Review for Math 111 Final Exam

The final exam is worth 30% (150/500 points). It consists of 26 multiple choice questions, 4 graph

matching questions, and 4 short answer questions. Partial credit will be awarded on the short answer

questions for quality of work shown. Calculators are not allowed. The following formula sheet will

be given on the final exam.

Trigonometric Formulas and Identities

Pythagorean Identity

sin2 𝜃 + cos2 𝜃 = 1

Sum and Difference Identities

sin(𝑢 + 𝑣) = sin 𝑢 cos 𝑣 + cos 𝑢 sin 𝑣

sin(𝑢 − 𝑣) = sin 𝑢 cos 𝑣 − cos 𝑢 sin 𝑣

cos(𝑢 + 𝑣) = cos 𝑢 cos 𝑣 − sin 𝑢 sin 𝑣

cos(𝑢 − 𝑣) = cos 𝑢 cos 𝑣 + sin 𝑢 sin 𝑣

Double Angle Identities

sin(2𝑢) = 2 sin 𝑢 cos 𝑢

cos(2u) = cos 𝑢2 − sin 𝑢2

cos(2𝑢) = 1 − 2 sin 𝑢2

cos(2𝑢) = 2 cos 𝑢2 − 1

Law of Sines

sin 𝐴

𝑎=

sin 𝐵

𝑏=

sin 𝐶

𝑐

Law of Cosines

𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴

𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑐 cos 𝐵

𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶

You should know other formulas such as: arc length of a circle, area of a sector of a circle,

Pythagorean identities, and Complimentary Angle Theorem.

Part 1 Multiple Choice (4 points each) Each question has one correct answer. Partial credit is NOT

possible.

1. Convert and angle measuring 165° into radians.

(A) 11𝜋

6

(B) 8𝜋

9

(C) 7𝜋

9

(D) 11𝜋

12

(E) None of these

2. Convert an angle measuring 13𝜋

10 radians into degrees.

(A) 234°

(B) 240°

(C) 246°

(D) 252°

(E) None of these

3. If angle 𝜃 is in standard position and 𝜃 = 445°, the terminal side of 𝜃 would lie in which quadrant?

(A) Quadrant 1

(B) Quadrant 2

(C) Quadrant 3

(D) Quadrant 4

4. If angle 𝜃 is in standard position and 𝜃 = −535°, the terminal side of 𝜃 would lie in which

quadrant?

(A) Quadrant 1

(B) Quadrant 2

(C) Quadrant 3

(D) Quadrant 4

5. If angle 𝜃 is in standard position and 𝜃 =19𝜋

12, the terminal side of 𝜃 would lie in which quadrant?

(A) Quadrant 1

(B) Quadrant 2

(C) Quadrant 3

(D) Quadrant 4

6. If angle 𝜃 is in standard position and 𝜃 = 7, the terminal side of 𝜃 would lie in which quadrant?

(A) Quadrant 1

(B) Quadrant 2

(C) Quadrant 3

(D) Quadrant 4

7. If angle 𝜃 is in standard position and 𝜃 = −160°, which one of the following is coterminal with 𝜃?

(A) 20°

(B) 160°

(C) 220°

(D) −520°

(E) −200°

8. If angle 𝜃 is in standard position and 𝜃 =11𝜋

8, which one of the following is coterminal with 𝜃?

(A) 3𝜋

8

(B) 5𝜋

8

(C) −5𝜋

8

(D) −11𝜋

8

(E) −19𝜋

8

9. A circle of radius 4 inches is intercepted by central angle 𝜃. If the intercepted arc length is 20

inches, determine 𝜃.

(A) 𝜃 = 5 𝑟𝑎𝑑𝑖𝑎𝑛𝑠

(B) 𝜃 = 5°

(C) 𝜃 =𝜋

36 𝑟𝑎𝑑𝑖𝑎𝑛𝑠

(D) 𝜃 =𝜋

36

°

10. A circle of radius 10 inches is intercepted by central angle 𝜃. If 𝜃 = 40°, determine the area of the

intercepted sector.

