103
CHAPTER 4 Trigonometry Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 335 Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . . 344 Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 350 Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 360 Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 373 Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 386 Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 397 Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 410 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

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Page 1: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

C H A P T E R 4Trigonometry

Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 335

Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . . 344

Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 350

Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 360

Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 373

Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 386

Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 397

Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 410

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

Page 2: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

335

C H A P T E R 4Trigonometry

Section 4.1 Radian and Degree Measure

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

■ Types of Angles:

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

(b) Right: Measure 90�.

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

(d) Straight: Measure 180�.

■ and are complementary if They are supplementary if

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

■ To convert degrees to radians, use radians.

■ To convert radians to degrees, use 1 radian

■ one minute of 1�.

■ one second of 1�.

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

■ Linear speed

■ Angular speed � ��t � s�rt

�arc length

time�

st

�s � r�

� 1�60 of 1� � 1�36001� �

� 1�601� �

� �180����.1� � ��180

� � � 180�.� � � 90�.�

Vocabulary Check

1. Trigonometry 2. angle

3. coterminal 4. radian

5. acute; obtuse 6. complementary; supplementary

7. degree 8. linear

9. angular 10. A �12r 2�

1.

The angle shown is approximately2 radians.

2.

The angle shown is approximately5.5 radians.

3.

The angle shown is approximatelyradians.3

4.

The angle shown is approximatelyradians.4

5.

The angle shown is approximately 1 radian.

6.

The angle shown is approximately6.5 radians.

Page 3: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

336 Chapter 4 Trigonometry

7. (a) Since lies in Quadrant I.

(b) Since lies in Quadrant III.� <7�

5<

3�

2 ;

7�

5

0 <�

5<

2 ;

5

9. (a) Since lies in Quadrant IV.

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

2; 2

2 <

12 < 0 ;

12

8. (a) Since lies in Quadrant III.

(b) Since lies in Quadrant III.� <9�

8<

3�

2;

9�

8

� <11�

8<

3�

2;

11�

8

10. (a) Since lies in Quadrant IV.

(b) Since lies in Quadrant II.11�

9

3�

2<

11�

9< �,

2< 1 < 0; 1

11. (a) Since lies in Quadrant III.

(b) Since lies in Quadrant II.�

2< 2.25 < � ; 2.25

� < 3.5 <3�

2 ; 3.5 12. (a) Since lies in Quadrant IV.

(b) Since 4.25 lies in Quadrant II.3�

2 < 4.25 < � ;

3�

2< 6.02 < 2� ; 6.02

13. (a)

(b)

− 23π

x

y

2�

3

54π

x

y

5�

414. (a)

(b)

x

y

52π

5�

2

− 74π

y

x

7�

4 15. (a)

(b)

−3

x

y

3

116π

x

y

11�

6

16. (a) 4

4

x

y (b)

x

y7�

Page 4: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.1 Radian and Degree Measure 337

26.

The angle shown is approximately120�.

27.

The angle shown is approximately 60º.

28.

The angle shown is approximately330�.

17. (a) Coterminal angles for

(b) Coterminal angles for

5�

6 2� �

7�

6

5�

6� 2� �

17�

6

5�

6

6 2� �

11�

6

6� 2� �

13�

6

618. (a)

(b)

11�

6 2� �

23�

6

11�

6� 2� �

6

7�

6 2� �

5�

6

7�

6� 2� �

19�

619. (a) Coterminal angles for

(b) Coterminal angles for

12 2� �

23�

12

12� 2� �

25�

12

12

2�

3 2� �

4�

3

2�

3� 2� �

8�

3

2�

3

20. (a)

(b)

2�

15 2� �

32�

15

2�

15� 2� �

28�

15

9�

4� 4� �

7�

4

9�

4� 2� �

421. (a) Complement:

Supplement:

(b) Complement: Not possible, is greater than

Supplement: � 3�

4�

4

2.

3�

4

� �

3�

2�

3

2

3�

6

22. (a) Complement:

Supplement:

(b) Complement: Not possible, is greater than

Supplement: � 11�

12�

12

2.

11�

12

� �

12�

11�

12

2

12�

5�

1223. (a) Complement:

Supplement:

(b) Complement: Not possible, 2 is greater than

Supplement: � 2 � 1.14

2.

� 1 � 2.14

2 1 � 0.57

24. (a) Complement: Not possible, 3 is greater than

Supplement:

(b) Complement:

Supplement: � 1.5 � 1.64

2 1.5 � 0.07

� 3 � 0.14

2. 25.

The angle shown is approximately .210º

Page 5: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

338 Chapter 4 Trigonometry

31. (a) Since lies in Quadrant II.

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

90� < 130� < 180�, 130� 32. (a) Since lies in Quadrant I.

(b) Since lies inQuadrant III.

180� < 257� 30� < 270�, 257� 30�

0� < 8.3� < 90�, 8.3�

33. (a) Since lies in Quadrant III.

(b) Since lies inQuadrant I.

360� < 336� < 270�, 336�

180� < 132� 50� < 90�, 132� 50� 34. (a) Since lies inQuadrant II.

(b) Since lies in Quadrant IV.90� < 3.4� < 0�, 3.4�

270� < 260� < 180�, 260�

35. (a)

(b)

150°

x

y

150�

x30°

y

30� 36. (a)

(b)

−120°

x

y

120�

−270°

x

y

270�

37. (a)

(b)

480°

x

y

480�

405°

x

y

405�

29.

The angle shown is approximately .165�

30.

The angle shown is approximately 10�.

38. (a)

(b)

−600°

x

y

600�

−750°x

y

750�

Page 6: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.1 Radian and Degree Measure 339

54. (a)

(b)34�

15�

34�

15 �180�

� � � 408�

11�

6�

11�

6 �180�

� � � 330� 55.

� 2.007 radians

115� � 115� �

180� 56.

� 1.525 radians

87.4� � 87.4�� �

180��

57. 216.35� � 216.35� �

180� � 3.776 radians

59. 532� � 532� �

180� � 9.285 radians

58. radians48.27� � 48.27�� �

180�� � 0.842

60. 345� � 345�� �

180�� � 6.021 radians

39. (a) Coterminal angles for 45

45 � 360 � 405

45 360 � 315

(b) Coterminal angles for

36 � 360 � 324

36 360 � 396���

���

36�

���

���

� 40. (a)

(b)

420� � 360� � 60�

420� � 720� � 300�

120� 360� � 240�

120� � 360� � 480� 41. (a) Coterminal angles for

(b) Coterminal angles for

180� 360� � 540�

180� � 360� � 180�

180�

240� 360� � 120�

240� � 360� � 600�

240�

42. (a)

(b)

230� 360� � 130�

230� � 360� � 590�

420� � 360� � 60�

420� � 720� � 300� 43. (a) Complement: 90 18 � 72

Supplement:

(b) Complement: Not possible, 115 is greater than 90 .

Supplement: 1180� 115� � 65�

��

180� 18� � 162�

���

44. (a) Complement:

Supplement:

(b) Complement:

Supplement: 180� 64� � 116�

90� 64� � 26�

180� 3� � 177�

90� 3� � 87� 45. (a) Complement:

Supplement:

(b) Complement: Not possible, is greater than

Supplement: 180� 150� � 30�

90�.150�

180� 79� � 101�

90� 79� � 11�

46. (a) Complement: Not possible, is greater than

Supplement:

(b) Complement: Not possible, is greater than

Supplement: 180� 170� � 10�

90�.170�

180� 130� � 50�

90�.130� 47. (a)

(b) 150� � 150� �

180� �5�

6

30� � 30� �

180� ��

6

48. (a)

(b) 120� � 120�� �

180�� �2�

3

315� � 315�� �

180�� �7�

449. (a)

(b) 240� � 240� �

180� � 4�

3

20� � 20� �

180� � �

950. (a)

(b) 144� � 144�� �

180�� �4�

5

270� � 270�� �

180�� � 3�

2

51. (a)

(b)7�

6�

7�

6 �180

� �� � 210�

3�

2�

3�

2 �180

� �� � 270� 52. (a)

(b)�

9�

9�180�

� � � 20�

7�

12�

7�

12�180�

� � � 105� 53. (a)

(b) 11�

30�

11�

30 �180

� �� � 66�

7�

3�

7�

3 �180

� �� � 420�

Page 7: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

340 Chapter 4 Trigonometry

61. radian0.83� � 0.83� �

180� � 0.014 62. 0.54� � 0.54�� �

180�� � 0.009 radians

63.�

7�

7�180

� �� � 25.714� 64.5�

11�

5�

11�180�

� � � 81.818� 65.15�

8�

15�

8 �180

� �� � 337.500�

66.13�

2�

13�

2 �180�

� � � 1170.000� 67.

� 756.000�

4.2� � 4.2��180

� �� 68. 4.8� � 4.8��180�

� � � 864.000�

69. 2 � 2�180

� ��� 114.592� 70. 0.57 � 0.57�180�

� � � 32.659�

71. (a)

(b) 128� 30� � 128� �3060�� � 128.5�

54� 45� � 54� � �4560�� � 54.75�

73. (a)

(b) 330� 25� � �330 �25

3600�� � 330.007�

85� 18� 30� � �85 �1860 �

303600�� � 85.308�

75. (a)

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

240.6� � 240� � 0.6�60�� � 240� 36�

77. (a)

(b)

� 3� 34� 48�

� �3� � 34� � 0.8�60� ��

3.58� � �3� � �0.58��60� ��

2.5� � 2� 30�

72. (a)

(b) � 2� � 0.2� � 2.2�2�12� � 2� � �1260��

� 245.167�� 245� � 0.167�245�10� � 245� � �1060��

74. (a)

(b)

� 408.272�

� �408� � 0.2667� � 0.0056��

408� 16� 20� � �408� � �1660�� � � 20

3600�� � � 135� 0.01� � 135.01�

135� 36� � 135� � 363600��

76. (a)

(b) � 0� 27�� 0� � 27�0.45� � 0� � �0.45��60� �

� 345� 7� 12�

� �345� � 7� � 0.2�60� ��

345.12� � �345� � �0.12��60� ��

78. (a)

(b)

� 0� 47� 11.4� � 0� � 47� � 11.4�

� 0� � 47� � �0.19��60� �

0.7865 � 0� � �0.7865��60��

� 0� 21� 18� � �0� � 21� � 18� �

� �0� � 21� � �0.3��60� ��

0.355� � �0� � �0.355��60� ��

79.

� �65 radians

6 � 5�

s � r� 81.

� �327 � 4 4

7 radians

32 � 7�

s � r�80.

� �2910 radians

29 � 10 �

s � r�

82.

Because the angle represented isclockwise, this angle is radians.

45

� �60

75�

4

5 radians

60 � 75�

s � r� 83.

� �6

27�

2

9 radians

6 � 27�

s � r� 84. feet, feet

� �s

r�

8

14�

4

7 radians

s � 8r � 14

Page 8: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.1 Radian and Degree Measure 341

100. � �s

r�

24

5� 4.8 radians � 4.8�180�

� � � 275�

91.

� 8.38 square inches

A �12

�4�2��

3� �8�

3 square inches

A �12

r 2� 92.

� 56.55 mm2

� 18� mm2

A �12

r 2� �12

�12�2��

4�

r � 12 mm, � ��

493.

� 12.27 square feet

A �12

�2.5�2�225�� �

180�

A �12

r 2�

94.

square miles� 5.64�21.56

12�A �

12

�1.4�2�330�

180���

r � 1.4 miles, � � 330� 95.

miless � r� � 4000�0.14782� � 591.3

� 8.46972� � 0.14782 radian

� � 41� 15� 50� 32� 47� 39�

96.

s � r� � �4000��0.1715� � 686.2 miles

� 0.1715 radian

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

r � 4000 miles97. � �

s

r�

450

6378� 0.071 radian � 4.04�

98. kilometers

The difference in latitude is about radians � 3.59�.0.062716

0.062716 � �

4006378

� �

400 � 6378�

s � r�

r � 3189 99. � �s

r�

2.5

6�

25

60�

5

12 radian

85.

� �25

14.5�

50

29 radians

25 � 14.5�

s � r� 87. in radians

� 47.12 inches

s � 15�180�� �

180� � 15� inches

s � r�, �86. kilometers,kilometers

� �s

r�

160

80� 2 radians

s � 160r � 80

88. feet,

� 9.42 feet

s � r� � 9��

3� � 3� feet

� � 60� ��

3r � 9 89. in radians

s � 3�1� � 3 meters

s � r�, � 90. centimeters,

� 15.71 centimeters

s � r� � 20��

4� � 5� centimeters

� ��

4r � 20

101. (a) 65 miles per hour feet per minute

The circumference of the tire is feet.

The number of revolutions per minute is

revolutions per minuter �5720

2.5�� 728.3

C � 2.5�

�65�5280�

60� 5720 (b) The angular speed is

Angular speed radians per minute�4576 radians

1 minute� 4576

� �5720

2.5��2�� � 4576 radians

t.

Page 9: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

342 Chapter 4 Trigonometry

102. Linear velocity for either pulley: inches per minute

(a) Angular speed of motor pulley: radians per minute

Angular speed of the saw arbor: radians per minute

(b) Revolutions per minute of the saw arbor: revolutions per minute1700�

2�� 850

� �v

r�

3400�

2� 1700�

� �v

r�

3400�

1� 3400�

1700�2�� � 3400�

103. (a)

(b)

� 9869.84 feet per minute

� 314123

� feet per minute

Linear speed ��7.25

2 in.�� 1 ft

12 in.��5200��2��

1 minute

� 32,672.56 radians per minute

� 10,400� radians per minute

Angular speed ��5200��2�� radians

1 minute104. (a)

(b)

� 628.32 feet per minute

Linear speed � 25�25.13274� feet per minute

r�

t� 200� feet per minute

r � 25 ft

� 25.13 radians per minute

� 8� radians per minute

4 rpm � 4�2�� radians per minute

105. (a)

Interval:

(b)

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

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

�400�, 1000�� radians per minute

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

106.

A �12�

125180�� � �252 112� � 175� � 549.8 square inches

r � 25 14 � 11

R � 25 25125°

14r

A �12

��R2 r2� 107.

� 1496.62 square meters

� 476.39� square meters

�12

�35�2�140��� �

180��140°

35

A �12

r 2�

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

Therefore, the chain moves feet, as does the smaller rear sprocket. Thus, the angle of the smaller sprocket is

and the arc length of the tire in feet is:

—CONTINUED—

14� feet

3 seconds

3600 seconds1 hour

1 mile

5280 feet� 10 miles per hour

Speed �st

��14���31 second

�14�

3 feet per second

s � �4���1412� �

14�

3 feet

s � �r

� �sr

��2���3 feet2�12 feet

� 4�

�r � 2 inches � 2�12 feet�.�2��3

s �13

�2�� �2�

3 feet

s � r�

Page 10: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.1 Radian and Degree Measure 343

115. The arc length is increasing. In order for the angle toremain constant as the radius r increases, the arc length s must increase in proportion to r, as can be seen fromthe formula s � r�.

116. The area of a circle is The circumference of a circle is

For a sector, Thus, for a sector.A ��r��r

2�

1

2�r 2C � s � r�.

Cr

2� A

C �2A

r

C � 2�A

r 2� r

C � 2�r.A � �r2 ⇒ � �A

r2.

108. —CONTINUED—

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

Distance

(c)

(d) The functions are both linear.

� �14�

3 feet per second�� 1 mile

5280 feet��t seconds� �7�

7920 t miles

Distance � Rate � Time

� �14�

3

feetrevolutions��

1 mile5280 feet��n revolutions� �

7�

7920n miles

�14���3

109. False. An angle measure of radians corresponds totwo complete revolutions from the initial to the terminalside of an angle.

4�

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

113. Increases, since the linear speed is proportional to the radius.

110. True. If and are coterminal angles, thenwhere n is an integer. The difference

between is � � n�360�� � 2�n.� and � � � n�360��

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

(b) A negative angle is generated by a clockwise rotation of the terminal side.

(c) Two angles in standard position with the same terminal sides are coterminal.

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

114. 1 radian

so one radian is much larger than one degree.

� �180

� ��� 57.3�,

117.4

42�

4

42�22

�42

8�

22

118.55

210�

52 5

10�

52

12

�5

22�22

�52

4

119. 22 � 62 � 4 � 36 � 40 � 4 � 10 � 210 120.

� 208 � 16 � 13 � 413

172 92 � 289 81

Page 11: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

344 Chapter 4 Trigonometry

Section 4.2 Trigonometric Functions: The Unit Circle

■ You should know the definition of the trigonometric functions in terms of the unit circle. Let t be a real number and the point on the unit circle corresponding to t.

■ The cosine and secant functions are even.

■ The other four trigonometric functions are odd.

■ Be able to evaluate the trigonometric functions with a calculator.

cot��t� � �cot ttan��t� � �tan t

csc��t� � �csc tsin��t� � �sin t

sec��t� � sec tcos��t� � cos t

cot t �x

y, y � 0tan t �

y

x, x � 0

sec t �1

x, x � 0cos t � x

csc t �1

y, y � 0sin t � y

�x, y�

Vocabulary Check

1. unit circle 2. periodic

3. period 4. odd; even

121.

Graph of shifted to the right by two units

y � x5

432−2

3

2

1

−1

−2

−3

x

yy = x5

y = (x − 2)5

f �x� � �x � 2�5 122.

Vertical shift four unitsdownward

4

2

−2

−6

32−2−3 1

y = x5

y = x5 − 4

x

yf �x� � x5 � 4

123.

Graph of reflectedin x-axis and shiftedupward by two units

y � x5

321−2−3

6

5

4

3

1

−1

−2

−3

x

y

y = x5

y = 2 − x5

f �x� � 2 � x5 124.

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

3

2

1

−1

−2

−3

21−3−4−5x

y

y = x5

y = − (x + 3)5 f �x� � ��x � 3�5

Page 12: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.2 Trigonometric Functions: The Unit Circle 345

7. corresponds to ���3

2, �

1

2�.t �7�

68. t �

5�

4, �x, y� � ��

�22

, ��22 �

10. t �5�

3, �x, y� � �1

2, �

�32 �9. corresponds to ��1

2, �

�3

2 �.t �4�

3

11. corresponds to �0, �1�.t �3�

2

13. corresponds to

sin

cos

tan t �y

x� 1

t � x ��2

2

t � y ��2

2

��2

2, �2

2 �.t ��

4

15. corresponds to

sin

cos

tan t �y

x� �

1

�3� �

�3

3

t � x ��3

2

t � y � �1

2

��3

2, �

1

2�.t � ��

6

12. t � �, �x, y� � ��1, 0�

14.

tan �

3�

�3�21�2

� �3

cos �

3�

12

sin �

3�

�32

t ��

3, �x, y� � �1

2, �32 �

16.

tan���

4� ���2�2�2�2

� �1

cos���

4� ��22

sin���

4� � ��22

t � ��

4, �x, y� � ��2

2, �

�22 �

1.

sin csc

cos sec

tan cot � �x

y� �

8

15� �

y

x� �

15

8

� �1

x� �

17

8� � x � �

8

17

� �1

y�

17

15� � y �

15

17

x � �8

17, y �

15

17

3.

sin csc

cos sec

tan cot � �x

y� �

12

5� �

y

x� �

5

12

� �1

x�

13

12� � x �

12

13

� �1

y� �

13

5� � y � �

5

13

x �12

13, y � �

5

13

5. corresponds to ��2

2, �2

2 �.t ��

4

2.

cot � �xy

�125

tan � �y

x�

512

sec � �1x

�1312

cos � � x �1213

csc � �1y

�135

sin � � y �5

13

x �1213

, y �5

13

4.

cot � �xy

�43

tan � �y

x�

34

sec � �1x

� �54

cos � � x � �45

csc � �1y

� �53

sin � � y � �35

x � �45

, y � �35

6. t ��

3, �x y� � �1

2, �32 �

Page 13: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

346 Chapter 4 Trigonometry

19. corresponds to

sin

cos

tan t �y

x� �

1

�3� �

�3

3

t � x ��3

2

t � y � �1

2

��3

2, �

1

2�.t �11�

620.

tan 5�

3�

��3�21�2

� ��3

cos 5�

3�

12

sin 5�

3� �

�32

t �5�

3, �x, y� � �1

2, �

�32 �

21. corresponds to

sin

cos

tan is undefined.t �y

x

t � x � 0

t � y � 1

�0, 1�.t � �3�

222.

tan��2�� �01

� 0

cos��2�� � 1

sin��2�� � 0

t � �2�, �x, y� � �1, 0�

23. corresponds to

sin csc

cos sec

tan cot t �xy � �1t �

y

x � �1

t �1

x� ��2t � x � �

�2

2

t �1

y� �2t � y �

�2

2

���2

2, �2

2 � .t �3�

424.

cot 5�

6�

1tan t

� ��3tan 5�

6�

1�2

��3�2� �

�33

sec 5�

6�

1cos t

� �2�3

3cos

5�

6� �

�32

csc 5�

6�

1sin t

� 2sin 5�

6�

12

t �5�

6, �x, y� � ��

�32

, 12�

25. corresponds to

sin csc

cos sec is undefined.

tan is undefined. cot t �xy � 0t �

y

x

t �1

xt � x � 0

t �1

y� �1t � y � �1

�0, �1�.t � ��

226.

is undefined.

is undefined. cot 3�

2�

0�1

� 0tan 3�

2

sec 3�

2cos

3�

2� 0

csc 3�

2�

1sin t

� �1sin 3�

2� �1

t �3�

2, �x, y� � �0, �1�

17. corresponds to

sin

cos

tan t �y

x� 1

t � x ��2

2

t � y ��2

2

��2

2, �2

2 �.t � �7�

418.

tan��4�

3 � ��3�2�1�2

� ��3

cos��4�

3 � � �12

sin��4�

3 � ��32

t � �4�

3, �x, y� � ��

12

, �32 �

Page 14: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.2 Trigonometric Functions: The Unit Circle 347

29. sin 5� � sin � � 0 31. cos 8�

3� cos

2�

3� �

1

2

33. cos��15�

2 � � cos �

2� 0

35. sin��9�

4 � � sin�7�

4 � � ��2

237.

