Ch. 10 Polar Coordinates; Vectors
10.1 Polar Coordinates
1 Plot Points Using Polar Coordinates
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Match the point in polar coordinates with either A, B, C, or D on the graph.
1) 3, π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
2) 3, – π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
Page 1
3) –3, – π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
4) –3, π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
5) –3, 4π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
Page 2
6) –3, 5π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
7) –3, – 5π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
8) 3, – 5π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
AB
C D
A) A B) B C) C D) D
Page 3
Plot the point given in polar coordinates.
9) 2, – π
4
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 4
10) 2, 7π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page 5
11) –4, – 5π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page 6
12) 2, – 3π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page 7
13) –4, 7π
4
-5 5
5
-5
-5 5
5
-5
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page 8
14) (2
,
30°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 9
15) (2
,
135°)
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page 10
16) (–2
,
405°)
A)
-5 5
5
-5
-5 5
5
-5
B)
-5 5
5
-5
-5 5
5
-5
C)
-5 5
5
-5
-5 5
5
-5
D)
-5 5
5
-5
-5 5
5
-5
Page 11
17) (–2, 45°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 12
18) (2, 45°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 13
19) 3, 7π
6
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 14
20) (2, 0°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 15
21) (2, 360°)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 16
22) –3, – π
4
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 17
23) 5, 5π
3
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 18
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
24) Plot the point (–2
,
4π) and find other polar coordinates (r, θ) of the point for which:
(a) r > 0, –2π ≤ θ < 0
(b) r < 0, 0 ≤ θ < 2π
(c) r > 0, 2π ≤ θ < 4π
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
25) Plot the point 1, π
2 and find other polar coordinates (r, θ) of the point for which:
(a) r > 0, –2π ≤ θ < 0
(b) r < 0, 0 ≤ θ < 2π
(c) r > 0, 2π ≤ θ < 4π
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 19
26) Plot the point 4, π
6 and find other polar coordinates (r, θ) of the point for which:
(a) r > 0, –2π ≤ θ < 0
(b) r < 0, 0 ≤ θ < 2π
(c) r > 02π ≤ θ < 4π
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
27) Plot the point 4, 5π
6 and find other polar coordinates (r, θ) of the point for which:
(a) r > 0, –2π ≤ θ < 0
(b) r < 0, 0 ≤ θ < 2π
(c) r > 02π ≤ θ < 4π
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
2 Convert from Polar Coordinates to Rectangular Coordinates
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
1) 7, 2π
3
A) – 7
2, 73
2B) 7
2, 73
2C) – 7
2, –73
2D) 7
2, –73
2
Page 20
2) –9, 2π
3
A) 9
2, –93
2B) – 9
2, –93
2C) 9
2, 93
2D) – 9
2, 93
2
3) –5, 3π
4
A) 52
2, –52
2B) –52
2, 52
2C) –52
2, –52
2D) 52
2, 52
2
4) 3, 3π
4
A) –32
2, 32
2B) 32
2, –32
2C) 32
2, 32
2D) –32
2, –32
2
5) 5, – 4π
3
A) – 5
2, 53
2B) 5
2, – 53
2C) 53
2, 5
2D) – 53
2, – 5
2
6) (–3
,
120°)
A) 3
2, –33
2B) – 3
2, –33
2C) 3
2, 33
2D) – 3
2, 33
2
7) (–3
,
–135°)
A) 32
2, 32
2B) 32
2, –32
2C) –32
2, –32
2D) –32
2, 32
2
8) (–1
,
–180°)
A) (1
,
0) B) (0
,
1) C) (–1
,
0) D) (0
,
–1)
9) (400, 130°) Round the rectangular coordinates to two decimal places.
A) (–257.12, 306.42) B) (–257.12, –306.42) C) (306.42, –257.12) D) (306.42, 257.12)
10) (4, 70°) Round the rectangular coordinates to two decimal places.
A) (1.37, 3.76) B) (3.76, 1.37) C) (1.59, 4.01) D) (4.01, 1.59)
11) (7.1
,
4.5) Round the rectangular coordinates to two decimal places.
A) (–1.50
,
–6.94) B) (7.08
,
0.56) C) (7.08
,
–0.56) D) (–1.50
,
6.94)
3 Convert from Rectangular Coordinates to Polar Coordinates
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The rectangular coordinates of a point are given. Find polar coordinates for the point.
1) (5
,
0)
A) (5
,
0) B) 5, π
2C) (–5
,
0) D) 5, 3π
2
Page 21
2) (0, 7)
A) 7, π
2B) 7, –π
2C) (7
,
0) D) (7
,
π)
3) (4
,
–4)
A) 4 2, –π
4B) 4 2, π
4C) –42, –3π
4D) –42,π
4
4) ( 3, –1)
A) 2, –π
6B) 2, π
6C) 2, –5π
6D) 2, 5π
6
5) (–3
,
0.2) Round the polar coordinates to two decimal places, with θin radians.
A) (3.01
,
3.08) B) (3.01
,
1.5) C) (3.01
,
–1.5) D) (–3.01
,
1.5)
6) (0.6, –1.1) Round the polar coordinates to two decimal places, with θin degrees.
A) (1.25, –61.39°) B) (1.25, 61.39°) C) (1.25, –57.93°) D) (1.25, 57.93°)
Solve the problem.
7) A woman walks 100 yards eastward along a straight shoreline and then swims 30 yards southward into
the ocean on a line that is perpendicular to the shoreline. Using her starting point as the pole and the east
direction as the polar axis, give her current position (i) in rectangular coordinates and (ii) in polar
coordinates. Round the coordinates to the nearest hundredth. Express θ in degrees.
A) (i) (100, –30);
(ii) (104.40, –16.70°)
B) (i) (–100, 30);
(ii) (104.40, –88.28°)
C) (i) (100, 30);
(ii) (11.40, –16.70°)
D) (i) (100, –30);
(ii) (11.40, –88.28°)
8) A fire truck is en route to an address that is 6 blocks east and 11 blocks south of the fire station. Using the
fire station as the pole and the east direction as the polar axis, express the fire truck’s destination (i) in
rectangular coordinates and (ii) in polar coordinates. Round the coordinates to the nearest hundredth.
Express θ in degrees.
A) (i) (6, –11);
(ii) (12.53, –61.39°)
B) (i) (6, –11);
(ii) (12.53, –28.61°)
C) (i) (6, 11);
(ii) (4.12, –61.39°)
D) (i) (6, 11);
(ii) (4.12, –28.61°)
4 Transform Equations between Polar and Rectangular Forms
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ).
