Berkeley City College Precalculus w/ Analytic Geometry ...€¦ · 27) z = 10(cos 30° + i sin...
14
Berkeley City College Precalculus w/ Analytic Geometry - Math 2 - Chapter 9 Homework 6 Due:_________________ Polar Coordinates and Complex Numbers Name___________________________________ 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 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 A B C D r -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 A B C D 1) Objective: (9.1) Plot Points Using Polar Coordinates 2) 3, - 5π 3 r -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 A B C D r -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 A B C D 2) Objective: (9.1) Plot Points Using Polar Coordinates Instructor: K. Pernell 1
Transcript of Berkeley City College Precalculus w/ Analytic Geometry ...€¦ · 27) z = 10(cos 30° + i sin...
Berkeley City College Precalculus w/ Analytic Geometry - Math 2 - Chapter 9
Homework 6 Due:_________________ Polar Coordinates and Complex Numbers
Name___________________________________
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 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
r-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
1)
Objective: (9.1) Plot Points Using Polar Coordinates
2) 3, - 5π
3
r-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
r-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
A B
CD
2)
Objective: (9.1) Plot Points Using Polar Coordinates
Instructor: K. Pernell 1
Plot the point given in polar coordinates.
3) (-2, 45°)
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
3)
Objective: (9.1) Plot Points Using Polar Coordinates
4) (2, 360°)
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
4)
Objective: (9.1) Plot Points Using Polar Coordinates
2
Solve the problem.
5) 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 > 0 2π ≤ θ < 4π
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
5)
Objective: (9.1) Plot Points Using Polar Coordinates
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
6) 7, 2π
36)
Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates
7) 5, - 4π
37)
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 + y2 - 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) r = 5
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
14)
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
15) r = 2 cos θ
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
15)
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
5
16) r sin θ = 5
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
16)
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
Match the graph to one of the polar equations.
17)
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) θ = π
3B) r = -
π
3C) θ = -
π
3D) r =
π
3
17)
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
6
Plot the complex number in the complex plane.
18) 6 + 5i
R-6 -4 -2 2 4 6
i6
4
2
-2
-4
-6
R-6 -4 -2 2 4 6
i6
4
2
-2
-4
-6
18)
Objective: (9.3) Plot Points in the Complex Plane
19) -8 + 7i
R-10 -5 5
i10
5
-5
-10
R-10 -5 5
i10
5
-5
-10
19)
Objective: (9.3) Plot Points in the Complex Plane
Write the complex number in rectangular form.
20) 8 cos π
6 + i sin
π
620)
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
w as specified. Leave your answer in polar form.
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°)
Find z
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
4 + i sin
3π
4
430)
Objective: (9.3) Use De Moivreʹs Theorem
31) 3 cos 5π
6 + i sin
5π
6
431)
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 33)
Objective: (9.3) Find Complex Roots
10
Answer KeyTestname: 13SPR_CH9_MATH2_HW_6
1) D
2) B
3)
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
4)
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
11
Answer KeyTestname: 13SPR_CH9_MATH2_HW_6
5)
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) 4, - 7π
6
(b) -4, 11π
6
(c) 4, 17π
6
6) - 7
2, 7 3
2
7) - 5
2, 5 3
2
8)3 2
2, 3 2
2
9) 8, -π
2
10) 2, -5π
6
11) r = 4 cos θ
12) r2 sin 2θ = 2
13) x2 + y2 = x2 + y2 + 2y
12
Answer KeyTestname: 13SPR_CH9_MATH2_HW_6
14)
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
x2 + y2 = 25; circle, radius 5,
center at pole
15)
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
(x - 1)2 + y2 = 1; circle, radius 1,
center at (1, 0) in rectangular coordinates
16)
r-5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
r-5 -4 -3 -2 -1 1 2 3 4 5 6
654321
-1-2-3-4-5-6
y = 5; horizontal line 5 units
above the pole
17) A
13
Answer KeyTestname: 13SPR_CH9_MATH2_HW_6
18)
R-6 -4 -2 2 4 6
i6
4
2
-2
-4
-6
R-6 -4 -2 2 4 6
i6
4
2
-2
-4
-6
19)
R-10 -5 5
i10
5
-5
-10
R-10 -5 5
i10
5
-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
31) - 9
2 -
9 3
2i
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