(A) 10𝜋

9𝑖𝑛.2

(B) 200𝑖𝑛.2

(C) 100𝜋

9𝑖𝑛.2

(D) 200𝜋

9𝑖𝑛.2

11. The area of a sector of a circle is 20 𝑖𝑛2. If the radius of the circle is 8 inches, what is the length of

the intercepted arc?

(A) 𝜋

36𝑖𝑛.

(B) 5𝑖𝑛.

(C) 10𝑖𝑛.

(D) 𝜋

18𝑖𝑛.

12. When a 6 foot tall person is standing 48 feet from the base of a flagpole, the angle of elevation to the

top of the flagpole is 30°. What is the height of the flagpole?

(A) The building is 16√3 + 6 feet tall.

(B) The building is 16√3 feet tall.

(C) The building is 48√3 feet tall.

(D) The building is 48√3 + 6 feet tall.

13. The right triangle below has a height of 16 meters. Write a function 𝑃(𝜃) representing the perimeter

of the triangle in terms of angle 𝜃.

(A) 𝑃(𝜃) = 16 + 𝑏 + ℎ

(B) 𝑃(𝜃) = 16 + 16 tan 𝜃 + 16 sin 𝜃

(C) 𝑃(𝜃) = 16 +tan 𝜃

16+

sin 𝜃

16

(D) 𝑃(𝜃) = 16 +16

tan 𝜃+

16

sin 𝜃

16

𝜃

14. If tan 𝜃 > 0 and csc 𝜃 < 0, then 𝜃 lies in which quadrant?

(A) Quadrant 1

(B) Quadrant 2

(C) Quadrant 3

(D) Quadrant 4

15. If sec 𝜃 > 0 and sin 𝜃 < 0, then 𝜃 lies in which quadrant?

(A) Quadrant 1

(B) Quadrant 2

(C) Quadrant 3

(D) Quadrant 4

16. The terminal side of angle 𝜃, in standard position, passes through the point (5, −12). Find the exact

value of csc 𝜃 and sec 𝜃.

(A) csc 𝜃 =13

12 and sec 𝜃 =

13

5

(B) csc 𝜃 =12

13 and sec 𝜃 =

13

5

(C) csc 𝜃 =13

12 and sec 𝜃 =

5

13

(D) csc 𝜃 = −13

12 and sec 𝜃 =

13

5

17. Given csc 𝜃 = 4 and cot 𝜃 < 0, determine: sin2 𝜃 + cos2 𝜃.

(A) −15

16

(B) 16

16

(C) 14

16

(D) −14

16

18. Use the complimentary angle theorem and fundamental identities to find the exact value of:

sin 50° cos 40° + cos 50° sin 40°.

(A) 1

(B) 2

(C) 0

(D) −1

19. Use the complimentary angle theorem and fundamental identities to find the exact value of:

cot2(40°) + 1 − sec2(50°) .

(A) −1

(B) 0

(C) 1

(D) 2

20. Given that cos(36°) = .81, determine the value of: sin(36°) ∙ csc(36°) +1

csc(54°).

(A) 1 − sin 9°

(B) 1.09

(C) 1.81

(D) 1 + sin(54°)

21. Find the exact value of sec (7𝜋

4) .

(A) −2

(B) 2

(C) √2

(D) −√2

(E) √2

2

22. Find the exact value of: tan (7𝜋

2) .

(A) 0

(B) 1

(C) −1

(D) √2

(E) Undefined

23. Find the exact value of: sec2 (7𝜋

4) + 3 sin2 (−

10𝜋

3).

(A) −7

4

(B) 17

4

(C) 3

(D) 25

4

24. Find the exact value of: sin (−5𝜋

2) − tan (−

5𝜋

4).

(A) −2

(B) −1

(C) 0

(D) 1

(E) Undefined

25. Given the function, 𝑦 = 3 − 2 cos(4𝑥 − 5), determine the amplitude and vertical shift.

(A) Amplitude: 2, Vertical shift: 3

(B) Amplitude: -2, Vertical shift: 3

(C) Amplitude: 3, Vertical shift: -2

(D) Amplitude: -3, Vertical shift: 2

26. Given the function, 𝑦 = 3 − 2 cos(4𝑥 − 5), determine the period and phase shift.

(A) Period: 𝜋

2, Phase shift: left

5

4

(B) Period: 𝜋

2, Phase shift: right

5

4

(C) Period: 𝜋

4, Phase shift: left

4

5

(D) Period: 𝜋

4, Phase shift: right

4

5

27. Using the graph (right) determine the period of the function.

The period is

(A) −4

(B) 2𝜋

3

(C) 𝜋

2

(D) −𝜋

6

28. Using the same graph as #27, determine the value of A in the equation modeling the function

𝑦 = 𝐴 cos (𝐵 (𝑥 −𝜋

6)) + 𝐷.

