Download Berkeley City College Precalculus - Math 1

Berkeley City College
Homework 6 Due:_________________
Precalculus - Math 1 - Chapter 10
Polar Coordinates and Complex Numbers
Name___________________________________
Match the point in polar coordinates with either A, B, C, or D on the graph.
π
1) -3, 3
1)
5
4
A
3
B
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
-1
-2
D
C
-3
-4
-5
Objective: (9.1) Plot Points Using Polar Coordinates
2) 3, - 5π
3
2)
5
4
A
3
B
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
-1
-2
D
-3
C
-4
-5
Objective: (9.1) Plot Points Using Polar Coordinates
Instructor: K. Pernell
1
Plot the point given in polar coordinates.
3) (-2, 45°)
3)
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
-1
-2
-3
-4
-5
Objective: (9.1) Plot Points Using Polar Coordinates
4) (2, 360°)
4)
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
-1
-2
-3
-4
-5
Objective: (9.1) Plot Points Using Polar Coordinates
2
Solve the problem.
5) Plot the point 4, (a) r > 0, (b) r < 0, (c) r > 0
5π
and find other polar coordinates (r, θ) of the point for which:
6
5)
-2π ≤ θ < 0
0 ≤ θ < 2π
2π ≤ θ < 4π
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
-1
-2
-3
-4
-5
Objective: (9.1) Plot Points Using Polar Coordinates
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
2π
6) 7, 3
6)
Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates
7) 5, - 4π
3
7)
Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates
8) (-3, -135°)
8)
Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates
3
The rectangular coordinates of a point are given. Find polar coordinates for the point.
9) (0, -8)
9)
Objective: (9.1) Convert from Rectangular Coordinates to Polar Coordinates
10) (- 3, -1)
10)
Objective: (9.1) Convert from Rectangular Coordinates to Polar Coordinates
The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ).
11) x2 + y 2 - 4x = 0
11)
Objective: (9.1) Transform Equations between Polar and Rectangular Forms
12) xy = 1
12)
Objective: (9.1) Transform Equations between Polar and Rectangular Forms
The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y).
13) r = 1 + 2 sin θ
13)
Objective: (9.1) Transform Equations between Polar and Rectangular Forms
4
Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
14)
14) r = 5
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6
r
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
15) r = 2 cos θ
15)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6 r
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
5
16) r sin θ = 5
16)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
1 2 3 4 5 6 r
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
Match the graph to one of the polar equations.
17)
17)
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5r
-2
-3
-4
-5
A) θ = π
3
B) r = -
π
3
C) θ = -
π
3
D) r = Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
6
π
3
Plot the complex number in the complex plane.
18) 6 + 5i
6
18)
i
4
2
-6
-4
-2
2
4
6 R
-2
-4
-6
Objective: (9.3) Plot Points in the Complex Plane
19) -8 + 7i
19)
10
i
5
-10
-5
5
R
-5
-10
Objective: (9.3) Plot Points in the Complex Plane
Write the complex number in rectangular form.
π
π
20) 8 cos + i sin 6
6
20)
Objective: (9.3) Plot Points in the Complex Plane
7
21) 4(cos 300° + i sin 300°)
21)
Objective: (9.3) Plot Points in the Complex Plane
22) 9(cos 180° + i sin 180°)
22)
Objective: (9.3) Plot Points in the Complex Plane
Write the complex number in polar form . Express the argument in degrees, rounded to the nearest tenth, if necessary.
23) 2 + 2i
23)
Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form
24) -6
24)
Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form
25) -12 + 16i
25)
Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form
Find zw or z
as specified. Leave your answer in polar form.
w
26) z = 5(cos 35° + i sin 35°)
w = 2(cos 40° + i sin 40°)
Find zw.
26)
Objective: (9.3) Find Products and Quotients of Complex Numbers in Polar Form
8
27) z = 10(cos 30° + i sin 30°)
w = 5(cos 10° + i sin 10°)
z
Find .
w
27)
Objective: (9.3) Find Products and Quotients of Complex Numbers in Polar Form
Write the expression in the standard form a + bi.
28) 2(cos 15° + i sin 15°) 3
28)
Objective: (9.3) Use De Moivreʹs Theorem
29) 2(cos 75° + i sin 75°) 3
29)
Objective: (9.3) Use De Moivreʹs Theorem
30)
2 cos 3π
3π
+ i sin 4
4
4
30)
Objective: (9.3) Use De Moivreʹs Theorem
31)
3 cos 5π
5π
+ i sin 6
6
4
31)
Objective: (9.3) Use De Moivreʹs Theorem
32) (1 + i)20
32)
Objective: (9.3) Use De Moivreʹs Theorem
9
Find all the complex roots. Leave your answers in polar form with the argument in degrees.
33) The complex fourth roots of -16
Objective: (9.3) Find Complex Roots
10
33)
Answer Key
Testname: 13SPR_CH10_MATH1_HW_6
1) D
2) B
3)
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
1
2
3
4
5 r
-1
-2
-3
-4
-5
4)
5
4
3
2
1
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
11
Answer Key
Testname: 13SPR_CH10_MATH1_HW_6
5)
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 r
-1
-2
-3
-4
-5
(a) 4, - (b) -4, (c)
4, 7π
6
11π
6
17π
6
7 7 3
6) - , 2 2
5 5 3
7) - , 2 2
8)
3 2 3 2
, 2
2
9) 8, 10) 2, -
π
2
5π
6
11) r = 4 cos θ
12) r2 sin 2θ = 2
13) x2 + y2 = x2 + y2 + 2y
12
Answer Key
Testname: 13SPR_CH10_MATH1_HW_6
14)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6
r
x2 + y2 = 25; circle, radius 5, center at pole
15)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6 r
(x - 1)2 + y2 = 1; circle, radius 1,
center at (1, 0) in rectangular coordinates
16)
6
5
4
3
2
1
-5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
1 2 3 4 5 6r
y = 5; horizontal line 5 units above the pole
17) A
13
Answer Key
Testname: 13SPR_CH10_MATH1_HW_6
18)
6
i
4
2
-6
-4
-2
2
4
6 R
-2
-4
-6
19)
10
i
5
-10
-5
5
R
-5
-10
20) 4 3 + 4i
21) 2 - 2 3i
22) -9
23) 2 2(cos 45° + i sin 45°)
24) 6(cos 180° + i sin 180°)
25) 20(cos 126.9° + i sin 126.9°)
26) 10(cos 75° + i sin 75°)
27) 2(cos 20° + i sin 20°)
28) 4 2 + 4 2i
29) -4 2 - 4 2i
30) -4
9 9 3
31) - - i
2
2
32) -1024
33) 2(cos 45° + i sin 45°), 2(cos 135° + i sin 135°), 2(cos 225° + i sin 225°), 16(cos 315° + i sin 315°)
14