Final Exam Review
Semester 1 – 2015
Precalculus
1. Convert 240° to radians.
2π
A.
3
7π
to degrees.
4
A. 225°
Name: ____________________________________
B.
4π
3
C.
5π
6
D.
7π
6
2. Convert
3. Evaluate sin
A.
B.
4. Evaluate sin ( −π ) .
A. 0
A.
1
2
C. 315°
D. 330°
13π
.
6
1
2
5. Evaluate csc
B. 270°
−1
2
C.
B. 1
7π
6
B.
−1
2
6. For what numbers θ, is f(θ) = cot θ defined?
A. All real numbers except odd multiples of
3
2
− 3
2
C. -1
D. undefined
C. 2
D. −2
π
2
B. All real numbers except integer multiples of π
C. All real numbers
D. All real numbers except odd multiples of π
7. For what number(s) x, does cos x =
D.
2
; 0 ≤ x ≤ 2π
2
8. If sin θ =
−7
3π
with
< θ < 2π, find tan θ.
2
9
9. Given the question: What is the measure of an angle in quadrant I that has a sine value of
1
?
2
Bill said that the answer was 30. Bob said that the answer was 30°. Technically who
is correct and why? Did Bill or Bob give the larger answer?
π

10. Graph one period of y =−1 + 3cos 4  x +  Label the x and y axis clearly then state the amplitude and
8

period.
Amp: ____________
Period: __________
Vertical Shift: _________
Phase Shift: _________
11. Graph one period of y = 2 cos ( x ) Label the x and y axis clearly then state the amplitude and period.
Amp: ____________
Period: __________
12. Find the period given y = -4sin (6 π x + 4)
π
1
A.
B.
3
3
π

13. Find the phase shift for y = 10 sin  2 x − 
2

π
A.
right
B. 5 π up
4
C. 6 π
D.
1
6
C. 2 π down
D.
π
left
2
14. Find the amplitude, period, and phase shift of y = - ½ cos (2x - 2 π )
A. 2, π , π
B. ½ , π , π
C. 2, 2 π , 2 π
D. ½, π ,
π
2
15. Write the equation of a sine function with amplitude of 3, period of π , phase shift
shift of 4
π

A. y =
4 + 3sin 2  x − 
3

π

C. y =
3 + 4sin  x − 
3

π

B. y =
4 + 3sin 2  x + 
3

π

D. y =
3 + 4sin 2  x + 
3

16. Which is the equation for the graph pictured?
π 
A. y = 2sin  x 
3 
1 
B. y = 2 cos  x 
3 
π 
C. y = 2 cos  x 
3 
1 
D. y = 2sin  x 
3 
π
to the left and vertical
3
17. Which is the equation for the graph pictured?
A. y = 3 tan ( 2 x )
B. y = 3cot ( 2 x )
C. y = 3 tan ( 4 x )
D. y = 3cot ( 4 x )
Find the exact value. NO CALCULATORS. Use Radians.
1
 −1 
=
=
y cos −1  
y cos −1  
18.
19.
2
 2 
21. y
=
y sin −1 (−1)
20.
 2
− 3
−1
=
sin
22. y sin −1  =
23. y cos −1 π



 2 
 2 
24. y =
tan −1 (1)
25. y =
tan −1 (−1)
−1
26. y =
sin −1 ( )
2
− 2
=
y sin −1 
27.

 2 
− 2
=
y cos −1 
28.

