Lial Trig Chapter 3 10e - · PDF filecircle of radius 1 by definition. (See exercise ... 49....

Copyright © 2013 Pearson Education, Inc. 22

Chapter 3 Radian Measure and the Unit Circle

Section 3.1 Radian Measure

1. 1

2. 2

3. 3

4. –1

5. –3

6. –2

7. 3

π

8. 6

π

9. 2

π

10. 2

3

π

11. 5

6

π

12. 3

2

π

13. 5

3

π-

14. 7

4

π-

15. 5

2

π

16. 8

3

π

17. 10 π

18. 20 π

19. 0

20. π

21. 5 π-

22. 10 π-

For exercises 23−28, answers may vary.

23. Multiply the degree measure by radian180

π

and reduce. Your answer will be in radians. Leave the answer as a multiple of ,π unless otherwise directed.

24. Multiply the radian measure by 180

π

and

reduce. Your answer will be in degrees.

25. One radian is the measure of an angle, with its vertex at the center of a circle, that intercepts an arc on the circle equal in length to the radius of the circle.

26. The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. A measure of one degree is equivalent to a rotation of 1/360 of a complete revolution. Thus, degree measure is based on the rotation of the terminal side of the angle, while radian measure is based on the length of the arc that is intercepted by the angle. (See exercise 25.)

27. A right angle measures 90° and intercepts an arc that is one-quarter of the circumference of a circle, or 2 4 2.π π= Since the angle

measures are equal, we have 180

90 .2

x xππ

æ ö ÷ç= =÷ç ÷çè ø

28. An angle of radian measure t in standard position intercepts an arc of length t on a circle of radius 1 by definition. (See exercise 25.)

29. 60

30. 480

31. 315

32. 120

33. 330

34. 675

35. 30-

36. 288-

Section 3.1 Radian Measure 23

Copyright © 2013 Pearson Education, Inc.

37. 126

38. 132

39. 48-

40. 63-

41. 153

42. 66

43. 900-

44. 2700

45. 0.68

46. 1.3

47. 0.742

48. 4.623

49. 2.43

50. 3.05

51. 1.122

52. 1.484

53. 0.9847

54. 2.140

55. 0.832391 -

56. 0.401675 -

57. 114 35¢

58. 286 29¢

59. 99 42¢

60. 175 20¢

61. 19 35¢

62. 564 14¢

63. 287 06¢-

64. 198 55¢-

65. Without the degree symbol on the 30, it is assumed that 30 is measured in radians. Thus, the approximate value of sin 30 is −0.98803,

not 12

.

66. An angle of one radian is the measure of an angle in standard position that intercepts an arc that is equal to the length of the radius of the circle. In a unit circle, that length is 1.

67. 3

2

68. 3

2

69. 1

70. 3

3

71. 2 3

3

72 2

73. 1

74. 1

75. 3-

76. 3

3-

77. 1

2

78. 3

3-

79. 1-

80. 1-

81. 3

2-

82. 3

3

83. 1

2

84. 3

2

85. 3

86. 2 3

3-

24 Chapter 3 Radian Measure and the Unit Circle


87. Begin the calculation with the blank next to 30º, and then proceed counterclockwise from there.

;6

π45 ; ;

3

π120 ; 135 ;

5;

6

π;π 7

;6

π

5;

4

π240 ; 300 ;

7 11;

4 6

π π

88. 2

180

π

89. (a) 4π

(b) 2

3

π

90. (a) 24π

(b) 6π

91. (a) 5π

(b) 8

3

π

92. (a) 16π

(b) 60π

93. 24π

94. (a) 200

π

(b) 3.15 ; 0.055 radian

Section 3.2 Applications of Radian Measure

1. 2π

2. 4π

3. 20π

4. 8

5. 6

6. 8

7. 1

8. 1.5

9. 2

10. To find the degree measure of a central angle in a circle if both the raidus and the length of the intercepted arc are known, first apply the formula s rθ= to find the radian measure.

Then multiply the radian measure by 180π to

find the degree measure.

11. 25.8 cm

12. 3.08 cm

13. 3.61 ft

14. 11.9 mi

15. 5.05 m

16. 169 cm.

17. 55.3 in.

18. 71.4ft

19. The length of the arc is doubled.

20. 180

rπ θ

21. 3500 km

22. 1500 km

23. 5900 km

24. 8800 km

25. 44° N.

26. 43° N.

27. 156º

28. 213º

29. 38.5°

30. 82.3°

31. 18.7 cm.

32. 29.2 in.

33. (a) 11.6 in.

(b) 37 5¢

34. 12.7 cm.

35. 146 in

36. (a) 39,616 rotations (b) 62.9 mph; Yes

37. 3 inπ

38. 4 inπ

39. 27 inπ

40. 39 inπ

41. 0.20 km

42. 850 ft

43. 6π

Section 3.3 The Unit Circle and Circular Functions 25


44. 16π

45. 72π

46. 75π

47. 60º

48. 240º

49. 1.5

50. 1

In Exercises 51−58, we will be rounding to the nearest tenth. 51. 1116.1 m2

52. 3744.8 km2

53. 706.9 ft2

54. 10,602.9 yd2

55. 114.0 cm2

56. 365.3m2

57. 1885.0 mi2

58. 19,085.2 km2

59. 3.6

60. 16 m

61. 28060 yd

62. the new area is twice the original area

63. 20 in

64. 64

65. (a) 2

27

π

(b) 478 ft.