(a)

(b) csc��t� � �csc t � �3

sin��t� � �sin t � �1

3

sin t �1

3

39.

(a)

(b) sec��t� �1

cos��t�� �5

cos t � cos��t� � �1

5

cos��t� � �1

5

30. cos 5� � cos � � �1

32. sin 9�

4� sin

4�

�22

34. sin 19�

6� sin

7�

6� �

12

36. cos��8�

3 � � cos 4�

3� �

12

38.

(a)

(b) csc t �1

sin t� �

83

sin t � �sin��t� � �38

sin��t� �38

40.

(a)

(b) sec��t� � sec t �1

cos t� �

43

cos��t� � cos t � �34

cos t � �34

41.

(a)

(b) sin�t � �� � �sin t � �4

5

sin�� � t� � sin t �4

5

sin t �4

543. sin

4� 0.7071

45. csc 1.3 �1

sin 1.3� 1.0378

42.

(a)

(b) cos�t � �� � �cos t � �45

cos�� � t� � �cos t � �45

cos t �45

44. tan �

3� 1.7321 46. cot 1 �

1tan 1

� 0.6421

47. cos��1.7� � �0.1288 49. csc 0.8 �1

sin 0.8� 1.3940

51. sec 22.8 �1

cos 22.8� �1.4486

53. (a)

(b) cos 2 � x � �0.4

sin 5 � y � �1

48. cos��2.5� � �0.8011

50. sec 1.8 �1

0051.8� �4.4014 52. sin��0.9� � �0.7833

54. (a)

(b) cos 2.5 � x � �0.8

sin 0.75 � y � 0.7

27. corresponds to

sin csc

cos sec

tan cot t �xy �

�3

3t �

y

x � �3

t �1

x� �2t � x � �

1

2

t �1

y� �

2�3

3t � y � �

�3

2

��1

2, �

�3

2 �.t �4�

328.

cot 7�

4�

1tan t

� �1 tan 7�

4�

��2�2�2�2

� �1

sec 7�

4�

1cos t

� �2 cos 7�

4�

�22

csc 7�

4�

1sin t

� ��2 sin 7�

4� �

�22

t �7�

4, �x, y� � ��2

2, �

�22 �

Page 15: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

348 Chapter 4 Trigonometry

57.

(a)

(b) From the table feature of a graphing utility we see that seconds.

(c) As t increases, the displacement oscillates but decreases in amplitude.

y � 0 when t � 5

y�t� �14 e�t cos 6t

t 0 1

y 0.25 0.0138 0.0883�0.0249�0.1501

34

12

14

58.

(a)

(b)

(c) y�1

2� �1

4 cos 3 � �0.2475 feet

y�1

4� �1

4 cos

3

2� 0.0177 feet

y�0� �1

4 cos 0 � 0.2500 feet

y�t� �1

4 cos 6t 59. False. means the function is odd, not

that the sine of a negative angle is a negative number.

For example:

Even though the angle is negative, the sine value is positive.

sin��3�

2 � � �sin�3�

2 � � ���1� � 1.

sin��t� � �sin t

60. True. since the period of tan is �.tan a � tan�a � 6� � 61. (a) The points have y-axis symmetry.

(b) since they have the same y-value.

(c) since the x-values have theopposite signs.cos�� � t1� � �cos t1

sin t1 � sin �� � t1�

62.

So and are even.

So and are odd.

So and are odd.cot �tan �

cot���� �x

�y� �cot �

cot � �xy

tan���� ��yx

� �tan �

tan � �yx

csc �sin �

csc���� � �1y

� �csc �

x

y

θθ−

(x, y)

(x, −y)

csc � �1y

sin���� � �y � �sin �

sin � � y

cos �sec �

sec � �1x

� sec����

cos � � x � cos���� 63.

f �1�x� �23

x �23

�23

�x � 1�

23

x �23

� y

2x � 3y � 2

x �12

�3y � 2�

y �12

�3x � 2�

f �x� �12

�3x � 2�

55. (a)

(b)

t � 1.82 or 4.46

cos t � �0.25

t � 0.25 or 2.89

sin t � 0.25 56. (a)

(b)

t � 0.7 or t � 5.6

cos t � 0.75

t � 4.0 or t � 5.4

sin t � �0.75

Page 16: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.2 Trigonometric Functions: The Unit Circle 349

64.

f�1�x� � 3�4�x � 1�

y � 3�4�x � 1�

4�x � 1� � y3

x � 1 �14

y3

x �14

y3 � 1

y �14

x3 � 1

f �x� �14

x3 � 1 66.

f �1�x� �2�2x � 1�

x � 1

y �2�2x � 1�

x � 1

y�x � 1� � 4x � 2

xy � y � 4x � 2

xy � 4x � y � 2

x�y � 4� � y � 2

x �y � 2y � 4

y �x � 2x � 4

f �x� �x � 2x � 465.

f �1�x� � �x2 � 4, x ≥ 0

±�x2 � 4 � y

x2 � y2 � 4

x � �y2 � 4

y � �x2 � 4

f �x� � �x2 � 4, x ≥ 2

67.

Intercept:

Vertical asymptote:

Horizontal asymptote: y � 2

x � 3

�0, 0�

y

x−2−4−6 2 4 6 8 10

−2

−4

−6

−8

4

6

8

2

f �x� �2x

x � 3

x 0 1 2 4 5 6

y 0 8 5 4�4�112

�1

68.

Horizontal asymptote:

Vertical asymptote:

Intercept: �0, 0�

x � �3, x � 2

x � 0

y

x−4 4 6 8

−2

−4

−6

−8

2

4

6

8

2−2

f �x� �5x

x2 � x � 6�

5x�x � 3��x � 2�, x � �3, 2

x 0 1 3 5

y 02524

52

�54

52

�103

�54

�2�4�6

69.

Intercepts:

Vertical asymptote:

Horizontal asymptote:

Hole in the graph at �2, 78�

y �12

x � �2

��5, 0�, �0, 54�

y

x−1−5−6 1 2

−1

−2

−3

−4

1

2

3

4

−2

f �x� �x2 � 3x � 10

2x2 � 8�

�x � 5��x � 2�2�x � 2��x � 2� �

x � 52�x � 2�, x � 2

x 0 1 3

y 0 2 145

54

�1�14

�1�3�4�5

Page 17: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

350 Chapter 4 Trigonometry

70.

Vertical asymptote:

Slant asymptote:

intercept:

intercept: ��5.86, 0�x-

�0, 18�y-

y �12

x �74

x �5 ± �89

4; x � �1.11, x � 3.61

y

x−4 46 82−2−6

−2

x �5 ± ���5�2 � �4��2���8�

2�2�

2x2 � 5x � 8 � 0

f �x� �x3 � 6x2 � x � 1

2x2 � 5x � 8�

12

x �74

�15�x � 4�

4�2x2 � 5x � 8�

Section 4.3 Right Triangle Trigonometry

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

(a) (b) (c)

(d) (e) (f)

■ You should know the following identities.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k)

■ You should know that two acute angles are complementary if and that cofunctions of complemen-tary angles are equal.

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

���

� � � � 90�,� and �

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

sin2 � � cos2 � � 1cot � �cos �

sin �tan � �

sin �

cos �

cot � �1

tan �tan � �

1

cot �sec � �

1

cos �

cos � �1

sec �csc � �

1

sin �sin � �

1

csc �

cot � �adj

oppsec � �

hyp

adjcsc � �

hyp

opp

tan � �opp

adjcos � �

adj

hypsin � �

opp

hyp

0 2 3 4 7

9 5 1�294

32

18�

15532�

175�

154y

�1�32�3�4x

Vocabulary Check

1. (i) (v) (ii) (iv) (iii) (vi)

(iv) (iii) (v) (i) (vi) (ii)

2. opposite; adjacent; hypotenuse

3. elevation; depression

oppositeadjacent

� tan �opposite

hypotenuse� sin �

adjacenthypotenuse

� cos �

hypotenuseopposite

� csc �adjacentopposite

� cot �hypotenuse

adjacent� sec �

Page 18: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.3 Right Triangle Trigonometry 351

5.

tan � �opp

adj�

1

2�2�

�2

4

cos � �adj

hyp�

2�2

3

sin � �opp

hyp�

1

3θ 3

1

adj � �32 � 12 � �8 � 2�2

cot � �adj

opp� 2�2

sec � �hyp

adj�

3

2�2�

3�2

4

csc � �hyp

opp� 3

tan � �opp

adj�

2

4�2�

1

2�2�

�2

4

cos � �adj

hyp�

4�2

6�

2�2

3

sin � �opp

hyp�

2

6�

1

62

adj � �62 � 22 � �32 � 4�2

cot � �adj

opp�

4�2

2� 2�2

sec � �hyp

adj�

6

4�2�

3

2�2�

3�2

4

csc � �hyp

opp�

6

2� 3

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

1.

tan � �opp

adj�

6

8�

3

4

cos � �adj

hyp�

8

10�

4

5

sin � �opp

hyp�

6

10�

3

5

6

8

θ

hyp � �62 � 82 � �36 � 64 � �100 � 10

cot � �adj

opp�

8

6 �

4

3

sec � �hyp

adj�

10

8�

5

4

csc � �hyp

opp�

10

6�

5

3

2.

513

sin csc

cos sec

tan cot � �adj

opp�

12

5� �

opp

adj�

5

12

� �hyp

adj�

13

12� �

adj

hyp�

12

13

� �hyp

opp�

13

5� �

opp

hyp�

5

13

adj � �132 � 52 � �169 � 25 � 12

3.

tan � �opp

adj�

9

40

cos � �adj

hyp�

40

41

sin � �opp

hyp�

9

41θ9

41

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

cot � �adj

opp�

40

9

sec � �hyp

adj�

41

40

csc � �hyp

opp�

41

9

4.

4

4

θ

sin csc

cos sec

tan cot � �adj

opp�

4

4� 1� �

opp

adj�

4

4� 1

� �hyp

adj�

4�2

4� �2� �

adj

hyp�

4

4�2�

1�2

��2

2

� �hyp

opp�

4�2

4� �2� �

opp

hyp�

4

4�2�

1�2

��2

2

hyp � �42 � 42 � �32 � 4�2

Page 19: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

352 Chapter 4 Trigonometry

6.

cot � �adj

opp�

15

8tan � �

opp

adj�

8

15

sec � �hyp

adj�

17

15cos � �

adj

hyp�

15

17

csc � �hyp

opp�

17

8sin � �

opp

hyp�

8

17

hyp � �152 � 82 � �289 � 17

15

8

θ

cot � �adj

opp�

7.5

4�

15

8tan � �

opp

adj�

4

7.5�

8

15

sec � �hyp

adj�

17�2

7.5�

17

15cos � �

adj

hyp�

7.5

17�2�

15

17

csc � �hyp

opp�

17�2

4�

17

8sin � �

opp

hyp�

4

17�2�

8

17

hyp � �7.52 � 42 �17

2

7.5

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

7.

tan � �opp

adj�

3

4

cos � �adj

hyp�

4

5

sin � �opp

hyp�

3

5

θ4

5

opp � �52 � 42 � 3

cot � �adj

opp�

4

3

sec � �hyp

adj�

5

4

csc � �hyp

opp�

5

3

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

tan � �opp

adj�

0.75

1�

3

4

cos � �adj

hyp�

1

1.25�

4

5

sin � �opp

hyp�

0.75

1.25�

3

5θ1

1.25

opp � �1.252 � 12 � 0.75

cot � �adj

opp�

1

0.75�

4

3

sec � �hyp

adj�

1.25

1�

5

4

csc � �hyp

opp�

1.25

0.75�

5

3

8.

cot � �adj

opp�

2

1� 2tan � �

opp

adj�

1

2

sec � �hyp

adj�

�5

2cos � �

adj

hyp�

2�5

�2�5

5

csc � �hyp

opp�

�5

1� �5sin � �

opp

hyp�

1�5

��5

5

hyp � �12 � 22 � �5

1

2

θ

cot � �6

3� 2tan � �

3

6�

1

2

sec � �3�5

6�

�5

2cos � �

6

3�5�

2�5

�2�5

5

csc � �3�5

3� �5sin � �

3

3�5�

1�5

��5

5

hyp � �32 � 62 � 3�5

3

6

θ

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

Page 20: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.3 Right Triangle Trigonometry 353

13. Given:

cot � �adj

opp�

1

3

sec � �hyp

adj� �10

csc � �hyp

opp�

�10

3

cos � �adj

hyp�

�10

10

sin � �opp

hyp�

3�10

10

hyp � �10

32 � 12 � �hyp�2

3

1

10

θ

tan � � 3 �3

1�

opp

adj14. Given:

cot � �adjopp

�1

�35�

�3535

csc � �hypopp

�6

�35�

6�3535

tan � �oppadj

��35

1� �35

cos � �adjhyp

�16

sin � �opphyp

��35

6

opp � �62 � 12 � �356

1

θ

35

sec � �61

�hypadj

9. Given:

cot � �adj

opp�

�7

3

sec � �hyp

adj�

4�7

7

csc � �hyp

opp�

4

3

tan � �opp

adj�

3�7

7

cos � �adj

hyp�

�7

4

adj � �7

32 � �adj�2 � 42

θ

7

4 3

sin � �3

4�

opp

hyp

11. Given:

cot � �adj

opp�

�3

3

csc � �hyp

opp�

2�3

3

tan � �opp

adj� �3

cos � �adj

hyp�

1

2

sin � �opp

hyp�

�3

2

opp � �3

1

2 3

θ

�opp�2 � 12 � 22

sec � � 2 �2

1�

hyp

adj

10. Given:

cot � �adj

opp�

5

2�6�

5�6

12

sec � �hyp

adj�

7

5

csc � �hyp

opp�

7

2�6�

7�6

12

tan � �opp

adj�

2�6

5

sin � �opp

hyp�

2�6

7

opp � �72 � 52 � �24 � 2�6

5

7 2 6

θ

cos � �57

�adjhyp

12. Given:

sec � �hyp

adj�

�26

5

csc � �hyp

opp�

�26

1� �26

tan � �opp

adj�

1

5

cos � �adj

hyp�

5�26

�5�26

26

sin � �opp

hyp�

1�26

��26

26

hyp � �52 � 12 � �26

261

5

θcot � �51

�adjopp

Page 21: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

354 Chapter 4 Trigonometry

15. Given:

sec � �hyp

adj�

�13

3

csc � �hyp

opp�

�13

2

tan � �opp

adj�

2

3

cos � �adj

hyp�

3�13

�3�13

13

sin � �opp

hyp�

2�13

�2�13

13

hyp � �13

22 � 32 � �hyp�22

3

13

θ

cot � �3

2�

adj

opp16. Given:

cot � �adjopp

��273

4

sec � �hyp

adj�

17�273

�17�273

273

tan � �opp

adj�

4�273

�4�273

273

cos � �adj

hyp�

�273

17

sin � �opp

hyp�

4

17

adj � �172 � 42 � �273

417

273

θ

csc � �174

�hypopp

17.

sin 30� �opphyp

�12

30� � 30�� �

180�� ��

6 radian

30°

60°12

3

18.

cos 45� �adjhyp

�1�2

��22

45� � 45�� �

180�� ��

4 radian

21

1

45°

19.

tan �

3�

oppadj

��31

� �3

3�

3�180�

� � � 60�

1

23

π3

π6

20.

sec �

4�

hypadj

��21

� �2

4�

4�180�

� � � 45�

21

1

π4

21.

� � 60� ��

3 radian

cot � ��33

�1�3

�adjopp

1

3

30°

60°

22.

� � 45� � 45�� �

180�� ��

4 radian

csc � � �2 �hypopp

21

1

45°

23.

cos �

6�

adjhyp

��32

6�

6�180�

� � � 30�

12

3

π3

π6

24.

sin �

4�

opphyp

�1�2

��22

� ��

4�

4�180�

� � � 45�

21

1

π4

25.

��

4 radian

� � 45� � 45�� �

180��

cot � � 1 �11

�adjopp

21

1

45°

45°26.

��

6 radian

� � 30� � 30�� �

180��

tan � ��33

�1�3

�oppadj

30°

60°12

3

Page 22: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.3 Right Triangle Trigonometry 355

31.

(a)

(b)

(c)

(d) sin�90� � �� � cos � �1

3

cot � �cos �

sin ��

1

3

2�2

3

�1

2�2�

�2

4

sin � �2�2

3

sin2 � �8

9

sin2 � � �1

3�2

� 1

sin2 � � cos2 � � 1

sec � �1

cos �� 3

cos � �1

332.

(a)

(b)

(c)

(d)

��1 �1

25��26

25�

�26

5

��1 � �1

5�2

csc � � �1 � cot 2 �

tan�90º � �� � cot � �1

tan ��

1

5

�1

�1 � 52�

1

�26�

�26

26

cos � �1

sec ��

1�1 � tan2 �

cot � �1

tan ��

1

5

tan � � 5

33. tan � cot � � tan �� 1

tan �� � 1 34. cos � sec � � cos �� 1

cos �� � 1

27.

(a)

(b)

(c)

(d) cot 60� �cos 60�

sin 60��

1�3

��3

3

cos 30� � sin 60� ��3

2

sin 30� � cos 60� �1

2

tan 60� �sin 60�

cos 60�� �3

sin 60� ��3

2, cos 60� �

1

228.

(a)

(b)

(c)

(d) cot 30� �1

tan 30��

3�3

�3�3

3� �3

cos 30� �sin 30�

tan 30��

1

2

�3

3

�3

2�3�

�3

2

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

3

csc 30� �1

sin 30�� 2

sin 30� �1

2, tan 30� �

�3

3

29.

(a)

(b)

(c)

(d) sec�90� � �� � csc � ��13

2

tan � �sin �

cos ��

2�13

13

3�13

13

�2

3

cos � �1

sec ��

3�13

�3�13

13

sin � �1

csc ��

2�13

�2�13

13

csc � ��13

2, sec � �

�13

330.

(a)

(b)

(c)

(d) sin � � tan � cos � � �2�6 ��1

5� �2�6

5

cot�90º � �� � tan � � 2�6

cot � �1

tan ��

1

2�6�

�6

12

cos � �1

sec ��

1

5

sec � � 5, tan � � 2�6

Page 23: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

356 Chapter 4 Trigonometry

35. tan � cos � � � sin �

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

sin � sin � � cos �

37.

� sin2 �

� �sin2 � � cos2 �� � cos2 �

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

39.

� 1

� �1 � tan2 �� � tan2 �

�sec � � tan ���sec � � tan �� � sec2 � � tan2 �

41.

� csc � sec �

�1

sin �

1

cos �

�1

sin � cos �

sin �

cos ��

cos �

sin ��

sin2 � � cos2 �

sin � cos �

40.

� 2 sin2 � � 1

� sin2 � � 1 � sin2 �

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

42.

� 1 � cot2 � � csc2 �

� 1 �cot �

� 1

cot ��

tan � � cot �

tan ��

tan �

tan ��

cot �

tan �

43. (a)

(b)

Note: cos 80� � sin�90� � 80�� � sin 10�

cos 80� � 0.1736

sin 10� � 0.1736 44. (a)

(b) cot 66.5� �1

tan 66.5�� 0.4348

tan 23.5� � 0.4348

45. (a)

(b) csc 16.35� �1

sin 16.35�� 3.5523

sin 16.35� � 0.2815 46. (a)

(b) sin 73� 56 � sin�73 � 56

60��

� 0.9609

cos 16� 18 � cos�16 �18

60��

� 0.9598

47. (a)

(b) csc 48� 7 �1

sin �48 �760��

� 1.3432

sec 42� 12 � sec 42.2� �1

cos 42.2�� 1.3499

49. (a)

(b) tan 11� 15 � tan 11.25� � 0.1989

cot 11� 15 �1

tan 11.25�� 5.0273

51. (a)

(b) tan 44� 28 16� � tan 44.4711� � 0.9817

csc 32� 40 3� �1

sin 32.6675�� 1.8527

48. (a)

(b)

� 1.0036

sec 4� 50 15� �1

cos 4� 50 15�

� 0.9964

cos 4� 50 15� � cos�4 �50

60�

15

3600��

50. (a)

(b) cos 56� 8 10� � cos�56 �8

60�

103600�

�� 0.5572

sec 56� 8 10� � sec�56 �8

60�

103600�

�� 1.7946

52. (a)

(b) cot�95

30 � 32��� 0.0699

sec�95

20 � 32��� 2.6695

Page 24: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.3 Right Triangle Trigonometry 357

62.

r �20

sin 45��

20

�2�2� 20�2

sin 45� �20

r

63.

Height of the building:

Distance between friends:

� 323.34 meters

cos 82� �45y

⇒ y �45

cos 82�

123 � 45 tan 82� � 443.2 meters

x � 45 tan 82�

45 m

82°

xy

tan 82� �x

45

53. (a)

(b) csc � � 2 ⇒ � � 30� ��

6

sin � �1

2 ⇒ � � 30� �

655. (a)

(b) cot � � 1 ⇒ � � 45� ��

4

sec � � 2 ⇒ � � 60� ��

354. (a)

(b) tan � � 1 ⇒ � � 45º ��

4

cos � ��2

2 ⇒ � � 45� �

4

59.

x � 30�3

1�3

�30

x

tan 30� �30

x

30°

30

x

60.

y � 18 sin 60� � 18��3

2 � � 9�3

sin 60� �y

18

61.

x �32�3

�32�3

3

�3x � 32

�3 �32

x

tan 60� �32

x

60°

32

x

64. (a) (b)

(c)

h � 270 feet

2�135� � h

tan � �63

�h

135

h

3

6132

Not drawn to scale

65.

