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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

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Trig/Pre-Calculus. Today’s Lesson: 6.5b Complex Numbers in Trig Form. 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. From Trig Form to Standard Form. - PowerPoint PPT Presentation

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Page 1: Trig/Pre-Calculus

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

Page 2: Trig/Pre-Calculus

From Trig Form to Standard Form

3sin

3cos3

iz

z 31

23

2i

z 3

23 3

2i

Page 3: Trig/Pre-Calculus

From Trig Form to Standard Form

z 8 cos23

isin23

z 8 1

23

2i

z 4 4 3i

Page 4: Trig/Pre-Calculus

From Trig Form to Standard Form

135sin135cos6 iz

iz2

2

2

26

iz 2323

Page 5: Trig/Pre-Calculus

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

Page 6: Trig/Pre-Calculus

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

Page 7: Trig/Pre-Calculus

Find the product the complex numbers.

z1 3cis3

z2 4cis6

z1z2 3 4 cis 3

6

12cis26

6

12cis36

12cis2

12 0 i

12i

Page 8: Trig/Pre-Calculus

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

Page 9: Trig/Pre-Calculus

Find the quotient the complex numbers.

z1 cis40

z2 cis10

z1z2

cis 40 10

cis 30

3

21

2i

Page 10: Trig/Pre-Calculus

Evaluate completely.

1 3i 12

iiiiii

iiiiiii

313131313131

31313131313131

Page 11: Trig/Pre-Calculus

DeMoivre’s TheoremPowers of Complex Numbers

z r cos isin

zn rn cosn isinn

For

zn rncisn

Page 12: Trig/Pre-Calculus

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

Page 13: Trig/Pre-Calculus

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'

Page 14: Trig/Pre-Calculus

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

Page 15: Trig/Pre-Calculus

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

Page 16: Trig/Pre-Calculus

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Page 17: Trig/Pre-Calculus
Page 18: Trig/Pre-Calculus
Page 19: Trig/Pre-Calculus

[Honors: 64,66,70,72]

Page 20: Trig/Pre-Calculus

[Honors: 64,66,70,72]

Page 21: Trig/Pre-Calculus