J
an
O
19
P
E
N
E
R
Perform the indicated operation.
1. (1+i)(2+2i)
In the complex plane, x is the real axis and y
is the imaginary axis. To graph the complex
number z = a + bi, graph the vector <a, b>
imaginary
2. (−1+i√3)(−1+i√3)(−1+i√3)
6.5 Complex Numbers
Let z = a + bi be a complex number where a and
b are both real numbers.
real
2+2i
3. 1+i
The absolute value of a complex number
is is the distance from the origin to the
point in the complex plane.
Find the absolute value of z = 3 ­ 2i
The polar (also called trigonometric) form of the
complex number z = a + bi is
z = r(cosθ + sinθi)
a = rcosθ
b = rsinθ
r = √a2 + b2
θ = tan-1 b/a
(angle with positive x axis
aka reference angle)
r is the modulus of z
θ is the argument of z
r = √a2 + b2
θ = angle from
positive x axis
Write the complex number z = 3 + 2i in
polar form.
Let z = ­3 + 5i.
Graph z. Find r and θ.
Write the complex number z = ­4 ­ 7i in polar
form.
Write z = 5(cos 35 ­ sin 35 i )
in rectangular form.
Multiplying Complex Numbers in
Trigonometric Form
z1 = r1(cosθ1 + i sinθ1)
z2 = r2(cosθ2 + i sinθ2)
z1z2 = r1r2(cos(θ1 + θ2) + i sin(θ1 + θ2))
Write z =
in rectangular form.
Perform the indicated operation:
π
[2(cos π + i sin π
)][6(cos π
12 +i sin 12 )]
4
4
Dividing Complex Numbers in
Trigonometric Form
z1 = r1(cosθ1 + i sinθ1)
z2 = r2(cosθ2 + i sinθ2)
z1 r 1
z2 = r2 (cos(θ1 - θ2 ) + i sin(θ1 - θ2 ))
Perform the indicated operation.
12(cos 62 + isin 62)
18(cos 37 + isin 37)
MathXL 6.5 day 1