Q. 56 In Problems 53鈥�60, find all th... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 56 (page 592)

In Problems 53鈥�60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
56. The complex cube roots of $- 8 - 8 i$ .

Short Answer

Expert verified

The complex cube roots of $- 8 - 8 i$ are $2 \sqrt[6]{2} (\cos 75 + i \sin 75), 2 \sqrt[6]{2} (\cos 195 + i \sin 195), 2 \sqrt[6]{2} (\cos 315 + i \sin 315)$ .

Step by step solution

Step 1. Rewrite -8-8i in the polar form.

The magnitude of $- 8 - 8 i$ is

$\sqrt{{(- 8)}^{2} + {(- 8)}^{2}} = \sqrt{64 + 64} = \sqrt{128} = 8 \sqrt{2}$

which gives

role="math" localid="1646811578868" $- 8 - 8 i = 8 \sqrt{2} (- \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}) = \sqrt{2} 路 2^{3} (\cos 225 掳 - i \sin 225 掳)$

Step 2. Use the theorem of complex roots.

We get

$z_{k} = \sqrt[3]{\sqrt{2} 路 2^{3}} [\cos (\frac{225}{3} + \frac{360 k}{3}) + i \sin (\frac{225}{3} + \frac{360 k}{3})]; k = 0, 1, 2 = 2 \sqrt[6]{2} [\cos (75 + 120 k) + i \sin (75 + 120 k)]$

Step 3. Substitute k=0,1,2 in zk=226(cos(75+120k)+isin(75+120k)).

We get

$z_{0} = 2 \sqrt[6]{2} (\cos 75 + i \sin 75) z_{1} = 2 \sqrt[6]{2} (\cos 75 + 120) + i \sin (75 + 120) = 2 \sqrt[6]{2} (\cos 195 + i \sin 195) z_{2} = 2 \sqrt[6]{2} (\cos (75 + 240) + i \sin (75 + 240)) = 2 \sqrt[6]{2} (\cos 315 + i \sin 315)$

So we have found complex cube roots of $- 8 - 8 i$ .

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91影视

In Problems 53鈥�60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
56. The complex cube roots of $- 8 - 8 i$ .

Short Answer

Step by step solution

Step 1. Rewrite -8-8i in the polar form.

Step 2. Use the theorem of complex roots.

Step 3. Substitute k=0,1,2 in zk=226(cos(75+120k)+isin(75+120k)).

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91影视

In Problems 53鈥�60, find all the complex roots. Leave your answers in polar form with the argument in degrees.56. The complex cube roots of-8-8i.

Short Answer

Step by step solution

Step 1. Rewrite -8-8i in the polar form.

Step 2. Use the theorem of complex roots.

Step 3. Substitute k=0,1,2 in zk=226(cos(75+120k)+isin(75+120k)).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Statistics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

In Problems 53鈥�60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
56. The complex cube roots of $- 8 - 8 i$ .