Q. 55 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. 55 (page 592)

In Problems 53鈥�60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
55. The complex fourth roots of $4 - 4 \sqrt{3} i$ .

Short Answer

Expert verified

The complex fourth roots of $4 - 4 \sqrt{3} i$ are $\sqrt[4]{8} (\cos 75 + i \sin 75), \sqrt[4]{8} (\cos 165 + i \sin 165), \sqrt[4]{8} (\cos 255 + i \sin 255), \sqrt[4]{8} (\cos 345 + i \sin 345)$ .

Step by step solution

Step 1. Rewrite 4-43i in the polar form.

The magnitude of $4 - 4 \sqrt{3} i$ is

role="math" localid="1646848982162" $\sqrt{4^{2} + {(- 4 \sqrt{3})}^{2}} = \sqrt{16 + 48} = \sqrt{64} = 8$

which gives

$4 - 4 \sqrt{3} = 8 (\frac{1}{2} - \frac{\sqrt{3}}{2} i) = 8 (\cos 300 掳 - i \sin 300 掳)$

Step 2. Using the theorem of complex roots.

We get

$z_{k} = \sqrt[4]{8} [\cos (\frac{300}{4} + \frac{360 k}{4}) + i \sin (\frac{300}{4} + \frac{360 k}{4})]; k = 0, 1, 2, 3 = \sqrt[4]{8} [\cos (75 掳 + 90 掳 k) + i \sin (75 掳 + 90 掳 k)]$

Step 3. Substitute k=0,1,2,3 in zk=84[cos(75+90k)+isin(75+90k)].

We get

$z_{0} = \sqrt[4]{8} (\cos 75 + i \sin 75) z_{1} = \sqrt[4]{8} [\cos (75 + 90) + i \sin (75 + 90)] = \sqrt[4]{8} (\cos 165 + i \sin 165) z_{2} = \sqrt[4]{8} [\cos (75 + 180) + i \sin (75 + 180)] = \sqrt[4]{8} (\cos 255 + i \sin 255) z_{3} = \sqrt[4]{8} (\cos (75 + 270) + i \sin (75 + 270)) = \sqrt[4]{8} (\cos 345 + i \sin 345)$

So we have found the complex fourth roots of $4 - 4 \sqrt{3} i$ .

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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.
55. The complex fourth roots of $4 - 4 \sqrt{3} i$ .

Short Answer

Step by step solution

Step 1. Rewrite 4-43i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2,3 in zk=84[cos(75+90k)+isin(75+90k)].

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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.55. The complex fourth roots of4-43i.

Short Answer

Step by step solution

Step 1. Rewrite 4-43i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2,3 in zk=84[cos(75+90k)+isin(75+90k)].

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Decision Maths

Logic and Functions

Mechanics Maths

Applied Mathematics

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.
55. The complex fourth roots of $4 - 4 \sqrt{3} i$ .