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January 06, 2015
Section 6.5 Trigonometric Form of a Complex Number
Objective: Multiply and divide complex numbers in trigonometric form
and find powers and nth roots of complex numbers.
Imaginary
axis
Remember from Section 2.4...
Complex number z = a + bi
Real axis
Absolute Value of a Complex #
la + bil = √ a2 + b2
*distance between (0, 0) and (a, b)
Ex1: Find absolute value of z = 4 - 3i.
Trigonometric Form of the Complex # z = a + bi
is given by z = r(cos θ + i sin θ)
where a = r cos θ,
b = r sin θ,
r = √a + b , and
tan θ = b/a.
2
(a, b)
r
b
2
The number r is the modulus of z,
and θ is called an argument of z.
Imaginary
axis
θ
a
*usually, 0 ≤ θ < 2π
Real axis
lar form !
Po
Also called
Ex2: Write z = -2 + 2i in trigonometric form.
tan θ = 2/(-2) = -1
therefore θ = 3π/4
Ex3: Write 4(cos (5π/6) + i sin (5π/6)) in a + bi form.
or.
calculat
e
h
t
y
r
T
4(cos (5π/6) + i sin (5
January 06, 2015
Multiplication and Division
Let z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2) be
complex numbers. Then:
z1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]
Product
Multiply moduli and add arguments.
z 1 r1
= [cos(θ1 - θ2) + i sin(θ1 - θ2)], z2≠ 0
z 2 r2
Quotient
Divide moduli and subtract arguments.
Ex4: If z1 = 8(cos 120° + i sin 120°) and z2 = 6(cos 150° + i sin 150°),
then find the following:
a) z1z2
b)
a) z1 z 2 = 48(cos (12
= 48(cos (27
b) z1 /z2 = (8/6)[co
= (4/3)
z1
z2
Check this out!
Double angle formula
z = r(cos θ + i sin θ)
z = r(cos θ + i sin θ)r(cos θ + i sin θ) = r (cos 2θ + i sin 2θ)
z = r2(cos 2θ +i sin 2θ) r(cos θ + i sin θ) = r (cos 3θ + i sin 3θ)
2
2
3
3
z = r (cos 4θ +i sin 4θ)
4
4
Leads to...
DeMoivre’s Theorem
If z = r(cos θ + i sin θ) is a complex number and n is a positive
integer, then z = [r(cos θ + i sin θ)] = r (cos nθ + i sin nθ).
n
n
n
r!
of complex #s easie
Makes finding powers
Ex5: Evaluate (-2√3 - 2i)5 .
January 06, 2015
Roots of Complex Numbers
The complex number u = a + bi is an nth root of the complex
number z if z = un = (a + bi)n.
For a positive integer n, the complex number z = r(cos θ + i sin θ)
has exactly n distinct nth roots given by ...
n
√r (cos θ +n2πk + i sin θ +n2πk ) where k = 0, 1, 2, . . . , n - 1.
Ex6: Find the four fourth roots of 16.
Ex7: Find the three third roots of 4√3 - 4i.