� � 0� ��

6

sin � �15003000

�12

θ

1500 ft3000 ft

66.

w � 100 tan 54� � 137.6 feet

tan 54� �w

100

tan � �opp

adj

56. (a)

(b) cos � �1

2 ⇒ � � 60� �

3

tan � � �3 ⇒ � � 60� ��

3

57. (a)

(b) sin � ��2

2 ⇒ � � 45� �

4

csc � �2�3

3 ⇒ � � 60� �

3

58. (a)

tan � �3�3

� �3 ⇒ � � 60� ��

3

cot � ��3

3 (b)

cos � �1�2

��2

2 ⇒ � � 45� �

4

sec � � �2

Page 25: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

358 Chapter 4 Trigonometry

67.

(a)

x �145

sin 23�� 371.1 feet

sin 23� �145

x

150 ft

5 ft23°

x

y

(b)

y �145

tan 23�� 341.6 feet

tan 23� �145

y(c) Moving down the line:

feet per second

Dropping vertically:

feet per second145

6� 24.17

145�sin 23�

6� 61.85

68. Let the height of the mountain.

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

Then tan

Substitute into the expression for tan

The mountain is about 1.3 miles high.

1.2953 � h

13 tan 9� tan 3.5�

tan 9� � tan 3.5�� h

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

tan 3.5� �h tan 9�

h � 13 tan 9�

tan 3.5� �h

htan 9�

� 13

3.5�.x �h

tan 9�

tan 9� �h

x ⇒ x �

h

tan 9�

3.5� �h

x � 13 and tan 9� �

hx.

9�x �

h � 69.

�x2, y2� � �28, 28�3�

x2 � �cos 60°��56� � �12��56� � 28

cos 60° �x2

56

y2 � sin 60°�56� � ��32 ��56� � 28�3

sin 60� �y2

56

60°

56

( , )x y2 2

�x1, y1� � �28�3, 28�

x1 � cos 30��56� ��32

�56� � 28�3

cos 30� �x1

56

y1 � �sin 30���56� � �12��56� � 28

sin 30� �y1

56

30°

56

( , )x y1 1

70.

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

x � 15 tan 3�

tan 3� �x

15

Page 26: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.3 Right Triangle Trigonometry 359

72.

tan 20� �y

x� 0.36

cos 20� �x

10� 0.94

sin 20� �y

10� 0.34

x � 9.4, y � 3.4

csc 20� �10

y� 2.92

sec 20� �10

x� 1.06

cot 20� �x

y� 2.75

73. True,

csc x �1

sin x ⇒ sin 60� csc 60� � sin 60�� 1

sin 60�� � 1

75. False,�2

2�

�2

2� �2 � 1

77. False,

sin 2� � 0.0349

sin 60�

sin 30��

cos 30�

sin 30�� cot 30� � 1.7321;

74. True, because sec�90� � �� � csc �.sec 30� � csc 60�

76. True, because

cot2 � � csc2 � � �1.

cot2 � � csc2 � � 1

1 � cot2 � � csc2 �

cot2 10� � csc2 10� � �1

78. False,

tan2�5�� � �tan 5���tan 5�� � 0.0077

tan��5��2 � tan 25� � 0.4663

tan��5�� 2 � tan2�5��. 79. This is true because the corresponding sides of similar triangles are proportional.

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

81. (a) (b) In the interval

(c) As approaches 0, approaches �.sin ��

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

0.0998 0.1987 0.2955 0.3894 0.4794sin �

71. (a)

(b)

(c)

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

h � 20 sin 85� � 19.9 meters

sin 85� �h

20

h20

85°

(e)Angle, Height (in meters)

80 19.7

70 18.8

60 17.3

50 15.3

40 12.9

30 10.0

20 6.8

10 3.5�

�(f) The height of the balloon

decreases as decreases.

20 h

θ

82. (a)

—CONTINUED—

sin 0 0.3090 0.5878 0.8090 0.9511 1

cos 1 0.9511 0.8090 0.5878 0.3090 0�

90�72�54�36�18�0��

Page 27: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

360 Chapter 4 Trigonometry

Section 4.4 Trigonometric Functions of Any Angle

■ Know the Definitions of Trigonometric Functions of Any Angle.

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

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

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

■ You should be able to find reference angles.

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

■ You should know that the period of sine and cosine is 2�.

0, �

2, �, and

3�

2.

cot � �x

y, y � 0tan � �

y

x, x � 0

sec � �r

x, x � 0cos � �

x

r

csc � �r

y, y � 0sin � �

y

r

r � �x2 � y2 � 0,�x, y��

82. —CONTINUED—

(b) increases from 0 to 1 as increases from to

(c) decreases from 1 to 0 as increases from to

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

90�.0��cos �

90�.0��sin �

83.

�x

x � 2, x � ±6

x2 � 6x

x2 � 4x � 12�

x2 � 12x � 36

x2 � 36�

x�x � 6��x � 6��x � 2� �

�x � 6��x � 6��x � 6��x � 6�

84.

�2t � 3

4 � t, t � ±

3

2, �4�

�2t � 3��t � 4��3 � 2t��3 � 2t� �

�2t � 3��2t � 3��t � 4��t � 4�

� ��2t � 3��t � 4�

�2t 2 � 5t � 12

9 � 4t 2 �4t 2 � 12t � 9

t 2 � 16

2t 2 � 5t � 12

9 � 4t 2

t 2 � 16

4t 2 � 12t � 9

85.

�2x2 � 10x � 20

�x � 2��x � 2�2�

2�x2 � 5x � 10��x � 2��x � 2�2

�3�x2 � 4� � 2�x2 � 4x � 4� � x2 � 2x

�x � 2��x � 2�2

3

x � 2�

2

x � 2�

x

x2 � 4x � 4�

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

86.�3

x�

1

4��12

x� 1�

12 � x

4x

12 � x

x

�12 � x

4x�

x

12 � x�

1

4, x � 0, 12

Page 28: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.4 Trigonometric Functions of Any Angle 361

Vocabulary Check

1. 2. 3.

4. 5. 6.

7. reference

cot �xr

� cos �rx

tan � �yx

csc �sin � �yr

1. (a)

r � �16 � 9 � 5

�x, y� � �4, 3� (b)

r � �64 � 225 � 17

�x, y� � �8, �15�

tan � �y

x�

3

4

cos � �x

r�

4

5

sin � �y

r�

3

5

cot � �x

y�

4

3

sec � �r

x�

5

4

csc � �r

y�

5

3

tan � �y

x� �

15

8

cos � �x

r�

8

17

sin � �y

r� �

15

17

cot � �x

y� �

8

15

sec � �r

x�

17

8

csc � �r

y� �

17

15

2. (a)

cot � �x

y�

�12

�5�

12

5

sec � �r

x�

13

�12� �

13

12

csc � �r

y�

13

�5�

13

�5

tan � �y

x�

�5

�12�

5

12

cos � �x

r� �

12

13

sin � �y

r� �

5

13

r � ���12�2 � ��5�2 � 13

x � �12, y � �5 (b)

cot � �x

y�

�1

1� �1

sec � �r

x�

�2

�1� ��2

csc � �r

y�

�2

1� �2

tan � �y

x�

1

�1� �1

cos � �x

r�

�1�2

� ��2

2

sin � �y

r�

1�2

��2

2

r � ���1�2 � 12 � �2

x � �1, y � 1

3. (a)

r � �3 � 1 � 2

�x, y� � ���3, �1� (b)

r � �16 � 1 � �17

�x, y� � ��4, 1�

tan � �y

x�

�3

3

cos � �x

r� �

�3

2

sin � �y

r� �

1

2

cot � �x

y� �3

sec � �r

x� �

2�3

3

csc � �r

y� �2

tan � �y

x� �

1

4

cos � �x

r� �

4�17

17

sin � �y

r�

�17

17

cot � �x

y� �4

sec � �r

x� �

�17

4

csc � �r

y� �17

Page 29: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

362 Chapter 4 Trigonometry

4. (a)

cot � �x

y�

3

1� 3

sec � �r

x�

�10

3

csc � �r

y�

�10

1� �10

tan � �y

x�

1

3

cos � �x

r�

3�10

�3�10

10

sin � �y

r�

1�10

��10

10

r � �32 � 12 � �10

x � 3, y � 1 (b)

cot � �x

y�

4

�4� �1

sec � �r

x�

4�2

4� �2

csc � �r

y�

4�2

�4� ��2

tan � �y

x�

�4

4� �1

cos � �x

r�

4

4�2�

�2

2

sin � �y

r�

�4

4�2� �

�2

2

r � �42 � ��4�2 � 4�2

x � 4, y � �4

5.

cot � �x

y�

7

24

sec � �r

x�

25

7

csc � �r

y�

25

24

tan � �y

x�

24

7

cos � �x

r�

7

25

sin � �y

r�

24

25

r � �49 � 576 � 25

�x, y� � �7, 24� 6.

cot � �x

y�

8

15

sec � �r

x�

17

8

csc � �r

y�

17

15

tan � �y

x�

15

8

cos � �x

r�

8

17

sin � �y

r�

15

17

r � �82 � 152 � 17

x � 8, y � 15 7.

cot � �x

y� �

2

5

sec � �r

x� �

�29

2

csc � �r

y�

�29

5

tan � �y

x� �

5

2

cos � �x

r� �

2�29

29

sin � �y

r�

5�29

29

r � �16 � 100 � 2�29

�x, y� � ��4, 10�

8.

cot � �x

y�

�5

�2�

5

2

sec � �r

x�

�29

�5� �

�29

5

csc � �r

y�

�29

�2� �

�29

2

tan � �y

x�

�2

�5�

2

5

cos � �x

r�

�5�29

� �5�29

29

sin � �y

r�

�2�29

� �2�29

29

r � ���5�2 � ��2�2 � �29

x � �5, y � �2 9.

cot � �xy

� �3568

� �0.5

sec � �rx

� ��5849

35� �2.2

csc � �ry

��5849

68� 1.1

tan � �yx

� �6835

� �1.9

cos � �xr

� �35�5849

5849� �0.5

sin � �yr

�68�5849

5849� 0.9

r � �12.25 � 46.24 ��5849

10

�x, y� � ��3.5, 6.8�

Page 30: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.4 Trigonometric Functions of Any Angle 363

18.

cot � � �8

15tan � �

y

x�

�15

8� �

15

8

sec � �17

8cos � �

x

r�

8

17

csc � � �17

15sin � �

y

r�

�15

17� �

15

17

tan � < 0 ⇒ y � �15

cos � �x

r�

8

17 ⇒ y � �15�

15.

tan � �y

x� �

3

4

cos � �x

r� �

4

5

sin � �y

r�

3

5

� in Quadrant II ⇒ x � �4

sin � �yr

�35

⇒ x2 � 25 � 9 � 16

10.

cot � �xy

�7�2

�31�4� �

1431

� �0.5tan � �yx

��31�4

7�2� �

3114

� �2.2

sec � �rx

��1157�4

7�2�

�115714

� 2.4cos � �xr

�7�2

�1157�4�

14�11571157

� 0.4

csc � �ry

��1157�4

�31�4� �

�115731

� �1.1sin � �yr

��31�4�1157�4

� �31�1157

1157� �0.9

r ���72�

2

� ��314 �

2

��1157

4

x � 312

�72

, y � �734

� �314

11.

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

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

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

13.

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

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

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

12. and

and

Quadrant I

x

r> 0

y

r> 0

cos � > 0sin � > 0

14. and

and

Quadrant IV

x

y< 0

r

x> 0

cot � < 0sec � > 0

cot � �x

y� �

4

3

sec � �r

x� �

5

4

csc � �r

y�

5

3

16.

in Quadrant III

tan � �y

x�

3

4

cos � �x

r� �

4

5

sin � �y

r� �

3

5

⇒ y � �3�

cos � �x

r�

�4

5 ⇒ y2 � 25 � 16 � 9

cot � �4

3

sec � � �5

4

csc � � �5

3

17.

is in Quadrant IV

tan � �y

x� �

15

8

cos � �x

r�

8

17

sin � �y

r� �

15

17

x � 8, y � �15, r � 17

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

tan � �yx

��15

8

cot � �x

y� �

8

15

sec � �r

x�

17

8

csc � �r

y� �

17

15

Page 31: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

364 Chapter 4 Trigonometry

19.

cot � �x

y� �3tan � �

y

x� �

1

3

sec � �r

x�

�10

3cos � �

x

r�

3�10

10

csc � �r

y� ��10sin � �

y

r� �

�10

10

x � 3, y � �1, r � �10

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

cot � �x

y� �

3

1�

3

�120.

cot � � ��15tan � �y

x� �

�15

15

sec � � �4�15

15cos � �

x

r� �

�15

4

csc � � 4sin � �y

r�

1

4

cot � < 0 ⇒ x � ��15

csc � �r

y�

4

1 ⇒ x � ��15�

21.

cot � �x

y� �

�3

3tan � �

y

x� ��3

sec � �r

x� �2cos � �

x

r� �

1

2

csc � �r

y�

2�3

3sin � �

y

r�

�3

2

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

sec � �r

x�

2

�1 ⇒ y2 � 4 � 1 � 3 22.

is undefined

is undefinedcot � �xy

tan � �yx

� 0

sec � �rx

� �1cos � �xr

��rr

� �1

csc � �ry

sin � � 0

y � 0, x � �r

sec � � �1 ⇒ � � � � 2�n

sin � � 0 ⇒ � � 0 � �n

23.

cot � is undefinedtan � � 0

sec � � �1cos � � �1

csc � is undefinedsin � � 0

cot � is undefined, �

2≤ � ≤

3�

2 ⇒ y � 0 ⇒ � � � 24. tan is undefined

is undefined.

is undefined. cot � �x

y�

0

y� 0tan � �

y

x

sec � �r

xcos � �

x

r�

0

r� 0

csc � �r

y� �1sin � �

y

r�

�r

r� �1

� ≤ � ≤ 2� ⇒ � �3�

2, x � 0, y � �r

⇒ � � n� ��

2�

25. To find a point on the terminal side of use any point onthe line that lies in Quadrant II. is onesuch point.

cot � � �1

sec � � ��2

csc � � �2

tan � � �1

cos � � �1�2

� ��2

2

sin � �1�2

��2

2

x � �1, y � 1, r � �2

��1, 1�y � �x� 26. Let

Quadrant III

cot � �x

y�

�x

��1�3� x� 3

sec � �r

x�

��10x��3

�x� �

�10

3

csc � �r

y�

��10x��3

��1�3�x� ��10

tan � �y

x�

��1�3�x

�x�

1

3

cos � �x

r�

�x

��10x��3� �

3�10

10

sin � �y

r�

��1�3�x��10x��3

� ��10

10

r ��x2 �1

9x 2 �

�10x

3

��x, �1

3x�,

x > 0.

Page 32: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.4 Trigonometric Functions of Any Angle 365

27. use any point onthe line Quadrant III. is onesuch point.

cot � ��1

�2�

1

2

sec � ��5

�1� ��5

csc � ��5

�2� �

�5

2

tan � ��2

�1� 2

cos � � �1�5

� ��5

5

sin � � �2�5

� �2�5

5

x � �1, y � �2, r � �5

��1, �2�y � 2x that lies inTo find a point on the terminal side of �, 28. Let

Quadrant IV

tan � � �3

4tan � �

y

x�

��4�3� x

x� �

4

3

sec � �5

3cos � �

x

r�

x

�5�3�x�

3

5

csc � � �5

4sin � �

y

r�

��4�3�x

�5�3� x� �

4

5

r ��x2 �16

9x2 �

5

3x

�x, �4

3x�,

4x � 3y � 0 ⇒ y � �4

3x

x > 0.

29.

sin � �yr

� 0

�x, y� � ��1, 0�, r � 1 30.

since corresponds to �0, �1�.3�

2

csc 3�

2�

r

y�

1

�1� �1 31.

sec undefined3�

2�

r

x�

1

0 ⇒

�x, y� � �0, �1�, r � 1

32.

since corresponds to ��1, 0�.3�

2

sec � �r

x�

1

�1� �1 33.

sin �

2�

yr

� 1

�x, y� � �0, 1�, r � 1 34. (undefined)

since corresponds to ��1, 0�.�

cot � �x

y�

�1

0

35.

undefinedcsc � �r

y�

1

0 ⇒

�x, y� � ��1, 0�, r � 1 36.

since corresponds to �0, 1�.�

2

cot �

2�

x

y�

0

1� 0

37.

′θ

203°

x

y

� � 203� � 180� � 23�

� � 203� 38.

′θ

309°

x

y

� � 360� � 309� � 51�

� � 309�

Page 33: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

45. Quadrant III

tan 225� � tan 45� � 1

cos 225� � �cos 45� � ��2

2

sin 225� � �sin 45� � ��2

2

� � 225�, � � 360� � 225� � 45�,

47. is coterminal with

Quadrant I

tan 750� � tan 30� ��3

3

cos 750� � cos 30� ��3

2

sin 750� � sin 30� �1

2

� � 30�,

30�.� � 750�

46. Quadrant IV

sin

cos

tan 300� � �tan 60� � ��3

300� � cos 60� �1

2

300� � �sin 60� � ��3

2

� � 300�, � � 360� � 300� � 60�,

48. is coterminal with

Quadrant IV

sin

cos

tan��405�� � �tan 45� � �1

��405�� � cos 45� ��2

2

��405�� � �sin 45� � ��2

2

� � 360� � 315� � 45�,

315�.� � �405�

366 Chapter 4 Trigonometry

39.

′θ

−245°

x

y

� � 180� � 115� � 65�

360� � 245� � 115� �coterminal angle�� � �245� 40. is coterminal with

′θ−145°

x

y

� � 215� � 180� � 35�

215�.� � �145�

41.

� � � �2�

3�

3′θ

23π

x

y � �2�

342.

� � 2� �7�

4�

4

′θ

74π

x

y� �7�

4

43.

� � 3.5 � �

′θ

3.5

x

y � � 3.5 44. is coterminal

with

� � 2� �5�

3�

3

5�

3.

′θ

x

y

113π

� �11�

3

Page 34: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.4 Trigonometric Functions of Any Angle 367

49. is coterminal with

Quadrant III

tan��150�� � tan 30� ��33

cos��150�� � �cos 30� � ��32

sin��150�� � �sin 30� � �12

� � 210� � 180� � 30�,

210�.� � �150� 50. .

sin

cos

tan��840�� � tan 60� � �3

��840�� � �cos 60� � �1

2

��840�� � �sin 60� � ��3

2

� � 240� � 180� � 60�, Quadrant III

� � �840� is coterminal with 240�

51. Quadrant III

tan 4�

3� tan

3� �3

cos 4�

3� �cos

3� �

1

2

sin 4�

3� �sin

3� �

�3

2

� ��

3,� �

4�

3, 52. Quadrant I

sin

cos

tan �

4� 1

4�

�2

2

4�

�2

2

� ��

4, � �

4, 53. Quadrant IV

tan���

6� � �tan �

6� �

�3

3

cos���

6� � cos �

6�

�3

2

sin���

6� � �sin �

6� �

1

2

� � ��

6, � �

6,

54. is coterminal with

sin

cos

tan is undefined.���

2� � tan 3�

2

���

2� � cos 3�

2� 0

���

2� � sin 3�

2� �1

3�

2.� � �

255. is coterminal with

Quadrant II

tan 11�

4� �tan

4� �1

cos 11�

4� �cos

4� �

�2

2

sin 11�

4� sin

4�

�2

2

� � � �34

� ��

4,

34

�.� �11�

456. is coterminal with

Quadrant III

sin

cos

tan 10�

3� tan

3� �3

10�

3� �cos

3� �

1

2

10�

3� �sin

3� �

�3

2

� �4�

3� � �

3,

4�

3.� �

10�

3

57. is coterminal with

tan��3�

2 � � tan �

2 which is undefined.

cos��3�

2 � � cos �

2� 0

sin��3�

2 � � sin �

2� 1

� ��

2.

2,� � �

3�

258. .

tan��25�

4 � � �tan��

4� � �1

cos��25�

4 � � cos��

4� ��22

sin��25�

4 � � �sin��

4� � ��22

� � 2� �7�

4�

4 in Quadrant IV.

� � �25�

4 is coterminal with

7�

4

Page 35: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

74. cot 1.35 �1

tan 1.35� 0.2245

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

cos 0.29� 1.0436

79. cot��11�

8 � �1

tan��11�

8 �� �0.4142 80. csc��

15�

14 � �1

sin��15�

14 �� 4.4940

368 Chapter 4 Trigonometry

59.

in Quadrant IV.

cos � �4

5

cos � > 0

cos2 � �16

25

cos2 � � 1 �9

25

cos2 � � 1 � ��3

5�2

cos2 � � 1 � sin2 �

sin2 � � cos2 � � 1

sin � � �3

560. cot

sin � �1

csc ��

1�10

��10

10

csc � �1

sin �

�10 � csc �

csc � > 0 in Quadrant II.

10 � csc2 �

1 � ��3�2 � csc2 �

1 � cot2 � � csc2 �

� � �3 61.

in Quadrant III.

sec � � ��13

2

sec � < 0

sec2 � �13

4

sec2 � � 1 �9

4

sec2 � � 1 � �3

2�2

sec2 � � 1 � tan2 �

tan � �3

2

62.

cot � � ��3

cot � < 0 in Quadrant IV.

cot2 � � 3

cot2 � � ��2�2 � 1

cot2 � � csc2 � � 1

1 � cot2 � � csc2 �

csc � � �2 63.

sec � �1

5�8�

85

cos � �1

sec � ⇒ sec � �

1cos �

cos � �58

64.

