# The roots of the eighth root of 1 and sketch the roots in the complex plane. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter I, Problem 37E
To determine

## To find: The roots of the eighth root of 1 and sketch the roots in the complex plane.

Expert Solution

The roots of eighth root of 1 are wk=11n[cos(0+2kπ8)+isin(0+2kπ8)]=coskπ8+isinkπ8  where k=0,1,2,,7.

### Explanation of Solution

Theorem used:

Roots of a complex number:

Let z=r(cosθ+isinθ) and n be a positive integer. Then z has the n distinct nth roots wk=r1n[cos(θ+2kπn)+isin(θ+2kπn)] where k=0,1,2,,n1.

Calculation:

Rewrite the complex number 1 in polar form.

The polar form of the complex number z=a+bi is z=r(cosθ+isinθ) where r=|z|=a2+b2  and tanθ=ba.

Consider the complex number 1.

Obtain the argument of the complex number 1.

tanθ=01=0

Thus, the argument of argument of the complex number 1 is θ=tan1(0)=0

Obtain the modulus of the complex number 1.

r=|1|=12+(0)2=1=1

Thus, the value of r=1.

Therefore, the polar form of the complex number 1 is 1=(cos0+isin0).

By the above theorem, the roots of eighth root of 1 are wk=11n[cos(0+2kπ8)+isin(0+2kπ8)]=coskπ8+isinkπ8 where k=0,1,2,,7.

Use online calculator to sketch the roots in the complex plane as shown below in Figure 1. From figure 1, it is observed that all roots of eighth root of 1 form a circle on complex plane.

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