Stewart Precalc 6e Section 8.3: Polar Form of Complex Numbers; De Moivre's Theorem

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Stewart Precalc 6e Section 8.3: Polar Form of Complex Numbers; De Moivre's Theorem 0/101
1 Concepts · Level 1
A complex number \(z = a + b i\) has two parts: \(a\) is the _____ part, and \(b\) is the _____ part. To graph \(a + b i\), we graph the ordered pair in the complex plane.
2 Concepts · Level 1
Let \(z = a + b i\). (a) The modulus of \(z\) is \(r =\) _____, and an argument of \(z\) is an angle \(\theta\) satisfying \(\tan \theta =\) _____. (b) We can express \(z\) in polar form as \(z =\) _____, where \(r\) is the modulus of \(z\) and \(\theta\) is the argument of \(z\).
3 Concepts · Level 1
(a) The complex number \(z = -1 + i\) in polar form is \(z =\) _____. The complex number \(z = 2(\cos\left(\dfrac{\pi}{6}\right) + i \sin\left(\dfrac{\pi}{6}\right))\) in rectangular form is _____. (b) The complex number graphed in the figure can be expressed in rectangular form as _____ or in polar form as _____.
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4 Concepts · Level 1
How many different \(n\)th roots does a nonzero complex number have? _____ fourth roots. These roots are _____. In the complex plane these roots all lie on a circle of radius _____. Graph the roots on the following graph.
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5 Modulus of a Complex Number · Level 2
Graph the complex number \(-3 i\) and find its modulus.
6 Modulus of a Complex Number · Level 2
Graph the complex number \(5 + 2 i\) and find its modulus.
7 Modulus of a Complex Number · Level 2
Graph the complex number \(\sqrt{3} + i\) and find its modulus.
8 Modulus of a Complex Number · Level 2
Graph the complex number \(-1 - \dfrac{\sqrt{3}}{3} i\) and find its modulus.
9 Modulus of a Complex Number · Level 2
Graph the complex number \(\dfrac{3 + 4 i}{5}\) and find its modulus.
10 Modulus of a Complex Number · Level 2
Graph the complex number \(\dfrac{-\sqrt{2} + i \sqrt{2}}{2}\) and find its modulus.
11 Operations in Complex Plane · Level 2
Sketch the complex number \(z = 1 + i\), and also sketch \(2 z\), \(-z\), and \(\dfrac{1}{2} z\) on the same complex plane.
12 Operations in Complex Plane · Level 2
Sketch the complex number \(z = -1 + i \sqrt{3}\), and also sketch \(2 z\), \(-z\), and \(\dfrac{1}{2} z\) on the same complex plane.
13 Complex Conjugates · Level 2
Sketch the complex number \(z = 8 + 2 i\) and its complex conjugate \(\overline{z}\) on the same complex plane.
14 Complex Conjugates · Level 2
Sketch the complex number \(z = -5 + 6 i\) and its complex conjugate \(\overline{z}\) on the same complex plane.
15 Sum and Product of Complex Numbers · Level 2
Sketch \(z_1\), \(z_2\), \(z_1 + z_2\), and \(z_1 z_2\) on the same complex plane, where \(z_1 = 2 - i\) and \(z_2 = 2 + i\).
16 Sum and Product of Complex Numbers · Level 2
Sketch \(z_1\), \(z_2\), \(z_1 + z_2\), and \(z_1 z_2\) on the same complex plane, where \(z_1 = -1 + i\) and \(z_2 = 2 - 3 i\).
17 Sets in the Complex Plane · Level 2
Sketch the set \(\{z = a + b i : a \leq 0, b \geq 0\}\) in the complex plane.
18 Sets in the Complex Plane · Level 2
Sketch the set \(\{z = a + b i : a > 1, b > 1\}\) in the complex plane.
19 Sets in the Complex Plane · Level 2
Sketch the set \(\{z : |z| = 3\}\) in the complex plane.
20 Sets in the Complex Plane · Level 2
Sketch the set \(\{z : |z| \geq 1\}\) in the complex plane.
21 Sets in the Complex Plane · Level 2
Sketch the set \(\{z : |z| < 2\}\) in the complex plane.
22 Sets in the Complex Plane · Level 2
Sketch the set \(\{z : 2 \leq |z| \leq 5\}\) in the complex plane.
23 Sets in the Complex Plane · Level 2
Sketch the set \(\{z = a + b i : a + b < 2\}\) in the complex plane.
24 Sets in the Complex Plane · Level 2
Sketch the set \(\{z = a + b i : a \geq b\}\) in the complex plane.
25 Polar Form of Complex Numbers · Level 3
Write the complex number \(1 + i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
26 Polar Form of Complex Numbers · Level 3
Write the complex number \(1 + \sqrt{3} i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
27 Polar Form of Complex Numbers · Level 3
Write the complex number \(\sqrt{2} - \sqrt{2} i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
28 Polar Form of Complex Numbers · Level 3
Write the complex number \(1 - i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
29 Polar Form of Complex Numbers · Level 3
Write the complex number \(2 \sqrt{3} - 2 i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
30 Polar Form of Complex Numbers · Level 3
Write the complex number \(-1 + i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
31 Polar Form of Complex Numbers · Level 3
Write the complex number \(-3 - 3 \sqrt{3} i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
32 Polar Form of Complex Numbers · Level 3
Write the complex number \(5 + 5 i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
