Name: ________________________ Class: ___________________ Date: __________
ID: A
Pre-Calculus Chapter 6 Practice Test
Multiple Choice
Identify the choice that best completes the statement or answers the question.
4. Find the trigonometric form of the complex
1. Given that Force 1 = 70 pounds and Force 2 =
number shown below.
120 pounds, find the angle between the forces
if the magnitude of the resultant force is 90
pounds. Round answer to the nearest degree.
a 48°
c 53°
e 60°
b 40°
d 45°
a
2. Use the dot product to find the magnitude of u
if u = 5i + 6j.
a Ä u Ä = 30
c Ä u Ä = 2 61
b Ä u Ä = 11
e Ä uÄ =
61
d Ä u Ä = 31
b
3. Find the angle between the vectors u and v.
c
d
a
b
c
d
e
e
5. Find the trigonometric form of the complex
number shown below.
−7i
ÊÁ
ˆ˜
ÁÁ cos π + isin π ˜˜˜
a 7Á
Á
4
4 ˜¯
Ë
c 7 (cos π + isin π )
ÁÊ
Á
Ë
ÁÁ cos
e 7Á
ÊÁ
11π
11π ˆ˜˜˜
9 ÁÁÁ cos
+ isin
˜
ÁË
6
6 ˜¯
4 3 4
2 2 3
a
−
i b −
+ i
9
9
9
9
c
9 9 3
−
i
2
2
d
1
9 3 9
− i
2
2
e −
9 3 9
+ i
4
4
ÊÁ
Á
Á
Ë
d 7Á
Á cos
3π
3π ˜ˆ˜˜
+ isin
˜
2
2 ˜¯
6. Find the standard form of the complex number shown below.
b 7 (cos 0 + isin0)
π
2
+ isin
π ˆ˜˜˜
˜
2 ˜¯
Name: ________________________
ID: A
7. Perform the operation shown below and leave
8. Divide the complex numbers below and leave
the result in trigonometric form.
ÈÍÍ
˘È
˘
ÍÎ 8 (cos 10° + i sin10° ) ˙˙˙˚ ÍÍÍÎ 5 (cos 200° + isin200°) ˙˙˙˚
the result in trigonometric form.
7 (cos 150° + i sin150° )
9 (cos 190° + i sin190° )
7
a
(cos 320° + isin340° )
9
7
(cos 320° + isin320° )
b
9
c 63 (cos 320° + isin320° )
7
(cos 340° + isin340° )
d
9
e 63 (cos 340° + isin340° )
a 13 (cos 210° + isin210° )
b 13 (cos 190° + isin190° )
c 40 (cos 20° + isin20° )
d 40 (cos 210° + isin210° )
e 40 (cos 190° + isin190° )
9. Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.
(−4 − 4i) (−8 − 8i)
ÊÁ
ˆ˜
ÊÁ
ˆ˜
ÁÁ cos π + isin π ˜˜˜ b −32 ÁÁÁ cos 5π + isin 5π ˜˜˜
a 64 Á
Á
Á
2
2 ˜¯
4
4 ˜¯
Ë
Ë
ÊÁ
ˆ
Ê
ˆ
Á cos 3π + isin 3π ˜˜˜ e 32 ÁÁÁ cos π + isin π ˜˜˜
d −64 Á
ÁÁ
˜˜
ÁÁ
˜
2
2 ¯
2
2 ˜¯
Ë
Ë
10. Use DeMoivre's Theorem to find the indicated
ÊÁ
Á
Ë
ÁÁ cos
c 32 Á
3π
3π ˆ˜˜˜
+ isin
˜
4
4 ˜¯
12. Use DeMoivre's Theorem to find the indicated
power of the complex number. Write the result
in standard form.
Ê
ˆ7
5 ÁÁÁ 3 + i ˜˜˜
Ë
¯
a 320 − 320 3 i
b 5 − 5 3i
c −320 3 − 320i
d 5 3 + 5i
e 320 3 + 320i
power of the following complex number.
˘6
ÍÈÍ ÊÁ
π
π ˆ˜ ˙˙˙
ÍÍ ÁÁ
ÍÍ −2 ÁÁ cos + i sin ˜˜˜˜ ˙˙˙
ÍÍ Ë
3
3 ¯ ˙˙˚
Î
a 32 + 32 3
c –64
b 32
d 64 + 64 3
e 64
13. Find the cube roots of the following complex
number. Write each of the roots in standard
form.