1) x2 + 4y2 = 4
A) r2(cos2 θ + 4 sin2 θ) = 4B)r
2(4 cos2 θ + sin2 θ) = 4
C) cos2 θ + 4 sin2 θ = 4r D) 4 cos2 θ + sin2 θ = 4r
2) x2 + y2 – 4x = 0
A) r = 4 cos θB) r = 4 sin θC) r cos2 θ = 4 sin θD) r sin2 θ = 4 cos θ
3) x2 = 4y
A) r cos2 θ = 4 sin θB) r sin2 θ = 4 cos θC) 4 cos2 θ = r sin θD) 4 sin2 θ = r cos θ
Page 22
4) y2 = 16x
A) r sin2 θ = 16 cos θB) r2 sin2 θ = 16 cos θ
C) sin2 θ = 16r cos θD) sin2 θ = 16r2 cos θ
5) xy = 1
A) r2 sin 2θ = 2 B) 2r sin θcos θ=1 C) r sin 2θ=2D)2r
2 sin θ cos θ = 1
6) 2xy = 3
A) r2 sin 2θ = 3 B) 2r cos θsin θ=3C)r
2 sin 2θ = 6D)r
2 cos θ sin θ = 6
7) 2x + 3y = 6
A) r(2 cos θ + 3 sin θ) = 6 B) r(2 sin θ+3 cos θ) = 6
C) 2 cos θ + 3 sin θ =6r D) 2 sin θ+3 cos θ= 6r
8) x = –3
A) r cos θ = –3 B) r cos θ=3 C) r sin θ=3 D) r sin θ= –3
9) y = 5
A) r sin θ = 5 B) r cos θ=5C)r
=5 D) sin θcos θ=5
10) y = x
A) sin θ = cos θB) r = sin θC) r =cos θD) sin θ= – cos θ
The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y).
11) r = cos θ
A) x2 + y2 = xB)x
2 + y2 = y C) (x + y)2 = x D) (x + y)2 = y
12) r = 1 + 2 sin θ
A) x2 + y2 = x
2 + y2 + 2y B) x2 + y2 = x
2 + y2 + 2x
C) x2 + y2 = x2 + y2 + 2y D) x2 + y2 = x2 + y2 + 2x
13) r = 10 sin θ
A) x2 + y2 = 10y B) x2 + y2 = 10x C) x2 + y2 = 10y D) x2 + y2 = 10x
14) r = 2(sin θ – cos θ)
A) x2 + y2 = 2y – 2x B) 2x2 + 2y2 = y – xC)x
2 + y2 = 2x – 2y D) 2x2 + 2y2 = x – y
15) r = 5
A) x2 + y2 = 25 B) x2 – y2 = 25 C) x +y =25 D) x + y =5
16) r = 5
1 + cos θ
A) y2 = 25 – 10x B) y2 = 10x – 25 C) x2 = 25 – 10y D) x2 = 10y – 25
17) r sin θ = 10
A) y = 10 B) x = 10 C) y =10x D) x =10y
18) r(1 – 2 cos θ) = 1
A) x2 + y2 = 1 + 2x B) x2 + y2 = 1 + 2x C) x2 + y2 = 2 + xD)x
2 + y2 = 2 + x
Page 23
10.2 Polar Equations and Graphs
1 Identify and Graph Polar Equations by Converting to Rectangular Equations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
1) r = 6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2 + y2 = 36; circle, radius 6,
center at pole
B)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2 + y
– 3 2 = 9; circle,
radius 3, center at 0, 3 in
rectangular coordinates
C)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x – 3 2 + y2 = 9; circle,
radius 3, center at 3, 0 in
rectangular coordinates
D)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x = 6; vertical line 6 units to the right
of the pole
Page 24
2) r = 4 sin θ
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2 + (y – 2)2 = 4; circle, radius 2,
center at (0, 2) in rectangular coordinates
B)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x – 2)2 + y2 = 4; circle, radius 2,
center at (2, 0) in rectangular coordinates
C)
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2 + (y + 2)2 = 4; circle, radius 2,
center at (0, –2) in rectangular coordinates
D)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x + 2)2 + y2 = 4; circle, radius 2,
center at (–2, 0) in rectangular coordinates
Page 25
3) r = 4 cos θ
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x – 2)2 + y2 = 4; circle, radius 2,
center at (2, 0) in rectangular coordinates
B)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2 + (y – 2)2 = 4; circle, radius 2,
center at (0, 2) in rectangular coordinates
C)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2 + (y + 2)2 = 4; circle, radius 2,
center at (0, –2) in rectangular coordinates
D)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x + 2)2 + y2 = 4; circle, radius 2,
center at (–2, 0) in rectangular coordinates
Page 26
4) r sin θ = 4
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
y = 4; horizontal line 4 units
above the pole
B)
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x = 4; vertical line 4 units
to the right of the pole
C)
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x = –4; vertical line 4 units
to the left of the pole
D)
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
y = –4; horizontal line 4 units
below the pole
Page 27
5) θ = π
3
r
-5 5
5
-5
r
-5 5
5
-5
A)
r
-5 5
5
-5
r
-5 5
5
-5
y = 3x; line through the pole making
an angle of π
3 with the polar axis
B)
r
-5 5
5
-5
r
-5 5
5
-5
y = – π
3; horizontal line π
3 units
below the pole
C)
r
-5 5
5
-5
r
-5 5
5
-5
y = – 3
3x; line through the pole making
an angle of π
3 with the polar axis
D)
r
-5 5
5
-5
r
-5 5
5
-5
x – π
3
2 + y2 = π2
9; circle, radius π
3,
center at π
3, 0 in rectangular coordinates
Page 28
6) r sec θ = –6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
A)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
(x + 3)2 + y2 = 9; circle, radius 3
center (–3, 0) in rectangular coordinates
B)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x = –6; vertical line 6 units
to the left of the pole
C)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
y = –6; horizontal line 6 units
below the pole
D)
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
r
–6–5–4–3–2–1 123456
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x2 + (y + 3)2 = 9; circle, radius 3,
center at (0, –3) in rectangular coordinates
Page 29
Match the graph to one of the polar equations.