33 Polar Form of Complex Numbers · Level 3
Write the complex number \(4 \sqrt{3} - 4 i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
34 Polar Form of Complex Numbers · Level 3
Write the complex number \(\sqrt{3} + i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
35 Polar Form of Complex Numbers · Level 3
Write the complex number \(3 + 4 i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
36 Polar Form of Complex Numbers · Level 3
Write the complex number \(i(2 - 2 i)\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
37 Polar Form of Complex Numbers · Level 3
Write the complex number \(3 i(1 + i)\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
38 Polar Form of Complex Numbers · Level 3
Write the complex number \(2(1 - i)\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
39 Polar Form of Complex Numbers · Level 3
Write the complex number \(4(\sqrt{3} + i)\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
40 Polar Form of Complex Numbers · Level 3
Write the complex number \(-3 - 3 i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
41 Polar Form of Complex Numbers · Level 3
Write the complex number \(2 + i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
42 Polar Form of Complex Numbers · Level 3
Write the complex number \(3 + \sqrt{3} i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
43 Polar Form of Complex Numbers · Level 3
Write the complex number \(\sqrt{2} + \sqrt{2} i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
44 Polar Form of Complex Numbers · Level 3
Write the complex number \(-\pi i\) in polar form with argument \(\theta\) between \(0\) and \(2 \pi\).
45 Products and Quotients in Polar Form · Level 3
Find the product \(z_1 z_2\) and the quotient \(\dfrac{z_1}{z_2}\) in polar form, where \(z_1 = \cos \pi + i \sin \pi\) and \(z_2 = \cos\left(\dfrac{\pi}{2}\right) + i \sin\left(\dfrac{\pi}{2}\right)\).
46 Products and Quotients in Polar Form · Level 3
Find the product \(z_1 z_2\) and the quotient \(\dfrac{z_1}{z_2}\) in polar form, where \(z_1 = \cos\left(\dfrac{\pi}{4}\right) + i \sin\left(\dfrac{\pi}{4}\right)\) and \(z_2 = \cos\left(3 \dfrac{\pi}{4}\right) + i \sin\left(3 \dfrac{\pi}{4}\right)\).
47 Products and Quotients in Polar Form · Level 3
Find the product \(z_1 z_2\) and the quotient \(\dfrac{z_1}{z_2}\) in polar form, where \(z_1 = 3(\cos\left(\dfrac{\pi}{6}\right) + i \sin\left(\dfrac{\pi}{6}\right))\) and \(z_2 = 5(\cos\left(4 \dfrac{\pi}{3}\right) + i \sin\left(4 \dfrac{\pi}{3}\right))\).
48 Products and Quotients in Polar Form · Level 3
Find the product \(z_1 z_2\) and the quotient \(\dfrac{z_1}{z_2}\) in polar form, where \(z_1 = 7(\cos\left(9 \dfrac{\pi}{8}\right) + i \sin\left(9 \dfrac{\pi}{8}\right))\) and \(z_2 = 2(\cos\left(\dfrac{\pi}{8}\right) + i \sin\left(\dfrac{\pi}{8}\right))\).
49 Products and Quotients in Polar Form · Level 3
Find the product \(z_1 z_2\) and the quotient \(\dfrac{z_1}{z_2}\) in polar form, where \(z_1 = 4(\cos 120^{\circ} + i \sin 120^{\circ})\) and \(z_2 = 2(\cos 30^{\circ} + i \sin 30^{\circ})\).
50 Products and Quotients in Polar Form · Level 3
Find the product \(z_1 z_2\) and the quotient \(\dfrac{z_1}{z_2}\) in polar form, where \(z_1 = \sqrt{2}(\cos 75^{\circ} + i \sin 75^{\circ})\) and \(z_2 = 3 \sqrt{2}(\cos 60^{\circ} + i \sin 60^{\circ})\).
51 Products and Quotients in Polar Form · Level 3
Find the product \(z_1 z_2\) and the quotient \(\dfrac{z_1}{z_2}\) in polar form, where \(z_1 = 4(\cos 200^{\circ} + i \sin 200^{\circ})\) and \(z_2 = 25(\cos 150^{\circ} + i \sin 150^{\circ})\).
52 Products and Quotients in Polar Form · Level 3
Find the product \(z_1 z_2\) and the quotient \(\dfrac{z_1}{z_2}\) in polar form, where \(z_1 = \left(\dfrac{4}{5}\right)(\cos 25^{\circ} + i \sin 25^{\circ})\) and \(z_2 = \left(\dfrac{1}{5}\right)(\cos 155^{\circ} + i \sin 155^{\circ})\).
53 Polar Form Operations · Level 4
Write \(z_1 = \sqrt{3} + i\) and \(z_2 = 1 + \sqrt{3} i\) in polar form, and then find the product \(z_1 z_2\) and the quotients \(\dfrac{z_1}{z_2}\) and \(\dfrac{1}{z_1}\).
54 Polar Form Operations · Level 4
Write \(z_1 = \sqrt{2} - \sqrt{2} i\) and \(z_2 = 1 - i\) in polar form, and then find the product \(z_1 z_2\) and the quotients \(\dfrac{z_1}{z_2}\) and \(\dfrac{1}{z_1}\).
55 Polar Form Operations · Level 4
Write \(z_1 = 2 \sqrt{3} - 2 i\) and \(z_2 = -1 + i\) in polar form, and then find the product \(z_1 z_2\) and the quotients \(\dfrac{z_1}{z_2}\) and \(\dfrac{1}{z_1}\).
56 Polar Form Operations · Level 4
Write \(z_1 = -\sqrt{2} i\) and \(z_2 = -3 - 3 \sqrt{3} i\) in polar form, and then find the product \(z_1 z_2\) and the quotients \(\dfrac{z_1}{z_2}\) and \(\dfrac{1}{z_1}\).
57 Polar Form Operations · Level 4
Write \(z_1 = 5 + 5 i\) and \(z_2 = 4\) in polar form, and then find the product \(z_1 z_2\) and the quotients \(\dfrac{z_1}{z_2}\) and \(\dfrac{1}{z_1}\).
58 Polar Form Operations · Level 4