 2 
=
y cos −1 (−1)
29.
30. cos ( cos −1 ( −0.76 ) )
31. cos ( cos −1 2.6 )

3
32. cos −1  −

 2 
5π
A.
6
B.
2π
3
C.
π
D. −
6
3π
4

− 3
33. sec  sin −1

2 

A.
2
2
B. 0
 2 
34. csc −1  −

3

2π
A.
3
C. 2
B. −
π
3
D. 1
C.
5π
6
D. −
π
6
Verify the identity.
(
)
1
35. tan 2 θ 1 + cot 2 θ =
cos 2 θ
37.
1 − sin θ
cos θ
=
cos θ
1 + sin θ
36.
cos 2 θ
= csc θ − sin θ
sin θ
15
π
, 0 <θ <
17
2
161
A. −
289
161
C.
289
θ
=
38. sin
20
3π
,
< θ < 2π
29
2
840
A.
841
41
C. −
841
cos θ
39.=
4
− , tan θ > 0
40. csc θ =
3
=
θ
41. tan
7
3π
, π <θ <
24
2
42. 4sin 2 θ = 1
π 5π
,
A.
6 6
π 5π 7π 11π
,
C. , ,
6 6 6 6
Find Cos ( 2θ )
240
289
163
D. −
289
B.
Find Sin ( 2θ )
840
841
41
D.
841
B. −
Find Cos ( 2θ )
Find Cos ( 2θ )
B.
D.
π 2π
3
,
3
π 3π 4π 5π
, ,
,
3 3 3 3
1
43. 1 − sin θ =
2
π 5π
,
A.
6 6
π 2π
C. ,
3 3
44.
B.
π 4π
3
,
3
π 11π
D. ,
6 6
0
45. 2 cos θ + 1 =
3 cot θ − 1 =0
46. sin 2 θ + sin θ =
0
4π 5π
,
A. 0, π ,
3 3
3π
C. 0, π ,
2
47. sin 4 x = −
π
π 5π
,
π 2π
3
,
3
π 5π
D. 0, π , ,
3 3
3
2
A. 0, π ,
C.
B. 0, π ,
4 4
4
B. 0
D.
π π 2π 7π 7π 13π 5π 19π
, ,
,
,
,
, ,
12 6 3 12 6 12 3 12
48. 2 cos ( 2θ ) = 3
49. sin 2 2 x = 1
50. Solve the right triangle given A = 33.70 and b = 1.8 m
A. B = 56.30 a = 0.1 m c = 2.4 m
B. B = 56.30 a = 0.1 m c = 1.8 m
C. B = 56.30 a = 2.4 m c = 3.0 m
D. B = 56.30 a = 1.2 m c = 2.2 m
51. On a sunny day, a flag pole and its shadow form the sides of a right triangle. If the hypotenuse is 52 meters
long and the shadow is 48 meters, how tall is the flag pole?
A. 20 m
B. 32 m
C. 71 m
D. 100 m
52. When sitting atop a tree and looking down at his pal Joey, the angle of depression of Mack’s line of sight is
37.90. If Joey is known to be standing 23 feet from the base of the tree, how tall is the tree (to the nearest
foot)?
A. 18ft.
B. 20 ft.
C. 22 ft.
D. 24 ft.
53. Use the law of sines and/or the law of cosines to solve the triangle
A.
B.
C.
D.
B = 37.20
B = 37.20
B = 37.20
B = 36.80
b = 384.2 ft.
b = 310.7 ft.
b = 32 ft.
b = 307.8 ft.
c = 310.7 ft.
c = 384.2 ft.
c = 26 ft.
c = 384.2 ft.
54. Use the law of sines and/or the law of cosines to solve the triangle
A. B = 600
C = 900
b = 15.58
0
0
B. B = 90
C = 60
b = 15.58
0
0
C = 90
b=9
C. B = 60
D. No such triangle
55. Use the law of sines and/or the law of cosines to solve the triangle ABC given A = 790, a = 32 yd, b = 65 yd
A.
B.
C.
D.
B = 290
C = 720
c = 81 yd
0
0
C = 72
c = 75 yd
B = 29
0
0
B = 72
C = 29
c = 79 yd
No possible triangle exists
56. Use the law of sines and/or the law of cosines to solve the triangle ABC given
C = 112.50, a = 5.3 m, b = 9.66m
A.
B.
C.
D.
c = 12.7 m A = 22.70
c = 15.6m A = 24.70
c = 18.5 m A = 20.70
No such triangle
B = 44.80
B = 42,80
B = 46.80
57. Use the law of sines and/or the law of cosines to solve the triangle ABC given a = 27 ft, b = 32 ft, c = 41 ft
A. A = 41.140 B = 51.240 C = 87.620
B. A = 41.140 B = 25.620 C = 113.240
C. A = 51.240 B = 41.140 C = 87.620
D. No such triangle
58. An airplane is sighted at the same time by two ground observers who are 5 miles apart and both directly
west of the airplane. They report the angles of elevation as 140 and 210. How high is the airplane?
A. 1.21 mi
B. 1.79 mi
C. 3.56 mi.
D. 5.27 mi
59. Two ships leave a harbor together traveling on a courses that have an angle of 1250 between them. If they
each travel 501 miles, how far apart are they (to the nearest mile)?
A. 40 mi
B. 463 mi
C. 889 mi
D. 1778 mi
60. Jack and Jill are having a great debate on the following problem:
Determine the number of triangles ABC possible given b = 24, c = 29, and B = 460
Jack believes the answer is 1 triangle where Jill is certain that there are 2 triangles.
Who is correct and why? Provide mathematical evidence to support your answer.
61. Find the corresponding rectangular coordinates for the point.
3 2 3 2 
A. 
,−