(A) 6

(B) −6

(C) −4

(D) 3

(E) −3

29. Which of the following equations does NOT model the graph in #27?

(A) 𝑦 = −3 cos(3𝑥 − .5𝜋) − 4

(B) 𝑦 = 3 cos (3 (𝑥 +𝜋

6)) − 4

(C) 𝑦 = −3 sin(3𝑥) − 4

(D) 𝑦 = 3 sin(3𝑥) − 4

(E) 𝑦 = 3 sin (3 (𝑥 −𝜋

3)) − 4

30. Which of the following equations does NOT represent the function graphed below?

(A) 𝑦 = 5 cos (𝜋

3(𝑥 − 1)) − 2

(B) 𝑦 = −5 cos (𝜋

3𝑥 +

2𝜋

3) − 2

(C) 𝑦 = 5 sin (𝜋

3(𝑥 −

1

2)) − 2

(D) 𝑦 = −5 sin (𝜋

3(𝑥 − 2.5)) − 2

31. Assume the amplitude of a sinusoidal function is 5 and the period is 8. If 𝑓(3) = 7 is a maximum

value of the function, then where would a minimum value occur? Where would another maximum

value occur?

(A) minimum at 𝑓(7) = 2, maximum at 𝑓(11) = 7

(B) minimum at 𝑓 (11

2) = −3, maximum at 𝑓(8) = 7

(C) minimum at 𝑓(7) = −3, maximum at 𝑓(11) = 7

(D) minimum at 𝑓(5.5) = 2, maximum at 𝑓(11) = 7

32. A weight, attached to the end of a very long spring, is bouncing up and down. For a small period of

time, this motion can be modeled by a sinusoidal function. When your stopwatch reads 1.3 seconds,

the weight is at a minimum height of 2.4 feet above the floor. When your stop watch reads 1.9

seconds, the weight reaches the next maximum height of 3.2 feet. Determine the equation modeling

the height of the weight, h, in terms of time, t.

(A) ℎ(𝑡) = −0.4 cos (2𝜋

1.2(𝑥 − 1.3)) + 2.8

(B) ℎ(𝑡) = −0.6 cos (2𝜋

1.2(𝑥 − 1.3)) + 2.4

(C) ℎ(𝑡) = −0.4 cos (2𝜋

0.6(𝑥 − 1.3)) + 2.4

(D) ℎ(𝑡) = −0.6 cos (2𝜋

0.6(𝑥 − 1.9)) + 2.8

33. Find the exact value of: cos−1 (−√2

2).

(A) 𝜋

4

(B) −𝜋

4

(C) 3𝜋

4

(D) 5𝜋

4

(E) 7𝜋

4

34. Find the exact value of: sin−1 (√3

2).

(A) 𝜋

3

(B) 𝜋

4

(C) 𝜋

6

(D) 𝜋

2

35. Find the exact value of: tan−1 (−1

√3).

(A) 2𝜋

3

(B) 5𝜋

6

(C) −𝜋

3

(D) −𝜋

6

(E) −2𝜋

3

36. Find the exact value of: cos−1 (cos (4𝜋

3)).

(A) 4𝜋

3

(B) −𝜋

3

(C) 𝜋

3

(D) 2𝜋

3

37. Find the exact value of: sin−1(tan(𝜋)).

(A) Undefined

(B) 0

(C) 𝜋

(D) −1

(E) 𝜋

2

38. Find the exact value of: sin−1 (cos (4𝜋

3)).

