 0 ⇒

138.

The distance is 1221 miles and the bearing is 85.6 .�

sec 4.4� �D

1217 ⇒ D � 1217 sec 4.4� � 1221

tan � �93

1217 ⇒ � � 4.4�

sin 25� �d4

810 ⇒ d4 � 342

cos 48� �d3

650 ⇒ d3 � 435

cos 25� �d2

810 ⇒ d2 � 734

sin 48� �d1

650 ⇒ d1 � 483

48°

48°

65°25°

d3d4

d d1 2

810650

D

B

CA θ

N

S

W E

d1 � d2 � 1217

d3 � d4 � 93

137.

h � 25 tan 21� � 9.6 feet

tan 21� �h

25

21°

h

25 feet

Page 97: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Review Exercises for Chapter 6 627

145.

Matches graph �d�.

Period: 2�

Amplitude: 3

y � 3 sin x 147.

Matches graph �b�.

Period: 2

Amplitude: 2

y � 2 sin �x146.

Amplitude:

Period:

graph is reflected in the x-axis

Matches graph (a).

a < 0 ⇒

2�

3

y � �3 sin x

148.

Amplitude:

Period:

Matches graph (c).

4�

2

y � 2 sin x2

149. is undefined at the zeros of

since sec � �1

cos � .

g��� � cos �f ��� � sec �

150. (a)

(b) tan�� ��

2� � �cot �

151. The ranges for the other four trigonometric functions arenot bounded. For the range is

For the range is���, �1 � �1, ��.

y � sec x and y � csc x,���, ��.y � tan x and y � cot x,

152.

(a) If is changed from to the amplitude of eachoscillation is increased.

(b) If k is changed from to the oscillations aredamped more quickly.

(c) If b is changed from 6 to 9, the frequency of theoscillations increases.

13,1

10

13,1

5A

y � Ae�kt cos bt �15 e�t�10 cos 6t

0.1 0.4 0.7 1.0 1.3

�0.2776�0.6421�1.1872�2.3652�9.9666�cot �

�0.2776�0.6421�1.1872�2.3652�9.9666tan�� ��

2��

153. (a)

12θ

x

� 72�tan � � ��

� 72 tan � � 72�

�1

2�12��12 tan �� �

1

2�122����

� �1

2bh� � �1

2r 2��

Area � Area of triangle � Area of sector

x � 12 tan �

tan � �x

12(b)

As .

The area increases without bound as � approaches �

2.

� ⇒ �2

, � ⇒ �

�2

00

800

Page 98: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

628 Chapter 6 Trigonometry

154. (a)

The area function increases more rapidly than thearc length function because it is a function of thesquare of the radius, while the arc length function isa function of the radius.

00 6

4

A s

s � r� � 0.8r, r > 0

A �12

r 2� �12 r 2�0.8� � 0.4r 2, r > 0 (b)

03

30

A s

0

s � r� � 10�, � > 0

A �12

r 2� �12�10 2�� � 50�, � > 0

155. Answers will vary.

Problem Solving for Chapter 6

1. (a)

revolutions

radians or

(b) s � r� � 47.25�5.5�� � 816.42 feet

990�� � �114 ��2�� �

11�

2

13248

�114

8:57 � 6:45 � 2 hours 12 minutes � 132 minutes 2. Gear 1:

Gear 2:

Gear 3:

Gear 4:

Gear 5:2419

�360�� � 454.737� � 7.94 radians

4032

�360�� � 450� �5�

2 radians

2422

�360�� � 392.727� � 6.85 radians

2426

�360�� � 332.308� � 5.80 radians

2432

�360�� � 270� �3�

2 radians

3. (a)

d �3000

sin 39�� 4767 feet

sin 39� �3000

d(b)

x �3000

tan 39�� 3705 feet

tan 39� �3000

x(c)

w � 3000 tan 63� � 3705 � 2183 feet

3000 tan 63� � w � 3705

tan 63� �w � 3705

3000

4. (a) are all similar triangles since they all have the same angles. is part of all three triangles and Thus,

(b) Since the triangles are similar, the ratios of corresponding sides are equal.

(c) Since the ratios: it does not matter which triangle is used to calculate sin A.

Any triangle similar to these three triangles could be used to find sin A. The value of sin A would not change.

(d) Since the values of all six trigonometric functions can be found by taking the ratios of the sides of a right triangle,similar triangles would yield the same values.

opphyp

�BCAB

�DEAD

�FGAF

� sin A

BCAB

�DEAD

�FGAF

�B � �D � �F.�C � �E � �G � 90�.�A�ABC, �ADE, and �AFG

Page 99: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Problem Solving for Chapter 6 629

5. (a)

h is even.

−1

−2

3

2� �

h�x� � cos2 x

6. Given: f is an even function and g is an odd function.

(a)

since f is even

Thus, h is an even function.

� h�x�

� � f �x��2

h��x� � � f ��x��2

h�x� � � f �x��2

7. If we alter the model so that we can use either a sine or a cosine model.

For the cosine model we have:

For the sine model we have:

Notice that we needed the horizontal shift so that the sine value was one when .

Another model would be:

Here we wanted the sine value to be 1 when t � 0.

h � 51 � 50 sin�8� t �3�

2 �t � 0

h � 51 � 50 sin�8� t ��

2�h � 51 � 50 cos�8� t�

b � 8�

d �12

�max � min� �12

�101 � 1� � 51

a �12

�max � min� �12

�101 � 1� � 50

h � 1 when t � 0,

8.