.

tan � ��65

4

tan � > 0 in Quadrant III

tan2 � �65

16

tan2 � � ��9

4�2

� 1

tan2 � � sec2 � � 1

1 � tan2 � � sec2 �

sec � � �9

4

65. sin 10� � 0.1736 66. sec 225� �1

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

68. csc��330�� �1

sin��330�� � 2.0000 69. tan 304� � �1.4826 70. cot 178� �1

tan 178�� �28.6363

71. sec 72� �1

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

75. tan �

9� 0.3640 76. tan��

9� � �0.3640

Page 36: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.4 Trigonometric Functions of Any Angle 369

81. (a) reference angle is 30 or is in Quadrant I or Quadrant II.

Values in degrees: 30 , 150

Values in radians:

(b) reference angle is 30 or is in Quadrant III or Quadrant IV.

Values in degrees: 210 , 330

Values in radians:7�

6,

11�

6

��

6 and ��sin � � �

1

2 ⇒

6,

5�

6

��

6 and ��sin � �

1

2 ⇒

82. (a) cos reference angle is 45 or and is in Quadrant I or IV.

Values in degrees: 45 , 315

Values in radians:

(b) cos reference angle is 45 or and is in Quadrant II or III.

Values in degrees: 135 , 225

Values in radians:3�

4,

5�

4

��

��

4�� � �

�2

2 ⇒

4,

7�

4

��

��

4�� �

�2

2 ⇒

83. (a) reference angle is 60 or and is in Quadrant I or Quadrant II.

Values in degrees: 60 , 120

Values in radians:

(b) reference angle is 45 or and is in Quadrant II or Quadrant IV.

Values in degrees: 135 , 315

Values in radians:3�

4,

7�

4

��

��

4�cot � � �1 ⇒

3,

2�

3

��

��

3�csc � �

2�3

3 ⇒

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

Quadrant I or IV.

Values in degrees: 60 , 300

Values in radians:

(b) sec reference angle is 60 or and is

in Quadrant II or III.

Values in degrees: 120 , 240

Values in radians:2�

3,

4�

3

��

��

3�� � �2 ⇒

3,

5�

3

��

��

3�� � 2 ⇒ 85. (a) reference angle is 45 or and is in

Quadrant I or Quadrant III.

Values in degrees: 45 , 225

Values in radians:

(b) reference angle is 30 or and

is in Quadrant II or Quadrant IV.

Values in degrees: 150 , 330

Values in radians:5�

6,

11�

6

��

��

6�cot � � ��3 ⇒

4,

5�

4

��

��

4�tan � � 1 ⇒

Page 37: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

370 Chapter 4 Trigonometry

86. (a) sin reference angle is 60 or and is

in Quadrant I or II.

Values in degrees: 60 , 120

Values in radians:

(b) sin reference angle is 60 or and

is in Quadrant III or IV.

Values in degrees: 240 , 300

Values in radians:4�

3,

5�

3

��

��

3�� � �

�3

2 ⇒

3,

2�

3

��

��

3�� �

�3

2 ⇒ 87. (a) New York City:

Fairbanks:

(b)

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

F � 36.641 sin�0.502t � 1.831� � 25.610

N � 22.099 sin�0.522t � 2.219� � 55.008

88.

(a) For February 2006,

(b) For February 2007,

(c) For June 2006,

(d) For June 2007,

S � 23.1 � 0.442�18� � 4.3 cos ��18�

6� 26,756 units

t � 18.

S � 23.1 � 0.442�6� � 4.3 cos ��6�

6� 21,452 units

t � 6.

S � 23.1 � 0.442�14� � 4.3 cos ��14�

6� 31,438 units

t � 14.

S � 23.1 � 0.442�2� � 4.3 cos ��2�

6� 26,134 units

t � 2.

S � 23.1 � 0.442t � 4.3 cos �t

689.

(a)

(b)

(c) y�1

2� � 2 cos 3 � �1.98 centimeters

y�1

4� � 2 cos�3

2� � 0.14 centimeter

y�0� � 2 cos 0 � 2 centimeters

y�t� � 2 cos 6t

Month New York City Fairbanks

February

March

May

June

August

September

November 6.5�46.8�

41.7�68.6�

55.6�75.5�

59.5�72.5�

48.6�63.4�

13.9�41.6�

�1.4�34.6�

90.

(a)

centimeters

(b)

centimeters

(c)

centimetersy�12� � 2e�1�2 cos�6 � 1

2� � �1.2

t �12

y�14� � 2e�1�4 cos�6 � 1

4� � 0.11

t �14

y�0� � 2e�0 cos 0 � 2

t � 0

y�t� � 2e�t cos 6t 91.

ampere I�0.7� � 5e�1.4 sin 0.7 � 0.79

I � 5e�2t sin t

Page 38: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.4 Trigonometric Functions of Any Angle 371

92. sin

(a)

miles

(b)

miles

(c)

milesd �6

sin 120�� 6.9

� � 120�

d �6

sin 90º�

6

1� 6

� � 90�

d �6

sin 30��

6

1�2� 12

� � 30�

� �6

d ⇒ d �

6

sin �93. False. In each of the four quadrants, the sign of the secant

function and the cosine function will be the same sincethey are reciprocals of each other.

94. False. For example, if and but is not the reference angle. The reference angle would be For in Quadrant II, For in Quadrant III,For in Quadrant IV, � � 360� � �.�

� � � � 180�.�� � 180� � �.�45�.360�n � � � 135�0 ≤ 135 ≤ 360,� � 225�,n � 1

95. As increases from to x decreases from 12 cm to 0 cm and y increases from 0 cm to 12 cm.

Therefore, increases from 0 to 1 and decreases from 1 to 0. Thus,

increases without bound, and when the tangent is undefined.� � 90�tan � �yx

cos � �x

12sin � �

y12

90�,0��

96. Determine the trigonometric function of the reference angle and prefix the appropriate sign.

97.

x-intercepts:

y-intercept:

No asymptotes

Domain: All real numbers x

�0, �4�

−2−6−8 2 4 6 8−2

−4

−8

2

4

6

8

y

x(1, 0)(−4, 0)

(0, −4)

��4, 0�, �1, 0�

y � x2 � 3x � 4 � �x � 4��x � 1� 98.

x-intercepts:

y-intercepts:

No asymptotes

Domain: All real numbers x

�0, 0�

−1−2−3 1 2 3 4 5−1

−2

−3

−4

1

2

y

x(0, 0) , 05

2( (

�0, 0�, �52, 0�

y � 2x2 � 5x � x�2x � 5�

99.

x-intercept:

y-intercept:

No asymptotes

Domain: All real numbers x

�0, 8�

��2, 0�

−6 −4−8 2 4 6 8

−4

10

12

y

x

(0, 8)

(−2, 0)

f �x� � x3 � 8 100.

x-intercepts:

y-intercepts:

No asymptotes

Domain: All real numbers x

�0, �3�

−2−3−4 2 3 4

−3

−4

1

2

3

4

y

x(−1, 0)

(0, −3)

(1, 0)

��1, 0�, �1, 0�

� �x2 � 3��x � 1��x � 1�

g�x� � x4 � 2x2 � 3��x2 � 3��x2 � 1�

Page 39: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

372 Chapter 4 Trigonometry

101.

x-intercept:

y-intercept:

Vertical asymptote:

Horizontal asymptote:

Domain: All real numbers except x � �2

y � 0

x � �2

�0, �74�

�7, 0�−8 −2 2 4 6 8

2

4

y

x

74

0, ( (−

(7, 0)

f �x� �x � 7

x2 � 4x � 4�

x � 7�x � 2�2

102.

x-intercepts:

To find the y-intercept, let

y-intercept:

Vertical asymptote:

To find the slant asymptote, use long division:

Slant asymptote:

Domain: All real numbers except

−8−12 4

−8

−16

−24

8

y

x(−1, 0)

(1, 0)

0, 15

−( (

x � �5

y � x � 5

x2 � 1x � 5

� x � 5 �24

x � 5

x � �5

�0, �15�

x � 0: 02 � 10 � 5

� �15

��1, 0�, �1, 0�,

h�x� �x2 � 1x � 5

��x � 1��x � 1�

x � 5103.

y-intercept:

Horizontal asymptote:

Domain: All real numbers

−2 −1 1 2 3 4−1

2

3

4

5

10, 12 ))

x

y

x

y � 0

�0, 12�

y � 2x�1

x 0 1 2 3

y 1 2 412

14

�1

104.

This is an exponential function (always positive) translated two units upward. There are no x-intercepts.

To find the y-intercept, let

y-intercepts:

The horizontal asymptote is the horizontal asymptote oftranslated two units upward.

Horizontal asymptote:

Domain: All real numbers x

y � 2

y � 3x�1

�0, 5�

y � 30�1 � 2 � 3 � 2 � 5

x � 0:

x321−1−2−3−4−5

7

6

5

3

1

2

(0, 5)

yy � 3x�1 � 2

Page 40: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.5 Graphs of Sine and Cosine Functions 373

Section 4.5 Graphs of Sine and Cosine Functions

■ You should be able to graph

■ Amplitude:

■ Period:

■ Shift: Solve

■ Key increments: (period)1

4

bx � c � 0 and bx � c � 2�.

2�

b

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

Vocabulary Check

1. cycle 2. amplitude

3. 4. phase shift

5. vertical shift

2�

b

105.

Domain: All real numbers except

x-intercepts:

Vertical asymptote: x � 0

�±1, 0�

x � 0

−12 −9 −6 −3 3 6 9 12

6

9

12

(−1, 0) (1, 0)x

yy � ln x4

106.

To find the x-intercept, let

x-intercepts:

To find the y-intercept, let

y-intercepts:

The vertical asymptote is the horizontal asymptote of translated two units to the left.

Vertical asymptote:

Domain: All real numbers such that x > �2x

x � �2

y � log10 x

�0, 0.301�

y � log10�x � 2� � log10 2 � 0.301

x � 0:

��1, 0�

0 � log10�x � 2� ⇒ 10� � x � 2 ⇒ x � �1

y � 0: 3

2

−1

−2

−3

321−1−3

(−1, 0)x

yy � log10�x � 2�

Page 41: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

374 Chapter 4 Trigonometry

7.

Period:

Amplitude: ��2� � 2

2�

1� 2�

y � �2 sin x 8.

Period:

Amplitude: �a� � ��1� � 1

2�

b�

2�

2�3� 3�

y � �cos 2x

39.

Period:

Amplitude: �3� � 3

2�

10�

5

y � 3 sin 10x

10.

Period:

Amplitude: �a� �1

3

2�

b�

2�

8�

4

y �1

3 sin 8x 11.

Period:

Amplitude: �12� �1

2

2�

2�3� 3�

y �1

2 cos

2x

312.

Period:

Amplitude: �a� �5

2

2�

b�

2�

1�4� 8�

y �5

2 cos

x

4

13.

Amplitude: �14� �14

Period: 2�

2�� 1

y �14

sin 2�x 14.

Period:

Amplitude: �a� �2

3

2�

b�

2�

��10� 20

y �2

3 cos

�x

1015.

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

g�x� � sin�x � ��

f �x� � sin x

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

f�x� � cos x, g�x� � cos�x � �� 17.

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

g�x� � �cos 2x

f �x� � cos 2x 18.

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

f �x� � sin 3x, g�x� � sin��3x�

19.

The period of f is twice that of g.

g�x� � cos 2x

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

f�x� � sin x, g�x� � sin 3x 21.

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

f �x� � 3 � sin 2x

f �x� � sin 2x

1.

Period:

Amplitude: �3� � 3

2�

2� �

y � 3 sin 2x 2.

Period:

Amplitude: �a� � 2

2�

b�

2�

3

y � 2 cos 3x 3.

Period:

Amplitude: �52� �5

2

2�

1�2� 4�

y �5

2 cos

x

2

4.

Period:

Amplitude: �a� � ��3� � 3

2�

b�

2�

1�3� 6�

y � �3 sin x

35.

Period:

Amplitude: �12� �1

2

2�

��3� 6

y �1

2 sin

�x

36.

Period:

Amplitude: �a� �3

2

2�

b�

2�

��2� 4

y �3

2 cos

�x

2

Page 42: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.5 Graphs of Sine and Cosine Functions 375

28.

Period:

Amplitude: 1

Symmetry: origin

Key points: Intercept Maximum Intercept Minimum Intercept

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

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

f �x�,g�x�g�x� � sin�x3� � f �x

3�,

�2�, 0��3�

2, �1���, 0���

2, 1��0, 0�

2�

b�

2�

1� 2�

− 2

2

π6

fg

x

yf �x� � sin x

29.

Period:

Amplitude: 1

Symmetry: axis

Key points: Maximum Intercept Minimum Intercept Maximum

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

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

�2�, 1��3�

2, 0���, �1���

2, 0��0, 1�

y-

2�

b�

2�

1� 2�

−1

g

f

xππ 2

yf �x� � cos x

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

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

27.

Period:

Amplitude: 2

Symmetry: origin

Key points: Intercept Minimum Intercept Maximum Intercept

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

g�x�g�x� � 4 sin x � ��2� f �x�,

�2�, 0��3�

2, 0���, 0���

2, �2��0, 0�

2�

b�

2�

1� 2�

x

−π π32 2

5

43

−5

f

g

yf �x� � �2 sin x

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

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

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

Page 43: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

376 Chapter 4 Trigonometry

31.

Period:

Amplitude:

Symmetry: origin

Key points: Intercept Minimum Intercept Maximum Intercept

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

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

f �x�,g�x�g�x� � 3 �12

sin x2

� 3 � f �x�,

�4�, 0��3�, 12��2�, 0���, �

12��0, 0�

12

2�

b�

2�

1�2� 4�

−1

1

2

3

4

5

f

g

x−π π3

yf �x� � �12

sin x2

32.

Period:

Amplitude: 4

Symmetry: origin

Key points: Intercept Maximum Intercept Minimum Intercept

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

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

�2, 0��32

, �2��1, 0��12

, 2��0, 0�

2�

b�

2�

�� 2

−8

2

4f

g

1 x

yf �x� � 4 sin �x

30.

Period:

Amplitude: 2

Symmetry: axis

Key points: Maximum Intercept Minimum Intercept Maximum

Since the graph of is the graph of but

i) shrunk horizontally by a factor of 2,

ii) shrunk vertically by a factor of and

iii) reflected about the axis.

Generate key points for the graph of by

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

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

f �x�x-

g�x�

x-

12,

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

��, 2��3�

4, 0���

2, �2���

4, 0��0, 2�

y-

2�

b�

2�

2� �

−2

2

π

f

g

x

yf �x� � 2 cos 2x

Page 44: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.5 Graphs of Sine and Cosine Functions 377

35.

Period:

Amplitude:

Key points:

�3�

2, �3�, �2�, 0�

�0, 0�, ��

2, 3�, ��, 0�,

3

2�

y

xπ π32 2

−− π2

π32

−4

1

2

3

4

y � 3 sin x

37.

Period:

Amplitude:

Key points:

�3�

2, 0�, �2�,

1

3�

�0, 1

3�, ��

2, 0�, ��, �

1

3�,

1

3

2�

y

xπ π2

π2

1

−1

23

43

1323

43

y �1

3 cos x 38.

Period:

Amplitude:

Key points:

�3�

2, 0�, �2�, 4�

�0, 4�, ��

2, 0�, ��, �4�,

4

2�

y

xππ 2π−2 −π

−2

−4

4

y � 4 cos x

36.

Period:

Amplitude:

Key points:

�3�

2, �

1

4�, �2�, 0�

�0, 0�, ��

2,

1

4�, ��, 0�,

1

4

2�

y

xππ 2π−2 −π

−1

−2

1

2

y �1

4 sin x

33.

Period:

Amplitude: 2

Symmetry: axis

Key points: Maximum Intercept Minimum Intercept Maximum

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

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

�2�, 2��3�

2, 0���, �2���

2, 0��0, 2�

y-

2�

b�

2�

1� 2�

−3

3

f

g

xππ 2

yf �x� � 2 cos x

34.

Period:

Amplitude: 1

Symmetry: axis

Key points: Minimum Intercept Maximum Intercept Minimum

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

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

�2�, �1��3�

2, 0���, 1���

2, 0��0, �1�

y-

2�

b�

2�

1� 2�

−2

2

ππ 2

f

g

x

yf �x� � �cos x

Page 45: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

378 Chapter 4 Trigonometry

45.

Period:

Amplitude: 1

Shift: Set

Key points: ��

4, 0�, �3�

4, 1�, �5�

4, 0�, �7�

4, �1�, �9�

4, 0�

x ��

4 x �

9�

4

x ��

4� 0 and x �

4� 2�

2�

−3

−2

1

2

3

ππ−x

yy � sin�x ��

4�; a � 1, b � 1, c ��

4

44.

Period:

Amplitude: 10

Key points:

1284−4−12

12

8

4

−12

x

y

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

2�

��6� 12

y � �10 cos �x

643.

Period:

Amplitude: 1

Key points:

−1 2 3

−3

−2

2

3

x

y

�0, 0�, �3

4, �1�, �3

2, 0�, �9

4, 1�, �3, 0�

2�

2��3� 3

y � �sin 2�x

3; a � �1, b �

2�

3, c � 0

39.

Period:

Amplitude: 1

Key points:

�3�, 0�, �4�, 1�

�0, 1�, ��, 0�, �2�, �1�,

4�

y

xπ4π2π−2

−1

−2

2

y � cos x

240.

Period:

Amplitude: 1

Key points:

�3�

8, �1�, ��

2, 0�

�0, 0�, ��

8, 1�, ��

4, 0�,

2

y

x

−2

1

2

π4

y � sin 4x

41.

Period:

Amplitude: 1

Key points:

�0, 1�, �14

, 0�, �12

, �1�, �34

, 0�

2�

2�� 1

1 2

−2

1

2

x

yy � cos 2�x 42.

Period:

Amplitude: 1

Key points:

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

2�

��4� 8

2

1

−2

62−2−6x

yy � sin �x

4

Page 46: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.5 Graphs of Sine and Cosine Functions 379

48.

Period:

Amplitude: 4

Shift: Set

Key points:

−6

−4

−2−

2

6

x

y

πππ 2

�5�

4, 0�, �7�

4, 4���

4, 4�, ��

4, 0�, �3�

4, �4�,

x � ��

4 x �

7�

4

x ��

4� 0 and x �

4� 2�

2�

y � 4 cos�x ��

4� 49.

Period: 3

Amplitude: 1

Key points:

–3 –2 –1 1 2 3−1

1

2

4

5

x

y

�0, 2�, �34

, 1�, �32

, 2�, �94

, 3�, �3, 2�

y � 2 � sin 2�x

3

50.

Period:

Amplitude: 5

Key points: �0, 2�, �6, �3�, �12, �8�, �18, �3�, �24, 2�

2�

��12� 24

−12 4 12

−24−20−16−12−8

48

1216

t

yy � �3 � 5 cos � t

12

46.

Period:

Amplitude: 1

Shift: Set

Key points:

2

−2

−1

x

y

−π π32 2

��, 0�, �3�

2, 1�, �2�, 0�, �5�

2, �1�, �3�, 0�

x � � x � 3�

x � � � 0 and x � � � 2�

2�

y � sin�x � �� 47.

Period:

Amplitude: 3

Shift: Set

Key points:

−6

−4

2

4

6

− ππx

y

���, 3�, ���

2, 0�, �0, �3�, ��

2, 0�, ��, 3�

x � �� x � �

x � � � 0 and x � � � 2�

2�

y � 3 cos�x � ��

Page 47: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

380 Chapter 4 Trigonometry

55.

Period:

Amplitude:

Shift:

Key points: ��

2,

2

3�, �3�

2, 0�, �5�

2,

�2

3 �, �7�

2, 0�, �9�

2,

2

3�

x ��

2 x �

9�

2

x

2�

4� 0 and

x

2�

4� 2�

2

3

4�

−4

−3

−2

−1

1

2

3

4

π π4x

y

y �2

3 cos�x

2�

4�; a �2

3, b �

1

2, c �

4

53.

Period:

Amplitude: 3

Shift: Set

Key points: ���, 0�, ���

2, �3�, �0, �6�, ��

2, �3�, ��, 0�

x � �� x � �

x � � � 0 and x � � � 2�

2�

−8

2

4

ππ 2x

yy � 3 cos�x � �� � 3

54.

Period:

Amplitude: 4

Shift: Set

Key points: ���

4, 8�, ��

4, 4�, �3�

4, 0�, �5�

4, 4�, �7�

4, 8�

x � ��

4 x �

7�

4

x ��

4� 0 and x �

4� 2�

2�

−4

2

4

6

10

x

y

π3π2π−2 − ππ

y � 4 cos�x ��

4� � 4

51.

Period:

Amplitude:

Vertical shift two units upward

Key points:

0.20.1−0.1 0

1.8

2.2

x

y

�0, 2.1�, � 1

120, 2�, � 1

60, 1.9�, � 1

40, 2�, � 1

30, 2.1�

1

10

2�

60��

1

30

y � 2 �1

10 cos 60�x 52.

Period:

Amplitude: 2

Key points:

−7

−6

−5

−4

1

−x

y

πππ 2

�0, �1�, ��

2, �3�, ��, �5�, �3�

2, �3�, �2�, �1�

2�

y � 2 cos x � 3

Page 48: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.5 Graphs of Sine and Cosine Functions 381

63.

Amplitude:

Vertical shift one unit upward of

Thus, f�x� � 2 cos x � 1.

g�x� � 2 cos x ⇒ d � 1

1

2�3 � ��1� � 2 ⇒ a � 2

f �x� � a cos x � d 64.

Amplitude:

a � 2, d � �1

d � 1 � 2 � �1

1 � 2 cos 0 � d

1 � ��3�2

� 2

f�x� � a cos x � d

65.

Amplitude:

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

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

g�x� � 4 cos xf �x�

1

2�8 � 0 � 4

f �x� � a cos x � d 66.

Amplitude:

Reflected in the axis:

a � �1, d � �3

d � �3

�4 � �1 cos 0 � d

a � �1x-

�2 � ��4�2

� 1

f�x� � a cos x � d

56.

