Trig/Pre-Calculus

You will: •Write complex numbers from trig form to standard form.•Find the product and quotient of complex numbers in trig form•Use DeMoivre’s Theorem to find powers of complex numbers

Today’s Lesson: 6.5b Complex Numbers in Trig Form

From Trig Form to Standard Form

3sin

3cos3

iz

z 31

23

2i

z 3

23 3

2i

From Trig Form to Standard Form

z 8 cos23

isin23

z 8 1

23

2i

z 4 4 3i

From Trig Form to Standard Form

135sin135cos6 iz

iz2

2

2

26

iz 2323

Product and Quotient of Two Complex Numbers

z1 r1 cos1 isin1

z2 r2 cos2 isin2

z1z2 r1r2 cos 1 2 isin 1 2

z1z2

r1r2cos 1 2 isin 1 2

Product

Quotient

For

r1cis1

r2cis 2

212121 cisrrzz

212

1

2

1 cisr

r

z

z

z1 2cis23

z2 8cis116

z1z2 2 8 cis 23

116

Find the product the complex numbers.

2 8 cis 46

116

2 8 cis 156

16cis52

16cis2

16 cos2

isin2

16 0 i

16i

Find the product the complex numbers.

z1 3cis3

z2 4cis6

z1z2 3 4 cis 3

6

12cis26

6

12cis36

12cis2

12 0 i

12i

Find the quotient the complex numbers.

z1 24cis300

z2 8cis75

z1z2

24

8cis 300 75

3cis 225

3 2

2

i

2

2

3 2

23 2

2i

Find the quotient the complex numbers.

z1 cis40

z2 cis10

z1z2

cis 40 10

cis 30

3

21

2i

Evaluate completely.

1 3i 12

iiiiii

iiiiiii

313131313131

31313131313131

DeMoivre’s TheoremPowers of Complex Numbers

z r cos isin

zn rn cosn isinn

For

zn rncisn

Use DeMoivre’s Theorem to find

3 cos150 isin150 4

3cis150 4

3 4cis 4 150

81cis600

81cis 600 360 81cis240

DeMoivre’s Theorem

81 cos240 isin240

81 1

23

2i

81

281 3

2i

Use DeMoivre’s Theorem to find

1 3i 12

r 1 2 3 2 2

Write in trig form (rcis )

QII

23

1 3i 12 2cis23

12

212cis 12 23

4096cis8

4096cis0

4096 cos0 isin0

4096 10i

4096

DeMoivre’s Theorem 4

31

3arctan'

Use DeMoivre’s Theorem to find

1 i 6

r 1 2 1 2 2

'arctan1

1

4

Write in trig form (rcis )

QI

4

1 i 6 2cis4

6

2 6cis 6 4

21

2

6

cis32

26

2cis32

DeMoivre’s Theorem 3

2

23cis32

8 cos32

isin32

8 0 1 i

8i

Use DeMoivre’s Theorem to find

1 i 8

r 1 2 1 2 2

'arctan1

1

4

Write in trig form (rcis )

QII

34

1 i 8 2cis34

8

2 8cis 8 34

21

2

8

cis6

24cis0DeMoivre’s Theorem 2

16 cos0 isin0

16 1 0 i

16

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