HERE - KNOWING IS NOT UNDERSTANDING
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Transcript HERE - KNOWING IS NOT UNDERSTANDING
Solving Complex Equations
π
of the form π§ = π + ππ
IMPORTANT POINT:
If z = r cis(ο±) then the value of r can
only be a positive number. It is the
length of the complex number!
z3 = 8 or z3 = 8 + 0i
let z = rcis(ο±)
1.
length = 8
angle = 00
so |8+0i| = 8 and arg(8+0i) = 00
r3cis (3ο±) = 8 + 0i
r3cis (3ο±) = 8 cis(0 + 360n)
r3 = 8
r =2
Solutions:
z1 = 2 cis(0)
z2 = 2 cis(120)
z3 = 2 cis(240)
3ο± = 0 + 360n
ο± = 120n
= 0, 120, 240
z3 = β 8
let z = rcis(ο±)
2.
or
z3 = β 8 + 0i
length = +8
angle = 1800
so | β 8 + 0i | = +8 and arg(β 8 + 0i ) = 1800
r3cis (3ο±) = 8 + 0i
r3cis (3ο±) = +8 cis(180 + 360n)
r3 = 8
r =2
Solutions:
z1 = 2 cis(60)
z2 = 2 cis(180)
z3 = 2 cis(30)
3ο± = 180 + 360n
ο± = 60 +120n
= 60, 180, 300
z3 = 8i or z3 = 0 + 8i
let z = rcis(ο±)
3.
length = +8
angle = 900
so |0 + 8i| = +8 and arg(0 + 8i) = 900
r3cis (3ο±) = 0 + 8i
r = +8 ο± = 900
r3cis (3ο±) = +8 cis(90 + 360n)
r3 = 8
r =2
Solutions:
z1 = 2 cis(30)
z2 = 2 cis(150)
z3 = 2 cis(270)
3ο± = 90 + 360n
ο± = 30 + 120n
= 30, 150, 270
z3 = β 8i or z3 = 0 β 8i
let z = rcis(ο±)
4.
length = +8
angle = 2700
so| 0 β 8i| = +8 and arg(0 β 8i ) = 2700
r3cis (3ο±) = 0 β 8i r = +8 ο± = 2700
r3cis (3ο±) = +8 cis(270 + 360n)
r3 = 8
r =2
Solutions:
z1 = 2 cis(90)
z2 = 2 cis(210)
z3 = 2 cis(330)
3ο± = 270 + 360n
ο± = 90 +120n
= 90, 210, 330
z3 = 3 + 3i
let z = rcis(ο±)
5.
length = β(9 + 9)= β18
angle = 450
so |3+3i| = 18 ½ and arg(3+3i) = 450
r3cis (3ο±) = (18 ½)cis( 45 + 360n)
r3 = 18 ½
r = 18 (1/6)
3ο± = 45 + 360n
ο± = 15 + 120n
= 15, 135, 255
Solutions:
z1 = 18 (1/6)cis(15)
z2 = 18 (1/6)cis(135)
z3 = 18 (1/6)cis(255)
z3 = β3 + 3i
let z = rcis(ο±)
6.
length = β(9 + 9)= β18
angle = 1350
so | β3 + 3i | = 18 ½ and arg(β3 + 3i ) = 1350
r3cis (3ο±) = (18 ½)cis( 135 + 360n)
r3 = 18 ½
r = 18 (1/6)
3ο± = 135 + 360n
ο± = 45 + 120n
= 45, 165, 285
Solutions:
z1 = 18 (1/6)cis(45)
z2 = 18 (1/6)cis(165)
z3 = 18 (1/6)cis(285)
z3 = k2 or
let z = rcis(ο±)
7.
z3 = k2 + 0i
length = +k2
angle = 00
so |k2 + 0i| = +k2 and arg(k2 + 0i) = 00
r3cis (3ο±) = k2+ 0i
r3cis (3ο±) = +k2 cis(0 + 360n)
r3 = k2
r = kβ
3ο± = 0 + 360n
ο± = 120n
= 0, 120, 240
Solutions:
z1 = k β
cis(0)
z2 = k β
cis(120)
z3 = k β
cis(240)
8.
z4 + k2 = 0 then z4 = βk2 + 0i
let z = rcis(ο±)
length = +k2
angle = 1800
so |βk2 + 0i| = +k2 and arg(βk2 + 0i) = 1800
r4cis (4ο±) = +k2 cis(180 + 360n)
r4 = k2
r = k½
4ο± = 180 + 360n
ο± = 45 + 90n
= 45, 135, 225, 315
Solutions:
z1 = k ½ cis(45)
z2 = k ½ cis(135)
z3 = k ½cis(225)
z4 = k ½cis(315)