10.5 Powers of Complex Numbers

Exploration!

Let z = r(cos θ + isin θ)

z2 = z • z = [r(cos θ + isin θ)] • [r(cos θ + isin θ)]

from yesterday… = r2(cos 2θ + isin 2θ)

how about z3 = z • z2 = [r(cos θ + isin θ)] • [r2(cos 2θ + isin 2θ)]

= r3(cos 3θ + isin 3θ)

this pattern keeps happening… let’s generalize!

DeMoivre’s Theorem If z = r(cos θ + isin θ) is a complex number in polar form, then for any integer n,

zn = rn(cos nθ + isin nθ)

Ex 1) Evaluate each power and express answer in rectangular form.

A)

3 54 4 2 29 9 5 5

3 54 4 2 29 9 5 5

4 43 3

312 2

5 cos sin cos sin

5 cos 3 sin 3 ( 1) cos 5 sin 5

125 cos sin 1 cos2 sin 2

125 125 31 0 1

2 2

i i

i i

i i

i i

B)

try on your own

1 0

2 2

123

2

7 7 7 76 6 6 6

3 1 3 1 3 1 41

2 2 2 2 4 4 4

tan6

3 11 cos sin 1 cos sin

2 2

x y r

i i i

Ex 2) Evaluate z7 for . Express answer in rectangular form.

*Hint: Convert to polar do DeMoivre’s convert back

3

2 2

iz

2 2 11 4 4 4

2 3 4

2 3 4

4 4 4 4 4 4

3 32 2 4 4

1 1 2 tan 2 cos sin

2 cos 2 sin 2 2 cos 3 sin 3 2 cos 4 sin 4

2 cos sin 2 2 cos sin 4 cos sin

2 22(0 ) 2 2 4( 1 0 )

2 22 2 2 4

r z i

z z z

i i i

i i i

i i i

i i

A nautilus shell has been traced on the complex plane.

z (1, 1) 1 + i rect complex

Ex 3) Calculate z2, z3, and z4 and see if they lie on the curve of the shell wall.

(1, 1) polar

(divide class into 3 groups, just do one)

Yes, they lie on the sketch of the shell wall!

Ex 4) Evaluate z4 for z = 2(cos 40° – isin 40°)

Do DeMoivre’s … WAIT is something wrong?

it has to be in complex polar can’t subtract!

change to z = 2(cos (–40°) + isin (–40°))

z4 = (2)4(cos (–160°) + isin (–160°))

z4 ≈ –15.0351 – 5.4723i

(calculator! )

*Remember: z0 = 1 and

*DeMoivre’s Theorem doesn’t just have to be positive powers.

Ex 5) Use DeMoivre’s Thm to evaluate and express in rectangular form.

A) B)

1nn

zz

523

5 2 23 3

10 103 3

6 cis

6 cos 5 sin 5

1cos sin

7776

1 1 1 3

7776 2 7776 2

1 3

15552 15552

i

i

i

i

412

2

412

2

116

116

cis 10

5 cis 25

cis 40

5 cis 50 cis 40

25 cis 50

cis (40 50 )251

cis ( 10 )400

try on your own

2 23 3

Homework

#1006 Pg 520 #1, 3, 5, 9, 11, 13, 16, 20, 21, 23, 29, 31, 32