27
3 3 3 3 3 3
a 3
i, +
i c 3,−3
b −3, −
2
2
2
2
3 3 3
3 3 3
i,− −
i
d 3,− +
2
2
2
2
3 3 3 3 3 3
e 3, +
i, −
i
2
2
2
2
11. Use DeMoivre's Theorem to find the indicated
power of the complex number. Write the result
in standard form.
ÈÍÍ
˘2
ÍÎ 5 (cos 330° + isin330° ) ˙˙˙˚
25 3 25
25 25 3
a
−
i b
+
i
2
2
2
2
25 25 3
25 25 3
c
−
i d − +
i
2
2
2
2
25 3 25
e −
+
i
2
2
2
Name: ________________________
ID: A
Short Answer
17. A straight road makes an angle, A, of 20° with
14. Given A = 14°, b = 9, and a = 7, use the Law
the horizontal. When the angle of elevation, B,
of the sun is 54°, a vertical pole beside the road
casts a shadow 6 feet long parallel to the road.
Approximate the length of the pole. Round
answer to two decimal places.
of Sines to solve the triangle (if possible) for
the value of c. If two solutions exist, find both.
Round answer to two decimal places.
15. Determine the area of a triangle having the
following measurements. Round your answer
to two decimal places.
A = 126°, b = 9, and c = 10
16. Determine the area of a triangle having the
following measurements. Round your answer
to two decimal places.
C = 82°34' , a = 11.0, and b = 12.5
18. Given a = 6, b = 9, and c = 12, use the Law of
Cosines to solve the triangle for the value of C.
Round answer to two decimal places.
19. Given a = 10, b = 5, and c = 11, use the Law of
Cosines to solve the triangle for the value of A.
Round answer to two decimal places.
20. In the figure below, a = 10, b = 5, and θ = 51° . Use this information to solve the parallelogram for c. The
diagonals of the parallelogram are represented by c and d. Round answer to two decimal places.
3
Name: ________________________
ID: A
21. In the figure below, a = 6, c = 18, and d = 14. Use this information to solve the parallelogram for b. The
diagonals of the parallelogram are represented by c and d. Round answer to two decimal places.
26. Find the vector v that has a magnitude of 4 and
22. A vertical pole 32 feet tall stands on a hillside
that makes an angle of 17° with the horizontal.
Determine the approximate length of cable that
would be needed to reach from the top of the
pole to a point 56 feet downhill from the base
of the pole. Round answer to two decimal
places.
is in the same direction as u, where
u = −3, −6 .
27. Let w be a vector with initial point (1, 5) and
terminal point (−6, −5) . Write w as a linear
combination of the standard unit vectors i and
j.
23. Find the component form of vector v with
initial point (−6, 2) and terminal point (7, −3) .
28. Find the magnitude and direction angle of
v = 7 ÁÊË cos 140°i + sin140°j ˜ˆ¯ .
24. Find the magnitude of vector v with initial
point (3, −6) and terminal point (0, 6).
29. Find the magnitude and direction angle of
25. Using the figure below, sketch a graph of the
v = 5i − 5j. Round direction angle to nearest
degree.
given vector. [The graphs in the answer choices
are drawn to the same scale as the graph
below.]
30. Find the component form of v if ÄvÄ = 2 and the
angle it makes with the x-axis is 120°.
u+ v
4
ID: A
Pre-Calculus Chapter 6 Practice Test
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
A
E
D
C
E
D
D
B
A
C
C
E
D
SHORT ANSWER
14.
15.
16.
17.
18.
19.
20.
21.
22.
c = 2.08 and 15.39
36.41 sq. units
68.17 sq. units
5.71 feet
104.48°
65.28°
13.71
14.97
72.17 feet
23. v = 13, −5
24. ÄvÄ = 3
17
1
ID: A
25.
4
8
,−
5
5
27. w = −7i − 10j
28. ÄvÄ = 7; θ = 140°
26. v = −
29. ÄvÄ = 5
30.
−1,
2 ; θ = 315°
3
2