7)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r = 3B)r = 6 cos θC) r =6 sin θD) r sin θ=3
8)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A) θ = π
6B) θ = –π
6C) r = π
6D) r = –π
6
9)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r = –4 sin θB) r = –4 cos θC) r = –2 D) r sin θ= –2
Page 30
10)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r = 2 cos θB) r = 2 sin θC) r =1 D) r sin θ=1
11)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r sin θ = –1B)r
= –2 cos θC) r = –1D)r
= –2 sin θ
12)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r = 4 + sin θB) r = 8 cos θC) r =8 sin θD) r = 4 +cos θ
Page 31
13)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A) r = 4 + cos θB) r = 8 cos θC) r =8 sin θD) r = 4 +sin θ
Page 32
2 Graph Polar Equations Using a Graphing Utility
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use a graphing utility to graph the polar equation.
1) r = 5
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 33
2) r sin θ = 4
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 34
3) r cos θ = 5
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 35
4) r sin θ = –5
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 36
5) r cos θ = –2
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 37
6) r = 4 + sin θ
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9
9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 38
7) r = 3 + cos θ
6
–9 9
–6
A) 6
–9
9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 39
8) r = 2 – sin θ
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9
9
–6
D) 6
–9 9
–6
Page 40
9) r = 3 – cos θ
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9
9
–6
Page 41
10) r = 2 + 4 sin θ
8
–12 12
–8
A) 8
–12 12
–8
B) 8
–12 12
–8
C) 8
–12 12
–8
D) 8
–12 12
–8
Page 42
11) r = 2 + 2 cos θ
8
–12 12
–8
A) 8
–12 12
–8
B) 8
–12 12
–8
C) 8
–12 12
–8
D) 8
–12 12
–8
Page 43
12) r = 2 – 2 sin θ
8
–12 12
–8
A) 8
–12 12
–8
B) 8
–12 12
–8
C) 8
–12 12
–8
D) 8
–12 12
–8
Page 44
13) r = 2 – 2 cos θ
8
–12 12
–8
A) 8
–12 12
–8
B) 8
–12 12
–8
C) 8
–12 12
–8
D) 8
–12 12
–8
Page 45
14) r2 = 9 sin(2θ)
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 46
15) r2 = 25 cos(2θ)
6
–9 9
–6
A) 6
–9 9
–6
B) 6
–9 9
–6
C) 6
–9 9
–6
D) 6
–9 9
–6
Page 47
16) r = 4 sin(2θ)
4
–6 6
–4
A) 4
–6 6
–4
B) 4
–6 6
–4
C) 4
–6 6
–4
D) 4
–6 6
–4
Page 48
17) r = 4 cos(4θ)
4
–6 6
–4
A) 4
–6 6
–4
B) 4
–6 6
–4
C) 4
–6 6
–4
D) 4
–6 6
–4
3 Test Polar Equations for Symmetry
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Test the equation for symmetry with respect to the given axis, line, or pole.
1) r = –2 cos θ; the polar axis
A) Symmetric with respect to the polar axis
B) May or may not be symmetric with respect to the polar axis
Page 49
2) r = –4 cos θ; the line θ = π
2
A) May or may not be symmetric with respect to the line θ = π
2
B) Symmetric with respect to the line θ = π
2
3) r = 4 sin θ; the pole
A) May or may not be symmetric with respect to the pole
B) Symmetric with respect to the pole
4) r = 4 + 4 sin θ; polar axis
A) May or may not be symmetric with respect to the polar axis
B) Symmetric with respect to the polar axis
5) r = 2 + 2 cos θ; the line θ = π
2
A) May or may not be symmetric with respect to the line θ = π
2
B) Symmetric with respect to the line θ = π
2
6) r = 6 + 2 sin θ; the line θ = π
2
A) Symmetric with respect to the line θ = π
2
B) May or may not be symmetric with respect to the line θ = π
2
7) r = 6 + 2 cos θ; the pole
A) May or may not be symmetric with respect to the pole
B) Symmetric with respect to the pole
8) r = 2 – 4 sin θ; the polar axis
A) May or may not be symmetric with respect to the polar axis
B) Symmetric with respect to the polar axis
9) r2 = sin(2θ); the pole
A) Symmetric with respect to the pole
B) May or may not be symmetric with respect to the pole
10) r = 3 sin(3θ); the line θ = π
2
A) Symmetric with respect to the line θ = π
2
B) May or may not be symmetric with respect to the line θ = π
2
Page 50
4 Graph Polar Equations by Plotting Points
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Identify and graph the polar equation.
1) r = 1 – cos θ
r
-5 5
5
-5
r
-5 5
5
-5
A)
r
-5 5
5
-5
r
-5 5
5
-5
cardioid
B)
r
-5 5
5
-5
r
-5 5
5
-5
cardioid
C)
r
-5 5
5
-5
r
-5 5
5
-5
cardioid
D)
r
-5 5
5
-5
r
-5 5
5
-5
cardioid
Page 51
2) r = 4 – 3 cos θ
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
A)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limacon without inner loop
B)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limacon without inner loop
C)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limacon with inner loop
D)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limacon with inner loop
Page 52
3) r = 3 + 4 cos θ
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
A)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limacon with inner loop
B)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limacon with inner loop
C)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limacon without inner loop
D)
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
r
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
limacon without inner loop
Page 53
4) r = 4 sin(2θ)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
rose with four petals
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
rose with two petals
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
circle
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
lemniscate
Page 54
5) r2 = 5 sin(2θ)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
A)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
lemniscate
B)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
lemniscate
C)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
rose with four petals
D)
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
r
-6 -4 -2 2 4 6
6
4
2
-2
-4
-6
rose with four petals
Page 55
6) r = 3θ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
logarithmic spiral
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
logarithmic spiral
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
logarithmic spiral
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
logarithmic spiral
Page 56
The polar equation of the graph is either r = a +b cos θ or r =a +b sin θ, a >0, b >0. Match the graph to one of the
equations.
7)
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
A) r = 3 + 4 cos θB) r = 4 +3 cos θC) r =3+4 sin θD) r = 4 +3 sin θ
8)
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
A) r = 3 + 2 cos θB) r = 2 +3 cos θC) r =2+3 sin θD) r = 3 +2 sin θ
9)
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
A) r = 4 + 3 sin θB) r = 3 +4 cos θC) r =3+4 sin θD) r = 4 +3 cos θ
Page 57
10)
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
r
–8–7–6–5–4–3–2–1 12345678
8
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7
-8
A) r = 3 + 4 sin θB) r = 3 +4 cos θC) r =4+3 sin θD) r = 4 +3 cos θ
Page 58
Graph the polar equation.