Period:

Amplitude: 3

Shift: Set

Key points: ���

6, �3�, ��

12, 0�, �0, 3�, � �

12, 0�, ��

6, �3�

x � ��

6 x �

6

6x � � � 0 and 6x � � � 2�

2�

6�

3

2

3

x

y

π

y � �3 cos�6x � ��

57.

−6 6

−4

4

y � �2 sin�4x � �� 58.

−12

−8

12

8

y � �4 sin�2

3x �

3� 59.

−3

−1

3

3

y � cos�2�x ��

2� � 1

60.

−6

−6

6

2

y � 3 cos��x

2�

2� � 2 61.

−20

−0.12

20

0.12

y � �0.1 sin��x

10� �� 62.

−0.03

−0.02

0.03

0.02

y �1

100 sin 120 � t

Page 49: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

382 Chapter 4 Trigonometry

74.

(a) Period

(b)1 cycle

4 seconds�

60 seconds

1 minute� 15 cycles per minute

�2�

��2� 4 seconds

v � 1.75 sin � t

2

71.

In the interval

when x � �5�

6, �

6,

7�

6,

11�

6.sin x � �

1

2

��2�, 2�,

y2 � �1

2

−2

2

2

�−2�

y1 � sin x 72.

y1 � y2 when x � �, ��

y2 � �1

−2

2

2

�−2�

y1 � cos x

73.

(a) Time for one cycle

(b) Cycles per min cycles per min

(c) Amplitude: 0.85; Period: 6

Key points: �0, 0�, �3

2, 0.85�, �3, 0�, �9

2, �0.85�, �6, 0�

�60

6� 10

�2�

��3� 6 sec

t2 4 8 10

0.25

0.50

0.75

1.00

−0.25

−1.00

vy � 0.85 sin �t

3

(c)

1 3 5 7

−2

−3

1

2

3

t

v

69.

Amplitude:

Period:

Phase shift:

Thus, y � 2 sin�x ��

4�.

�1����

4 � � c � 0 ⇒ c � ��

4

bx � c � 0 when x � ��

4

2� ⇒ b � 1

a � 2

y � a sin�bx � c� 70.

Amplitude:

Period: 2

Phase shift:

a � 2, b ��

2, c � �

2

c

b� �1 ⇒ c � �

2

2�

b� 4 ⇒ b �

2

2 ⇒ a � 2

y � a sin�bx � c�

67.

Amplitude:

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

Period:

Phase shift:

Thus, y � �3 sin 2x.

c � 0

2�

b� � ⇒ b � 2

a � �3.

�a� � �3�y � a sin�bx � c� 68.

Amplitude:

Period:

Phase shift:

a � 2, b �12

, c � 0

c � 0

2�

b� 4� ⇒ b �

1

2

4�

2 ⇒ a � 2

y � a sin�bx � c�

Page 50: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.5 Graphs of Sine and Cosine Functions 383

78. (a) and (c)

Reasonably good fit

(d) Period is 29.6 days.

(e) March 12

The Naval observatory says that 50% ofthe moon’s face will be illuminated onMarch 12, 2007.

y � 0.44 � 44% ⇒ x � 71.

y

x10 20 30 40

0.2

0.4

0.6

0.8

1.0

Perc

ent o

f m

oon’

sfa

ce il

lum

inat

ed

Day of the year

(b)

y �12

�12

sin�0.21x � 0.92�

C � 0.92

Horizontal shift: 0.21�3 � 7.4� � C � 0

b �2�

29.6� 0.21

2�

b� 4�7.4� � 29.6

Period: 8 � 8 � 7 � 6 � 85

� 7.4 (average length of interval in data)

Amplitude: 12

⇒ a �12

Vertical shift: 12

⇒ d �12

75.

(a) Period:

(b) f �1

p� 440 cycles per second

2�

880��

1

440 seconds

y � 0.001 sin 880�t 76.

(a) Period:

(b)1 heartbeat

6�5 seconds�

60 seconds

1 minute� 50 heartbeats per minute

2�

�5���3�

6

5 seconds

P � 100 � 20 cos 5� t

3

77. (a)

(b)

The model is a good fit.

00

12

100

C�t� � 56.55 � 26.95 cos�� t6

� 3.67�

d �12

�high � low �12

�83.5 � 29.6 � 56.55

cb

� 7 ⇒ c � 7��

6� � 3.67

b �2�

p�

2�

12�

6

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

a �12

�high � low �12

�83.5 � 29.6 � 26.95 (c)

The model is a good fit.

(d) Tallahassee average maximum:

Chicago average maximum:

The constant term, gives the average maximum temperature.

(e) The period for both models is months.

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

(f ) Chicago has the greater variability in temperaturethroughout the year. The amplitude, a, determines thisvariability since it is .1

2�high temp � low temp

2�

��6� 12

d,

56.55�

77.90�

00

12

100

Page 51: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

384 Chapter 4 Trigonometry

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

.sin�x� � cos�x ��

2�2

1

−2

f = g

xπ π32 2

− π32

y

86. f�x� � sin x, g�x� � �cos�x ��

2�

x 0

0 1 0 0

0 1 0 0�1�cos�x ��

2��1sin x

2�3�

2�

2

83. True.

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

2�.cos x � sin�x ��

2�, y � �cos x � �sin�x ��

2�,

84. Answers will vary.

Conjecture: sin x � �cos�x ��

2�2

1

−2

f = g

x

y

π π32 2

π32

79.

(a) Period

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

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

(c)

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

00

365

60

�2�

2�

365

� 365

C � 30.3 � 21.6 sin�2� t

365� 10.9� 80. (a) Period

The wheel takes 12 minutes to revolve once.

(b) Amplitude: 50 feet

The radius of the wheel is 50 feet.

(c)

00 20

110

�2�

��

6�� 12 minutes

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

sin(x)sin(x � 2�) 82. False. has an amplitude that is half that of For y � a cos bx, the amplitude is �a�.y � cos x.

y �12 cos 2x

Page 52: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.5 Graphs of Sine and Cosine Functions 385

87. (a)

The graphs are nearly the same for

(b)

The graphs are nearly the same for ��

2< x <

2.

−2

−2 2� �

2

��

2< x <

2.

−2

−2 2� �

2 (c)

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

4< x <

3�

4.

−2

−2 2� �

2

−2

2�−2�

2

cos x � 1 �x2

2!�

x4

4!�

x6

6!

sin x � x �x3

3!�

x5

5!�

x7

7!

88. (a)

(by calculator)

(c)

(by calculator)

(e)

(by calculator)cos 1 � 0.5403

cos 1 � 1 �1

2!�

1

4!� 0.5417

sin �

6� 0.5

sin �

6� 1 �

���6�3

3!�

���6�5

5!� 0.5000

sin 1

2� 0.4794

sin 1

2�

1

2�

�1�2�3

3!�

�1�2�5

5!� 0.4794 (b)

(by calculator)

(d)

(by calculator)

(f)

(by calculator)cos �

4� 0.7071

cos �

4� 1 �

���4�2

2!�

���4�2

4!� 0.7074

cos��0.5� � 0.8776

cos��0.5� � 1 ���0.5� 2

2!�

��0.5�4

4!� 0.8776

sin 1 � 0.8415

sin 1 � 1 �1

3!�

1

5!� 0.8417

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

89. log10 x � 2 � log10�x � 2�1�2 �12 log10�x � 2�

91. ln t3

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

93.

� log10 xy

12 �log10 x � log10 y� �

12 log10�xy�

90.

� 2 log2 x � log2�x � 3�

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

92.

�1

2 ln z �

1

2 ln�z2 � 1�

ln z

z2 � 1�

1

2 ln � z

z2 � 1� �1

2�ln z � ln�z2 � 1�

94.

� log2 x3y

� log2 x2(xy)

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

95.

� ln�3xy4�

ln 3x � 4 ln y � ln 3x � ln y4

Page 53: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

386 Chapter 4 Trigonometry

96.

� ln�x2�2x �

� ln�x3�2xx2 �

� ln �2xx2 � ln x3

�12 �ln

2xx2� � ln x3

12

�ln 2x � 2 ln x� � 3 ln x �12

�ln 2x � ln x2� � ln x3

Section 4.6 Graphs of Other Trigonometric Functions

■ You should be able to graph

■ When graphing or you should first graph orbecause

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

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

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

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

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

y � a csc�bx � c�y � a sec�bx � c�

y � a cot�bx � c�y � a tan�bx � c�

97. Answers will vary.

1.

Period:

Matches graph (e).

2�

2� �

y � sec 2x 3.

Period:

Matches graph (a).

�� 1

y �1

2 cot � x2.

Period:

Asymptotes:

Matches graph (c).

x � ��, x � �

b�

1�2� 2�

y � tan x

2

4.

Period:

Matches graph (d).

2�

y � �csc x 5.

Period:

Asymptotes:

Matches graph (f).

x � �1, x � 1

2�

b�

2�

��2� 4

y �1

2 sec

�x

26.

Period:

Asymptotes:

Reflected in x-axis

Matches graph (b).

x � �1, x � 1

2�

b�

2�

��2� 4

y � �2 sec �x

2

Vocabulary Check

1. vertical 2. reciprocal 3. damping 4.

5. 6. 7. 2����, �1� � �1, ��x � n�

Page 54: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.6 Graphs of Other Trigonometric Functions 387

7.

Period:

Two consecutive asymptotes:

x � ��

2 and x �

2

1

2

3

xπ π

y

y �1

3 tan x

9.

Period:

Two consecutive asymptotes:

3x ��

2 ⇒ x �

6

3x � ��

2 ⇒ x � �

6

3

x

4

3

2

1

− π3

π3

yy � tan 3x

x 0

y 013

�13

4�

4

x 0

y 0 1�1

12�

12

8.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

−3

1

2

3

ππ−x

yy �1

4 tan x

x 0

y 01

4�

1

4

4�

4

10.

Period:

Two consecutiveasymptotes:

x � �1

2, x �

1

2

�� 1

2

−8

−4

x

yy � �3 tan �x

x 0

y 3 0 �3

1

4�

1

4

13.

Period:

Two consecutiveasymptotes:

x � 0, x � 1

2�

�� 2

x

4

3

2

1

−3

−4

21−1−2

yy � csc �x

11.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

2�

1

2

3

y

− π

y � �1

2 sec x 12.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

2�

π

3

2x

yy �

1

4 sec x

14.

Period:

Two consecutiveasymptotes:

x � 0, x ��

4

2�

4 �

2

8

6

4

2

−2− π4

π4

x

yy � 3 csc 4x

0

�1�1

2�1y

3�

3x 0

12

1

412

y

3�

3x

2 1 2y

5

612

1

6x

6 3 6y

5�

24�

8�

24x

Page 55: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

388 Chapter 4 Trigonometry

20.

Period:

Two consecutive asymptotes:

x � 0, x � 2

��2� 2

2

4

6

2−2x

yy � 3 cot �x

219.

Period:

Two consecutive asymptotes:

x

2� � ⇒ x � 2�

x

2� 0 ⇒ x � 0

1�2� 2�

1

2

3

x

y

π−2 2π

y � cot x

2

x

y 1 0 �1

3�

2�

2

21.

Period:2�

2� �

3

xπ π

y

y �1

2 sec 2x 22.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

�2

3

ππ−

y

x

y � �1

2 tan x

x 0

y 0 �1

2

1

2

4�

4

1

3 0 �3y

3

2

1

2x

0

1 11

2y

6�

6x

15.

Two consecutiveasymptotes:

x � �12

, x �12

Period: 2�

�� 2

1−1−2−3 2 3−1

x

yy � sec �x � 1 16.

Period:

Two consecutiveasymptotes:

x � ��

8, x �

8

2�

4�

2

2

4

6

x

y

π4

π4

π2

y � �2 sec 4x � 2

17.

Period:

Two consecutiveasymptotes:

x � 0, x � 2�

2�

1�2� 4�

2

4

6

yy � csc x

2

0

1 0 1y

1

3�

1

3x 0

0 �2�2y

12�

12x

2 1 2y

5�

3�

3x

18.

Period:

Two consecutiveasymptotes:

x � 0, x � 3�

2�

1�3� 6�

2

4

6

ππ 2x

yy � csc x

3

2 1 2y

5�

23�

2�

2x

Page 56: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.6 Graphs of Other Trigonometric Functions 389

23.

Period:

Two consecutive asymptotes:

�x4

��

2 ⇒ x � 2

�x4

� ��

2 ⇒ x � �2

��4� 4

−4 4

2

4

6

x

y

y � tan �x

4

x 0 1

y 0 1�1

�1

24.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

�4

3

2

1

πx

yy � tan�x � ��

x 0

y 10�1

4�

4

25.

Period:

Two consecutive asymptotes:

x � 0, x � �

2�

1

2

3

4

x

− ππ322

yy � csc�� � x�

2 1 2y

5�

6�

2�

6x

26.

Period:

Two consecutive asymptotes:

x � 0, x ��

2

2�

2� �

y

x

−1

−2

π32

π2

2ππ

y � csc�2x � ��

�2�1�2y

5�

12�

4�

12x

27.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

2�

y

x

−1

1

2

3

4

π 2π−π 3π

y � 2 sec�x � �� 28.

Period:

Two consecutive asymptotes:

x � �12

, x �12

2�

�� 2

y

x1 2 3 4

1

2

3

y � �sec �x � 1

0

�4�2�4y

3�

3x

0

0 1�1y

1

3�

13

x

29.

Period:

Two consecutive asymptotes:

x � ��

4, x �

3�

4

2�

xπ2

1

2

yy �1

4 csc�x �

4� 30.

Period:

Two consecutive asymptotes:

x � ��

2, x �

2

−2

xπ32

− π32

yy � 2 cot�x ��

2�

x 0

y 2 0 �2

4�

4

12

1

412

y

7�

12�

4�

12x

Page 57: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

390 Chapter 4 Trigonometry

39.

−6

−0.6

6

0.6y � 0.1 tan��x4

��

4�

41.

xπ π2

2

y

x � �7�

4, �

3�

4,

4,

5�

4

tan x � 1

40.

−6

−2

6

2

y �1

3 cos��x

2�

2� ⇒ y �

1

3 sec��x

2�

2�

42.

xπ π2

2

1

y

x � �5�

3, �

2�

3,

3,

4�

3

tan x � �3 43.

x

y

π2

3π2

1

2

3

−3

x � �4�

3, �

3,

2�

3,

5�

3

cot x � ��3

3

31.

−5

5�−5�

5

y � tan x

333.

−4

�2

�2

4

y � �2 sec 4x ��2

cos 4x32.

−3

3

�4

3�4

3−

y � �tan 2x

34.

−3

−2

3

2

y � sec �x ⇒ y �1

cos��x� 35.

−3

− �2

3�2

3

3

y � tan�x ��

4� 36.

−3

�2

3�2

3−

3

�1

4 tan�x ��

2�

y �1

4 cot�x �

2�

37.

y ��1

sin�4x � ��

−3

− �2

�2

3y � �csc�4x � �� 38.

y �2

cos�2x � ��

−4

−� �

4 y � 2 sec(2x � �) ⇒

Page 58: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.6 Graphs of Other Trigonometric Functions 391

44.

xπ2

π2

3π2

3π2

− −

2

3

−3

y

x � �7�

4, �

3�

4,

4,

5�

4

cot x � 1 45.

xππ 2π π2

1

−−

y

x � ±2�

3, ±

4�

3

sec x � �2 46.

xππ π2π2

1

−−

y

x � �5�

3, �

3,

3,

5�

3

sec x � 2

47.

x � �7�

4, �

5�

4,

4,

3�

4

xπ 3ππ3π

2 2 2 2− −

1

2

3

−1

ycsc x � �2 48.

x � �2�

3, �

3,

4�

3,

5�

3

xπ 3ππ3π

2 2 2 2− −

1

2

3

y csc x � �2�3

3

49.

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

f �x� � sec x

� f �x�

�1

cos x

�1

cos��x�

f ��x� � sec��x�

y

x

3

4

ππ π2−

f �x� � sec x �1

cos x 50.

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

y � tan x

tan��x� � �tan x

x

2

3

−3

3π2

− π2− π

23π2

y f�x� � tan x

51.

(a)

(b) on the interval,

(c) As andsince is

the reciprocal of f �x�.g �x�g�x� �

12 csc x → ±�

f �x� � 2 sin x → 0x → �,

6< x <

5�

6f > g

1

−1

2

3

f

g

xπ ππ32 4

π4

y

g�x� �1

2 csc x

f �x� � 2 sin x 52.

(a)

(b) The interval in which

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

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

−3

g

f

3

1−1

f�x� � tan �x

2, g�x� �

1

2 sec

�x

2

Page 59: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

392 Chapter 4 Trigonometry

57.

As

Matches graph (d).

x → 0, f �x� → 0 and f �x� > 0.

f �x� � x cos x 58.

Matches graph (a) as x → 0, f�x� → 0.

f�x� � x sin x

60.

Matches graph (c) as x → 0, g�x� → 0.

g�x� � x cos x59.

As

Matches graph (b).

x → 0, g�x� → 0 and g�x� is odd.

g�x� � x sin x

61.

The graph is the line y � 0.

f �x� � g�x�

g�x� � 0

−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

x

yf �x� � sin x � cos�x ��

2� 62.

It appears that That is,

sin x � cos�x ��

2� � 2 sin x.

f�x� � g�x�.

g�x� � 2 sin x

−4

2

4

xππ−

y

f�x� � sin x � cos�x ��

2�

63.

f �x� � g�x�

g�x� �1

2�1 � cos 2x�

–1

2

3

y

xππ−

f �x� � sin2 x 64.

It appears that That is,

cos2 �x

2�

1

2�1 � cos �x�.

f�x� � g�x�.

g�x� �1

2�1 � cos �x�

−1

−3 3 6−6

2

3

x

yf�x� � cos2 �x

2

53.

The expressions are equivalent except when and y1 is undefined.

sin x � 0

sin x csc x � sin x� 1

sin x� � 1, sin x � 0

−2

3−3

2

y1 � sin x csc x and y2 � 1 54.

The expressions are equivalent.

sin x sec x � sin x 1

cos x�

sin x

cos x� tan x

−4

−2 2

4

� �

y1 � sin x sec x, y2 � tan x

55.

The expressions are equivalent.

−4

2�−2�

4cot x �cos x

sin x

y1 �cos x

sin x and y2 � cot x �

1

tan x56.

The expressions are equivalent.

tan2 x � sec2 x � 1

1 � tan2 x � sec2 x

−1

3

�2

3�2

3

y1 � sec2 x � 1, y2 � tan2 x

Page 60: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.6 Graphs of Other Trigonometric Functions 393

65.

The damping factor is

As x → �, g�x� →0.

y � e�x2�2.

�e�x2�2 ≤ g�x� ≤ e�x2�2

−1

8−8

1g�x� � e�x2�2 sin x

66.

Damping factor:

As x → �, f�x� → 0.

−3 6

−3

3

e�x

f�x� � e�x cos x 67.

Damping factor:

As x→�, f �x� → 0.

−9 9

−6

6

y � 2�x�4.

�2�x�4 ≤ f �x� ≤ 2�x�4

f �x� � 2�x�4 cos �x 68.

Damping factor:

As x → �, h �x� → 0.

−8

−1

8

1

2�x2�4

h�x� � 2�x2�4 sin x

69.

As x → 0, y → �.

0

−2

8�

6

y �6

x� cos x, x > 0 71.

As x → 0, g�x� → 1.

−1

6�−6�

2

g�x� �sin x

x70.

As x → 0, y → �.

0

−2

6�

6

y �4

x� sin 2x, x > 0

72.

As x → 0, f (x) → 0.

−1

−6� 6�

1

f (x) � 1 � cos x

x73.

As oscillates between and 1.�1

x → 0, f �x�

−2

−� �

2

f �x� � sin 1

x74.

As oscillates.x → 0, h(x)

−1

−� �

2

h(x) � x sin 1

x

75.

Gro

und

dist

ance

x

14

10

6

2

−2

−6

−10

−14Angle of elevation

d

π π π32 4

π4

d �7

tan x� 7 cot x

tan x �7

d76.

x

20

40

60

80

Angle of camera

Dis

tanc

e

d

0 π2

π4

π4

π2

− −

d �27

cos x� 27 sec x, �

2< x <

2

cos x �27

d

Page 61: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

394 Chapter 4 Trigonometry

77.

(a)

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

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

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

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

p �2�

��12� 24 months

00 100

50,000

R

C

R � 25,000 � 15,000 cos �t12

C � 5000 � 2000 sin �t12

,

78.

Month (1 ↔ January)

Law

n m

ower

sal

es(i

n th

ousa

nds

of u

nits

) 150135120105907560453015

2 4 6 8 10 12t

SS � 74 � 3t � 40 cos �t6

79.

(a) Period of

Period of sin

Period of months

Period of monthsL�t� : 12

H�t� : 12

� t

6:

2�

��6� 12

cos � t

6:

2�

��6� 12

L�t� � 39.36 � 15.70 cos � t

6� 14.16 sin

� t

6

H�t� � 54.33 � 20.38 cos � t

6� 15.69 sin

� t

6

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

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

80. (a)

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

y �12

e�t�4 cos 4t

0 4�

−0.6

0.6 81. True. Since

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

csc xyx

y � csc x �1

sin x,

Page 62: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.6 Graphs of Other Trigonometric Functions 395

83. As from the left,

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

2

f�x� � tan x → �.x → �

282. True.

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

y �1

sin�x ��

2��

1cos x

� sec x.

��2y � sin x

y � sec x �1

cos x

84. As from the left, .

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

f (x) � csc x → �x → �

85.

(a)

The zero between 0 and 1 occurs at x 0.7391.

3

−2

−3

2

f �x� � x � cos x

(b)

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

x9 � cos 0.7504 0.7314

x8 � cos 0.7221 0.7504

x7 � cos 0.7640 0.7221

x6 � cos 0.7014 0.7640

x5 � cos 0.7935 0.7014

x4 � cos 0.6543 0.7935

x3 � cos 0.8576 0.6543

x2 � cos 0.5403 0.8576

x1 � cos 1 0.5403

x0 � 1

xn � cos�xn�1�

86.

