Name _______________________________________ Date ___________________ Class __________________ CHAPTER

5

Review 5.1-5.4 Trigonometric Functions and Identities

Select the best answer. 1. The sides of right ABC have lengths 10 ft, 24 ft, and 26 ft. Which could be the cosine of angle B? 5 13 A C 13 5 5 13 B D 12 12 2. A carpenter is making a ramp for a ski jump. The ramp is made from a 25 ft board tilted at a 30° angle from the ground. Approximately how far off the ground is the end of the ramp? A 11.6 ft B 12.5 ft 3. Which is the value of cot ? (Picture is not to scale.)

3 4 4 B 3 A

4. Which is the angle of rotation shown below?

A 210° B 150° 5. Which two angles are both coterminal with   90°? A 270° and 90° C 0° and 180° B 270° and 450° D 450° and 270°

Name _______________________________________ Date ___________________ Class __________________

6. When viewed from the side, a nail is located at 90° on a tire. The tire turns counterclockwise through 8.5 complete rotations. What is the new position of the nail in the tire? A 0°

C 180°

B 90°

D 270°

7. What is the reference angle for  165°? A 15° B 65° 8. A circular baggage return has a diameter of 40 ft. A piece of luggage makes 3 revolutions in 12 minutes. Approximately how far does one piece of luggage travel in 1 minute? A 10 ft

C 62 ft

B 31 ft

D 93 ft

9. Which angle measure is closest to 8 radians? A 100°

C 150°

B 120°

D 360°

10. The sine of  is

3 1 . Which is the cotangent of  ? and the cosine of  is  2 2

A

3 2

C 

3 2

B

3

D  3

11. The point (3, 4) is on the terminal side of  in standard position. Which is the sine of ?

3 A  5 B

3 5

 C D

4 5

4 5

12. Kate leans a ladder against her house to get on to the roof. The house is 25 feet tall and the foot of the ladder is 15 feet away from the side of the house. Which is the angle that the ladder makes with the ground? A 31°

C 53°

B 37°

D 59°

Name _______________________________________ Date ___________________ Class __________________

13. Which value of  makes the equation 2(cos )  1 true? A 30°

C 120°

B 60°

D 135°

 2 ? 14. Which set of expressions, where n is an integer, gives all possible values of sin1   2    A (2)n,   (2n B C

 4

 6

 (2 )n,

3  (2 )n 4

 (2 )n,

5  (2 )n 6

D   (2n, 2 (2n 15. A blueprint of a table is shown below. What is the area of the table to the nearest tenth of a square foot? (Picture is not to scale.)

A 14.8 ft2 B 29.5 ft2 16. The Rodriguez family wants to tile the countertop in their kitchen shown below. The tile they want to use costs $5 per square foot. What is the approximate cost of tiling this area? (Picture is not to scale.)

A $37.42

C $47.43

B $42.86

D $56.00

17. The sides of right ABC have lengths 12 ft, 16 ft, and 20 ft. Which could be the cosine of angle C? 3 4 A C 5 3 5 5 B D 3 4

Name _______________________________________ Date ___________________ Class __________________

18. A camper creates a make-shift tent by leaning a branch against a tree and draping a tarp over it. The roof is constructed from a 15 ft branch that makes a 15 angle with the ground. Approximately how tall is the tent at its highest point? F 3.89 ft H 14.49 ft G 4.02 ft J 15.56 ft 19. What is the value of csc ? (Picture is not to scale.)

13 12 13 B 5 A

5 13 12 D 13 C

20. What is the angle of rotation shown below?

F 30 G 120

H 240 J 330

21. Which two angles are both coterminal with   50? A 230 and 130C 130 and 310 B 230 and 410 D 410 and 310 22. When viewed from the side, a nail is located at 85 on a tire. The tire turns counterclockwise through 7.25 complete rotations. What is the new position of the nail in the tire? F 175

H 5

G 5

J 175

23. What is the reference angle for   210? A 30

C 120

B 60

D 150

Name _______________________________________ Date ___________________ Class __________________

24. A Ferris wheel has a diameter of 100 ft. The passengers ride in cars that make 5 revolutions in 7.5 minutes. Approximately how far does a car on the Ferris wheel travel in 1 minute? F 209 ft