11) r = 1 – cos θ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 59
12) r = 2
1 – cos θ
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 60
13) r = 5
1 – sin θ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 61
14) r = 2
1 – 2 sin θ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 62
15) r = 4
4 – 2 cos θ
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 63
16) r = 5
5 – 4 sin θ
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 64
17) r = θ
,
θ ≥ 0
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 65
18) r = 4
θ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 66
19) r = csc θ – 2
,
0
<
θ
<
π
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 67
20) r = sin θ tan θ
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
Page 68
21) r = tan θ, – π
2 < θ < π
2
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
A)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
B)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
C)
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
r
–5–4–3–2–1 12345
5
4
3
2
1
-1
-2
-3
-4
-5
D)
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
r
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
Page 69
10.3 The Complex Plane; De Moivre’s Theorem
1 Plot Points in the Complex Plane
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Plot the complex number in the complex plane.
1) –1 + 4i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page 70
2) 6i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page 71
3) –6 + i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page 72
4) –2 – i
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
A)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
B)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
C)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
D)
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
R
-6 -4 -2 2 4 6
i
6
4
2
-2
-4
-6
Page 73
5) –9 + 7i
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
A)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
B)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
C)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
D)
R
-10 -5 5
i
10
5
-5
-10
R
-10 -5 5
i
10
5
-5
-10
2 Convert a Complex Number from Rectangular form to Polar Form
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if
necessary.
1) 1 – 3
i
A) 2(cos 300° + i sin 300°) B) 2(cos 330° +i sin 330°)
C) 4(cos 300° + i sin 300°) D) 4(cos 330° +i sin 330°)
2) 3 – 3i
A) 3 2(cos 315° + i sin 315°) B) 9(cos 315° +i sin 315°)
C) 3 2(cos 300° + i sin 300°) D) 9(cos 300° +i sin 300°)
Page 74
3) –3
A) 3(cos 180° + i sin 180°) B) 3(cos 0° +i sin 0°)
C) 3(cos 270° + i sin 270°) D) 3(cos 90° +i sin 90°)
4) –6i
A) 6(cos 270° + i sin 270°) B) 6(cos 90° +i sin 90°)
C) 6(cos 180° + i sin 180°) D) 6(cos 0° +i sin 0°)
5) 9 – 12i
A) 15(cos 306.9° + i sin 306.9°) B) 15(cos 126.9° +i sin 126.9°)
C) 15(cos 53.1° + i sin 53.1°) D) 15(cos 233.1° +i sin 233.1°)
6) 9 3 + 9
A) 18(cos 30° + i sin 30°) B) 18(cos 330° +i sin 330°)
C) 9(cos 30° + i sin 30°) D) 9(cos 330° +i sin 330°)
Write the complex number in rectangular form.
7) 8 cos π
6 + i sin π
6
A) 4 3 + 4i B) 4 + 43iC)
3
4 + 1
4iD)
1
4 + 3
4i
8) 3 cos π
3 + i sin π
3
A) 3
2 + 33
2iB)
3
2 + 33
2iC)3 + iD)
3
6 + 3
6i
9) 4 cos 11π
6 + i sin 11π
6
A) 2 3 – 2i B) –23 – 2i C) 2 + 23iD)2
– 23i
10) 4(cos 300° + i sin 300°)
A) 2 – 23
iB)
–2 + 23iC)23 – 2i D) –23 – 2i
11) 6(cos 330° + i sin 330°)
A) 3 3 – 3i B) 3 3 + 3i C) –33 + 3i D) –33 – 3i
12) 9(cos 180° + i sin 180°)
A) –9B)9 C)
–9i D) 9i
13) 0.6(cos 300° + i sin 300°)
A) 0.300 – 0.520i B) 1.100 –0.266i C) 0.520 +0.300i D) 1.466 +1.100i
Page 75
3 Find Products and Quotients of Complex Numbers in Polar Form
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find zw or z
w as specified. Leave your answer in polar form.
1) z = 10(cos 30° + i sin 30°)
w = 5(cos 10° + i sin 10°)
Find zw.
A) 50(cos 40° + i sin 40°) B) 50(cos 300° +i sin 300°)
C) 15(cos 300° + i sin 300°) D) 15(cos 40° +i sin 40°)
2) z = 10(cos 45° + i sin 45°)
w = 5(cos 15° + i sin 15°)
Find zw.
A) 50(cos 60° + i sin 60°) B) 5(cos 30° +i sin 30°)
C) 5(cos 60° + i sin 60°) D) 50(cos 30° +i sin 30°)
3) z = 5(cos 35° + i sin 35°)
w = 2(cos 40° + i sin 40°)
Find zw.
A) 10(cos 75° + i sin 75°) B) 7(cos 75° +i sin 75°)
C) 10(cos 50.9° + i sin 50.9°) D) 7(cos 50.9° +i sin 50.9°)
4) z = 8 cos π
6 + i sin π
6
w = 3 cos π
2 + i sin π
2
Find zw.
A) 24 cos 2π
3 + i sin 2π
3B) 12 cos 2π
3 + i sin 2π
3
C) 24 cos π
3 + i sin π
3D) 12 cos π
3 + i sin π
3
5) z = 6 cos 3π
2 + i sin 3π
2
w = 12 cos 5π
6 + i sin 5π
6
Find zw.
A) 72 cos π
3 + i sin π
3B) 36 cos π
3 + i sin π
3C) 72 cos π
6 + i sin π
6D) 36 cos π
6 + i sin π
6
6) z = 2 + 2i
w = 3
– i
Find zw.
A) 4 2 cos π
12 + i sin π
12 B) 4 2 cos 23π
12 + i sin 23π
12
C) 4 cos π
12 + i sin π
12 D) 4 cos 23π
12 + i sin 23π
12
Page 76
7) z = 10(cos 30° + i sin 30°)
w = 5(cos 10° + i sin 10°)
Find z
w.
A) 2(cos 20° + i sin 20°) B) 5(cos 3° +i sin 3°)
C) 5(cos 20° + i sin 20°) D) 2(cos 3° +i sin 3°)
8) z = 10(cos 45° + i sin 45°)
w = 5(cos 15° + i sin 15°)
Find z
w.
A) 2(cos 30° + i sin 30°) B) 2(cos 45° +i sin 45°)
C) 1
2(cos 30° + i sin 30°) D) 1
2(cos 45° + i sin 45°)
9) z = 5(cos 200° + i sin 200°)
w = 4(cos 50° + i sin 50°)
Find z
w.
