Chapter 9: Q. 53 (page 592)

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
53. The complex cube root of $1 + i$ .

Short Answer

Expert verified

The complex cube roots of $1 + i$ arerole="math" localid="1646812515277" $\sqrt[6]{2} (\cos 15 + i \sin 15), \sqrt[6]{2} (\cos 135 + i \sin 135), \sqrt[6]{2} (\cos 255 + i \sin 255)$ .

Step by step solution

Step 1. Rewrite 1+i in the polar form.

The magnitude of $1 + i$ is

$\sqrt{1^{2} + 1^{2}} = \sqrt{2}$

which gives

$1 + i = \sqrt{2} (\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}) = \sqrt{2} (\cos 45 ° + i \sin 45 °)$

Step 2. Using the theorem of complex roots.

We get

$z_{k} = \sqrt[3]{\sqrt{2}} [\cos (\frac{45}{3} + \frac{360 k}{3}) + i \sin (\frac{45}{3} + \frac{360 k}{3})]; k = 0, 1, 2 = \sqrt[6]{2} [\cos (15 + 120 k) + i \sin (15 + 120 k)]$

Step 3. Substitute k=0,1,2 in zk=26(cos(15°+120°k)+isin(15°+120°k))

We get

$z_{0} = \sqrt[6]{2} (\cos 15 + i \sin 15) z_{1} = \sqrt[6]{2} (\cos (15 + 120) + i \sin (15 + 120)) = \sqrt[6]{2} (\cos 135 + i \sin 135) z_{2} = \sqrt[6]{2} (\cos (15 + 240) + i \sin (15 + 240)) = \sqrt[6]{2} (\cos 255 + i \sin 255)$

So we have found the complex cube roots of $1 + i$ .

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In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
53. The complex cube root of $1 + i$ .

Short Answer

Step by step solution

Step 1. Rewrite 1+i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2 in zk=26(cos(15°+120°k)+isin(15°+120°k))

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In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.53. The complex cube root of1+i.

Short Answer

Step by step solution

Step 1. Rewrite 1+i in the polar form.

Step 2. Using the theorem of complex roots.

Step 3. Substitute k=0,1,2 in zk=26(cos(15°+120°k)+isin(15°+120°k))

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Statistics

Geometry

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.
53. The complex cube root of $1 + i$ .