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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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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
HW: p440 #33-39 odds, 51-61 odds, 73-83 odds [Honors: 64,66,70,72]
[Honors: 64,66,70,72]
[Honors: 64,66,70,72]