(a)

(c) Amplitude: 20

The blood pressure ranges between and

(e) Period

64 �60

�2��b� ⇒ b �6460

� 2� �3215

�6064

�1516

sec

100 � 20 � 120.100 � 20 � 80

070

5

130

P � 100 � 20 cos�8�

3t�

(b)

This is the time between heartbeats.

(d) Pulse rate �60 sec�min34 sec�beat

� 80 beats�min

Period �2�

�8�

3 ��

68

�34

sec

(b)

h is even.

−1

−2

3

2� �

h�x� � sin2 x

(b)

since g is odd

Thus, h is an even function.

� h�x�

� �g�x��2

� ��g�x��2

h��x� � �g��x��2

h�x� � �g�x��2

Conjecture: The square of either an even function or an odd function is an even function.

Page 100: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

630 Chapter 6 Trigonometry

9. Physical (23 days):

Emotional (28 days):

Intellectual (33 days):

(a)

(b) Number of days since birth until September 1, 2006:

5 11 31 1

20 years leap years remaining August days day inJuly days September

All three drop early in the month, then peak toward the middle of the month, and drop againtoward the latter part of the month.

(c) For September 22, 2006, use

I � 0.945

E � 0.901

P � 0.631

t � 7369.

−2

7349 7379

2

P

I

E

t � 7348

���

����t � 365 � 20

−2

7300 7380

2

P IE

I � sin 2� t33

, t ≥ 0

E � sin 2� t28

, t ≥ 0

P � sin 2� t23

, t ≥ 0

10.

(a)

(b) The period of

The period of

(c) is periodic since the sineand cosine functions are periodic.h�x� � A cos x � sin �x

g�x� is �.

f �x� is 2�.

−6

6

� �

g

f

g�x� � 2 cos 2x � 3 sin 4x

f �x� � 2 cos 2x � 3 sin 3x 11. (a) Both graphs have a period of 2 and intersect when They should also intersect when

and

(b) The graphs intersect when

(c) Since and the graphs will intersect again at these values. Therefore f �13.35� � g��4.65�.

�4.65 � 5.35 � 5�2�13.35 � 5.35 � 4�2�

x � 5.35 � 3�2� � �0.65.

x � 5.35 � 2 � 7.35.x � 5.35 � 2 � 3.35x � 5.35.

Page 101: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

Problem Solving for Chapter 6 631

12. (a) is true since this is a two period horizontal shift.

(b) is not true.

is a horizontal translation of

is a doubling of the period of

(c) is not true.

is a horizontal

translation of by half a period.

For example, sin�12

�� � 2��� � sin�12

��.

f �12

t�f �1

2 �t � c�� � f �1

2t �

12

c�

f �12

�t � c�� � f �12

t�

f �t�.f �12

t�

f �t�.f �t �12

c�

f �t �12

c� � f �12

t�

f �t � 2c� � f �t� 13.

(a)

(b)

(c) feet

(d) As you more closer to the rock, decreases, whichcauses y to decrease, which in turn causes d todecrease.

�1

d � y � x � 3.46 � 1.71 � 1.75

tan �1 �y2

⇒ y � 2 tan 60� � 3.46 feet

tan �2 �x2

⇒ x � 2 tan 40.52� � 1.71 feet

�2 � 40.52�

sin �2 �sin �1

1.333� sin 60�

1.333� 0.6497

sin �1

sin �2� 1.333

2 ft

xy

d

θθ

1

2

14.

(a)

The graphs are nearly the same for �1 < x < 1.

−2

2

�2

−�2

arctan x � x �x3

3�

x5

5�

x7

7

(b)

The accuracy of the approximation improved slightly byadding the next term �x9�9�.

−2

2

�2

−�2

Page 102: CHAPTER 6 Trigonometry - De La Salle High School · 180 0.009 radians 59. 7 5 7 180 11 25.714 60. 5 11 180 81.818 61. 15 8 15 8 180 337.500 ... 538 Chapter 6 Trigonometry 83. s r

632 Chapter 6 Trigonometry

Chapter 6 Practice Test

1. Express 350° in radian measure. 2. Express in degree measure.�5���9

3. Convert to decimal form.135�14�12� 4. Convert form.�22.569� to D�M�S�

5. If use the trigonometric identities to find tan .�cos � �23, 6. Find given .sin � � 0.9063�

7. Solve for in the figure below.

20°

35

x

x 8. Find the magnitude of the reference angle for .� � �6���5

9. Evaluate .csc 3.92 10. Find .sec � given that � lies in Quadrant III and tan � � 6

11. Graph y � 3 sin x

2. 12. Graph y � �2 cos�x � ��.

13. Graph y � tan 2x. 14. Graph .y � �csc�x ��

4�

15. Graph using a graphing calculator.y � 2x � sin x, 16. Graph using a graphing calculator.y � 3x cos x,

17. Evaluate .arcsin 1 18. Evaluate arctan��3�.

19. Evaluate sin�arccos 4

�35�. 20. Write an algebraic expression for .cos�arcsin x

4�

For Exercises 21–23, solve the right triangle.

A C

B

ca

b

21. A � 40�, c � 12 22. B � 6.84�, a � 21.3 23. a � 5, b � 9

24. A 20-foot ladder leans against the side of a barn. Find the height of the top of the ladder if the angle of elevation of the ladder is .67°

25. An observer in a lighthouse 250 feet above sea level spots a ship off the shore. If the angle of depression to the ship is , how far out is the ship?5°