A) 5
4(cos 150° + i sin 150°) B) 5
4(cos 40° + i sin 40°)
C) 4
5(cos 150° + i sin 150°) D) 4
5(cos 40° + i sin 40°)
10) z = 8 cos π
2 + i sin π
2
w = 3 cos π
6 + i sin π
6
Find z
w.
A) 8
3cos π
3 + i sin π
3B) 8
3cos π
12 + i sin π
12
C) 5 cos π
3 + i sin π
3D) 5 cos π
12 + i sin π
12
11) z = 3cos 7π
4 + i sin 7π
4
w = 6cos 9π
4 + i sin 9π
4
Find z
w.
A) 2
2cos 3π
2 + i sin 3π
2B) 2
2cos π
2 + i sin π
2
C) 3 2 cos 3π
2 + i sin 3π
2D) 3 2 cos π
2 + i sin π
2
Page 77
12) z = 6 cos 3π
2 + i sin 3π
2
w = 12 cos 5π
6 + i sin 5π
6
Find z
w.
A) 1
2cos 2π
3 + i sin 2π
3B) 1
2cos π
3 + i sin π
3
C) 1
6cos 2π
3 + i sin 2π
3D) 1
6cos π
3 + i sin π
3
13) z = 1 + i
w = 3
– i
Find zw.
A) 2 2(cos 15° + i sin 15°) B) 2
2(cos 15° + i sin 15°)
C) 2 2(cos 75° + i sin 75°) D) 2 2(cos 345° + i sin 345°)
14) z = 1 + i
w = 1 – 3
i
Find z
w.
A) 2
2(cos 105° + i sin 105°) B) 1
2(cos 15° + i sin 15°)
C) 1
2(cos 105° + i sin 105°) D) 2
2(cos 15° + i sin 15°)
15) z = 1 – i
w = 1 – 3
i
Find z
w.
A) 2
2(cos 15° + i sin 15°) B) 1
2(cos 75° + i sin 75°)
C) 1
2(cos 15° + i sin 15°) D) 2
2(cos 75° + i sin 75°)
4 Use De Moivre’s Theorem
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the expression in the standard form a + bi.
1) 2(cos 15° + i sin 15°) 3
A) 4 2 + 42iB)32 + 32iC)4
+4i D) 3 + 3i
2) 2 cos 3π
4 + i sin 3π
4
4
A) –4B)4 C)
–4i D) 4i
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3) 3 cos 5π
6 + i sin 5π
6
4
A) – 9
2 – 93
2iB)
– 93
2 – 9
2iC)
– 9
2 + 93
2iD)
– 93
2 + 9
2i
4) (1 + i)20
A) –1024 B) 1024i C) –1024i D) 1024
5) (1 – i)10
A) –32i B) 32 C) 32 –32i D) –32 +32i
6) (–3 + i)6
A) –64 B) 64i C) –64 3 + 64i D) 64 – 64 3i
7) ( 3 + i)5
A) –16 3 + 16i B) 9 3 + 5i C) 16 – 16 3iD)163 – 16i
8) – 1
2 – 3
2i10
A) – 1
2 – 3
2iB)
– 1
2 + 3
2iC)
1
2 + 3
2iD)
1
2 – 3
2i
9) 2(cos 75° + i sin 75°) 3
A) –42
– 42iB)42 – 42iC)
–4 – 42iD)42 + 42i
10) 2(cos 105° + i sin 105°) 3
A) 4 2 – 42iB)
–42 – 42iC)4
– 42iD)
–42 + 42i
5 Find Complex Roots
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find all the complex roots. Leave your answers in polar form with the argument in degrees.
1) The complex cube roots of –8i
A) 2(cos 90° + i sin 90°), 2(cos 210° +i sin 210°), 2(cos 330° +i sin 330°)
B) 8(cos 90° + i sin 90°), 8(cos 210° +i sin 210°), 8(cos 330° +i sin 330°)
C) 512(cos 90° + i sin 90°), 512(cos 210° +i sin 210°), 512(cos 330° +i sin 330°)
D) 2(cos 180° + i sin 180°), 2(cos 300° +i sin 300°), 2(cos 60° +i sin 60°)
2) The complex fourth roots of –16
A) 2(cos 45° + i sin 45°), 2(cos 135° +i sin 135°), 2(cos 225° +i sin 225°), 16(cos 315° + i sin 315°)
B) 42(cos 45° + i sin 45°), 42(cos 135° + i sin 135°), 42(cos 225° + i sin 225°), 42(cos 315° + i sin 315°)
C) 16(cos 45° + i sin 45°), 16(cos 135° +i sin 135°), 16(cos 225° +i sin 225°), 16(cos 315° + i sin 315°)
D) 2(cos 90° + i sin 90°), 2(cos 180° +i sin 180°), 2(cos 270° +i sin 270°), 2(cos 360° + i sin 360°)
Page 79
3) The complex fifth roots of 3 + i
A) 52(cos 6° + i sin 6°), 52(cos 78° + i sin 78°), 52(cos 150° + i sin 150°), 52(cos 222° + i sin 222°),
52(cos 294° + i sin 294°)
B) 32(cos 6° + i sin 6°), 32(cos 78° +i sin 78°), 32(cos 150° +i sin 150°), 32(cos 222° + i sin 222°),
32(cos 294° + i sin 294°)
C) 52(cos 30° + i sin 30°), 52(cos 102° + i sin 102°), 52(cos 174° + i sin 174°), 52(cos 246° + i sin 246°),
52(cos 318° + i sin 318°)
D) 32(cos 30° + i sin 30°), 32(cos 102° +i sin 102°), 32(cos 174° +i sin 174°), 32(cos 246° + i sin 246°),
32(cos 318° + i sin 318°)
4) The complex fifth roots of –2i
A) 52(cos 54° + i sin 54°), 52(cos 126° + i sin 126°), 52(cos 198° + i sin 198°), 52(cos 270° + i sin 270°),
52(cos 342° + i sin 342°)
B) 52(cos 45° + i sin 45°), 52(cos 117° + i sin 117°), 52(cos 189° + i sin 189°), 52(cos 261° + i sin 261°),
52(cos 333° + i sin 333°)
C) 42(cos 54° + i sin 54°), 42(cos 126° + i sin 126°), 42(cos 198° + i sin 198°), 42(cos 270° + i sin 270°),
42(cos 342° + i sin 342°)
D) 32(cos 54° + i sin 54°), 32(cos 126° +i sin 126°), 32(cos 198° +i sin 198°), 32(cos 270° + i sin 270°),
32(cos 342° + i sin 342°)
Page 80
10.4 Vectors
1 Graph Vectors
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the vectors in the figure below to graph the following vector.