2 
 2
3 2 3 2 
B. 
,

2 
 2
( −3, −135 )
 3 2 3 2
C.  −
,

2
2 

°
 3 2 3 2
D.  −
,−

2
2 

62. Find the corresponding rectangular coordinates for the point ( 400,130° )
A. (306.42, 257.12)
C. (306.42, -257.12)
B. (-257.12, -306.42)
D. (-257.12, 306.42)
 π
63. Plot the point P with polar coordinates  4,  and find other polar coordinates of this same point for which:
 6
(a) r > 0 , −2π ≤ θ < 0 and (b) r < 0 , 0 ≤ θ < 2π
(
64. Find the corresponding polar coordinates, in degrees, for the point −1, 3
A. ( 2,120° )
B. ( 2,150° )
C. ( −2,120° )
)
D. ( −2,150° )
65. Find the corresponding polar coordinates, in radians, for the point ( 2, −2 ) .
7π 

A.  2 2,

4 

π

B.  2 2, − 
4

3π 

C.  −2 2, 
4 

D. All of these
4
66. Change the rectangular equation to polar form. x 2 + 4 y 2 =
A. cos 2 θ + 4sin 2 θ =
B. r 2 ( 4 cos 2 θ + sin 2 θ ) =
4r
4
4r
C. 4 cos 2 θ + sin 2 θ =
D. r 2 ( cos 2 θ + 4sin 2 θ ) =
4
67. Change the polar equation to rectangular form. r = 4sin θ
A. x 2 + y 2 =
B. x 2 + y 2 =
C. x = 4
4x
4y
68. Match the graph to the correct polar equation.
A. r = 1
C. r = 2 cos θ
B. r = 2 sin θ
D. r sin θ = 1
69. Match the graph to the correct polar equation.
A. r = 8 cos θ
B. r = 8 sin θ
C. r = 4 + sin θ
D. r = 4 + cos θ
D. y = 4
70. Identify the polar equation. r= 6 + 6sin θ
A. Cardioid
B. Limacon with a loop
C. Limacon without a loop
D. Lemniscate
71. Find the magnitude v of the vector.
A. 25
B. 7
C.
v = 3i – 4j
5
D. 5
72. A vector has initial point P = (-3, 4) and terminal point Q = (7, -1). Find its position vector and write in the
form ai + bj
73. Given v = -i +4j and w = 3i + 2j , find the following:
i. v + w
ii. v • w
iii. the angle between the vectors in degrees
74. Given v =−2i − 5 j and w= 4i + 10 j
i. find the dot product v w
ii. find the angle between the vectors.
iii. determine if the vectors are parallel, orthogonal or neither.
75. Given v= 2i + 2 j and w =−i + j
i. find the dot product v w
ii. find the angle between the vectors.
iii. determine if the vectors are parallel, orthogonal or neither.
76. Given v= 4i − 3 j and w= 2i + 5 j
i. find the dot product v w
ii. find the angle between the vectors.
iii. determine if the vectors are parallel, orthogonal or neither.
For questions 77 – 79:
A football is thrown through the air. The parametric equations describing the motion of the ball as a
=
function of time
are x 80t cos 35° and y =−16t 2 + 80t sin(35°) + 7 .
77. What is the maximum height of the football?
A. 100.9 ft.
B. 1.45 ft.
C. 39.9 ft.
D. 200 ft.
78. How long was the football in the air?
A. 5 sec.
B. 3 sec.
C. 2 sec.
D. 6 sec.
79. How far did the football get thrown?
A. 100.9
B. 197 ft.
C. 300 ft.
D. 50 ft.
80. Jim and Cheryl are trying to figure out how many different ways the Swine Queen and one alternate will be
chosen from 9 contestants in the Miss Lake County Fair pageant. Jim says that they should use a
combination to count the number of ways. Cheryl disagrees. She says you should use a permutation. Do
you agree with Jim or Cheryl? Explain your reason.
81. If n(B) = 60, n(A ∩ B) = 11, and n(A ∪ B) = 105, find n(A).
A. 58
B. 56
C. 54
82. How many are in B but not in A?
A. 23
B. 26
C. 27
D. 20
D. 45
83. In a student survey, 118 students indicated that they speak Spanish, 31 students indicated that they speak
French, 9 students indicated that they speak both Spanish and French, and 123 students indicated that they
speak neither. How many students participated in the survey?
A. 272
B. 263
C. 140
D. 254
84. A restaurant offers a choice of 5 salads, 7 main courses, and 4 desserts. How many possible 3 course meals
are there?
A. 280
B. 35
C. 16
D. 140
