Tennessee State University
College of Engineering
Department of Mathematical Sciences
Pre Calculus II
Final Exam Review Package
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Final Exam Review Package
1.
If the angle 15◦ is in standard position, find
two positive coterminal angles and two negative
coterminal angles.
1
1
1
(a) sin θ = , cos θ = , tan θ
5
12
13
cot θ = 5, sec θ = 12, csc θ = 13
(a) 375◦ , 735◦ , −345◦ , −705◦
(b) 285◦ , 555◦ , −255◦ , −525◦
(d) 105◦ , 195◦ , −75◦ , −165◦
2.
12
, cos θ =
13
12
cot θ = , sec θ =
5
5
12
, tan θ =
13
5
5
12
, csc θ =
13
13
12
, cos θ =
13
5
cot θ = , sec θ =
12
5
12
, tan θ =
13
5
13
13
, csc θ =
5
2
(b) sin θ =
(c) 195◦ , 375◦ , −165◦ , −345◦
Find the values of the six trigonometric functions for the angle θ.
(c) sin θ =
(d) sin θ = 5, cos θ = 12, tan θ = 13
cot θ = 13, sec θ = 12, csc θ = 5
Page 1 of 11
Review Package · Pre-Calculus II
3.
Find the exact values of
x
and
y.
Department of Mathematical Sciences
6.
Find y by referring to the graph of the trigonometric function.
As x → π − , cot x → y
(a) 2
(b) −∞
(c) 1
7.
(a)
(b)
(c)
(d)
4.
(a) x = π, x = 4π
√
√
x = 6 3, y = 12 3
√
√
x = 12 3, y = 6 3
√
x = 12, y = 6 3
√
x = 12 3, y = 6
(b) x = π, x =
8.
Find the exact values of the trigonometric functions for the acute angle θ.
15
sin θ =
17
1
1
15
(a) sin θ = , cos θ = , tan θ =
17
17
8
cot θ = 15, sec θ = 17, csc θ = 8
17
1
15
(b) sin θ = , cos θ = , tan θ =
17
15
8
8
cot θ = 15, sec θ = 17, csc θ =
15
17
1
15
(c) sin θ = , cos θ = , tan θ =
17
8
15
17
cot θ = 15, sec θ = , csc θ = 8
17
15
8
15
(d) sin θ = , cos θ = , tan θ =
17
17
8
17
17
8
cot θ = , sec θ = , csc θ =
15
8
15
5.
Refer to the graph of y = cos x to find the exact
values of x in the interval [π, 4π] that satisfy the
equation.
cos x = 1
(a) III
(c) I
(b) IV
(d) II
π
2
(d) x = 2π, x = 4π
Refer to the graph of y = cos x to find the exact
values of x in the interval [0, 4π] that satisfy the
equation.
1
cos x = −
2
4π
10π
7π
π
,x=
,x=
,x=
3
3
3
2
2π
4π
10π
8π
(b) x =
,x=
,x=
,x=
3
3
3
3
2π
4π
11π
7π
,x=
,x=
,x=
(c) x =
3
3
3
2
2π
7π
10π
8π
(d) x =
,x=
,x=
,x=
3
3
3
3
(a) x =
9.
Find the reference angle θR if θ equal
π
3
π
(b) θR =
2
(a) θR =
Find the quadrant containing θ if the given conditions are true.
csc θ > 0 and sec θ < 0
π
2
(c) x = 2π, x =
10.
Page 2 of 11
2π
3
π
(d) θR =
6
(c) θR =
Find the period. y = sin 8x
π
8
(b) 8π
(c) 8
(a)
(d) 4π
(e)
π
4
2π
.
3
Review Package · Pre-Calculus II
11.
Sketch the graph of the equation. y = 4sin(x −
π
)
3
Department of Mathematical Sciences
Find the exact value. tan 45◦ + tan 210◦
13.
1
(a) 1 + √
3
1
1 + √3
(b)
2
14.
(a)
15.
(a) cos (−7)
(c) sin (3)
(b) cos (−3)
(d) sin (−7)
13
If α and β are acute angles such that csc α =
12
4
and cot β = , find:
3
tan (α + β)
(b)
16.
63
16
6
7
44
125
13
(b) −
44
(c)
7
16
63
(d) −
2
(c)
7
3
If tan α = −
and cot β =
for a second24
4
quadrant angle α and a third-quadrant angle β,
find:
cos (α − β)
(a)
17.