Write \(z_1 = 4 \sqrt{3} - 4 i\) and \(z_2 = 8 i\) in polar form, and then find the product \(z_1 z_2\) and the quotients \(\dfrac{z_1}{z_2}\) and \(\dfrac{1}{z_1}\).
59 Polar Form Operations · Level 4
Write \(z_1 = -20\) and \(z_2 = \sqrt{3} + i\) in polar form, and then find the product \(z_1 z_2\) and the quotients \(\dfrac{z_1}{z_2}\) and \(\dfrac{1}{z_1}\).
60 Polar Form Operations · Level 4
Write \(z_1 = 3 + 4 i\) and \(z_2 = 2 - 2 i\) in polar form, and then find the product \(z_1 z_2\) and the quotients \(\dfrac{z_1}{z_2}\) and \(\dfrac{1}{z_1}\).
61 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \((1 + i)^{20}\).
62 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \((1 - \sqrt{3} i)^5\).
63 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \((2 \sqrt{3} + 2 i)^5\).
64 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \((1 - i)^8\).
65 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \(\left(\dfrac{\sqrt{2}}{2} + \dfrac{\sqrt{2}}{2} i\right)^{12}\).
66 De Moivre's Theorem · Level 4
Find the indicated power using De Moivre's Theorem: \((\sqrt{3} - i)^{-10}\).
67 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \((2 - 2 i)^8\).
68 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \(\left(-\dfrac{1}{2} - \dfrac{\sqrt{3}}{2} i\right)^{15}\).
69 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \((-1 - i)^7\).
70 De Moivre's Theorem · Level 4
Find the indicated power using De Moivre's Theorem: \((2 \sqrt{3} + 2 i)^{-5}\).
71 De Moivre's Theorem · Level 3
Find the indicated power using De Moivre's Theorem: \((1 - i)^{-8}\).
72 nth Roots of Complex Numbers · Level 4
Find the square roots of \(4 \sqrt{3} + 4 i\), and graph the roots in the complex plane.
73 nth Roots of Complex Numbers · Level 4
Find the cube roots of \(4 \sqrt{3} + 4 i\), and graph the roots in the complex plane.
74 nth Roots of Complex Numbers · Level 4
Find the fourth roots of \(-81 i\), and graph the roots in the complex plane.
75 nth Roots of Complex Numbers · Level 4
Find the fifth roots of \(32\), and graph the roots in the complex plane.
76 nth Roots of Complex Numbers · Level 4
Find the eighth roots of \(1\), and graph the roots in the complex plane.
77 nth Roots of Complex Numbers · Level 4
Find the cube roots of \(1 + i\), and graph the roots in the complex plane.
78 nth Roots of Complex Numbers · Level 4
Find the cube roots of \(i\), and graph the roots in the complex plane.
79 nth Roots of Complex Numbers · Level 4
Find the fifth roots of \(i\), and graph the roots in the complex plane.
80 nth Roots of Complex Numbers · Level 4
Find the fourth roots of \(-1\), and graph the roots in the complex plane.
81 nth Roots of Complex Numbers · Level 4
Find the fifth roots of \(-16 - 16 \sqrt{3} i\), and graph the roots in the complex plane.
82 Solving Equations with Complex Roots · Level 4
Solve the equation \(z^4 + 1 = 0\).
83 Solving Equations with Complex Roots · Level 4
Solve the equation \(z^8 - i = 0\).
84 Solving Equations with Complex Roots · Level 4
Solve the equation \(z^3 - 4 \sqrt{3} - 4 i = 0\).
85 Solving Equations with Complex Roots · Level 4
Solve the equation \(z^6 - 1 = 0\).
86 Solving Equations with Complex Roots · Level 4
Solve the equation \(z^3 + 1 = -i\).
87 Solving Equations with Complex Roots · Level 4
Solve the equation \(z^3 - 1 = 0\).
88 Roots of Unity (Proof) · Level 5
(a) Let \(w = \cos\left(2 \dfrac{\pi}{n}\right) + i \sin\left(2 \dfrac{\pi}{n}\right)\) where \(n\) is a positive integer. Show that \(1, w, w^2, w^3, ..., w^{n - 1}\) are the \(n\) distinct \(n\)th roots of \(1\). (b) If \(z \neq 0\) is any complex number and \(s^n = z\), show that the \(n\) distinct \(n\)th roots of \(z\) are \(s, s w, s w^2, s w^3, ..., s w^{n - 1}\).
89 Discovery: Sums of Roots of Unity · Level 5
Find the exact values of all three cube roots of \(1\) (see Exercise 97) and then add them. Do the same for the fourth, fifth, sixth, and eighth roots of \(1\). What do you think is the sum of the \(n\)th roots of \(1\) for any \(n\)?
90 Discovery: Products of Roots of Unity · Level 5
Find the product of the three cube roots of \(1\) (see Exercise 97). Do the same for the fourth, fifth, sixth, and eighth roots of \(1\). What do you think is the product of the \(n\)th roots of \(1\) for any \(n\)?
91 Complex Quadratic Formula · Level 5
The quadratic formula works whether the coefficients of the equation are real or complex. Solve these equations using the quadratic formula and, if necessary, De Moivre's Theorem. (a) \(z^2 + (1 + i) z + i = 0\). (b) \(z^2 - i z + 1 = 0\). (c) \(z^2 - (2 - i) z - \left(\dfrac{1}{4}\right) i = 0\).
92 Example - Graphing Complex Numbers · Level 2
Graph the complex numbers \(z_1 = 2 + 3 i\), \(z_2 = 3 - 2 i\), and \(z_1 + z_2\).
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93 Example - Graphing Sets of Complex Numbers · Level 2
Graph each set of complex numbers.
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(a) \(S = \{ a + b i | a \geq 0 \}\)
(b) \(T = \{ a + b i | a < 1, b \geq 0 \}\)