1) u + z
A) B)
C) D)
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2) 3w
A) B)
C) D)
Page 82
3) v – w
A) B)
C) D)
Page 83
4) z – v
A) B)
C) D)
Page 84
5) 2u – z – w
A) B)
C) D)
Use the figure below. Determine whether the given statement is true or false.
6) A + H = F
A) True B) False
7) G + H = F
A) True B) False
8) H + I + J = B
A) True B) False
Page 85
9) A + B + C + D + E = 0
A) True B) False
10) C + D + G + I + J = 0
A) True B) False
2 Find a Position Vector
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The vector v has initial position P and terminal point Q. Write v in the form ai +bj; that is, find its position vector.
1) P = (0, 0); Q = (–2
,
–3)
A) v = –2i – 3jB) v = –3i–3jC) v=2i+3jD) v =3i+2j
2) P = (1
,
2); Q = (–6
,
–1)
A) v = –7i – 3jB) v = –3i–7jC) v=7i+3jD) v =3i+7j
3) P = (–6
,
–5); Q = (6
,
–2)
A) v = 12i + 3jB) v = 3i+12jC) v=11i+4jD) v =4i+11j
3 Add and Subtract Vectors Algebraically
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) If u = 12i – 2j and v = –7i + 9j
,
find u +v.
A) 5i + 7jB) 4i + 7jC) 19i+4jD) –19i+7j
2) If u = –7i – 2j and v = 5i + 7j
,
find u –v.
A) –12i – 9jB) –2i +5jC) –13i+5jD) –14i+5j
4 Find a Scalar Multiple and the Magnitude of a Vector
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) If w = 4i + 4j
,
find 2w.
A) 8i + 8jB) 8i + 4jC) 6i+6jD) 6i +4j
2) If v = 3i – 5j and w = –7i + 4j, find 3v –4w.
A) 37i – 31jB) –4i –jC) –19i +jD) 17i–10j
3) If v = 3i + 4j
,
find v.
A) 5 B) 25 C) 7 D) 5
4) If v = –3i + 2j
,
find v.
A) 13 B) 5 C) 5 D) 13
5) If v = –i – j
,
find v.
A) 2 B) 2 C) 0 D) 1
6) If v = 5i + 7j
,
what is 6v?
A) 6 74 B) 12i 6 C) 12 6 D) –674
Page 86
7) If v = –10i + j and w = –4i + j
,
find v +w.
A) 10 2 B) 6 C) 118 D) 205
Find the quantity if v = 5i – 7j and w = 3i + 2j.
8) v + w
A) 74 + 13 B) 87 C) 2 6 + 13 D) 39
9) v – w
A) 74 – 13 B) 85 C) 2 6 – 13 D) 11
10) v + w
A) 89 B) 74 + 13 C) 39 D) 2 6 + 13
11) v – w
A) 85 B) 74 – 13 C) 77 D) 2 6 – 13
5 Find a Unit Vector
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the unit vector having the same direction as v.
1) v = 4i
A) u = iB) u = 4iC) u=16iD) u = 1
4i
2) v = –7j
A) u = –jB) u = –7jC) u=49jD) u = – 1
7j
3) v = –3i – 4j
A) u = – 3
5i – 4
5jB) u = –15i–20jC) u = 4
5i + 3
5jD) u = – 5
3i – 5
4j
4) v = 12i – 5j
A) u = 12
13i – 5
13jB) u = 156i–65jC) u = 5
13i – 12
13jD) u = 13
12i – 13
5j
5) v = –3i + j
A) u = – 310
10 i + 10
10 jB) u = –310
i + 10j
C) u = – 310
10 i – 10
10 jD) u = – 10
3i + 10j
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6 Find a Vector from Its Direction and Magnitude
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Find a vector v whose magnitude is 13 and whose component in the idirection is three times the
component in the j direction.
A) v = 39
10 10 i + 13
10 10 jor v = – 39
10 10 i – 13
10 10 j
B) v = 13
10 10 i + 39
10 10 jor v = – 13
10 10 i – 39
10 10 j
C) v = – 39
10 10 i + 13
10 10 jor v = 39
10 10 i – 13
10 10 j
D) v = 13
10 10 i – 39
10 10 jor v = – 13
10 10 i + 39
10 10 j
2) If P = (–2, 5) and Q = (x, –27), find all numbers x such that the vector represented by PQ has length –40.
A) {22
,
–26} B) {26
,
–26} C) {26
,
22} D) {22
,
–28}
Write the vector v in the form ai + bj, given its magnitude v and the angle
α
it makes with the positive x–axis.
3) v = 3
,
α = 60°
A) v = 3
2i + 33
2jB) v = 33
2i + 3
2jC) v = 32
2i + 32
2jD) v = – 3
2i – 33
2j
4) v = 13
,
α = 225°
A) v = – 13 2
2i – 13 2
2jB) v = – 13 3
2i – 13
2j
C) v = – 13
2i – 13 3
2jD) v = 13 2
2i + 13 2
2j
5) v = 11
,
α = 210°
A) v = – 11 3
2i – 11
2jB) v = – 11
2i – 11 3
2j
C) v = – 11 2
2i – 11 2
2jD) v = 11 3
2i – 11
2j
6) v = 12
,
α = 90°
A) v = 12jB) v = 12iC) v=12i–12jD) v =12i+12j
7) v = 15
,
α = 180°
A) v = –15iB) v = –15jC) v=15jD) v = –15i–15j
Find the direction angle of the vector v. Round to the nearest tenth if necessary.
8) v = 8i – 8j
A) 315° B) 135° C) 330° D) 225°
9) v = –8i + 8j
A) 135° B) 315° C) 150° D) 120°
10) v = –i + 3
j
A) 120° B) 150° C) 135° D) 300°
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11) v = 43
i – 4j
A) 330° B) 300° C) 315° D) 150°
12) v = 3i – 6j
A) 296.6° B) 243.4° C) 333.4° D) 206.6°
13) v = –i – 6j
A) 260.5° B) 99.5° C) 170.5° D) 189.5°
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
14) A truck pushes a load of 45 tons up a hill with an inclination of 35°. Express the force vector Fin terms of i
and j. Round the components of F to two decimal places.