H 471 ft

G 418 ft

J 942 ft

25. Which angle measure is closest to 4.7 radians? A 90

C 180

B 135

D 270

26. The tangent of  is 1 and the cosine of  is  F  2

H

2 2

2 2

J

2

G 

2 . Which is the cosecant of  ? 2

27. The point (4, 6) is on the terminal side of  in standard position. Which is the cosine of ? A 

2 13 13

C

2 13

B 

2 13

D

2 13 13

28. Charles is trying to get his kite out of a tree. The kite is 12 feet above him and he’s standing 10 feet from the tree. If he can throw a ball in a straight line, at what angle should he throw it in order to hit the kite? F 33.6 H 50.2 G 39.8 J 56.4

29. Which value of  makes the equation 2 (sin  )  1 true? 3 A 150 C 300 B 225 D 330

Name _______________________________________ Date ___________________ Class __________________

30. The Ford family wants to carpet a section of their basement with the dimensions shown below. The carpet they want to use costs $1.50 per square foot. What is the approximate cost of carpeting this area? (Picture is not to scale.)

A $15.78

C $26.98

B $17.99

D $29.68

31. The legs of right XYZ have lengths 3 ft and 2 ft. Which could be the cosine of angle C? 2 13 3 A C 13 2 5 2 B D 3 3 32. A camper creates a make-shift tent by leaning a branch against a tree and draping a tarp over it. The roof is constructed from a 13 ft branch that makes a 19 angle with the ground. Approximately how tall is the tent at its highest point? F 4.23 ft H 8.32 ft G 4.48 ft J 12.29 ft

33. What is the value of cot ? (Picture is not to scale.)

A B

2 2 1 2

C

2

D 2 2

34. What is the angle of rotation shown below?

F 330 G 30

H 30 J 330

Name _______________________________________ Date ___________________ Class __________________

35. Which two angles are both coterminal with   400? A 320 and 40 C 40 and 320 B 40 and 320 D 40 and 320 36. The tip of a needle on a gauge is located at 170. The needle turns clockwise through 6.35 complete rotations. What is the new position of the tip of the needle? F 44

H 205

G 135

J 296

37. What is the reference angle for   210? A 30

C 120

B 60

D 150

38. A Ferris wheel has a diameter of 97 ft. The passengers ride in cars that make 6 revolutions in 10 minutes. Approximately how far does a car on the Ferris wheel travel in 1 minute? F 91 ft

H 183 ft

G 152 ft

J 304 ft

39. Which angle measure is closest to A 0

C 180

B 75

D 270

4



radians?

40. The cotangent of  is 1 and the cosine of  is

F G

2 2 2 2



2 2

J 

2

H

2 . Which is the cosecant of ? 2

2

41. The point (2, 3) is on the terminal side of  in standard position. Which is the cosine of ? A 

2 13 13

C

2 13

B 

2 13

D

2 13 13

Name _______________________________________ Date ___________________ Class __________________

42. Eileen uses a laser pointer to identify a point 3.6 feet above her on a screen that is 6.7 feet across the room from her. At what angle is she holding the laser pointer? F 28.2 H 41.6 G 32.5 J 57.5 43. Which value of  makes the equation A 30 B 60

2 3

(csc  ) 

4 true? 3

C 135 D 240

44. The Jackson family wants to cover a section of a wall with fabric. The wall has the dimensions shown below. The fabric they want to use costs $4.50 per square foot. What is the approximate cost of covering this area? (Picture is not to scale.)

A $56.57 B $119.65

C $191.06 D $254.56

45. The sides of right ABC have lengths 3, 4, and 5. What are the three possible values of the cosine of angle A?

46. A truck drives 11.5 feet up an incline to a loading dock. The incline makes a 10° angle with the parking lot. Find the approximate vertical height of the incline. Round your answer to the nearest foot.

47. What is the value of sec ? (Picture is not to scale.)

Name _______________________________________ Date ___________________ Class __________________

48.What is the angle of rotation shown below?

49. What are the smallest positive and the largest negative angles that are coterminal with   90°?

50. The indicator on a circular volume knob is located at 100°. Becky hears a song that she liked and turns the knob clockwise through 2.5 rotations. What is the new position of the indicator on the volume knob?

51. What is the reference angle for   200°?

52. A circular plate has a diameter of 10 inches. An ant walking along the edge makes 2 revolutions in 4 minutes. Approximately how many inches does the ant travel in one minute? Round your answer to the nearest tenth of an inch.