(A) Undefined

(B) 2𝜋

3

(C) −𝜋

3

(D) 𝜋

6

(E) −𝜋

6

39. Find the exact value of: cos(sin−1(1)).

(A) Undefined

(B) 1

(C) 0

(D) −1

(E) 𝜋

2

40. Find the exact value of: tan(sin−1(−1)).

(A) Undefined

(B) 1

(C) 0

(D) −1

(E) −𝜋

2

41. Find the exact value of: sin(tan−1(−1)).

(A) √2

2

(B) −√2

2

(C) 𝜋

4

(D) −𝜋

4

(E) 7𝜋

4

42. Find the exact value of: cos(tan−1(2)).

(A) Undefined

(B) 1

√5

(C) 2

√3

(D) 2

√5

43. Find the exact value of: cot (cos−1 (−3

2)).

(A) Undefined

(B) −3

√5

(C) −√5

3

(D) −2

√5

(E) −√5

2

44. Express tan(cos−1 𝑢) as an algebraic expression involving u.

(A) √1−𝑢2

𝑢

(B) √1+𝑢2

𝑢

(C) 𝑢

√1−𝑢2

(D) 𝑢

√1+𝑢2

45. The length of the shadow of a building 34 meters tall is 37 meters. Which of the following would

give the angle of elevation of the sun?

(A) 𝜃 = tan−1 (37

34)

(B) 𝜃 = tan−1 (34

37)

(C) 𝜃 = sin−1 (34

37)

(D) 𝜃 = sin−1 (37

34)

46. Simplify the expression: tan 𝜃 − sec 𝜃 csc 𝜃. The result is

(A) − tan 𝜃

(B) cot 𝜃

(C) − cot 𝜃

(D) tan 𝜃

47. Simplify the expression: (sec 𝜃 − 1)(sec 𝜃 + 1). The result is

(A) cot2 𝜃

(B) tan2 𝜃

(C) − tan2 𝜃

(D) − cot2 𝜃

48. Simplify the expression: cos 𝜃(tan 𝜃 + cot 𝜃). The result is

(A) 1

(B) cos 𝜃

(C) sin 𝜃

(D) sec 𝜃

(E) csc 𝜃

49. Simplify the expression: tan 𝛼+tan 𝛽

cot 𝛼+cot 𝛽. The result is

(A) sin2(𝛼𝛽)

cos2(𝛼𝛽)

(B) 2

(C) tan 𝛼 tan 𝛽

(D) cot 𝛼 cot 𝛽

50. Use the sum and difference identities to determine the exact value of: cos (17𝜋

12).

(A) √6+√2

4

(B) √6−√2

4

(C) −√6−√2

4

(D) √2−√6

4

51. Use the sum and difference identities to determine the exact value of: sin (𝜋

12).

(A) √6+√2

4

(B) √6−√2

4

(C) −√6−√2

4

(D) √2−√6

4

52. Find the exact value of the expression: sin (𝜋

4) cos (

𝜋

12) + cos (

𝜋

4) sin (

𝜋

12).

(A) 1

2

(B) √2

2

(C) √3

2

(D) 1

53. If 𝛼 = tan−1 (−4

3), determine the exact value of: sin (𝛼 +

3𝜋

4).

(A) −√2

10

(B) 7√2

10

(C) √2

10

(D) −7√2

10

54. Use the sum and difference identities to simplify: cos (𝜃 −𝜋

2).

(A) sin 𝜃

(B) cos 𝜃

(C) − sin 𝜃

(D) − cos 𝜃

55. If 𝛽 = tan−1 (−2

3), determine the exact value of: sin(2𝛽).

(A) 12

13

(B) 12

√13

(C) −12

13

(D) −12

√13

56. If the terminal side of angle 𝜃, in standard position, passes through the point (−5, 3), determine the

exact value of: cos(2𝜃).