Enter your answer directly below each part above.

94 Example - Calculating the Modulus · Level 1
Find the moduli of the complex numbers \(3 + 4 i\) and \(8 - 5 i\).
95 Example - Absolute Value of Complex Numbers · Level 2
Graph each set of complex numbers.
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(a) \(C = \{ z | |z| = 1 \}\)
(b) \(D = \{ z | |z| \leq 1 \}\)

Enter your answer directly below each part above.

96 Example - Polar Form · Level 3
Write each complex number in polar form.
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(a) \(1 + i\)
(b) \(-1 + \sqrt{3} i\)
(c) \(-4 \sqrt{3} - 4 i\)
(d) \(3 + 4 i\)

Enter your answer directly below each part above.

97 Example - Multiplying and Dividing · Level 3
Let \(z_1 = 2 (\cos\left(\dfrac{\pi}{4}\right) + i \sin\left(\dfrac{\pi}{4}\right))\) and \(z_2 = 5 (\cos\left(\dfrac{\pi}{3}\right) + i \sin\left(\dfrac{\pi}{3}\right))\). Find (a) \(z_1 z_2\) and (b) \(\dfrac{z_1}{z_2}\).
98 Example - De Moivre's Theorem · Level 3
Find \((\dfrac{1}{2} + \left(\dfrac{1}{2}\right) i)^{10}\).
99 Example - nth Roots of Complex Numbers · Level 4
Find the six sixth roots of \(z = -64\), and graph these roots in the complex plane.
100 Example - Cube Roots of Complex Number · Level 3
Find the three cube roots of \(z = 2 + 2 i\), and graph these roots in the complex plane.
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101 Example - Solving Equations with nth Roots · Level 3
Solve the equation \(z^6 + 64 = 0\).

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