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Section 6.5
Complex Numbers in Polar Form
Complex Plane
b
• z = a + bi
Complex number
z = a + bi
a
Plotting Complex Numbers & Finding the Absolute
Value of Complex Numbers
The distance from 0 to a number a on a number line
is |a|.
|a|
The distance from the origin to the point z in the
complex plane is the absolute value of z = a + bi
denoted by |z|
|z|= a 2 + b2
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Plot each complex number and find its absolute
value.
z = 3i
Same as
z = 0 + 3i
z=4
z = 4 + 0i
z = 2 + 5i
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z = ‐3 – 4i
Polar Form of a Complex Number
z= a + bi is in rectangular form.
b r
(a b)
•(a,b)
r = a 2 + b2
θ
a
cos θ =
a
r
a = rcosθ
sin θ =
b
r
tan θ =
b
a
b = rsinθ
z = a + bi
Z = rcosθ + (rsinθ)i
Z = r(cosθ + isinθ) This is the polar form of a
complex number.
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The value of r is called the modulus of z and the
angle θ is called the argument of z with
0 ≤ θ < 2π. Do not forget the interval in which θ
mustt lie
li when
h writing
iti in
i polar
l form.
f
Plot each complex number. Then write the
complex number in polar form. Argument may
be expressed in degrees or radians.
•
1+ 3 i
Z = 1+ 3 i
a=1
b= 3
Find r
r=
Findθ
tanθ = b/a=
( (1) ) + ( 3 )
2
2
= 1+ 3 = 4 = 2
3
Since θ is in quadrant I, θ = π/3
z = 1 + 3 i = 2(cos
π
π
+ i sin )
3
3
o
o
or 2(cos 60 + i sin 60 )
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1− 5 i
Change from polar form to rectangular form or
writing a complex number in rectangular form.
p number in rectangular
g
form.
Write each complex
Round to nearest tenth.
12(cos60o + isin60o)
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4(cos
5π
5π
+ i sin )
6
6
30(cos 2.3 + i sin 2.3)
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How to Find the Product of Two Complex Numbers
z1 = r1(cosθ1 + i sin θ1)
z2 = r2 (cos θ2 + i sinθ2)
z1z2 = r1r2[cos(θ1+ θ2) +isin(θ1 +θ2)]
Find products. Leave answers in polar form.
z1 = 4(cos 15o + i sin 15o)
z2 = 7(cos 25o + i sin 25o)
z1 = 3(cos120o+isin120o)
z2 = 6(cos250o +isin250o)
Find z1z2
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How to Find the Quotient of Two Complex Numbers
z1 = r1(cosθ1 + isinθ2)
z2 = r2(cosθ2 + isinθ2)
z1 r1
= [ cos(θ1 − θ 2 ) + i sin(θ1 − θ 2 ) ]
z2 r2
z1 = 50(cos80o + i sin 80o )
z2 = 10(cos 20o + i sin 20o )
Power of a Complex Number
8