10.3 Polar Form of Complex Numbers

We have explored complex numbers as solutions.Now we connect to both the rectangular and polar planes.

Every complex number can be represented in the forma + bi

a

P

θ

br

/ absolute value of a + bi

horizontal axis = real axisvertical axis = imaginary axis

real part corresponds to

x-axis

a + bi

a = rcos θ

(a, b)

modulus

argumentθ

1i

imaginary part corresponds to

y-axis

(r, θ)

& b = rsin θ

a + bi = rcos θ + irsin θ = r(cos θ + isin θ)

2 2r a bi a b

Ex 1) Graph each complex number and find the modulus. A) 2 + 3i B) –2i

(2, 3)

2 2(2) (3) 4 9 13

B

modulus:

(0, –2)

A

2 2(0) ( 2) 0 4 2

modulus:

The expression r(cos θ + isin θ) is often abbreviated r cis θ. This is the polar form of the complex number.(a + bi is the rectangular form)We need to be able to convert between the forms.

Ex 2) Express the complex number in rectangular form.

2 cos sin3 3

2 62 2 cos 2 sin

3 3 2 2

1 2 3 62 2

2 2 2 2

i

r x y i

A)

38cis 8 cos sin 8cos 8 4 3

6 6 6 6 2

18sin 8 4 4 3 4

6 2

i x

y i

B)

22

1 3 1 3

1 3 ( 1) 3 1 3 2

2 22 cos sin

3 3

w i x y

r w i

i

2 2 4 4 8 2 2

7 72 2 cos sin

4 4

r z i

i

Ex 3) Express each complex number in polar form. Use θ [0, 2π) A) z = 2 – 2i x = 2 y = –2

in QIV

B)

2 2cos

22 2

2 2sin

22 27

4

x

r

y

r

in QII

1cos

2

3sin

22

3

On your

own

2 260 11 ( 60) ( 11) 3721 61

61 cos3.3229 sin 3.3229

61cis 3.3229

r z i

i

Ex 3) Express each complex number in polar form. Use θ [0, 2π) C) z = –60 – 11i x = –60 y = –11

in QIII

2 cos sin

2 cos sin

i

i

D)

1

11tan

6011

tan60

0.1813

3.3229

y

x

needs to be positive!remember: cos (–π) = cos π sin (–π) = –sin π

Ex 4) Show that the product of a complex number and its conjugate is always a real number.

Let and its conjugate

Then,

which is a real number.

2 2 2

2 2

( )( )

z a bi z a bi

z z a bi a bi

a abi abi b i

a b

Homework

#1004 Pg 506 #1–27 odd (skip 7 & 11), 33, 46, 47, 48