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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]