2
(d) 1 + √
3
Express as a trigonometric function of one angle. cos(2) sin(−5) − cos(5) sin(2)
(a) −
(b)
(c) 1
13
125
4
(d) −
13
(c)
Tell whether the following equation is an identity.
cos (u + v) · cos (u − v) = 2 cos u cos v
(a) The equation is an identity
(b) The equation is not an identity
18.
12.
π
π
+ cos
3
4
√
√
(c) 5 − 2
√
1+ 2
(d)
2
Find the exact value. cos
(a) 4
√
(b)
5
3
If the following expression is equated to one of
the expressions below, the resulting equation is an
identity.
x
x
6 cos sin
2
2
Select the correct expression.
(a) −3 cos x
(b) 6 sin x
Page 3 of 11
(c) 6 cos x
(d) 3 sin x
Review Package · Pre-Calculus II
19.
Find the solutions of the equation that are in
2π
the interval [0, ).
7
cos 7u + cos 14u = 0
π π 5π
,
,
7 21 21
π 4π 8π
(b) 0, ,
,
7 21 21
(a)
20.
π 2π 4π
,
,
7 21 21
2π 4π
(d) 0,
,
21 21
(c)
Department of Mathematical Sciences
25.
Find the exact value of the expression.
1
cos−1 (− )
2
2π
2π
(a) −
(c) −
5
3
2π
2π
(b)
(d)
5
3
26.
Find the exact value of the expression.
1
arctan √
3
π
π
(a)
(c)
4
3
π
(b)
6
27.
Find the exact value of the expression.
3
sin [arcsin (− )]
10
3
3
(c) −
(a) −
8
10
7
(b) −
8
28.
Find the exact value of the expression.
1
cos [cos−1 (− )]
8
1
1
(a) −
(c)
6
2
1
1
(b) −
(d)
8
10
29.
Find the exact value of the expression.
5π
cos−1 [cos ( )]
6
π
5π
(c) −
(a)
6
6
π
(b)
3
30.
Find the exact value of the expression.
2
sin [cos−1 (− )]
3
√
√
10
13
(c)
(a)
3
3
√
√
5
11
(b)
(d)
3
3
Express as a sum or difference.
sin 6t sin 3t
1
1
cos (4t) − cos (8t)
2
2
(b) cos (3t) − cos (9t)
1
1
(c) cos (3t) − cos (9t)
2
2
1
1
(d) sin (3t) − sin (9t)
2
2
(a)
21.
Express as a sum or difference.
2 sin 7θ cos 4θ
(a) cos (11θ) + cos (3θ)
1
1
(b) sin (11θ) + sin (3θ)
2
2
(c) cos (10θ) + cos (4θ)
(d) sin (11θ) + sin (3θ)
22.
Express as a sum or difference.
3 cos 4x sin 6x
3
3
sin (10x) + cos (2x)
2
2
3
3
(b) sin (2x) − sin (10x)
2
2
(c) cos (2x) + cos (10x)
1
1
(d) cos (3x) − cos (11x)
2
2
(a)
23.
24.
Express as a product.
sin 11θ + sin 3θ
(a) 2 cos (7θ) cos (4θ)
(c) 2 sin (7θ) cos (4θ)
(b) 2 sin (8θ) sin (3θ)
(d) 2 cos (8θ) sin (3θ)
31.
Express as a product.
sin 3t − sin 15t
(a) 2 cos (9t) cos (6t)
(b) 2 sin (10t) sin (5t)
(c) −2 sin (10t) cos (5t)
(d) −2 cos (9t) sin (6t)
Page 4 of 11
Complete the statement. As x → 1− , cos−1 x →
(a) −
(b) 0
π
2
(c) −
π
6
Review Package · Pre-Calculus II
32.
Find the solutions of the equation that are in
the given interval.
π π
6 sin3 θ + 18 sin2 θ − 5 sin θ − 15 = 0; (− , )
2 2
Department of Mathematical Sciences
(a) θ = 0.9503, θ = −1.3503
(b) θ = 1.0503, θ = −1.2503
(c) θ = 0.8503, θ = −1.4503
(d) θ = 1.1503, θ = −1.1503
33.
Find the solutions of the equation that are in
the given interval.
sin 2x = −1.5 cos x; [0, 2π)
37.
(a) x = 1.8708, x = 5.0124
x = 5.7351, x = 3.6897
(c) x = 1.7708, x = 4.9124
x = 5.2351, x = 3.7897
38.
34.