85. How many different license plates can be made using 2 letters followed by 3 digits selected from the digits 0
through 9, if neither letters nor digits may be repeated?
A. 468,000
B. 327,600
C. 676,000
D. 39,000
86. Mary finds 10 fish at a pet store that she would like to buy, but she can afford only 4 of them. In how many
ways can she make her selection? How many ways can she make her selection if he decides that one of the
fish is a must?
A. 2520; 252
B. 151,200; 60,480 C. 5040; 504
D. 210; 84
87. A committee is to be formed consisting of 4 men and 2 women. If the committee members are to be chosen
from 8 men and 12 women, how many different committees are possible?
A. 136
B. 38,760
C. 4,620
D. 221,760
88. An environmental organization has 32 members. Each member will be placed on exactly 1 of 4 teams. Each
team will work on a different issue. The first team has 9 members, the second has 6, the third has 7, and the
fourth has 10. In how many ways can these teams be formed?
A. 5.506588817 × 1017
B. 5.506588817 × 1016
D. 2.631308369 × 1035
C. 5.519149563 × 1027
89. How many different 10-letter words (real or imaginary) can be formed from the letters in the word
MANAGEMENT?
A. 453,600
B. 22,680
C. 226,800
D. 3,628,800
90. Use the binomial theorem to expand ( x + 4 )
91. Use the binomial theorem to expand
5
( 2 x + 1)
92. Use the binomial theorem to expand ( 3 x − 2 )
4
3
93. Find the 6th term in the expansion of ( 2 x + 3)
8
A. 217, 728x 3
B. 26,127,360x3
C. 3,888x 3
D. 108,864x3
94. Find the probability of having exactly 3 girls in a 5 child family.
A. 0.6
B. 0.063
C. 0.094
D. 0.313
95. A bag of marbles contains 5 purple, 7 green and 4 blue marbles. If you choose 3 marbles from the bag
without replacing them, what is the probability of choosing 3 green marbles?
A. 0.063
B. 0.084
C. 0.438
D. 0.188
96. Explain why this would be considered a probability model.
97. A 6-sided die is rolled. What is the probability of rolling a number less than 3?
1
2
1
1
A.
B.
C.
D.
3
6
3
2
98. The table below shows the results of a consumer survey of annual incomes in 100 households. What is the
probability that a household has an annual income less than $25,000?
A. 0.27
C. 0.73
B. 0.56
D. 0.20
99. The psychology lab at a college is staffed by 5 male doctoral students, 10 female doctoral students, 16
male undergraduates, and 11 female undergraduates. If a person is selected at random from the group, find
the probability that the selected person is an undergraduate or a female.
A.
37
42
B.
13
21
C.
9
14
D.
1
2
100. During July in Jacksonville, Florida, it is not uncommon to have afternoon thunderstorms. On average,
10.1 days have afternoon thunderstorms. What is the probability that a randomly selected day in July will
not have a thunderstorm? Round to two decimal places, if necessary.
A. 0.9
B. 0.67
C. 0.33
D. 0.66
101. A die is weighted so that an even-numbered face is three times as likely to occur as an odd-numbered face.
What is the probability of rolling an even-numbered face?
2
1
3
3
A.
B.
C.
D.
3
2
4
2
Answers
1.
2.
3.
4.
5.
6.
B
C
A
A
D
B
35.
π 7π
,
4 4
−7 2
8.
8
9. Bob is correct. Bill’s answer is
in radians, not degrees. Bill’s
answer is larger.
π
π
10. amp: 3 per:
VS: -1 PS: 2
8
7.
36.
37.
11. amp: 2 per: 2 π
12.B
13.A
14.B
15.B
16.D
17.A
π
18.
3
20. −
22. −
24.
π
4
26. −
19.
π
2
π
3
21.
2π
3
π
4
23.DNE
25. −
π
6
3π
4
30.-0.76
31.DNE
32.A
33.C
34.B
28.
1
1
=
tan 2 θ cos 2 θ
1
tan 2 θ + 1 = 2
cos θ
1
sec 2 θ =
cos 2 θ
1
1
=
2
cos θ cos 2 θ
cos 2 θ
1
=
− sin θ
sin θ
sin θ
cos 2 θ
1
sin 2 θ
=
−
sin θ
sin θ sin θ
cos 2 θ 1 − sin 2 θ
=
sin θ
sin θ
cos 2 θ cos 2 θ
=
sin θ
sin θ
tan 2 θ + tan 2 θ
π
27. −
29. π
4
π
4
cos θ
 1 + sin θ  1 − sin θ
=