(c) 17.8 ft

(d) 2672 ft

66. 275.4 in.

67. (a) 140 ft

(b) 102 ft

(c) 2622 ft .

68. 550 m; 1800 m

69. 21900 yd

70. 1.15 mi

71. radius: 3950 mi;circumference 24,800 mi

72. (a) The longitude at Greenwich is 0º.

(b) Answers will vary.

73. the area is quadrupled.

74. 2

sector ,360

rπ θ= is in degrees.θ

75. 21

2V r hθ= or

2

2

r hV θ=

76. ( )2 21 2

1

2V r r hθ= -

77. Lrθ

=

78. cos2

h r θ=

79. 1 cos2

d r θæ ö÷ç= - ÷ç ÷çè ø

80. 1 cos2

Ld θθæ ö÷ç= - ÷ç ÷çè ø

Section 3.3 The Unit Circle and Circular Functions

1. (a) 1

(b) 0

(c) undefined

2. (a) 0

(b) 1-

(c) 0

3. (a) 0

(b) 1

(c) 0

4. (a) 0

(b) 1-

(c) 0

5. (a) 0

(b) 1-

(c) 0

6. (a) 1

(b) 0

(c) undefined



7. 1

2-

8. 1

2

9. 1-

10. 2-

11. 2-

12. 3-

13. 1

2-

14. 3

15. 2

2

16. 2-

17. 3

2

18. 1

2-

19. 2 3

3

20. 2 3

3

21. 3

3-

22. 2

2-

23. 0.5736

24. 0.7314

25. 0.4068

26. 0.5397

27. 1.2065

28. 0.1944

29. 14.3338

30. 1.0170

31. 1.0460-

32. 2.1291-

33. 3.8665-

34. 1.1848

35. cos 0.8 ≈ 0.7 36. cos 0.6 0.8»

37. sin 2 0.9» 38. sin 4 ≈ –0.75

39. sin 3.8 0.6»- 40. cos3.2 1.0»-

41. 2.3 radiansθ » or 4 radiansθ »

42. 4.4 radiansθ » or 5.0 radiansθ »

43. 0.8 radianθ » or 2.4 radiansθ »

44. 1.3 radiansθ » or 5.0 radiansθ »

45. negative

46. negative

47. negative

48. positive

49. positive

50. negative

51. 2 2

sin ; cos2 2

tan 1; cot 1

θ θ

θ θ

= =

= =

sec 2;csc 2θ θ= =

52. 8 15 8

sin ; cos ; tan17 17 15

θ θ θ=- =-

15 17 17cot ;sec ; csc

8 15 8θ θ θ=- =- =

53. 12 5 12

sin ; cos ; tan13 13 5

θ θ θ=- = =-

5 13 13cot ;sec ; csc

12 5 12θ θ θ=- = =-

54. 1 3 3

sin ; cos ; tan2 2 3

θ θ θ=- =- =

2 3cot 3;sec ;csc 2

3θ θ θ= =- =-

55. 0.2095

56. 0.6720

57. 1.4426

58. 1.2799

59. 0.3887

60. 1.3634

Chapter 3 Quiz 27


61. 5

.6

s π=

62. 2

3s π=

63. 4

3s π=

64. 7

6s π=

65. 7

4s π=

66. 11

6s π=

67. 4

3

π;

5

3

π

68. 2

3

π;

4

3

π

69. ,4

π 3 5 7

, ,4 4 4

π π π

70. ,3

π

2 4 5, ,

3 3 3

π π π

71. 11 7

, ,6 6

π π- -

5 5, , ,

6 6 6 6

π π π π- -

72. 3 3

, , ,4 4 4 4

π π π π- -

73. ( ) ( ), 0.8011, 0.5985x y = -

74. ( ) ( ), 0.9668, 0.2555x y = - -

75. ( ) ( ), 0.4385, 0.8987x y = -

76. ( ) ( ), 0.7259, 0.6878x y = -

77. quadrant I

78. quadrant IV

79. quadrant II

80. quadrant III

81. 0.9846

82. 2.7824

83. (a) 32.4 (b) Answers will vary.