(a) β = 64◦ , b ≈ 11.4, c ≈ 12.4
39.
(c) β = 63◦ , b ≈ 11.4, c ≈ 12.5
(d) β = 63◦ , b ≈ 11.3, c ≈ 12.5
35.
(a) no triangle exists
40.
(c) α = 40◦ , β = 55◦ , a = 10
(d) α = 40◦ , β = 55◦ , a = 8
36.
A straight road makes an angle of 15◦ with the
horizontal. When the angle of elevation of the sun
is 57◦ , a vertical pole at the side of the road casts
a shadow d = 87 feet long directly down the road,
as shown in the figure. Approximate the length of
the pole.
(d) 106.79ft
(a) 29◦ , 104◦ , 46◦
(c) 35◦ , 101◦ , 44◦
(b) 29◦ , 101◦ , 44◦
(d) 35◦ , 104◦ , 46◦
(a) 26◦
(c) 25◦
(b) 23◦
(d) 29◦
Approximate the area of a parallelogram that
has sides of lengths a and b if one angle at a vertex
has measure θ
a = 9.0, b = 16.0, θ = 40◦
Solve △ABC.
y = 85◦ , c = 11, b = 12
(b) α = 38◦ , β = 57◦ , a = 10
(b) 106.89ft
A triangular plot of land has sides of lengths
420 feet, 310 feet, and 180 feet. Approximate the
smallest angle between the sides.
Solve △ABC.
α = 39◦ , y = 78◦ , a = 8.1
(b) β = 63◦ , b ≈ 11.5, c ≈ 12.6
(c) 107.19ft
Solve △ABC.
a = 10.0, b = 17.0, c = 12.0
(b) x = 1.5708, x = 4.7124
x = 5.4351, x = 3.9897
(d) x = 1.6708, x = 4.8124
x = 5.3351, x = 3.8897
(a) 106.69ft
41.
Page 5 of 11
(a) 89.3
(c) 92.6
(b) 95.2
(d) 96.8
Find 3a + 5b. a = (6, −2), b = (3, 3)
(a) 3a + 5b = (33, 9)
(c) 3a + 5b = (27, 15)
(b) 3a + 5b = (39, 12)
(d) 3a + 5b = (15, 15)
Find ||a||. a = (−5, −2)
(a) ||a|| =
(b) ||a|| =
√
√
26
29
(c) ||a|| =
(d) ||a|| =
√
√
30
27
Review Package · Pre-Calculus II
42.
Sketch the vector corresponding to a + b.
a = (−2, 4), b = (−4, 2)
Department of Mathematical Sciences
7π
5π
(a) θ =
(c) θ =
4
4
3π
(b) θ =
4
44.
Find the magnitude of the vector a.
−6i + 7j
√
√
(a) 86
(c) 87
√
√
(b) 88
(d) 85
a =
45.
Find a unit vector that has the same direction
as the vector a. a = (0, 3)
(a)
46.
(a) (1, 0)
(c) (0, 1)
(b) (1, 1)
(d) (0, −3)
Find a vector that has the opposite direction of
6i − 2j and two times the magnitude.
(a) −12i + 4j
(b) −10i + 2j
47.
(b)
(b) 54
49.
(c) −10
(d) 46
Find the dot product of the two vectors 4i and
2i + 10j.
(a) 8
(c) 18
(b) 24
(d) 0
Given that a = (5, −3), b = (1, 4), and c =
(−1, 3), find the number a · (b + c).
(c)
50.
(a) 21
(c) 31
(b) −11
(d) −21
Find the absolute value |9 − 12i|
(a) 15
(b) −15
51.
43.
(d) −13i + 5j
Find the dot product of the two vectors (−4, 1)
and (5, 10).
(a) 30
48.
(c) −15i + 7j
Find the smallest positive angle θ from the positive x-axis to the vector OP that corresponds to
a. a = (6, −6)
Page 6 of 11
(c) −12
(d) 9
Find the absolute value |9i|
(a) 9
(b) −i
(c) −9
(d) i
Review Package · Pre-Calculus II
52.
Express the complex number in trigonometric
form
√ with 0 ≤ θ ≤ 2π.
9 3 + 9i
π
3
7π
(b) 18cis
6
(a) 9cis
53.
(d) 18cis
π
6
π
6
Express the complex number in trigonometric
form with 0 ≤ θ ≤ 2π.
(a) 8cisπ
π
(b) 8cis
2
54.
(c) −18cis
59.
