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Chapter 9: Q. 60 (page 592)

In Problems 53鈥�60, find all the complex roots. Leave your answers in polar form with the argument in degrees.

60. The complex fifth roots of-i.

Short Answer

Expert verified

The complex fifth roots of-iarecos54+isin54,cos126+isin126,cos198+isin198,cos270+isin270,cos342+sin342.

Step by step solution

01

Step 1. Rewrite -i in the polar form.

The magnitude of -iis

0+(-1)2=1=1

Which gives

-i=1(0-i)=(cos270掳-isin270掳)

02

Step 2. Using De Moivre's Theorem.

We get

zk=15[cos(2705+360k5)+isin(2705+360k5)]=[cos(54掳+72掳k)+isin(54掳+72掳k)];k=0,1,2,3,4

03

Step 3. Substitute k=0,1,2,3,4 in zk=cos(54+72k)+isin(54+72k).

We get

z0=cos54+isin54z1=cos126+isin126z2=cos198+isin198z3=cos270+isin270z4=cos342+isin342

So we have found the complex fifth roots of-i.

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