(A) 16

√34

(B) 19

34

(C) 1

(D) 8

17

57. Solve the equation: 2 cos2 𝑥 − 1 = 0, for x on the interval from [0, 2𝜋).

(A) 𝑥 =𝜋

4,

7𝜋

4

(B) 𝑥 =𝜋

4,

3𝜋

4,

5𝜋

4,

7𝜋

4

(C) 𝑥 =𝜋

3,

2𝜋

3,

4𝜋

3,

5𝜋

3

(D) No solution

58. Solve the equation: 2 sin(2𝑥) + 1 = 0, for x on the interval from [0, 2𝜋).

(A) 𝑥 =2𝜋

3,

4𝜋

3

(B) 𝑥 =7𝜋

6,

11𝜋

6

(C) 𝑥 =7𝜋

12,

11𝜋

12,

19𝜋

12,

23𝜋

12

(D) 𝑥 =𝜋

3,

2𝜋

3,

4𝜋

3,

5𝜋

3

59. Solve the equation: 2 sin2 𝑥 = sin 𝑥 + 1, for x on the interval from [0, 2𝜋).

(A) 𝑥 = −1

2, 1

(B) 𝑥 =1

2, −1

(C) 𝑥 =2𝜋

3, 𝜋,

4𝜋

3

(D) 𝑥 =𝜋

2,

7𝜋

6,

11𝜋

6

60. Solve the equation: 4(1 + sin 𝑥) = cos2 𝑥, for x on the interval from [0, 2𝜋).

(A) 𝑥 =3𝜋

2

(B) 𝑥 = −𝜋

2

(C) 𝑥 = −1

(D) 𝑥 = 𝜋

61. Solve the equation: sin(2𝑥) = cos 𝑥, for x on the interval from [0, 2𝜋).

(A) 𝑥 = 0,1

2

(B) 𝑥 =𝜋

6,

𝜋

2,

5𝜋

6,

3𝜋

2

(C) 𝑥 =𝜋

3,

𝜋

2,

3𝜋

2,

5𝜋

2

(D) 𝑥 = 0,𝜋

2, 𝜋,

5𝜋

2

(E) No solution

62. Solve the equation: cos(2𝑥) + 5 cos 𝑥 + 3 = 0, for x on the interval from [0, 2𝜋).

(A) No solution

(B) 𝑥 =2𝜋

3,

4𝜋

3

(C) 𝑥 = −1

2, −2

(D) 𝑥 =7𝜋

6,

11𝜋

6

63. Two runners, approaching the finish line, in a marathon determine that the angles of elevation of a

news helicopter covering the race are 45° and 60°. If the helicopter is 300 feet directly above the

finish line, how far apart are the runners?

(A) The runners are 300 − 100√3 feet apart.

(B) The runners are 300 feet apart.

(C) The runners are 100√3 feet apart.

(D) The runners are 300√3 feet apart.

64. A loading ramp 10 feet long that makes an angle of 45° with the horizontal is to be replaced by one

that makes an angle of 30° with the horizontal. How long is the new ramp?

(A) The new ramp is 20 feet long.

(B) The new ramp is 10√6

3 feel long.

(C) The new ramp is 10√2 feet long.

(D) The new ramp is 10√3 feet long.

65. If a triangle does NOT have any right angles, then the triangle is called

(A) obtuse

(B) scalene

(C) oblique

(D) acute

66. Which of the following might result in two possible triangles?

(A) 𝑏 = 10, 𝐵 = 120°, 𝐶 = 125°

(B) 𝑎 = 2, 𝑐 = 1, 𝐴 = 120°

(C) 𝐴 = 50°, 𝑏 = 3, 𝐶 = 85°

(D) 𝐴 = 100°, 𝐵 = 30°, 𝑏 = 6

67. Clint is building a swing set for his children. Each supporting end of the swing set is to be an A-

frame constructed with two 10-foot-long 4 by 4’s joined at a 45° angle. To prevent the swing set

from tipping over, Clint wants to secure the base of each A-frame in concrete footings. How far

apart should the footings for each A-frame be?

(A) The footings should be 10 feet apart.

(B) The footings should be 200 − 100√3 feet apart.

(C) The footings should be 20 − 10√2 feet apart.

(D) The footings should be √200 − 100√2 feet apart.

68. Determine the length of side c of the oblique triangle if 𝑎 = 2, 𝑏 = 3, 𝐶 = 60°.

(A) 𝑐 = √13 − 6√3

(B) 𝑐 = 2√2

(C) 𝑐 = √7

(D) 𝑐 = 13 − 6√2

Part 2: Match the trigonometric function with its correct graph below. Write the appropriate

letter in the space provided. Select each letter at most once.