(c) 8cis0
√
7
113cis(tan−1 )
8
√
π
(b) 113cis
2
(d) −8cisπ
√
7
113cis(tan )
8
√
7
(d) 113cis
8
(c)
55.
Express in the form a + bi, where a and b are
real numbers.
π
π
14(cos + i sin )
4
4
√
√
√
√
(a) 7 2 − 7 2i
(c) 7 2 + 7 2i
√
√
√
(b) 7 3 + 7i
(d) 14 2 + 14 2i
56.
Express in the form a + bi, where a and b are
real numbers.
7(cos π + i sin π)
(a) 7i
(b) −7 + 0i
z1 = −5, z2 = −2
(b) 256i
(d) 256
(b) 4096
(c) −2048i
(d) 2048
60.
Use De Moivre’s theorem to change the given
complex number to the form a + bi, where a and b
are√real numbers.
√
2
2
+
i)15
(−
2
2
√
√
√
√
2
2i
2
2i
(c) −
−
−
(a)
2
2
2
2
√
√
√
√
2
2i
2
2i
(b)
(d) −
+
+
2
2
2
2
61.
Use De Moivre’s theorem to change the given
complex number to the form a + bi, where a and b
are
√ real numbers.
( 3 + i)11
√
(a) −1024 3 + 1024i
√
(b) −1024 3 − 1024i
(d) 7 + 0i
z1
.
z2
(c) 128i
(a) −4096i
(c) −7i
Use trigonometric forms to find z1 z2 and
(a) 128
Use De Moivre’s theorem to change the given
complex number to the form a + bi, where a and b
are real numbers.
(−1 + i)24
Express the complex number in trigonometric
form with 0 ≤ θ ≤ 2π. 8 + 7i
(a)
57.
Department of Mathematical Sciences
58.
Use De Moivre’s theorem to change the given
complex number to the form a + bi, where a and b
are real numbers.
62.
z1
5
= + 0i
z2
4
z1
5
(b) z1 z2 = 10 + 0i,
= + 0i
z2
2
z1
5
(c) z1 z2 = 10 + 0i,
= − + 0i
z2
2
z1 5
(d) z1 z2 = −10 + 0i,
+ + 0i
z2 2
(a) z1 z2 = 10 + 0i,
Find the eight eighth roots of five.
(a)
(b)
(c)
(d)
Page 7 of 11
√
(c) 1024 3 − 1024i
√
(d) 1024 3 + 1024i
√
2
2i
±
)
5(
2 √ 2
√
1
3i
8
5( ±
)
2
2
√
±85
√
√
3
i
8
5(
± )
2
2
√
8
√
(e) 1
√
(f) ± 8 5i
(g)
√
8
(h)
√
8
√
2
2i
±
)
5(−
2
2
7i
√
Review Package · Pre-Calculus II
63.
Find the solutions of the equation. x4 −625 = 0
(a) 10i, −10i
(b) 10, −10, 10i, −10i
64.
65.
69.
1.
(c) 5, −5, 5i, −5i
Department of Mathematical Sciences
x2 y 2
Find the one of the foci of the ellipse
+
=
49 9
(d) 5, −5
Find the focus of the parabola. 4y = x2
(a) F(2, 1)
(c) F(0, 1)
(b) F(0, 4)
(d) F(1, 1)
√
(a) ( 40, 0)
√
(b) ( 58, 0)
Find an equation for the parabola shown in the
figure.
70.
Sketch the graph of the ellipse. y 2 + 4x2 = 4.
(a)
(a) y 2 = 4(x − 1)
(b) x2 = 4(y − 1)
66.
(b) y 2 = 24(x − 2)
(d) y 2 = (x − 1)
(c) x2 = 24(y − 2)
(d) y 2 = (x − 2)
Find an equation of the parabola that satisfies
the condition. Vertex V(−7, 5), directrix y = 3
(a) (x − 7)2 = 8(y − 5)
(b) (x + 7)2 = (y − 5)
68.
(c) y 2 = 4(x + 1)
Find an equation of the parabola that satisfies
the condition. Focus F(8, 0), directrix x = −4
(a) y 2 = 24(x + 2)
67.
√
(c) (0, − 58)
√
(d) (0, 40)
(c) (y + 7)2 = 8(y − 5)
(d) (x + 7)2 = 8(y − 5)
Find the upper vertice of the ellipse
(a) (6, 0)
(c) (8, 0)
(b) (0, 2)
(d) (−6, 0)
x2 y 2
+ = 1.