Name _______________________________________ Date ___________________ Class __________________

53. Convert

 radians to the nearest degree. 4

54. The tangent of  is 1 and the cosine of  is 

2 . What is the cosecant of  ? 2

55. The point (6, 8) is on the terminal side of  in standard position. What is the cosine of ?

56. A 50 foot tall tower under construction has a rope tied to the top and attached to a post below. The post is 24 feet from the base of the tower. What angle does the rope make with the ground? Round your answer to the nearest degree.

57. What is the smallest value of  that makes the equation

2 3

(sin  )  1 true?

58. The measurements of a triangular patio are shown below. What is the area of the patio? Round your answer to the nearest tenth of a square foot. (Picture is not to scale.)

59. The Brady family is planting a new garden in their backyard with the dimensions shown below. They want to plant bushes that require 2 square feet of surface area each. How many bushes can they plant in their new garden? Round your answer down to the nearest whole bush. (Picture is not to scale.)

60. The sides of right ABC have lengths 10, 24, and 26. What are the possible values of the cosine of angle A?

61. A business is adding a wheelchair ramp to its front entrance. The ramp will be 12.5 feet long and makes a 5° angle with the sidewalk. Find the approximate vertical distance from the sidewalk to the front entrance. Round your answer to the nearest tenth of a foot.

62. What is the value of sec ? (Picture is not to scale.)

63. What is the angle of rotation shown below?

64. What are the smallest positive and the largest negative angles that are coterminal with   65°?

65. The indicator on a circular volume knob is located at 170°. Venuta likes a song that comes on the radio and turns the knob clockwise through 4.75 rotations. What is the new position of the indicator on the volume knob?

66. What is the reference angle for   235°?

67. A circular race track has a diameter of 1200 meters. A car makes 4 revolutions in 6 minutes. Approximately how many meters does the car travel in one minute? Round your answer to the nearest meter.

68. Convert

3 radians to the nearest tenth of a degree. 8

69. The tangent of  is

3 and the sine of  is 

3 . What is the secant of  ? 2

70. The point (12, 16) is on the terminal side of  in standard position. What is the cosine of ?

71. A sailboat has a taut line running from the top of the mast to the side of the boat. The mast is 30 feet tall and sits 8 feet from the side of the boat. What angle does the line make with the deck of the boat? Round your answer to the nearest degree.

72. What is the smallest value of  that makes the equation

2 3

(cos  )  1 true?

73. The legs of right ABC have lengths 3 and 1. What are the possible values of the cosine of angle A?

74. A business is adding a wheelchair ramp to its front entrance. The ramp will be 18.5 feet long and makes a 6.5° angle with the sidewalk. Find the approximate vertical distance from the sidewalk to the front entrance. Round your answer to the nearest tenth of a foot.

75. What is the value of csc ? (Picture is not to scale.)

76. What is the angle of rotation shown below?

77. What are the smallest positive and the largest negative angles that are coterminal with   301°?

78. The indicator on the handle of a water faucet is located at 37.5°. Brandy starts a bath by turning the handle clockwise through 3.45 rotations. What is the new position of the indicator on the volume knob?

79. What is the reference angle for   215°?

80. A circular track has a diameter of 0.16 miles. A jogger makes 2.5 revolutions in 40 minutes. Approximately how many miles will the jogger run in one hour? Round your answer to the nearest tenth of a mile.

81. Convert

16 radians to the nearest tenth of a degree. 3

82. The tangent of  is 

1 3

and the secant of  is 

2 3

. What is the sine of ?

83. The point (36, 15) is on the terminal side of  in standard position. What is the cosine of ?

Prove each trigonometric identity. 84. sin2  sin2 cot2  1

85. cot2 cos2  cot2  cos2

86. tan2  tan2 sin2  sin2

87.

sin   cos   sec   csc  sin  cos 

Rewrite each expression in terms of cos. Then simplify. 88. 2sin cos cot

89.

1  cot  cot  (sin   cos  )

90. cos4  sin4  sin2

Prove each trigonometric identity. cos  1  sin   cos  91. 1  sin 

92. tan  sec  

93. tan2  sin2  tan2 sin2

94. 1  cot4  csc4  2 cot2

sec   tan  cos   cot 

Rewrite each expression in terms of sin and cos. Then simplify. 95.

cot  sin 

96. cot2  cos cot2

97. tan2  4 sec2  1