a. Answer z 0 = 4 ( cos 60 0 + i sin 60 0 ) , z 1 = 4 ( cos 150 0 + i sin 150 0 ) , z 2 = 4 ( cos 240 0 + i sin 240 0 ) and z 3 = 4 ( cos 330 0 + i sin 330 0 ) . Explanation Given information: Write each of the roots in trigonometric form. Calculation: From the given information, write each of the roots in trigonometric form using the graph of roots are as shown below: Let’s note the roots: z i = r 1 ( cos θ i + i sin θ i ) , where i = 0 , 1 , 2 . From the given graph, there are three roots, then they are: 360 0 n = 360 0 3 = 120 0 Starting from the x − axis going counter clockwise, the roots are: Now, write the roots: z 0 = 2 ( cos 30 0 + i sin 30 0 ) z 1 = 2 ( cos 150 0 + i sin 150 0 ) z 2 = 2 ( cos 270 0 + i sin 270 0 ) Thus, the roots in trigonometric forms are z 0 = 2 ( cos 30 0 + i sin 30 0 ) , z 1 = 2 ( cos 150 0 + i sin 150 0 ) , and z 2 = 2 ( cos 270 0 + i sin 270 0 ) . b. To determine The complex number by using roots. Verify the results by using graphing utility. Answer 8 i . Explanation Given information: Identify the complex number whose roots are given.Use a graphing utility to verify your results. Calculation: From the given information, identify the complex number whose roots are given and by using graphing utility verify the result are as shown below: From the part (a) found the roots: z 0 = 2 ( cos 30 0 + i sin 30 0 ) z 1 = 2 ( cos 150 0 + i sin 150 0 ) z 2 = 2 ( cos 270 0 + i sin 270 0 ) These equations verify the equation: z 3 = ω Now find ω , by using the fact that any z i verifies the above equation: ω = z 0 4 = z 1 4 = z 2 4 = z 3 4 To find ω , ω = z 0 3 = [ 2 ( cos 30 0 + i sin 30 0 ) ] 3 z 0 3 = 2 3 [ cos ( 3 ⋅ 30 0 ) + i sin ( 3 ⋅ 30 0 ) ] z 0 3 = 8 ( cos 90 0 + i sin 90 0 ) z 0 3 = 8 ( 0 + i ) z 0 3 = z = 8 i Thus , the complex number is 8 i .

3rd Edition

ISBN: 9781285607160

Author: Larson

Publisher: CENGAGE CO

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Question

Chapter 6, Problem 147RE

To determine

The roots in the trigonometric form by using the graph of the roots of the complex number.

Expert Solution

Answer to Problem 147RE

$z0=4(cos600+isin600),z1=4(cos1500+isin1500),$

$z2=4(cos2400+isin2400)$ and $z3=4(cos3300+isin3300)$ .

Explanation of Solution

Given information:

Write each of the roots in trigonometric form.

EBK PRECALCULUS W/LIMITS, Chapter 6, Problem 147RE , additional homework tip 1

Calculation:

From the given information, write each of the roots in trigonometric form using the graph of roots are as shown below:

Let’s note the roots:

$zi=r1(cosθi+isinθi)$ , where

$i=0,1,2$ .

From the given graph, there are three roots, then they are:

$3600n=36003=1200$

Starting from the $x−$ axis going counter clockwise, the roots are:

Now, write the roots:

$z0=2(cos300+isin300)z1=2(cos1500+isin1500)z2=2(cos2700+isin2700)$

Thus, the roots in trigonometric forms are $z0=2(cos300+isin300),z1=2(cos1500+isin1500),$ and $z2=2(cos2700+isin2700)$ .

To determine

The complex number by using roots. Verify the results by using graphing utility.

Expert Solution

Answer to Problem 147RE

$8i$ .

Explanation of Solution

Given information:

Identify the complex number whose roots are given.Use a graphing utility to verify your results.

EBK PRECALCULUS W/LIMITS, Chapter 6, Problem 147RE , additional homework tip 2

Calculation:

From the given information, identify the complex number whose roots are given and by using graphing utility verify the result are as shown below:

From the part (a) found the roots:

$z0=2(cos300+isin300)z1=2(cos1500+isin1500)z2=2(cos2700+isin2700)$

These equations verify the equation:

$z3=ω$

Now find

$ω$ , by using the fact that any

$zi$ verifies the above equation:

$ω=z04=z14=z24=z34$

To find $ω$ ,

$ω=z03=[2(cos300+isin300)]3z03=23[cos(3⋅300)+isin(3⋅300)]z03=8(cos900+isin900)z03=8(0+i)z03=z=8i$

Thus , the complex number is $8i$ .

Chapter 6, Problem 146RE

Chapter 6, Problem 148RE

Chapter 6 Solutions

EBK PRECALCULUS W/LIMITS

Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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The roots in the trigonometric form by using the graph of the roots of the complex number .

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The roots in the trigonometric form by using the graph of the roots of the complex number.

Answer to Problem 147RE

Explanation of Solution

The complex number by using roots. Verify the results by using graphing utility.

Answer to Problem 147RE

Explanation of Solution

Chapter 6 Solutions