84. 8.6 hr; 15.4 hr 85. (a) 30

(b) 60

(c) 75

(d) 86

(e) 86

(f) 60

86. (a) 1 F (b) 19 F

(c) 53 F

(d) 58 F

(e) 48 F

(f) 12 F

87. (a) 1

2

(b) 3

2

(c) 3

(d) 2

(e) 2 3

3

(f) 3

3

88. (a) 0.7880

(b) 0.6157

(c) 0.7813

(d) 1.269

(e) 1.624

(f) 1.280

Chapter 3 Quiz (Sections 3.1−3.3)

1. 5

4

π

2. 11

6

π-

3. 300

4. 210-



5. 1.5

6. 267,500 in

7. 2

2

8. 1

2-

9. 0

10. 2

3

π

Section 3.4 Linear and Angular Speed

1. 2 secπ

2. 2 secπ

3. (a) 2

π radians

(b) 10 cmπ

(c) 5

3

π cm per sec

4. (a) 2

5

π radians

(b) 12 cmπ

(c) 3π cm per sec

5. 2π radians

6. 5

4 5 4tθ π θ πω θ= = = radians

7. 3

32

π radian per sec

8. 25

π radian per sec

9. 6

min5

10. 9 min

11. 0.1803 radian per sec

12. 2.078 radians per sec

13. 10.77 radians

14. 20.51 radians

15. 8 m per secπ

16. 72

cm per sec5

π

17. 9

radians per sec5

18. 6 radians per sec


20. 9.296 cm per sec

21. 18 cmπ

22. 216

yd5

π

23. 12 sec

24. 4 sec

25. 3

32

π radian per sec

26. 18

π radian per sec

27. radian per hr.6

π

28. 30

π radian per sec.

29. 30

π radian per min.

30. 600 radians per minπ

31. 7

cm per min30

π

32. 14

15

π mm per sec.

33. 168π m per min

34. 1260 cm per min.π

35. 1500 m per min.π

36. 112,880 cm per minπ

37. 16.6 mph

38. 24.62 hr

Chapter 3 Review Exercises 29


39. (a) 2

radian365

π

(b) radian per hr4380

π

(c) 67,000 mph

40. (a) 2 radians per day; radian per hr12

ππ

(b) 0

(c) 12,800 km per day;533 km per hrπ π

(d) 28,000 km per day;1200 km per hr

41. (a) 3.1 cm per sec

(b) 0.24 radian per sec

42. larger pulley: 125

cm per sec6

π

smaller pulley: 125

radians per sec48

π

43. 3.73 cm

44. 29 sec


46. 125 ft per sec

Chapter 3 Review Exercises

1. 2 radians

2. (a) quadrant II

(b) quadrant III

(c) quadrant III

(d) quadrant I

3. Three of the many possible answers are 1 2 , 1 4 , and 1 6 .π π π+ + +

4. 6

2 ,nπ π+ n represents any integer

5. 4

π

6. 2

3

π

7. 35

36

π

8. 11

6

π

9. 40

9

π

10. 17

3

π

11. 225

12. 162

13. 480

14. 216

15. 110-

16. 756-

17. inπ

18. 4

in3

π

19. 12 inπ

20. 2156 mi

21. 35.8 cm

22. 8.77 cm

23. 49.06

24. 22263 in.

25. 2273 m

26. 2

2

sθ

27. 4500 km

28. 12,000 km

29. 3

4 ;1.5 sq units

30. 1

2 ;16 sq units

31. (a) 3

π radians

(b) 2 inπ

32. Answers will vary, 22

r θæ ö÷ç ÷ç ÷çè ø



33. 3

34. 1

2-

35. 1

2-

36. 3-

37. 2

38. undefined

39. tan 1 > tan 2

40. tan1 sin1>

41. sin 2 > cos 2

42. (a) A

(b) C

(c) B

43. 0.8660

44. 2.7976

45. 0.9703

46. 11.4266-

47. 1.9513

48. 1.0080-

49. 0.3898

50. 1.3265

51. 0.5148

52. 0.9424

53. 1.1054

54. 1.3497

55. 4

s π=

56. 2

3

π

57. 7

6

π

58. 11

6

π

59. 15

sec32

60. 108 radians

61. 20

π radians per sec

62. 285.3 cm

63. 1260 cm per secπ

64. 36

π radian per sec.

65. 5 inches.

66. (a) 0; The face of the moon is not visible.

(b) 1

;2

Half the face of the moon is visible.

(c) 1; The face of the moon is completely visible.

(d) 1

;2

Half the face of the moon is visible.

Chapter 3 Test

1. 2

3

π

2. 4

π-

3. 0.09

4. 135

5. 210-

6. 229.18

7. (a) 4

3

(b) 215,000 cm

8. 2 radians

9. 2

2

10. 3

2-

11. undefined

Chapter 3 Test 31


12. 2-

13. 0

14. 0

15. 7 1 7 3

sin ;cos6 2 6 2

7 3 7tan ;csc 2

6 3 6

π π

π π

=- =-

= =-

7 2 3 7

sec ;cot 36 3 6

π π=- =

16. sine and cosine: ( ),-¥ ¥;

tangent and secant:

( )| 2 1 , where is any integer2

s s n nπì üï ïï ï¹ +í ýï ïï ïî þ;

cotangent and cosecant :

{ }| , where is any integers s n nπ¹

17. (a) 0.9716

(b) 3

π

18. (a) 2

3

π radians

(b) 40 cmπ

(c) 5π cm per sec

19. 8.127 mi per sec

20. (a) 75 ft

(b) 45

π radian per sec

Lial Trig Chapter 3 10e - · PDF filecircle of radius 1 by definition. (See exercise ... 49....

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Transcript of Lial Trig Chapter 3 10e - · PDF filecircle of radius 1 by definition. (See exercise ... 49....