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

y � x �2x3

3!�

16x5

5!

−6

− �2

3�2

3

6y � tan x 87.

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

y2 � 1 �x2

2!�

5x4

4!

−6

6

�2

3�2

3

y1 � sec x

88. (a)

—CONTINUED—

−3 3

−2

2

y2

−3 3

−2

2

y1

y2 �4��sin �x �

13

sin 3�x �15

sin 5�x�y1 �4��sin �x �

13

sin 3�x�

Page 63: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

396 Chapter 4 Trigonometry

88. —CONTINUED—

(b)

(c) y4 �4��sin �x �

13

sin 3�x �15

sin 5�x �17

sin 7�x �19

sin 9�x�

−3 3

−2

2

y3 �4��sin �x �

13

sin 3�x �15

sin 5�x �17

sin 7�x�

89.

x �ln 54

2 1.994

2x � ln 54

e2x � 54 91.

x � �ln 2 �0.693

ln 2 � � x

2 � e�x

3 � 1 � e�x

300100

� 1 � e�x

300

1 � e�x � 100

93.

1.684 � 1031

x �2 � e73

3

3x � 2 � e73

3x � 2 � e73

ln�3x � 2� � 73

95.

x � ±�e3.2 � 1 ±4.851

x2 � e3.2 � 1

x2 � 1 � e3.2

ln�x2 � 1� � 3.2

97.

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

x � 2log8 xx � �1

x � 2, �1

�x � 2��x � 1� � 0

x2 � x � 2 � 0

x2 � x � 2

x�x � 1� � 81�3

log8�x�x � 1�� �13

log8 x � log8�x � 1� �13

90.

x �ln 983 ln 8

0.735

3x � log8 98

83x � 98

92.

t �1

365 �log10 5

log10 1.00041096� 10.732

365t � log1.00041096 5

1.00041096365t � 5

1 �0.15365

1.00041096

�1 �0.15365 �

365t

� 5

94.

x �14 � e68

2 �1.702 � 1029

14 � e68 � 2x

14 � 2x � e68

ln(14 � 2x) � 68

96.

22,022.466

x � e10 � 4

x � 4 � e10

ln�x � 4� � 10

12 ln�x � 4� � 5

ln�x � 4 � 5 98.

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

log6 x��65

x � ±�65

x2 � 1 � 64

x�x2 � 1� � 64x

log6�x(x2 � 1�� � log6�64x�

log6 x � log6�x2 � 1� � log6�64x�

Page 64: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.7 Inverse Trigonometric Functions 397

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

Function Domain Range

y � arcsin x ⇒ x � sin y

y � arccos x ⇒ x � cos y

y � arctan x ⇒ x � tan y

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

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

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

2< y <

2

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

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

2 ≤ y ≤

2

��

2< x <

2�� < x < �

0 ≤ y ≤ ��1 ≤ x ≤ 1

��

2≤ y ≤

2�1 ≤ x ≤ 1

Vocabulary Check

Alternative Function Notation Domain Range

1.

2.

3. ��

2< y <

2�� < x < �y � tan�1 xy � arctan x

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

��

2≤ y ≤

2�1 ≤ x ≤ 1y � sin�1 xy � arcsin x

1. ��

2≤ y ≤

2 ⇒ y �

6y � arcsin

1

2 ⇒ sin y �

1

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

2≤ y ≤

2 ⇒ y � 0

3. 0 ≤ y ≤ � ⇒ y ��

3y � arccos

1

2 ⇒ cos y �

1

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

2

5.

��

2< y <

2 ⇒ y �

6

y � arctan �3

3 ⇒ tan y �

�3

3 for 6.

��

2< y <

2 ⇒ y � �

4

y � arctan��1� ⇒ tan y � �1 for

Section 4.7 Inverse Trigonometric Functions

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398 Chapter 4 Trigonometry

7.

0 ≤ y ≤ � ⇒ y �5�

6

y � arccos���3

2 � ⇒ cos y � ��3

2 for 8.

��

2≤ y ≤

2 ⇒ y � �

4

y � arcsin���2

2 � ⇒ sin y � ��2

2 for

9.

��

2< y <

2 ⇒ y � �

3

y � arctan���3� ⇒ tan y � ��3 for

11.

0 ≤ y ≤ � ⇒ y �2�

3

y � arccos��1

2� ⇒ cos y � �1

2 for

13.

��

2≤ y ≤

2 ⇒ y �

3

y � arcsin �3

2 ⇒ sin y �

�3

2 for

10.

��

2< y <

2 ⇒ y �

3

y � arctan��3 � ⇒ tan y � �3 for

12.

��

2≤ y ≤

2 ⇒ y �

4

y � arcsin �2

2 ⇒ sin y �

�2

2 for

14.

��

2< y <

2 ⇒ y � �

6

y � arctan���3

3 � ⇒ tan y � ��3

3 for

15. y � arctan 0 ⇒ tan y � 0 for ��

2< y <

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

17.

y � x

g�x� � arcsin x

−1

1.5−1.5

fg

1 f�x� � sin x 18.

Graph

Graph

Graph y3 � x.

y2 � tan�1 x.

y1 � tan x.

−2

g

f

2

�2

�2

f �x� � tan x and g�x� � arctan x

19. arccos 0.28 � cos�1 0.28 � 1.29

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

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

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

29. arctan 0.92 � tan�1 0.92 � 0.74

31. arcsin�34� � sin�1�0.75� � 0.85 33. arctan�7

2� � tan�1�3.5� � 1.29

20. arcsin 0.45 � 0.47

22. arccos��0.7� � 2.35

24. arctan 15 � 1.50

28. arcsin��0.125� � �0.13

26. arccos 0.26 � 1.31

30. arctan 2.8 � 1.23

32. arccos��13� � 1.91

34. arctan��95

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

.���3

3,�

6�, and �1, �

4����3, ��

3�,

y � arctan x.

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Section 4.7 Inverse Trigonometric Functions 399

36.

cos��

6� ��3

2

arccos��1

2� �2�

3

arccos��1� � � 37.

tan� � arctan x

4 θ4

x

tan � �x

4

38.

� � arccos 4

x

cos � �4

x39.

sin� � arcsin�x � 2

5 �θ

5x + 2

sin � �x � 2

5

40.

� � arctan�x � 1

10 �

tan � �x � 1

1041.

� � arccos�x � 32x �

θ

x + 3

2x

cos � �x � 3

2x

42.

x � 1

� � arctan 1

x � 1

tan � �x � 1

x2 � 1�

1

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

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

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

arcsin�sin 3�� � arcsin�0� � 0

48.

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

2

arccos�cos 7�

2 � � arccos 0 ��

249. Let

and sin y �3

5.

tan y �3

4, 0 < y <

2,

xy

53

4

y

y � arctan 3

4,

50. Let

sec�arcsin 4

5� � sec u �5

3.

sin u �4

5, 0 < u <

2,

3

45

u

u � arcsin 4

5, 51. Let

and cos y �1�5

��5

5.

tan y � 2 �2

1, 0 < y <

2,

x

y

5 2

1

yy � arctan 2,

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400 Chapter 4 Trigonometry

52. Let

1

25

u

sin�arccos �5

5 � � sin u �2

�5�

2�5

5.

cos u ��5

5, 0 < u <

2,

u � arccos �5

5, 53. Let

and

xy

5

12

13

y

cos y �12

13.sin y �

5

13, 0 < y <

2,

y � arcsin 5

13,

55. Let

and

xy

34−3

5

y

sec y ��34

5.tan y � �

3

5, �

2< y < 0,

y � arctan��3

5�,54. Let

13

12

−5u

cscarctan��5

12� � csc u � �13

5.

tan u � �5

12, �

2 < u < 0,

u � arctan��5

12�,

56. Let

4−3

7u

tanarcsin��3

4� � tan u � �3�7

� �3�7

7.

sin u � �3

4, �

2< u < 0,

u � arcsin��3

4�, 57. Let

and

xy

3

−2

5

y

sin y ��5

3.cos y � �

2

3,

2< y < �,

y � arccos��2

3�,

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Section 4.7 Inverse Trigonometric Functions 401

60. Let

x + 12

1

x

u

sin�arctan x� � sin u �x

�x2 � 1 .

tan u � x �x

1,u � arctan x, 61. Let

and cos y � �1 � 4x2.

sin y � 2x �2x

1,

1 − 4x2

2x

y

1

y � arcsin�2x�,

62. Let

sec�arctan 3x� � sec u � �9x2 � 1.

tan u � 3x �3x

1,

1

3x9 + 1x2

u

u � arctan 3x, 63. Let

and sin y � �1 � x2.

cos y � x �x

1, 1 − x2

xy

1

y � arccos x,

64. Let

x −1

2x x− 2

1

u

sec�arcsin�x � 1�� � sec u �1

�2x � x2.

sin u � x � 1 �x � 1

1,

u � arcsin�x � 1�, 65. Let

and tan y ��9 � x2

x.

cos y �x

3, 9 − x2

xy

3

y � arccos�x

3�,

66. Let

cot�arctan 1

x� � cot u � x.

tan u �1

x, 1

ux

x + 12

u � arctan 1

x, 67. Let

and csc y ��x2 � 2

x.

tan y �x�2

,x2 + 2

x

y

2

y � arctan x�2

,

59. Let

and cot y �1

x.

tan y � x �x

1,

x2 + 1x

y

1

y � arctan x,58. Let

cot�arctan 5

8� � cot u �8

5.

tan u �5

8, 0 < u <

2, 5

8

89

u

u � arctan 5

8,

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402 Chapter 4 Trigonometry

69.

They are equal. Let

and

The graph has horizontal asymptotes at y � ±1.

g�x� �2x

�1 � 4x2� f �x�

1 + 4x2

2x

y

1

sin y �2x

�1 � 4x2.

tan y � 2x �2x

1,

y � arctan 2x, −3 3

−2

2

f �x� � sin�arctan 2x�, g�x� �2x

�1 � 4x2

70.

Asymptote:

These are equal because:

Let

Thus, f �x� � g�x�.

��4 � x2

x� g�x�

f �x� � tan�arccos x

2� � tan u

u � arccos x

2. 2

ux

4 − x 2

x � 0

g�x� ��4 � x2

x

−3

−2

3

2

f�x� � tan�arccos x

2� 71. Let

Thus,

x2 + 81

x

y

9

arcsin y ��9

�x2 � 81, x < 0.

arcsin y �9

�x2 � 81, x > 0;

tan y �9

x and sin y �

9�x2 � 81

, x > 0; �9

�x2 � 81, x < 0.

y � arctan 9

x.

72. If

then

arcsin �36 � x2

6� arccos

x

6.

sin u ��36 � x2

6,

6

ux

36 − x 2

arcsin �36 � x2

6� u, 73. Let Then,

and

Thus,

(x − 1)2 + 9

3

y

x − 1

y � arcsin �x � 1��x2 � 2x � 10

.

sin y � �x � 1���x � 1�2 � 9

.

cos y �3

�x2 � 2x � 10�

3��x � 1�2 � 9

y � arccos 3

�x2 � 2x � 10.

68. Let

cos�arcsin x � h

r � � cos u ��r2 � �x � h�2

r.

sin u �x � h

r, r

x h−

r x h− −( )22

u

u � arcsin x � h

r,

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Section 4.7 Inverse Trigonometric Functions 403

75.

Domain:

Range:

This is the graph of with a factor of 2.

−2 −1 1 2

π

π2

x

y

f �x� � arccos x

0 ≤ y ≤ 2�

�1 ≤ x ≤ 1

y � 2 arccos x

76.

Domain:

Range:

This is the graph ofwith a

horizontal stretch of afactor of 2.

f �x� � arcsin x

��

2≤ y ≤

2

�2 ≤ x ≤ 2

1 2−2x

π

π

y

y � arcsin x

277.

Domain:

Range:

This is the graph of shifted

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

��

2≤ y ≤

2

0 ≤ x ≤ 2

−1 1 2 3

π

π

y

x

f �x� � arcsin�x � 1�

78.

Domain:

Range:

This is the graph ofshifted

two units to the left.y � arccos t

0 ≤ y ≤ �

�3 ≤ t ≤ �1

−4 −3 −2 −1t

π

yg�t� � arccos�t � 2� 79.

Domain: all real numbers

Range:

This is the graph ofwith a

horizontal stretch of a factor of 2.

g�x� � arctan�x�

��

2< y <

2−4 −2 2 4

π

π

y

x

f �x� � arctan 2x

80.

Domain: all real numbers

Range:

This is the graph ofshifted

upward units.��2y � arctan x

0 < y ≤ �

−4 −2 2 4x

π

y

f�x� ��

2� arctan x 81.

Domain:

Range: all real numbers

−2 1 2

π

y

v

1 − v2

vy

1

�1 ≤ v ≤ 1, v � 0

h�v� � tan�arccos v� ��1 � v2

v

74. If

then

2

ux − 2

4x x− 2

arccos x � 2

2� arctan

�4x � x2

x � 2.

cos u �x � 2

2,

arccos x � 2

2� u,

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404 Chapter 4 Trigonometry

82.

Domain:

Range: 0 ≤ y ≤ �

�4 ≤ x ≤ 4

−4 −2 2 4x

π

yf�x� � arccos

x

483.

0

2�

−1 1

f �x� � 2 arccos�2x�

84.

−0.5 0.5

2�

−2�

f�x� � � arcsin�4x� 85.

−2 4

�−

f �x� � arctan�2x � 3� 86.

−4 4

−2�

�2

f�x� � �3 � arctan��x�

87.

5

−2

−4

4

f �x� � � � arcsin�23� � 2.412 88.

5

−2

−4

4

f�x� ��

2� arccos�1

�� � 2.82

89.

The graph implies that the identity is true.

� 3�2 sin�2t ��

4� � 3�2 sin�2t � arctan 1�

−6

−2

6

� 2�

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

3�

90.

The graph implies that

is true.

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

B�−6

6−6

6 � 5 sin�� t � arctan 4

3�

� �42 � 32 sin�� t � arctan 4

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

(b)

s � 20: � � arcsin 5

20� 0.25

s � 40: � � arcsin 5

40� 0.13

sin� � arcsin 5

s

sin � �5

s

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Section 4.7 Inverse Trigonometric Functions 405

92. (a)

(b) When

When

� � arctan 1200

750� 1.01 � 58.0.

s � 1200,

� � arctan 300

750� 0.38 � 21.8.

s � 300,

� � arctan s

750

tan � �s

75093.

(a)

(b) is maximum when feet.

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

� 0.

x � 2

0

−0.5

6

1.5

� arctan 3x

x2 � 4

94. (a)

(b)

feeth � 20 tan � � 20 �1117

� 12.94

tan � �hr

�h

20

r �12

�40� � 20

� � arctan 11

17� 0.5743 � 32.9

tan � �11

1795.

(a)

(b)

h � 50 tan 26 � 24.39 feet

tan 26 �h

50

� � arctan�20

41� � 26.0

tan � �20

41

20 ft

41 ft

θ

96. (a)

(b)

� � arctan 6

1� 1.41 � 80.5

x � 1 mile

� � arctan 6

7� 0.71 � 40.6

x � 7 miles

� � arctan 6

x

tan � �6

x97. (a)

(b)

x � 12: � � arctan 12

20� 31.0

x � 5: � � arctan 5

20� 14.0

� � arctan x

20

tan � �x

20

98. False.

is not in the range of

arcsin 12

��

6

arcsin�x�.5�

6

99. False.

is not in the range of the arctangent function.

arctan 1 ��

4

5�

4

100. False.

is defined for all real x, but and require

Also, for example,

Since , but undefined.arcsin 1arccos 1

���2

0�arctan 1 �

4

arctan 1 �arcsin 1arccos 1

.

�1 ≤ x ≤ 1.arccos xarcsin xarctan x

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406 Chapter 4 Trigonometry

101.

Domain:

Range: 0 < x < �

x−1−2 21

π2

y

π

�� < x < �

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

103.

Domain:

Range: ��

2, 0� � �0,

2���, �1� � �1, ��

x−2 −1 21

y

π2

π2

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

102. if and only if where

and and

The domain of arcsec x is

and the range is

x−2 −1 1 2

y

π

π2

0, �

2� � ��

2, �.

���, �1� � �1, ��y �

2< y ≤ �.0 ≤ y <

2x ≤ �1 � x ≥ 1

sec y � xy � arcsec x

104. (a)

(b)

(c)

(d) y � arccsc 2 ⇒ csc y � 2 and ��

2≤ y < 0 � 0 < y ≤

2 ⇒ y �

6

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

6

y � arcsec 1 ⇒ sec y � 1 and 0 ≤ y <�

2�

2< y ≤ � ⇒ y � 0

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

2�

2< y ≤ � ⇒ y �

4

105.

(a)

(b)

(c)

(d)

� 1.25 � ���

4� � 2.03

Area � arctan 3 � arctan��1�

a � �1, b � 3

� 1.25 � 0 � 1.25

Area � arctan 3 � arctan 0

a � 0, b � 3

��

4� ��

4� ��

2

Area � arctan 1 � arctan��1�

a � �1, b � 1

Area � arctan 1 � arctan 0 ��

4� 0 �

4

a � 0, b � 1

Area � arctan b � arctan a 106.

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

3�,

00

6

g

f

12

g�x� � 6 arctan x

f �x� � �x

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Section 4.7 Inverse Trigonometric Functions 407

107.

(a)

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

over its entire domain, .

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

one-to-one and thus has an inverse.

��

2≤ x ≤

2f �1 � f � x

�1 ≤ x ≤ 1f � f �1 � x

y � x

−2

−� �

2

−2

−� �

2

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

f �x� � sin�x�, f �1�x� � arcsin�x�

108. (a) Let Then,

Therefore, arcsin arcsin x.

(c) Let

(e)

x1

y1

2y

1 − x 2

� arctan x

�1 � x2arcsin x � arcsin

x

1

x

1

y1

2y

� y1 � ��

2� y1� �

2

arctan x � arctan 1

x� y1 � y2

y2 ��

2� y1.

��x� � �

y � �arcsin x.

�y � arcsin x

sin��y� � x

�sin y � x

sin y � �x

y � arcsin��x�. (b) Let Then,

Thus,

(d) Let then andThus, which implies that and

are complementary angles and we have

arcsin x � arccos x ��

2.

� � ��

2

�sin � � cos cos � x.

sin � � x� � arcsin x and � arccos x,

arctan��x� � �arctan�x�.

y � �arctan x

�y � arctan x

arctan�tan��y�� � arctan x

tan��y� � x, ��

2< �y <

2

�tan y � x

tan y � �x, ��

2< y <

2

y � arctan��x�.

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

142�

10

196� 0.051

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408 Chapter 4 Trigonometry

111. �1.1�50 � 117.391 112. 16�2� �1

162�� 2.718 10�8

113.

csc � �43

sec � �4�7

�4�7

7

cot � ��73

tan � �3�7

�3�7

7

cos � ��74

adj � �7

�adj�2 � 7

�adj�2 � 9 � 16

�adj�2 � �3�2 � �4�24 3

θ

sin � �34

�opphyp

114.

csc � �12�5

sec � � �5

cot � �12

sin � �2�5

cos � �1�5

hyp � �12 � 22 � �52

1

θ

tan � � 2

115.

csc � �6

�11�

6�1111

sec � �65

cot � �5

�11�

5�1111

tan � ��11

5

sin � ��11

6

opp � �11

�opp�2 � 11

�opp�2 � 25 � 36

�opp�2 � �5�2 � �6�2 6

5

θ

cos � �56

�adjhyp

116.

cot � �1

2�2�

�24

csc � �3

2�2�

3�24

sec � � 3

tan � � 2�2

sin � �2�2

3

cos � �13

� 2�2

� �8

opp � �32 � 12

3

1

θ

sec � � 3

Page 76: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.7 Inverse Trigonometric Functions 409

117. Let the number of people presently in the group. Each person’s share is now If two more join the group, each person’s share would then be

There were 8 people in the original group.

x � �10 is not possible.

x � �10 or x � 8

6250�x � 10��x � 8� � 0

6250�x2 � 2x � 80� � 0

6250x2 � 12500x � 500,000 � 0

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

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

250,000

x � 2�

250,000

x� 6250

Share per person with

two more people �

Original share

per person � 6250

250,000��x � 2�.250,000�x.x �

118. Rate downstream:

Rate upstream:

The speed of the current is 3 miles per hour.

x � ±3

x2 � 9

315 � 324 � x2

1260 � 4�324 � x2�

630 � 35x � 630 � 35x � 4�324 � x2�

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

35

18 � x�

35

18 � x� 4

�Time to go upstream� � �Time to go downstream� � 4

rate time � distance ⇒ t �d

r

18 � x

18 � x

119. (a)

(b)

(c)

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

A � 15,000�1 �0.035365 ��365��10�

� $21,285.66

A � 15,000�1 �0.035

12 ��12��10�� $21,275.17

A � 15,000�1 �0.035

4 ��4��10�� $21,253.63 120. Data:

To find:

Assume:

Then:

� 458,504.31

� 632,000 � �632742�

2

� 632,000 � �e�r�2�2

y � P0e�r�8 � P0e

�r�4 � e�r�4

e�r�2 �P0e�r�4

P0e�r�2 �632742

632,000 � P0e�r�4

742,000 � P0e�r�2

P � P0 � e�rt

�8, y�

�4, 632,000��2, 742,000�,

Page 77: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.8 Applications and Models

410 Chapter 4 Trigonometry

■ You should be able to solve right triangles.

■ You should be able to solve right triangle applications.