1 + sin θ
 1 + sin θ  cos θ
1 − sin 2 θ
cos θ
=
(1 + sin θ ) cos θ 1 + sin θ
cos 2 θ
cos θ
=
(1 + sin θ ) cos θ 1 + sin θ
cos θ
cos θ
=
1 + sin θ 1 + sin θ
38. A
39. B
1
40. −
8
527
41.
625
42. C
43. A
π 4π
44. ,
3 3
2π 4π
45.
,
3 3
46. C
47. D
π 11π 13π 23π
48.
,
,
,
12 12 12 12
π 3π 5π 7π
49. , , ,
4 4 4 4
50. D
51. A
52. A
53. B
54. A
55. D
56. A
57. A
58. C
59. C
60. 2 triangles – Jill is correct
61. B
62. D
7π 
 −11π  
63.  4,
  −4,

6 
6 

64. A
65. D
66. D
67. C
68. B
69. D
70. A
71. D
72. -5i+10j
73. 2i+6j, 5, 70.35°
74. -58, π , parallel
75. 0,
π
, orthogonal
2
76. -7, 1.83, neither
77. C
78. B
79. B
80. Permutation – Cheryl’s correct
81. B
82. A
83. B
84. D
85. A
86. D
87. C
88. B
89. C
90. x + 20 x + 160 x + 640 x + 1280 x + 1024
91. 16 x 4 + 32 x 3 + 24 x 2 + 8 x + 1
92. 27 x3 − 54 x 2 + 36 x − 8
93. D
94. D
95. A
96. Each of the probabilities is
between 0 and 1 and the sum
of probabilities is 1.
97. D
98. A
99. A
100. C
101. C
5
4
3
2