69. 𝑦 = sin 𝑥 ________

70. 𝑦 = cos 𝑥 ________

71. 𝑦 = tan 𝑥 ________

72. 𝑦 = csc 𝑥 ________

73. 𝑦 = sec 𝑥 ________

74. y = cot 𝑥 ________

75. 𝑦 = sin−1 𝑥 ______

76. 𝑦 = cos−1 𝑥 ______

77. 𝑦 = tan−1 𝑥 ______

(A) (B) (C)

(D) (E) (F)

(G) (H) (I)

Part 3 Short Answer Partial credit is possible on these short answer exercises. Show your work for

full credit. Answers given without clear supporting work or reasonable explanation may receive

little or no credit.

77. Sketch the graph of 𝑦 = 5 sin (3𝑥 −𝜋

4) − 2 below. Be sure to graph at least one period and label 5

significant ordered pairs on your graph.

78. Sketch the graph of 𝑦 = −2 cos (𝜋

3𝑥 − 𝜋) + 4 below. Be sure to graph at least one period and label

5 significant ordered pairs on your graph.

79. Solve the following equations for 𝜃 on the interval [0, 2𝜋). Show your work clearly and BOX your

solution(s).

a. cos(2𝜃) = −1

2 b. cos 𝜃2 − sin 𝜃2 + sin 𝜃 = 0

c. sin 𝜃2 = 6(cos 𝜃 + 1) d. cos(2𝜃) + 6 sin 𝜃2 = 4

e. sin(2𝜃) sin 𝜃 = cos 𝜃

80. A sinusoidal function has a maximum at the point (3, 38) and the next minimum at the point (21,4).

The next maximum would occur at the point (______, ______).

Graph two complete periods of this function.

Vertical Shift: Amplitude: Period: Phase Shift:

Write the equation in the form: 𝑑(𝑡) = 𝐴 cos(𝐵(𝑡 − 𝐶)) + 𝐷

𝑑(𝑡) = __________________________________________

81. The Ferris Wheel: The tallest Ferris wheel in the world is the High Roller located on the Las

Vegas strip. You decide to take a ride. 18 minutes into your ride, you reach the highest point at 550

feet above the ground. 33 minutes into your ride, you are at the lowest point 30 feet above the

ground.

Graph two complete periods of your ride showing distance above the ground against time.

Vertical Shift: Amplitude: Period: Phase Shift:

Write the equation in the form: 𝑑(𝑡) = 𝐴 cos(𝐵(𝑡 − 𝐶)) + 𝐷

𝑑(𝑡) =_________________________________________

Answers

1. D

2. A

3. A

4. C

5. D

6. A

7. C

8. C

9. A

10. C

11. B

12. A

13. D

14. C

15. D

16. D

17. B

18. A

19. B

20. C

21. C

22. E

23. B

24. C

25. A

26. B

27. B

28. E

29. D

30. C

31. C

32. A

33. C

34. A

35. D

36. D

37. B

38. E

39. C

40. A

41. B

42. B

43. A

44. A

45. B

46. B

47. B

48. E

49. C

50. D

51. B

52. C

53. B

54. A

55. C

56. D

57. B

58. C

59. D

60. A

61. B

62. B

63. A

64. C

65. C

66. B

67. D

68. C

69. E

70. A

71. B

72. F

73. I

74. D

75. H

76. C

77. G

ANSWERS

CONTINUE…

78.

79.

80.

a. 𝜃 =𝜋

3,

2𝜋

3,

4𝜋

3,

5𝜋

3

b. 𝜃 =7𝜋

6,

11𝜋

6,

𝜋

2

c. 𝜃 = 𝜋

d. 𝜃 =𝜋

3,

2𝜋

3,

4𝜋

3,

5𝜋

3

e. 𝜃 =𝜋

2,

3𝜋

2,

𝜋

4,

3𝜋

4,

5𝜋

4,

7𝜋

4

81.

𝑑(𝑡) = 17 cos (2𝜋

36(𝑡 − 3)) + 21

(many equations possible)

82.

𝑑(𝑡) = 260 cos (2𝜋

30(𝑡 − 18)) + 290

(many equations possible)