36 4
Page 8 of 11
(b)
Review Package · Pre-Calculus II
Department of Mathematical Sciences
(y + 5)2
74.
Find the foci of the hyperbola
−
4
(x + 5)2
=1
7
√
√
(a) F(−2, −2 + 11), F’(−2, −2 − 11)
√
√
(b) F(−5, −5 + 17), F’(−5, −5 − 17)
√
√
(c) F(−5, −5 + 11), F’(−5, −5 − 11)
(c)
75.
71.
Find a vertex of the ellipse
(y − 4)2
= 1.
9
(a) (8, −4)
(b) (0, −4)
72.
Find an equation for the hyperbola shown in
the figure
(x − 4)2
+
16
(c) (0, 4)
(d) (8, 4)
Find an equation for the ellipse shown in the
figure.
x2
y2
−
=1
49 33
x2
y2
(b)
−
=1
16 33
(a)
76.
(c)
x2
y2
−
=1
16 49
Identify the graph of the equation as a parabola
(with vertical or horizontal axis), circle, ellipse, or
hyperbola
1
(x + 3) = y 2
3
(a) Parabola with horizontal axis
(b) Hyperbola
(c) Circle
x2 y 2
+
=1
8
6
x2 y 2
(b)
+
=1
6
8
(a)
73.
x2 y 2
+
=1
16
9
x2
y2
(d)
+
=1
9
16
Find the vertices of the hyperbola
(a) V(3, 0), V’(−3, 0)
(b) V(9, 0), V’(−9, 0)
77.
(c)
Identify the graph of the equation as a parabola
(with vertical or horizontal axis), circle, ellipse, or
hyperbola
−x2 = y 2 − 25
x2 y 2
−
=1
81 25
(c) V(5, 0), V’(−5, 0)
Page 9 of 11
(a) Ellipse
(b) Parabola with vertical axis
(c) Circle
Review Package · Pre-Calculus II
78.
Use the graph of f to find the simplest expression g(x) such that the equation f (x) = g(x) is an
identity. Verify the identity.
sin2 x − sin4 x
f (x) =
(1 − sec2 x) cos4 x
(a) g(x) = cos x
(d) g(x) = tan x
(b) g(x) = 1
(c) g(x) = sin x
(e) g(x) = −1
79.
Complete the identity selecting its right side
from five expressions given in the choices.
sin 4x + cos 4x cot 4x =
(a)
(b)
(c)
(d)
(e)
80.
Department of Mathematical Sciences
81.
Complete the identity selecting its right side
from five expressions given in the choices.
cot2 4α − cos2 4α =
(a) 2 csc2 4α
(b) csc2 4α + cot2 4α
(c) cot2 4α + cos2 4α
82.
(e) cos2 8α
Complete the identity selecting its right side
from five expressions given in the choices.
sec2 u − 1
=
sec2 u
(a) csc2 u
(b) sin2 u
(c) cos2 u
sec 4x csc 4x
sec 4x csc 4x + tan 4x
csc 4x
cot 4x cos 4x
sec 4x
(d) cot2 4α cos2 4α
Complete the identity selecting its right side
from five expressions given in the choices.
cot y − tan y
sin y cos y
(a) sin2 y − cos2 y
(d) csc2 y + sec2 y
(b) cos2 y − sin2 y
(c) sin2 y + cos2 y
(e) csc2 y − sec2 y
Page 10 of 11
(d) sec2 u
(e) cot2 u
Review Package · Pre-Calculus II
Department of Mathematical Sciences
Answer Key
Problem
Answer
Problem
Answer
Problem
Answer
Problem
Answer
Problem
Answer
1
A
18
D
35
A
52
D
69
A
2
C
19
A
36
B
53
A
70
B
3
B
20
C
37
?
54
A
71
D
4
D
21
D
38
B
55
C
72
D
5
D
22
B
39
C
56
B
73
B
6
B
23
C
40
A
57
B
74
C
7
D
24
D
41
B
58
D
75
B
8
B
25
D
42
A
59
B
76
A
9
A
26
B
43
A
60
C
77
C
10
E
27
C
44
D
61
C
78
E
11
B
28
B
45
C
62
A,C,F,G
79
C
12
D
29
A
46
A
63
C
80
3
13
A
30
B
47
C
64
C
81
D
14
D
31
B
48
A
65
A
82
B
15
A
32
D
49
D
66
B
16
A
33
B
50
A
67
D
17
B
34
B
51
A
68
B
Last Version: November 20, 2013
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