15) Two forces, F1 of magnitude 35 newtons (N) and F2of magnitude 55 newtons, act on an object at angles of
45° and –60° (respectively) with the positive x–axis. Find the direction and magnitude of the resultant
force; that is, find F1 + F2. Round the direction and magnitude to two decimal places.
16) Two forces, F1 of magnitude 60 newtons (N) and F2of magnitude 70 newtons, act on an object at angles of
40° and 130° (respectively) with the positive x–axis. Find the direction and magnitude of the resultant
force; that is, find F1 + F2. Round the direction and magnitude to two decimal places.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
17) Two forces of magnitude 25 pounds and 40 pounds act on an object. The force of 40 lb acts along the
positive x–axis, and the force of 25 lb acts at an angle of 80° with the positive x–axis. Find the direction and
magnitude of the resultant force. Round the direction and magnitude to the nearest whole number.
A) Direction: 29°; magnitude: 51 lb B) Direction: 40°; magnitude: 47 lb
C) Direction: 51°; magnitude: 51 lb D) Direction: 4°; magnitude: 65 lb
7 Model with Vectors
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) An audio speaker that weighs 50 pounds hangs from the ceiling of a restaurant from two cables as shown
in the figure. To two decimal places, what is the tension in the two cables?
A) Tension in right cable: 35.90 lb; tension in left cable: 41.59 lb
B) Tension in right cable: 35.90 lb; tension in left cable: 14.10 lb
C) Tension in right cable: 41.59 lb; tension in left cable: 35.90 lb
D) Tension in right cable: 14.10 lb; tension in left cable: 41.59 lb
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2) A box of supplies that weighs 1500 kilograms is suspended by two cables as shown in the figure. To two
decimal places, what is the tension in the two cables?
A) Tension in right cable: 1098.08 kg; tension in left cable: 776.46 kg
B) Tension in right cable: 723.54 kg; tension in left cable: 776.46 kg
C) Tension in right cable: 776.46 kg; tension in left cable: 1098.08 kg
D) Tension in right cable: 776.46 kg; tension in left cable: 723.54 kg
3) A tightrope walker located at a certain point deflects the rope as indicated in the figure. If the weight of the
tightrope walker is 100 pounds, how much tension is in each part of the rope? Round your answers to the
nearest tenth.
4.8° 3.2°
100 pounds
A) tension in the left part: –140.9 lb;
tension in the right part: –105.2 lb
B) tension in the left part: –318.7 lb;
tension in the right part: 296.8 lb
C) tension in the left part: –160.6 lb;
tension in the right part: –120.0 lb
D) tension in the left part: –236.7 lb;
tension in the right part: 220.4 lb
Page 90
4) At a state fair truck pull, two pickup trucks are attached to the back end of a monster truck as illustrated in
the figure. One of the pickups pulls with a force of 1400 pounds and the other pulls with a force of 3400
pounds with an angle of 45° between them. With how much force must the monster truck pull in order to
remain unmoved? HINT: Find the resultant force of the two trucks. Round your answer to the nearest
tenth.
1400 lb
45°
3400 lb
A) The truck must pull with a force of 4500.2 lb. B) The truck must pull with a force of 2605.4 lb.
C) The truck must pull with a force of 4276.9 lb. D) The truck must pull with a force of 2197.3 lb.
10.5 The Dot Product
1 Find the Dot Product of Two Vectors
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the dot product v · w.
1) v = i – j
,
w = i + j
A) 0 B) 2 C) 1 D) –1
2) v = –3i
,
w = 2j
A) 0 B) –6C)13
D) –1
3) v = i + 2j, w = 6i – j
A) 4 B) 8 C) –4D)0
4) v = 7i – 3j, w = 9i + j
A) 60 B) 66 C) –20 D) 0
5) v = –13i + 9j
,
w = 14i – 6j
A) –236 B) –128 C) –182 D) –54
6) v = 5i – 8j, w = i – j
A) 5 + 8B)5 – 8C)13 D)
–13
7) v = i – 3j, w = 7i + j
A) 7 – 3 B) 7 + 3 C) –7 – 3 D) –10
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2 Find the Angle Between Two Vectors
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the angle between v and w. Round your answer to one decimal place, if necessary.
1) v = –5i + 2j, w = –8i + 2j
A) 7.8° B) 17.8° C) 3.9° D) 353.9°
2) v = –5i + 7j
,
w = –6i – 4j
A) 88.2° B) 90.9° C) 20.7° D) 110.8°
3) v = 3i
,
w = j
A) 90° B) 180° C) 0° D) 270°
Solve the problem.
4) An airplane has an air speed of 550 miles per hour bearing N30°W. The wind velocity is 50 miles per hour
in the direction N30°E. To the nearest tenth, what is the ground speed of the plane? What is its direction?
A) 576.6 mph; N25.7°W B) 526.8 mph; N55.3°W
C) 552.3 mph; N54.8°W D) 552.3 mph; N24.8°W
5) A DC–10 jumbo jet maintains an airspeed of 600 miles per hour in a southeasterly direction. The velocity
of the jet stream is a constant 50 miles per hour from the west. Find the actual speed and direction of the
aircraft. (Round the speed and direction to the nearest tenth.)
A) 636.3 mph; S48.2°E B) 565.8 mph; N41.4°E
C) 565.8 mph; S48.2°E D) 636.3 mph; S41.8°E
6) A power boat in still water maintains a speed of 40 miles per hour. The boat heads directly across a river
perpendicular to the current which has a speed of 7 miles per hour. Find the actual speed and direction of
the boat.
A) 41 mph; 10° off course B) 40 mph; 10° off course
C) 19 mph; 22° off course D) 16 mph; 26° off course
3 Determine Whether Two Vectors Are Parallel
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Which of the following vectors is parallel to v =i–j?
A) w = 2i – 2jB) w = i–2jC) w=i+jD) w =2i+2j
2) Which of the following vectors is parallel to v =i+2j?
A) w = –i – 2jB) w = i–2jC) w=i+jD) w =2i+2j
3) Which of the following vectors is parallel to v =i+2j?
A) w = 1
2i + jB) w = i – 1
2jC) w=i+jD) w =2i+2j
4) Which of the following vectors is parallel to v =–10i–8j?
A) w = 20i + 16jB) w = 3i–5jC) w=4i+4jD) w = –20i+25j
5) Which of the following vectors is parallel to v=9i+5j?
