Sec. 6.6b

One reason for writing complex numbers inOne reason for writing complex numbers intrigonometric form is the convenience for multiplyingtrigonometric form is the convenience for multiplyingand dividing:and dividing:

• The The product product involves the product of the involves the product of the moduli moduli and the sum of the and the sum of the argumentsarguments

• The The quotientquotient involves the quotient of the involves the quotient of the moduli moduli and the difference of the and the difference of the argumentsarguments

2 2 2 2cosθ sin θz r i

1 2 1 2 1 2 1 2cos θ θ sin θ θz z r r i

1 11 2 1 2

2 2

cos θ θ sin θ θz r

iz r

2 0r

1 1 1 1cosθ sin θz r i Let

and

1.

. Then

2. ,

1

π π25 2 cos sin

4 4z i

2

π π14 cos sin

3 3z i

π π

350 2 cos sin 478.109 128.10912 12

i i

Express the product of the given complex numbers instandard form:

Product:

1 2 2 cos135 sin135z i

2 6 cos300 sin 300z i

2cos 165 sin 165 0.455 0.122

3i i

Express the quotient of the given complex numbers instandard form:

Quotient:

1

3 33 cos sin

4 4z i

2

1cos sin

3 6 6z i

1 2

3 11 11cos sin

3 12 12z z i

Express the product of the given complex numbers intrigonometric form:

1 5 cos 220 sin 220z i

2 2 cos115 sin115z i

1

2

5cos105 sin105

2

zi

z

Express the quotient of the given complex numbers intrigonometric form:

Find the product and quotient of the given complex numbersin two ways, (a) using standard forms, and (b) using trig. forms.

1 2 3z i 2 1 3z i The product:

1 2 2 3 1 3z z i i 22 2 3 3 3 3i i i

2 3 3 2 3 3 i 3.196 6.464i

The quotient:

1

2

2 3

1 3

z i

z i

1 3

1 3

i

i

2 3 3 2 3 3

4

i

1.799 0.116i

Find the product and quotient of the given complex numbersin two ways, (a) using standard forms, and (b) using trig. forms.

1 2 3z i 2 1 3z i Next, find the trigonometric forms:

1 13 cos 0.983 sin 0.983z i 2 2 cos 3 sin 3z i

The product:

1 2 2 13 cos 0.983 3 sin 0.983 3z z i 3.196 6.464i The quotient:

1

2

13cos 0.983 3 sin 0.983 3

2

zi

z 1.799 0.116i

First, let’s look at a problem: cosθ sin θz r i

2 cosθ sin θ cosθ sin θz z z r i r i

2 cos θ θ sin θ θr i 2 cos 2θ sin 2θr i

RealAxis

ImaginaryAxis

z

z2

Graphically:Graphically: 2θ

θrr 2

Now, let’s find the cube of z:3 2z z z

2cosθ sin θ cos 2θ sin 2θr i r i

3 cos θ 2θ sin θ 2θr i

3 cos3θ sin 3θr i

4 4 cos 4θ sin 4θz r i

5 5 cos5θ sin5θz r i

And the pattern continues for higher powers:

De Moivre’s TheoremDe Moivre’s TheoremThis pattern is generalized to give:

Let

and let n be a positive integer. Then

cosθ sin θz r i

cos sinnnz r i

cos θ sin θnr n i n

31 3i

πθ

3

Find using De Moivre’s Theorem.

Begin witha graph:

Modulus r = 2

Argument

2 cos sin3 3

z i

31 3i

31 3 8i

Find using De Moivre’s Theorem.

Verify with your calculator!!! Verify with your calculator!!!

2 cos sin3 3

z i

3 32 cos 3 sin 3

3 3z i

8 cos sini 8 1 0i 8

8

2 2

2 2i

Find using De Moivre’s Theorem.

Convert to trig. form:3 3

cos sin4 4

z i

8 3 3cos 8 sin 8

4 4z i

cos6 sin 6i 1 0i 1

31 3 8i

8

2 21

2 2i

1 3i2 2

2 2i

The complex number

is a third root of –8

The complex number

is an eighth root of 1

15 15243 cos sin 243

2 2i i

53 3

3 cos sin2 2

i

Use De Moivre’s Theorem to find the indicated powerof the given complex number. Write your answer instandard form.

10 101296 cos sin 648 648 3

3 3i i

45 5

6 cos sin6 6

i

Use De Moivre’s Theorem to find the indicated powerof the given complex number. Write your answer instandard form.

20 205 cos18.546 sin18.546 5 0.954 0.299i i

203 4i

Use De Moivre’s Theorem to find the indicated powerof the given complex number. Write your answer instandard form.

205 cos0.927 sin 0.927i

cos sin 1i

3

1 3

2 2i

Use De Moivre’s Theorem to find the indicated powerof the given complex number. Write your answer instandard form.

3

cos sin3 3i