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

Vocabulary Check

1. elevation; depression 2. bearing

3. harmonic motion

1. Given:

20°b = 10

a

AC

B

c

B � 90� � 20� � 70�

cos A �b

c ⇒ c �

b

cos A�

10

cos 20�� 10.64

tan A �a

b ⇒ a � b tan A � 10 tan 20� � 3.64

A � 20�, b � 10 2. Given:

cos B �a

c ⇒ a � c cos B � 15 cos 54� � 8.82

� 15 sin 54� � 12.14

sin B �b

c ⇒ b � c sin B

� 90� � 54� � 36�

A � 90� � B 54°

b

a

AC

B

c = 15

B � 54�, c � 15

3. Given:

71°

b = 24

a

AC

B

c

A � 90� � 71� � 19�

sin B �b

c ⇒ c �

b

sin B�

24

sin 71�� 25.38

tan B �b

a ⇒ a �

b

tan B�

24

tan 71�� 8.26

B � 71�, b � 24 4. Given:

8.4°

b

a = 40.5 AC

Bc

sin A �a

c ⇒ c �

a

sin A�

40.5

sin 8.4�� 277.24

�40.5

tan 8.4�� 274.27

tan A �a

b ⇒ b �

a

tan A

� 90� � 8.4� � 81.6�

B � 90� � A

A � 8.4�, a � 40.5

5. Given:

b = 10

a = 6

AC

B

c

B � 90� � 30.96� � 59.04�

tan A �a

b�

6

10 ⇒ A � arctan

3

5� 30.96º

� 2�34 � 11.66

c2 � a2 � b2 ⇒ c � �36 � 100

a � 6, b � 10 6. Given:

cos B �a

c ⇒ B � arccos

a

c� arccos

25

35� 44.42�

� arcsin 25

35� 45.58�

sin A �a

c ⇒ A � arcsin

a

c

� �600 � 24.49

� �352 � 252

b � �c2 � a2

b

a = 25 c = 35

AC

Ba � 25, c � 35

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Section 4.8 Applications and Models 411

7. Given:

B � 90� � 72.08� � 17.92�

A � arccos 16

52� 72.08º

cos A �16

52

� �2448 � 12�17 � 49.48

a � �522 � 162

b = 16

c = 52a

AC

Bb � 16, c � 52 8. Given:

� 8.03�

� arcsin 1.32

9.45 a

b = 1.32

c = 9.45

AC

Bsin B �

b

c ⇒ B � arcsin

b

c

cos A �b

c ⇒ A � arccos

b

c � arccos

1.32

9.45� 81.97�

a � �c2 � b2 � �87.5601 � 9.36

b � 1.32, c � 9.45

9. Given:

b

c = 430.5a

AC

B

12°15′

b � 430.5 cos 12�15� � 420.70

cos 12�15� �b

430.5

a � 430.5 sin 12�15� � 91.34

sin 12�15� �a

430.5

B � 90� � 12� 15� � 77� 45�

A � 12� 15�, c � 430.5 10. Given:

a = 14.2

b

c

AC

B

65°12′

tan B �b

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

cos B �a

c ⇒ c �

a

cos B�

14.2

cos 65� 12�� 33.85

A � 90� � B � 90� � 65� 12� � 24� 48�

B � 65� 12�, a � 14.2

11.

12 b 1

2 b

b

h

θ θ

h �1

2�4� tan 52� � 2.56 inches

tan � �h

�1�2�b ⇒ h �

1

2b tan � 12.

12 b 1

2 b

b

h

θ θ

h �12

�10� tan 18� � 1.62 meters

tan � �h

�1�2�b ⇒ h �12

b tan �

13.

12 b 1

2 b

b

h

θ θ

h �12

�46� tan 41� � 19.99 inches

tan � �h

�1�2�b ⇒ h �12

b tan � 14.

12 b 1

2 b

b

h

θ θ

h �12

�11� tan 27� � 2.80 feet

tan � �h

�1�2�b ⇒ h �12

b tan �

Page 79: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

412 Chapter 4 Trigonometry

18.

� 81.2 feet

h � 125 tan 33

33°

h

125

tan 33� �h

125

19. (a)

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

(c) feeth � 19.9

h � y � x � 50�tan 47�40� � tan 35��.

x � 50 tan 35� and y � 50 tan 47�40�

tan 35� �x

50 and tan 47�40� �

y

50

� y� x

50 ft

47° 40′

35°

h

x

y

20.

� 123.5 feet

h � 100 tan 51�

51°

h

100

tan 51� �h

100

21.

34°

4000x

� 2236.8 feet

x � 4000 sin 34º

sin 34� �x

400022.

θ

50 ft

75 ft

� � arctan 32

� 56.3�

tan � �7550

23. (a)

(b)

(c)

The angle of elevation of the sum is 35.8�.

� � arctan 121

2

1713

� 35.8�

tan � �121

2

1713

θ

17 ft

12 ft

1

1

3

2

24.

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

� � 14.03�

� � arcsin� 400016,500�

sin � �4000

16,500

16,500 mi

4,000 mi

θα

Not drawn to scale

12,500 � 4000 � 16,500

17.

16 si 74 h � 19.7 feet

20 sin 80� � h

80°

h20 ft

16 sin 80� �h

20

15.

� 107.2 feet

x �50

tan 25�25°

50

x

tan 25� �50x

16.

� 1648.5 feet

x �600

tan 20

600

20°x

tan 20� �600

x

Page 80: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.8 Applications and Models 413

26. (a) Since the airplane speed is

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

18°

16500a

a � 16,500 sin 18� � 5099 ft

sin 18� �a

16,500

�275ft

sec��60sec

min� � 16,500ft

min,

(b)

275s 10,000feet

18°

� 117.7 seconds

s �10,000

275(sin 18�)

sin 18� �10,000

275s

27.

x � 4 sin 10.5� � 0.73 mile

10.5°4 x sin 10.5� �

x

4

28.

Angle of grade:

Change in elevation:

� 2516.3 feet

� 21,120 sin�arctan 0.12�

y � 21,120 sin �

sin � �y

21,120

� � arctan 0.12 � 6.8�

tan � �12x

100x

100x12 =x y4 miles = 21,120 feet

θ 29. The plane has traveled

N

S

EW

90052°

38°a

b

cos 38� �b

900 ⇒ b � 709 miles east

sin 38� �a

900 ⇒ a � 554 miles north

1.5�600� � 900 miles.

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

Reno is miles W of Miami.

(b) The return heading is

N

S

EW

280°

10°

280�.

2472 cos 10� � 2434

N

S

EW

100°

80°10° Miami

Reno

2472 mi

2472 sin 10� � 429

25.

� � arctan� 950

26,400� � 2.06�

tan � �950

26,400

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

5 miles

950feet θ

θ

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

Page 81: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

414 Chapter 4 Trigonometry

32.

(a) hours

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

miles E

miles S

(c) Bearing from N is 178.6�.

240 cos 1.4� � 239.9

240 sin 1.4� � 5.9

t �42820

� 21.4

S

EW

N

Not drawn to scale

88.6°

1.4°

428

(b)

�d

50 ⇒ d � 68.82 meters

tan C �d

50 ⇒ tan 54�

C � � � � 54�

� 90� � � 22�

� � � � 32�

33.

(a)

Bearing from A to C: N 58º E

N

S

EW

C

B

d

A

θ α

φγ

β

β50

� � 90� � 32� � 58�

� � 32�, � 68�

34.

� 5.46 kilometers

d �30

cot 34� � cot 14�

d cot 34� � 30 � d cot 14�

cot 34� �30 � d cot 14�

d

�d

30 � d cot 14�

tan 34� �d

y�

d

30 � x

tan 14� �d

x ⇒ x � d cot 14� 35.

Bearing: N 56.3 W

N

S

EW

Port

Ship

45

30θ

tan � �45

30 ⇒ � � 56.3� 36.

N

S

EW

160

85

Plane

Airport

� 208.0� or 528� W

Bearing � 180� � arctan� 85160�

37.

Distance between ships: D � d � 1933.3 ft

tan 4� �350

D ⇒ D � 5005.23 ft

350

6.5°4°

dD

S1S2

Not drawn to scale

tan 6.5� �350

d ⇒ d � 3071.91 ft

31.

(a) nautical miles south

nautical miles west

(b)

Bearing: S W

Distance:

nautical miles from port� 130.9

d � �104.952 � 78.182

36.7�

tan � �20 � b

a�

78.18104.95

⇒ � � 36.7�

sin 29� �b

120 ⇒ b � 58.18

cos 29� �a

120 ⇒ a � 104.95

N

S

EW

120

20

29°

b

a

Page 82: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.8 Applications and Models 415

39.

� 17,054 ft

a cot 16� � a cot 57� �55

6 ⇒ a � 3.23 miles

cot 16� �a cot 57� � �55�6�

a

tan 16� �a

a cot 57� � �55�6�

tan 16� �a

x � �55�6�

57°16°

x55060

H

P1 P2

a

tan 57� �a

x ⇒ x � a cot 57�

41.

� � arctan 5 � 78.7�

tan � � � �1 � �3�2�1 � ��1��3�2�� � ��5�2

�1�2� � 5

L2: 3x � y � 1 ⇒ y � �x � 1 ⇒ m2 � �1

L1: 3x � 2y � 5 ⇒ y � 3

2x �

5

2 ⇒ m1 �

3

240.

� 5410 feet

h �17

� 1tan 2.5�

�1

tan 9��� 1.025 miles

htan 2.5�

�h

tan 9�� 17

x �h

tan 9�� 17

tan 9� �h

x � 17

x �h

tan 2.5�

x − 17172.5° 9°

x

h

Not drawn to scale

tan 2.5� �hx

42.

� arctan�97� � 52.1�

� � arctan� m2 � m1

1 � m2m1� � arctan� �1�5� � 2

1 � �1�5��2�� tan � � � m2 � m1

1 � m2m1�L2 � x � 5y � �4 ⇒ m2 �

1

5

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

� � 35.3�.

� � arctan 1

�2

a

2a

θ

tan � �a

�2a�

1�2

�a2 � a2 � �2a.

44.

� � arctan �2 � 54.7º

θa

a 2

tan � �a�2

a� �2 45.

Length of side:

36°25

d

2d � 29.4 inches

sin 36� �d

25 ⇒ d � 14.69

38.

Distance between towns:

dD

28°

28°

55°

55°10 km

T2T1

D � d � 18.8 � 7 � 11.8 kilometers

cot 28� �D

10 ⇒ D � 18.8 kilometers

cot 55 �d

10 ⇒ d � 7 kilometers

Page 83: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

416 Chapter 4 Trigonometry

46.

� 25 inches

Length of side � 2a � 2�12.5�

a � 25 sin 30� � 12.5

sin 30� �a

25

2530°

a 47.

y � 2b � 2��3r

2 � � �3r

b ��3r

2

b � r cos 30�

cos 30� �b

r

br

r

30°

2

x

y

48.

Distance � 2a � 9.06 centimeters

� 4.53

a � c sin 15� � 17.5 sin 15�

sin 15� �a

c

c �35

2� 17.5

c

15°a

49.

a �10

cos 35�� 12.2

cos 35� �10

a

b � 10 tan 35� � 7

tan 35� �b

10

35° 35°ba

10 10

10

10

10

10

50.

c � �10.82 � 7.22 � 13 feet

b �6

sin � 7.2 feet

sin �6

b

� 90 � 33.7 � 56.3�

f �21.6

2� 10.8 feet

a �18

cos �� 21.6 feet

6

6bca

f

θ φ9

36

cos � �18

a

� � arctan 2

3� 0.588 rad � 33.7�

tan � �12

18

51.

Use since

Thus, d � 4 sin�� t�.

2�

�� 2 ⇒ � � �

d � 0 when t � 0.d � a sin �t

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

Amplitude:

Period:

d � 3 sin�� t

3 �

2�

�� 6 ⇒ � �

3

�a� � 3

0 ⇒ d � a sin � t.t � 0

53.

Use since

Thus, d � 3 cos�4�

3t� � 3 cos�4�t

3 �.

2�

�� 1.5 ⇒ � �

4�

3

d � 3 when t � 0.d � a cos �t

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

Amplitude:

Period:

d � 2 cos�� t

5 �

2�

�� 10 ⇒ � �

5

�a� � 2

2 ⇒ d � a cos � t.t � 0

Page 84: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.8 Applications and Models 417

55.

(a) Maximum displacement amplitude

(b)

cycles per unit of time

(c)

(d) 8� t ��

2 ⇒ t �

1

16

d � 4 cos 40� � 4

� 4

Frequency ��

2��

8�

2�

� 4�

d � 4 cos 8�t 56.

(a) Maximum displacement:

(b) Frequency: cycles per unit of time

(c)

(d) Least positive value for t for which

t ��

2

1

20��

1

40

20� t ��

2

20� t � arccos 0

cos 20� t � 0

1

2 cos 20� t � 0

d � 0

t � 5 ⇒ d �12

cos 100� �12

2��

20�

2�� 10

�a� � �12� �1

2

d �1

2 cos 20� t

57.

(a) Maximum displacement amplitude

(b)

cycles per unit of time

(c)

(d) 120�t � � ⇒ t �1

120

d �1

16 sin 600� � 0

� 60

Frequency ��

2��

120�

2�

�1

16�

d �1

16 sin 120�t 58.

(a) Maximum displacement:

(b) Frequency: cycles per unit of time

(c)

(d) Least positive value for t for which

t ��

792��

1

792

792� t � �

792� t � arcsin 0

sin 792� t � 0

1

64 sin 792� t � 0

d � 0

t � 5 ⇒ d �1

64 sin�3960�� � 0

2��

792�

2�� 396

�a� � � 1

64� �1

64

d �1

64 sin 792�t

59.

� � 2��264� � 528�

264 ��

2�

Frequency ��

2�

d � a sin �t 60. At buoy is at its high point

Returns to high point every 10 seconds:

Period:

d �7

4 cos

� t

5

2�

�� 10 ⇒ � �

5

�a� �7

4

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

⇒ d � a cos � t.t � 0,

Page 85: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

418 Chapter 4 Trigonometry

61.

(a)

t

1

−1

y

π π38 8

π4

π2

y �1

4 cos 16t, t > 0

(b) Period:

(c)1

4cos 16t � 0 when 16t �

2 ⇒ t �

32

2�

16�

8

62. (a)

(c) L � L1 � L2 �2

sin ��

3

cos �

(b)

The minimum length of the elevator is 7.0 meters.

(d)

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

−12

−2� 2�

12

0.1 23.0

0.2 13.1

0.3 9.9

0.4 8.43

cos 0.4

2

sin 0.4

3

cos 0.3

2

sin 0.3

3

cos 0.2

2

sin 0.2

3

cos 0.1

2

sin 0.1

L1 � L2L2L1�

0.5 7.6

0.6 7.2

0.7 7.0

0.8 7.13

cos 0.8

2

sin 0.8

3

cos 0.7

2

sin 0.7

3

cos 0.6

2

sin 0.6

3

cos 0.5

2

sin 0.5

L1 � L2L2L1�

63. (a) and (b)

Base 1 Base 2 Altitude Area

8 22.1

8 42.5

8 59.7

8 72.7

8 80.5

8 83.1

8 80.7

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

� � 60�

8 sin 70º8 � 16 cos 70º

8 sin 60º8 � 16 cos 60º

8 sin 50º8 � 16 cos 50º

8 sin 40º8 � 16 cos 40º

8 sin 30º8 � 16 cos 30º

8 sin 20º8 � 16 cos 20º

8 sin 10º8 � 16 cos 10º

(c)

(d)

The maximum of 83.1 square feet occurs when

� ��

3� 60�.

0900

100

� 64�1 � cos ���sin ��

� �16 � 16 cos ���4 sin ��

A��� � 8 � �8 � 16 cos �� �8 sin �

2 �

Page 86: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Section 4.8 Applications and Models 419

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

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

64. (a)

(c) Period:

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

2�

��6� 12

Month (1 January)↔

Ave

rage

sale

s(i

n m

illio

ns o

f do

llars

)

t

S

2 4 121086

3

6

9

12

15

(b)

Shift:

Note: Another model is

The model is a good fit.

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

S � 8 � 6.3 sin��t6

��

2�.

S � 8 � 6.3 cos�� t

6 �S � d � a cos bt

d � 14.3 � 6.3 � 8

2�

b� 12 ⇒ b �

6

t2 4 121086

3

6

9

12

15

Ave

rage

sal

es(i

n m

illio

ns o

f do

llars

)

Month (1 ↔ January)

Sa �

1

2�14.3 � 1.7� � 6.3

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

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

69. passes through

y � 4x � 6

y � 2 � 4x � 4

y � 2 � 4�x � ��1��

y

x−1−2−3−4 1 2 3 4

−1

1

2

3

5

6

7

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

y � �12x �

16

b �16

0 � �16 � b

0 � �12�1

3� � b

y

x−1−2−3 2 3

−1

−2

−3

1

2

3

y � �12x � b

�13, 0�m � �

12

71. Passes through and

y � �45

x �225

y � 6 � �45

x �85

y � 6 � �45

x � ��2�

y

x−1−2 1 2 3 4 5

−1

1

2

3

4

6

7m �

2 � 63 � ��2� � �

45

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

y � �43

x �13

y �23

� �43�x �

14�

� �43

�1

�3�4

y

x−1−2−3 2 3

−1

−2

−3

1

2

3

m ��1�3� � ��2�3���1�2� � �1�4�

��12

, 13��1

4, �

23�

Page 87: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Review Exercises for Chapter 4

420 Chapter 4 Trigonometry

1. � � 0.5 radian 2. � � 4.5 radians

3.

(a)

(b) The angle lies in Quadrant II.

(c) Coterminal angles:

3�

4� 2� � �

5�

4

11�

4� 2� �

3�

4

114π

x

y

� �11�

44.

(a)

(b) Quadrant I

(c)

2�

9� 2� � �

16�

9

2�

9� 2� �

20�

9

x

y

29π

� �2�

95.

(a)

(b) The angle lies in Quadrant II.

(c) Coterminal angles:

�4�

3� 2� � �

10�

3

�4�

3� 2� �

2�

3

− 43π

x

y

� � �4�

3

6.

(a)

(b) Quadrant I

(c)

�23�

3� 2� � �

17�

3

�23�

3� 8� �

3

x

y

233π

� � �23�

37.

(a)

(b) The angle lies in Quadrant I.

(c) Coterminal angles:

70� � 360� � �290�

70� � 360� � 430�

70°

y

x

� � 70� 8.

(a)

(b) Quadrant IV

(c)

280� � 360� � �80�

280� � 360� � 640�

x

280°

y

� � 280�

Page 88: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Review Exercises for Chapter 4 421

9.

(a)

(b) The angle lies in Quadrant III.

(c) Coterminal angles:

�110� � 360� � �470�

�110� � 360� � 250�

−110°

x

y

� � �110� 10.

(a)

(b) Quadrant IV

(c)

�405� � 360� � �45�

�405� � 720� � 315�

x

−405°

y

� � �405� 11.

� 8.378 radians

�8�

3 radians

480� � 480� �� rad

180�

12. �127.5� ��

180�� �2.225 13.

� �3�

16 radian � �0.589 radian

�33� 45� � �33.75� � �33.75� �� rad

180�

14. 196� 77� � �196 �7760�

��

180�� 3.443 15.

5� rad

7�

5� rad

7�

180�

� rad� 128.571�

16. �11�

6 �

180�

�� �330.000� 17. �3.5 rad � �3.5 rad �

180�

� rad� �200.535�

18. 5.7 �180�

�� 326.586� 19. radians

inchess � r� � 20�23�

30 � � 48.17

138� �138�

180�

23�

30 20.

meterss � 11.52

�113

� meters

s � r� � 11 � � 60180��

60� �60�

180 radians

21. (a)

(b)

� 400� inches per minute

Linear speed �6�66 23�� inches

1 minute

� 66 23� radians per minute

Angular speed ��331

3��2�� radians1 minute

22.

� 12.05 miles per hour

� 212.1 inches per second

� 67.5� inches per second

� �5� rad�s� � �13.5 inches�

�linear speed� � �angular speed� � �radius�

23. radians

square inchesA �12

r 2� �12

�18�2�2�

3 � � 339.29

120� �120�

180�

2�

324.

A � 55.31 square millimeters

A �12

�r 2 �12�

5�

6 �6.5 2

26. t �3�

4, �x, y� � ��

�22

, �22 �25. corresponds to the point .��

12

, �32 �t �

2�

3

Page 89: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

422 Chapter 4 Trigonometry

27. corresponds to the point ���32

, 12�.t �

5�

628. t � �

4�

3, �x, y� � ��

12

, �32 �

29. corresponds to the point

cot 7�

6�

xy

� �3tan 7�

6�

y

x�

1�3

��3

3

sec 7�

6�

1x

� �2�3

3cos

7�

6� x � �

�3

2

csc 7�

6�

1y

� �2sin 7�

6� y � �

1

2

���3

2, �

1

2�.t �7�

630. corresponds to the point

cot �

4�

xy

� 1tan �

4�

yx

� 1

sec �

4�

1x

� �2cos �

4� x �

�22

csc �

4�

1y

� �2sin �

4� y �

�22

��22

, �22 �.t �

4

31. corresponds to the point

cot��2�

3 � �xy

��33

tan��2�

3 � �y

x� �3

sec��2�

3 � �1x

� �2cos��2�

3 � � x � �1

2

csc��2�

3 � �1y

� �2�3

3sin��2�

3 � � y � ��3

2

��1

2, �

�3

2 �.t � �2�

332. corresponds to the point

is undefined.

is undefined.cot 2� �xy

tan 2� �yx

� 0

sec 2� �1x

� 1cos 2� � x � 1

csc 2� �1y

sin 2� � y � 0

�1, 0�.t � 2�

33. sin 11�

4� sin

3�

4�

�22

34. cos 4� � cos 0 � 1 35. sin��17�

6 � � sin��5�

6 � � �12

36. cos��13�

3 � � cos�5�

3 � �12 37. tan 33 � �75.3130 38. csc 10.5 �

1sin 10.5

� �1.1368

39. sec�12�

5 � �1

cos�12�

5 �� 3.2361 40. sin��

9� � �0.3420

41.

cot � �adjopp

�54

tan � �oppadj

�45

cos � �adjhyp

�5

�41�

5�4141

sec � �hypadj

��41

5

sin � �opphyp

�4

�41�

4�4141

csc � �hypopp

��41

4

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

cot � �adjopp

�66

� 1tan � �oppadj

�66

� 1

sec � �hypadj

�6�2

6� �2cos � �

adjhyp

�6

6�2�

�22

csc � �hypopp

�6�2

6� �2sin � �

opphyp

�6

6�2�

�22

hyp � �62 � 62 � 6�2

adj � 6, opp � 6

43.

cot � �adjopp

�4

4�3�

�33tan � �

oppadj

�4�3

4� �3

cos � �adjhyp

�48

�12

sec � �hypadj

�84

� 2

sin � �opphyp

�4�3

8�

�32

csc � �hypopp

�8

4�3�

2�33

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

Page 90: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Review Exercises for Chapter 4 423

44.

cot � �adjopp

�2�14

5

sec � �hypadj

�9

2�14�

9�1428

csc � �hypopp

�95

tan � �oppadj

�5

2�14�

5�1428

cos � �adjhyp

�2�14

9

sin � �opphyp

�59

adj � �92 � 52 � 2�14

opp � 5, hyp � 9 45.