A) w = 3
2i + 5
6jB) w = 45i –25jC) w = – 15
2i + 25
6jD) w = 15
2i – 25
6j
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6) The amount of energy collected by a solar panel depends on the intensity of the sun’s rays and the area of
the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of
the sun’s rays. Let the vector A represent the area, in square centimeters, whose direction is the orientation
of a solar panel. See the figure. The total number of watts collected by the panel is given by W = |I · A|.
Suppose I = –0.06 –0.05 and A = 200, 400 . (i) Find I and Aand interpret the meaning of each; (ii)
Compute W and interpret its meaning; (iii) If the solar panel is to collect the maximum number of watts,
what must be true about I and A?
A) (i) I= 0.078; the intensity of the sun’s rays is approximately 0.078 W/cm2. A = 447; the area of the
solar panel is approximately 447 cm2.
(ii) W = 32; 32 watts of energy is collected.
(iii) Vectors I and A should be parallel with the solar panels facing the sun.
B) (i) I= 0.006; the intensity of the sun’s rays is approximately 0.006 W/cm2. A = 200,000; the area of
the solar panel is approximately 200,000 cm2.
(ii) W = 32; 32 watts of energy is collected.
(iii) Vectors I and A should be perpendicular with the solar panels at a 90° angle to the sun.
C) (i) I= 0.078; the intensity of the sun’s rays is approximately 0.078 W/cm2. A = 447; the area of the
solar panel is approximately 447 cm2.
(ii) W = 34; 34 watts of energy is collected.
(iii) Vectors I and A should be parallel with the solar panels facing the sun.
D) (i) I= 0.033; the intensity of the sun’s rays is approximately 0.033 W/cm2. A = 346; the area of the
solar panel is approximately 346 cm2.
(ii) W = 34; 34 watts of energy is collected.
(iii) Vectors I and A should be orthogonal with the solar panels at a 45° angle to the sun.
4 Determine Whether Two Vectors Are Orthogonal
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Which of the following vectors is orthogonal to 20i–8j?
A) w = –10i – 25jB) w = 20i+4jC) w=15i–6jD) w =4i+3j
State whether the vectors are parallel, orthogonal, or neither.
2) v = 3i + j
,
w = i – 3j
A) Orthogonal B) Parallel C) Neither
3) v = 2i + 3j
,
w = 3i – 2j
A) Orthogonal B) Parallel C) Neither
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4) v = 3i – j
,
w = 6i – 2j
A) Parallel B) Orthogonal C) Neither
5) v = 3i + 4j
,
w = 6i + 8j
A) Parallel B) Orthogonal C) Neither
6) v = 4i – 2j
,
w = 4i + 2j
A) Parallel B) Orthogonal C) Neither
7) v = i + 2
j,w = i – 4j
A) Orthogonal B) Parallel C) Neither
Solve the problem.
8) Find a so that the vectors v = i – aj and w=4i+2j are orthogonal.
A) 2 B) – 2C)
– 1
2D) 1
2
5 Decompose a Vector into Two Orthogonal Vectors
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Decompose v into two vectors v1 and v2, where v1is parallel to w and v2is orthogonal to w.
1) v = i + 9j
,
w = i + j
A) v1 = 5i + 5j
,
v2 = –4i + 4jB) v1 = 11
2i + 11
2j, v2 = – 9
2i + 7
2j
C) v1 = 5i + 5j
,
v2 = 4i – 4jD) v1=10i+10j
,
v2 = –8i + 8j
2) v = i – 4j
,
w = 3i + j
A) v1 = – 3
10i – 1
10j, v2 = 13
10i – 39
10jB) v1 = – 3
10i – 1
10j, v2 = 11
10i – 39
10j
C) v1 = – 1
3i – 1
9j, v2 = 4
3i – 35
9jD) v1 = – 3
10i – 1
10j, v2 = – 11
10i – 47
10j
3) v = –3i + 5j
,
w = 2i + j
A) v1 = – 2
5i – 1
5j, v2 = – 13
5i + 26
5jB) v1 = – 2
5i – 1
5j, v2 = – 14
5i + 18
5j
C) v1 = – 1
2i – 1
4j, v2 = – 5
2i + 21
4jD) v1 = – 2
5i – 1
5j, v2 = 7
5i + 36
5j
4) v = –3i + 4j
,
w = 3i + j
A) v1 = – 3
2i – 1
2j, v2 = – 3
2i + 9
2jB) v1 = – 3
2i – 1
2j, v2 = – 17
10i + 9
2j
C) v1 = – 5
3i – 5
9j, v2 = 4
3i + 49
9jD) v1 = – 3
2i + 1
2j, v2 = – 3
2i – 9
2j
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Solve the problem.
5) An SUV weighing 6000 pounds is parked on a street which has an incline of 12°. Find the force required to
keep the SUV from rolling down the hill and the force of the SUV perpendicular to the hill. Round the
forces to the nearest hundredth.
A) 1247.47 lb and 5868.89 lb B) 1754.23 lb and 5737.83 lb
C) 731.22 lb and 5955.28 lb D) 623.74 lb and 2934.44 lb
6 Compute Work
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem. Round your answer to the nearest tenth.
1) Find the work done by a force of 9 pounds acting in the direction of 38° to the horizontal in moving an
object 5 feet from (0, 0) to (5, 0).
A) 35.5 ft–lb B) 27.7 ft–lb C) 70.9 ft–lb D) 37.7 ft–lb
2) A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the
horizontal. How much work is done in moving the wagon 50 feet?
A) 2718.9 ft–lb B) 1267.9 ft–lb C) 1617.4 ft–lb D) 2110.8 ft–lb
3) A person is pulling a freight cart with a force of 58 pounds. How much work is done in moving the cart 70
feet if the cart’s handle makes an angle of 27° with the ground?
A) 3617.5 ft–lb B) 1843.2 ft–lb C) 184.3 ft–lb D) 3764.4 ft–lb
4) Find the work done by a force of 200 pounds acting in the direction –i+2jin moving an object 75 feet from
(0, 0) to (–75, 0).
A) 6708.2 ft–lb B) 15,000.0 ft–lb C) 13,416.1 ft–lb D) 8944.9 ft–lb
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Ch. 10 Polar Coordinates; Vectors
Answer Key
10.1 Polar Coordinates
1 Plot Points Using Polar Coordinates
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10.2 Polar Equations and Graphs
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10.3 The Complex Plane; De Moivre’s Theorem
1 Plot Points in the Complex Plane
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10.4 Vectors
1 Graph Vectors
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10.5 The Dot Product
1 Find the Dot Product of Two Vectors
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