(a)

(b)

(c)

(d) tan � �sin �cos �

�1�3

�2�2��3�

1

2�2�

�24

sec � �1

cos ��

3

2�2�

3�24

cos � �2�2

3

cos � ��89

cos2 � �89

cos2 � � 1 �19

�13�

2

� cos2 � � 1

sin2 � � cos2 � � 1

csc � �1

sin �� 3

sin � �13

46.

(a)

(b)

(c)

(d) csc � � �1 � cot2 � ��1 �1

16�

�174

cos � �1

sec ��

1�17

��1717

sec � � �1 � tan2 � � �1 � 16 � �17

cot � �1

tan ��

14

tan � � 4 47.

(a)

(b)

(c)

(d) tan � �sin �cos �

�1�4

�15�4�

1�15

��1515

sec � �1

cos ��

4�15

�4�15

15

cos � ��15

4

cos � ��1516

cos2 � �1516

cos2 � � 1 �1

16

�14�

2

� cos2 � � 1

sin2 � � cos2 � � 1

sin � �1

csc ��

14

csc � � 4

48.

(a)

(b) cot � � �csc2 � � 1 � �25 � 1 � 2�6

sin � �1

csc ��

15

csc � � 5

(c)

(d) sec�90� � �� � csc � � 5

tan � �1

cot ��

12�6

��612

49. tan 33� � 0.6494 51. sin 34.2� � 0.562150. csc 11� �1

sin 11�� 5.2408

Page 91: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

424 Chapter 4 Trigonometry

55.

kilometer or 71.3 meters

1 10'°x

Not drawn to scale

3.5 km

x � 3.5 sin 1� 10� � 0.07

sin 1� 10� �x

3.556.

x �25

tan 52�� 19.5 feet

tan 52� �25x

52°

x

25

57.

cot � �xy

�34

tan � �yx

�43

sec � �rx

�53

cos � �xr

�35

csc � �ry

�54

sin � �yr

�45

x � 12, y � 16, r � �144 � 256 � �400 � 20 58.

cot � �xy

� �34

tan � �yx

� �43

sec � �rx

�53

cos � �xr

�35

csc � �ry

� �54

sin � �yr

� �45

r � �32 � ��4�2 � 5

�x, y� � �3, �4�

59.

cot � �xy

�2�35�2

�4

15tan � �

yx

�5�22�3

�154

sec � �rx

��241�6

2�3�

�2414

cos � �xr

�2�3

�241�6�

4�241

�4�241

241

csc � �ry

��241�6

5�2�

2�24130

��241

15sin � �

yr

�5�2

�241�6�

15�241

�15�241

241

r ���23�

2

� �52�

2

��241

6

x �23

, y �52

52. sec 79.3� �1

cos 79.3�� 5.3860 53.

� 3.6722

cot 15� 14� �1

tan�15 �1460� 54.

� 0.2045

cos 78� 11� 58 � cos�78 �1160

�58

3600��

60.

cot � �xy

��10�3�2�3

� 5tan � �y

x�

�2�3�10�3

�15

sec � �rx

��2�26��3

�10�3� �

�265

cos � �xr

��10�3

�2�26��3� �

5�2626

csc � �ry

��2�26��3

�2�3� ��26sin � �

y

r�

�2�3

�2�26��3� �

�2626

r ����103 �

2

� ��23�

2

�2�26

3

�x, y� � ��103

, �23�

Page 92: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Review Exercises for Chapter 4 425

61.

cot � �xy

��0.54.5

� �19

tan � �yx

�4.5

�0.5� �9

sec � �rx

��82�2�0.5

� ��82cos � �xr

��0.5�82�2

���82

82

csc � �ry

��82�2

4.5�

�829

sin � �yr

�4.5

�82�2�

9�8282

r � ���0.5�2 � �4.5�2 � �20.5 ��82

2

x � �0.5, y � 4.5

62.

cot � �xy

�0.30.4

�34

� 0.75tan � �y

x�

0.40.3

�43

� 1.33

sec � �rx

�0.50.3

�53

� 1.67cos � �xr

�0.30.5

�35

� 0.6

csc � �ry

�0.50.4

�54

� 1.25sin � �y

r�

0.40.5

�45

� 0.8

r � ��0.3�2 � �0.4�2 � 0.5

�x, y� � �0.3, 0.4�

64.

cot � �x�

y��

�2x�3x

�23

sec � �rx�

��13x�2x

� ��13

2

csc � �ry�

��13x�3x

� ��13

3

tan � �y�

x��

�3x�2x

�32

cos � �x�

r�

�2x�13x

� �2�13

13

sin � �y�

r�

�3x�13x

� �3�13

13

r � ���2x�2 � ��3x�2 � �13x

�x�, y� � � ��2x, �3x�, x > 0 65. is in Quadrant IV.

cot � � �5�11

11

sec � �6

5

csc � �r

y� �

6�11

11

tan � �y

x� �

�11

5

cos � �x

r�

5

6

sin � �y

r� �

�11

6

r � 6, x � 5, y � ��36 � 25 � ��11

sec � �6

5, tan � < 0 ⇒ �

63.

cot � �x�

y��

x4x

�14

tan � �y�

x��

4xx

� 4

sec � �rx�

��17x

x� �17cos � �

x�

r�

x�17x

��1717

csc � �ry�

��17x

4x�

�174

sin � �y�

r�

4x�17x

�4�17

17

r � �x2 � �4x�2 � �17x

x� � x, y� � 4x

�x, 4x�, x > 0

Page 93: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

426 Chapter 4 Trigonometry

66.

is in Quadrant II.

cot � �1

tan �� �

�52

sec � �1

cos �� �

3�55

tan � �sin �cos �

� �2�5

5

cos � � ��1 � sin2 � � ��53

sin � �1

csc ��

23

csc � �32

, cos � < 0 67. is in Quadrant II.

cot � � ��55

3

sec � � �8

�55� �

8�55

55

csc � �8

3

tan � �y

x� �

3�55

� �3�55

55

cos � �x

r� �

�55

8

sin � �y

r�

3

8

y � 3, r � 8, x � ��55

sin � �3

8, cos � < 0 ⇒ �

68.

is in Quadrant III.

cot � �1

tan ��

45

csc � �1

sin �� �

�415

sin � � ��1 � cos2 � � ��1 �1641

� �5�41

41

cos � �1

sec �� �

4�4141

sec � � ��1 � tan2 � � ��1 � �2516� � �

�414

tan � �54

, cos � < 0 69.

cot � �x

y�

�2�21

� �2�21

21

sec � �r

x�

5

�2� �

5

2

csc � �r

y�

5�21

�5�21

21

tan � �y

x� �

�21

2

sin � �y

r�

�21

5

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

cos � �xr

��25

⇒ y2 � 21

70.

is in Quadrant IV.

cot � �1

tan �� ��3

tan � �sin �cos �

� ��33

sec � �1

cos ��

2�33

cos � � �1 � sin2 � ��1 � �14� �

�32

csc � �1

sin �� �2

sin � � �24

� �12

, cos � > 0 71.

′θ

264°

x

y

�� � 264� � 180� � 84�

� � 264�

Page 94: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Review Exercises for Chapter 4 427

72.

′θ

635°

x

y

�� � 85�

� � 635� � 720� � 85� 73.

′θx

y

65π−

�� � � �4�

5�

5

�6�

5� 2� �

4�

5

� � �6�

574.

′θ

x

y

173π

�� ��

3

� 6� ��

3

� �17�

3�

18�

3�

3

75.

tan �

3� �3

cos �

3�

12

sin �

3�

�32

76.

tan �

4�

2

�2�2� 1

cos �

4�

�2

2

sin �

4�

�2

277.

tan��7�

3 � � �tan �

3� ��3

cos��7�

3 � � cos �

3�

12

sin��7�

3 � � �sin �

3� �

�32

78.

�2

��2�2� �1

tan�� 5�

4 � � �tan �

4

cos�� 5�

4 � � �cos �

4� �

�22

sin�� 5�

4 � � sin �

4�

�22

79.

tan 495� � �tan 45� � �1

cos 495� � �cos 45� � ��2

2

sin 495� � sin 45� ��2

280.

tan��150�� ��1�2

��3�2�

�33

cos��150�� � ��32

sin��150�� � �12

81.

tan��240�� � �tan 60� � ��3

cos��240�� � �cos 60� � �1

2

sin��240�� � sin 60� ��3

282.

tan�315�� ���2�2�2�2

� �1

cos�315�� ��22

sin�315�� � ��22

83. sin 4 � �0.7568 84. tan 3 � �0.1425 85. sin��3.2� � 0.0584

86. cot��4.8� �1

tan��4.8� � 0.0878 87. sec�12�

5 � �1

cos�12�

5 �� 3.2361 88. tan��25�

7 � � 4.3813

Page 95: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

428 Chapter 4 Trigonometry

89.

2

1

−2

x

− ππ322

y

Period: 2�

Amplitude: 1

y � sin x 91.

Amplitude: 5

Period:

−6

−2

2

4

6

xπ6

y

2�

2�5� 5�

f �x� � 5 sin 2x

590.

Amplitude: 1

Period:

x

2

−1

−2

2πππ−

y

2�

y � cos x

92.

Amplitude: 8

Period:

−8

−6

−4

8

x

y

π8π4

2�

1�4� 8�

f �x� � 8 cos��x4� 93.

Shift the graph of two units upward.

4

3

2

−1

−2

xπππ 2

y

y � sin x

y � 2 � sin x 94.

Amplitude: 1

Period:

321−2−3x

−1

−2

−3

−5

−6

−1

y

2�

�� 2

y � �4 � cos �x

96.

Amplitude: 3

Period:

−4

−3

1

2

3

4

y

2�

g�t� � 3 cos�t � � � 97.

(a)

(b)

� 264 cycles per second.

f �1

1�264

y � 2 sin�528�x�

2�

b�

1264

⇒ b � 528�

a � 2,

y � a sin bx

98. (a)

014

12

22

S�t� � 18.09 � 1.41 sin��t6

� 4.60�

95.

Amplitude:

Period:

−4

−3

−2

−1

1

3

4

y

2�

5

2

g�t� �5

2 sin�t � ��

(b)

so this is expected.

(c) Amplitude: 1.41

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

12 months � 1 year,

Period �2�

��6� �2��6� � 12

Page 96: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Review Exercises for Chapter 4 429

105.

Graph first.

4

3

2

1

−3

−4

x

− ππ322

y

y � sin x

f �x� � csc x 106.

2

y

f �t� � 3 csc�2t ��

4�

107.

and first.

As x →, f �x� →.

y � �xGraph y � x−9

−6

9

6f �x� � x cos x 108.

Damping factor:

As x →, f �x� →.

x 4

−2

−300

300

� 2�

g�x� � x 4 cos x

109. arcsin��12� � �arcsin

12

� ��

6111. arcsin 0.4 � 0.41 radian

113. sin�1��0.44� � �0.46 radian

115. arccos �32

��

6 117. cos�1��1� � �

110. arcsin��1� � ��

2

112. radianarcsin�0.213� � 0.21 114. radianssin�1�0.89� � 1.10

116. arccos��22 � �

4

99.

4

3

2

1

πx

y

f �x� � tan x 100.

t

1

2

3

y

π2

π2

f �t� � tan�t ��

4� 101.

x

4

3

2

1

−3

−4

ππ−

y

f �x� � cot x

102.

t

3

2

1

y

ππ−

g�t� � 2 cot 2t 103.

Graph first.

ππ−−1

−2

−3

−4

x

y

y � cos x

f �x� � sec x 104.

1

y

h�t� � sec�t ��

4�

Page 97: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

430 Chapter 4 Trigonometry

119. arccos 0.324 � 1.24 radians118. cos�1��32 � �

6120. radiansarccos��0.888� � 2.66

122. radianstan�1�8.2� � 1.45121. tan�1� � 1.5� � �0.98 radian

127.

Use a right triangle. Letthen

and cos � � 45.tan � � 34 � � arctan 34

θ

5

4

3

cos�arctan 34� �45

126.

−1.5 1.5

�2

−�2

f �x� � �arcsin 2x125.

−4 4

�2

−�2

f �x� � arctan�x2� � tan�1�x

2�

123.

−1.5 1.5

−�

f �x� � 2 arcsin x � 2 sin�1�x� 124.

−1.5 1.5

�3

0

y � 3 arccos x

128. Let

tan�arccos 35� � tan u �43

4

u

3

5

u � arccos 35.

129.

Use a right triangle. Let then and sec � �

135 .tan � �

125

� � arctan 125

θ

5

1213

sec�arctan 125 � �

135 130. Let

cot �arcsin��1213� � cot u � �

512

u

−1213

5u � arcsin��1213�.

131. Let Then

and

2

yx

4 − x 2

tan y � tan�arccos�x2�� �

�4 � x2

x.cos y �

x2

y � arccos�x2�. 132.

x −11

θ

12 − (x − 1)2

sec � �1

�x�2 � x�

cos � � �12 � �x � 1�2 � �x�2 � x�

sin � � x � 1

� � arcsin�x � 1� ⇒ ��

2≤ � ≤

2

sec�arcsin�x � 1��

Page 98: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Review Exercises for Chapter 4 431

133.

� � arctan�7030� � 66.8�

tan � �7030

134.

h � 25 tan 21� � 9.6 feeth

25

21°

tan 21� �h

25

135.

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

sec 4.4� �D

1217 ⇒ D � 1217 sec 4.4� � 1221

tan � �93

1217 ⇒ � � 4.4�

sin 25� �d4

810 ⇒ d4 � 342

cos 48� �d3

650 ⇒ d3 � 435

cos 25� �d2

810 ⇒ d2 � 734

sin 48� �d1

650 ⇒ d1 � 483

48°

48°

65°25°

d3d4

d d1 2

810650

D

B

CA θ

N

S

W E

d1 � d2 � 1217

d3 � d4 � 93

136. Amplitude:

Period: 3 seconds

d � 0.75 cos�2� t3 �

b �2�

3

a � 0.75

d � a cos bt

1.52

� 0.75 inches 137. False. The sine or cosine functionsare often useful for modeling simple harmonic motion.

138. True. The inverse sine,is defined whereand

��

2≤ y ≤

2.

�1 ≤ x ≤ 1y � arcsin x,

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

� 140. False. The range of arctan is

so arctan��1� � ��

4.��

2,

2�,

141.

Matches graph �d�

Period: 2�

Amplitude: 3

y � 3 sin x

142. matches graph (a).

Period:

Amplitude: 3

2�

y � �3 sin x 143.

Matches graph �b�

Period: 2

Amplitude: 2

y � 2 sin �x 144. matches graph (c).

Period:

Amplitude: 2

4�

y � 2 sin x2

145. is undefined at the zeros of since sec � �1

cos � .g��� � cos �f ��� � sec �

Page 99: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

432 Chapter 4 Trigonometry

Problem Solving for Chapter 4

1. (a)

revolutions

radians or

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

990�� � �114 ��2�� �

11�

2

13248

�114

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

Gear 2:

Gear 3:

Gear 4:

Gear 5:2419

�360�� � 454.737� � 7.94 radians

4032

�360�� � 450� �5�

2 radians

2422

�360�� � 392.727� � 6.85 radians

2426

�360�� � 332.308� � 5.80 radians

2432

�360�� � 270� �3�

2 radians

3. (a)

(b)

x �3000

tan 39�� 3705 feet

tan 39� �3000

x

d �3000

sin 39�� 4767 feet

sin 39� �3000

d(c)

w � 3000 tan 63� � 3705 � 2183 feet

3000 tan 63� � w � 3705

tan 63� �w � 3705

3000

146. (a) (b) tan�� ��

2� � �cot �0.1 0.4 0.7 1.0 1.3

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

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

2��

147. The ranges for the other four trigonometric functions are not bounded. For the rangeis For the range is���, �1� � �1, ��.

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

148.

(a) A is changed from The displacement isincreased.

(b) k is changed from The friction damps theoscillations more rapidly.

(c) b is changed from 6 to 9: The frequency of oscilla-tion is increased.

110 to 13 :

15 to 13 :

y � Ae�kt cos bt �15e�t�10 cos 6t

149.

(a)

As r increases, the area function increases more rapidly.

00 6

4

A s

s � r�0.8� � 0.8r, r > 0

A �12 r 2�0.8� � 0.4r 2, r > 0

A �12 r 2�, s � r�

150. Answers will vary.

(b)

03

30

A s

0

s � 10�, � > 0

A �12 �10�2� � 50�, � > 0

Page 100: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Problem Solving for Chapter 4 433

5. (a)

h is even.

(b)

h is even.

−1

−2

3

2� �

h�x� � sin2 x

−1

−2

3

2� �

h�x� � cos2 x

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

For the cosine model we have:

For the sine model we have:

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

Another model would be:

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

h � 51 � 50 sin�8� t �3�

2 �t � 0

h � 51 � 50 sin�8� t ��

2�h � 51 � 50 cos�8� t�

b � 8�

d �12

�max � min� �12

�101 � 1� � 51

a �12

�max � min� �12

�101 � 1� � 50

h � 1 when t � 0, 8.

(a)

(b)

This is the time between heartbeats.

(c) Amplitude: 20

The blood pressure ranges between and

(d) Pulse rate

(e) Period

64 �60

2��b ⇒ b �

6460

� 2� �3215

�6064

�1516

sec

�60 sec�min34 sec�beat

� 80 beats�min

100 � 20 � 120.100 � 20 � 80

Period �2�

8��3�

68

�34

sec

070

5

130

P � 100 � 20 cos�8�

3t�

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

(a)

since f is even

Thus, h is an even function.

(b)

since g is odd

Thus, h is an even function.

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

� h�x�

� �g�x��2

� ��g�x��2

h��x� � �g��x��2

h�x� � �g�x��2

� h�x�

� � f �x��2

h��x� � � f ��x��2

h�x� � � f �x��2

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

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

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

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

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

opphyp

�BCAB

�DEAD

�FGAF

� sin A

BCAB

�DEAD

�FGAF

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

Page 101: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

434 Chapter 4 Trigonometry

9. Physical (23 days):

Emotional (28 days):

Intellectual (33 days):

(a)

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

5 11 31 1

20 years leap years remaining August days day inJuly days September

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

(c) For September 22, 2006, use

I � 0.945

E � 0.901

P � 0.631

t � 7369.

−2

7349 7379

2

P

I

E

t � 7348

����

����t � 365 20

−2

7300 7380

2

P IE

I � sin 2� t33

, t ≥ 0

E � sin 2� t28

, t ≥ 0

P � sin 2� t23

, t ≥ 0

10.

(a)

(b) The period of

The period of

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

g�x� is �.

f �x� is 2�.

−6

6

� �

g

f

g�x� � 2 cos 2x � 3 sin 4x

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

and

(b) The graphs intersect when

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

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

x � 5.35 � 3�2� � �0.65.

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

Page 102: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Problem Solving for Chapter 4 435

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

(b) is not true.

is a horizontal translation of

is a doubling of the period of

(c) is not true.

is a horizontal

translation of by half a period.

For example, sin�12

�� � 2��� � sin�12

��.

f �12

t�f �1

2 �t � c�� � f �1

2t �

12

c�

f �12

�t � c�� � f �12

t�

f �t�.f �12

t�

f �t�.f �t �12

c�

f �t �12

c� � f �12

t�

f �t � 2c� � f �t� 13.

(a)

(b)

(c) feet

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

�1

d � y � x � 3.46 � 1.71 � 1.75

tan �1 �y2

⇒ y � 2 tan 60� � 3.46 feet

tan �2 �x2

⇒ x � 2 tan 40.52� � 1.71 feet

�2 � 40.52�

sin �2 �sin �1

1.333� sin 60�

1.333� 0.6497

sin �1

sin �2� 1.333

2 ft

xy

d

θθ

1

2

14.

(a)

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

−2

2

�2

−�2

arctan x � x �x3

3�

x5

5�

x7

7

(b)

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

−2

2

�2

−�2

Page 103: CHAPTER 4 Trigonometry - Burlington School Districtshubbard/pdfs/ch04/csg_ch04.pdf · Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . .344 ... Section 4.6 Graphs

Chapter 4 Practice Test

436 Chapter 4 Trigonometry

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

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

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

7. Solve for in the figure below.

20°

35

x

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

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

11. Graph y � 3 sin x

2. 12. Graph y � �2 cos�x � ��.

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

4�

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

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

19. Evaluate sin�arccos 4

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

4�

For Exercises 21–23, solve the right triangle.

A C

B

ca

b

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

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

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