Trigonometric (Polar) Form of

Complex Numbers

How is it Different?In a rectangular system, you

go left or right and up or down.

In a trigonometric or polar system, you have a direction to travel and a distance to travel in that direction.

2 2z i 2 cos45 sin 45

(2,45)

z i

Polar form

RealAxis

Imaginary Axis

Remember a complex number has a real part and an imaginary part. These are used to plot complex numbers on a complex plane.

z a bi

b

a

z

2 2z a b

1tanb

a

The angle formed from the real axis and a line from the origin to (a, b) is called the argument of z, with requirement that 0 < 2.

modified for quadrant and so that it is between 0 and 2

z a bi The absolute value or modulus of z denoted by z is the distance from the origin to the point (a, b).

Trigonometric Form of a Complex Number

2 2The modulus is r a b

1

The argument can be found

by using tan adjusting for

correct quadrant if necessary

b

a

b

a

cos sinz r i

Note: You may use any other trig functions and their relationships to the

right triangle as well as tangent.

r

RealAxis

Imaginary Axis

r

�

Plot the complex number and then convert to trigonometric form: iz 3

Find the modulus r

1

3

2413 22r

3

1tan 1 but in Quad II

6

5

6

5sin

6

5cos2

iz

Find the argument

It is easy to convert from trigonometric to rectangular form because you just work the trig functions and distribute the r through.

i 3

6

5sin

6

5cos2

iz

i

2

1

2

32

2

3 2

1

If asked to plot the point and it is in trigonometric form, you would plot the angle and radius.

2

6

5 Notice that is the same as plotting

3 i 31

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

Graphing Utility:Write the complex numberin standard form a + bi.

3 33.75 cos sin4 4

i

3 33.75 cos sin4 4

i

2.652 2.652i

[2nd] [decimal point]

Multiplying Complex NumbersTo multiply complex numbers in

rectangular form, you would FOIL and convert i2 into –1.

To multiply complex numbers in trig form, you simply multiply the rs and

and the thetas.

2

a bi c di

ac adi bci dbi

ac adi bci db

ac db ad bc i

1 1 1 2 2 2

1 2 1 2 1 2

cos sin cos sin

cos sin

r i r i

r r i

The formulas are scarier than they are.

use sum formula for sinuse sum formula for cos

Replace i 2 with -1 and group real terms and then imaginary terms

irr 2121212121 sincoscossinsinsincoscos

Must FOIL these

221121 sincossincos iirr

21 2 1 2 2 1 1 2 1 2cos cos sin cos sin cos sin sinr r i i i

Let's try multiplying two complex numbers in trigonometric form together.

1111 sincos irz 2222 sincos irz

1 2 1 1 1 2 2 2cos sin cos sinz z r i r i

212121 sincos irr

Look at where we started and where we ended up and see if you can make a statement as to what happens to the r 's and the 's when you multiply two complex numbers.

Multiply the Moduli and Add the Arguments

Example

1 2

1

2

2 3 2 4 cos30 sin 30

3 2 3 2 6 cos45 sin 45

multiply z z

Where z i i

z i i

1 2

2

2 3 2 3 2 3 2

6 6 6 6 6 2 6 2

6 6 6 6 6 2 6 2

6 6 6 2 6 6 6 2

z z

i i

i i i

i i

i

Rectangular form Trig form

1 2

4 cos30 sin 30 6 cos45 sin 45

4 6 cos 30 45 sin 30 45

24 cos75 sin 75

z z

i i

i

i

2 2

6 6 6 2 6 6 6 2

216 72 12 72 216 72 12 72

576 24

r

r

r

1 6 6 6 2tan 75

6 6 6 2

Dividing Complex Numbers

In rectangular form, you rationalize using the complex

conjugate.

2

2 2 2

2 2

2 2 2 2

a bi

c dia bi c di

c di c di

ac adi bci bdi

c d iac adi bci bd

c dac bd bc ad

ic d c d

In trig form, you just divide the rs and subtract the theta.

1 1 1

2 2 2

11 2 1 2

2

cos sin

cos sin

cos sin

r i

r i

ri

r

Then numbers.complex twobe

sincos and sincosLet 22221111 irzirz

zz

rr

i1

2

1

21 2 1 2 cos sin

then,0 If 2 z

(This says to multiply two complex numbers in polar form, multiply the moduli and add the arguments)

(This says to divide two complex numbers in polar form, divide the moduli and subtract the arguments)

21212121 sincos irrzz

wzzw

iwiz

(b) (a) :find

,120sin120cos6 and 40sin40cos4 If

4 cos 40 sin 40 6 cos120 sin120zw i i

12040sin12040cos64 i

24 cos160 sin160i

multiply the moduli add the arguments (the i sine term will have same argument)

If you want the answer in rectangular coordinates simply compute the trig functions and multiply the 24 through.

i34202.093969.024

i21.855.22

zw

i

i

4 40 40

6 120 120

cos sin

cos sin

46

40 120 40 120cos sin i

23

80 80cos sin i

23

280 280cos sin i

divide the moduli subtract the arguments

In polar form we want an angle between 0 and 360° so add 360° to the -80°

In rectangular coordinates:

ii 66.012.09848.01736.03

2

Example

1

2

1

2

3 2 3 2 6 cos45 sin 45

2 3 2 4 cos30 sin 30

zdivide

z

Where z i i

z i i

Rectangular form

2

2

3 2 3 2

2 3 2

3 2 3 2 2 3 2

2 3 2 2 3 2

6 6 6 2 6 6 6 2

12 4

6 6 6 2 6 6 6 2

12 4

6 6 6 2 6 6 6 2

16 16

i

i

i i

i i

i i i

i

i i

i

Trig form

6 cos45 sin 45

4 cos30 sin 30

6cos 45 30 sin 45 30

43

cos15 sin152

i

i

i

i

2 2

6 6 6 2 6 6 6 2

16 16

216 72 12 72 216 72 12 72

256

576 9 3

256 4 2

r

r

r

1

6 6 6 2

16tan 156 6 6 2

16

Powers of Complex NumbersThis is horrible in rectangular

form.

...

na bi

a bi a bi a bi a bi

The best way to expand one of these is using Pascal’s

triangle and binomial expansion.

You’d need to use an i-chart to simplify.

It’s much nicer in trig form. You just raise the r to the power and multiply theta by the exponent.

cos sin

cos sinn n

z r i

z r n i n

3 3

3

5 cos20 sin 20

5 cos3 20 sin 3 20

125 cos60 sin 60

Example

z i

z i

z i

Roots of Complex Numbers

• There will be as many answers as the index of the root you are looking for– Square root = 2 answers– Cube root = 3 answers, etc.

• Answers will be spaced symmetrically around the circle– You divide a full circle by the number of

answers to find out how far apart they are

General Process

1. Problem must be in trig form

2. Take the nth root of n. All answers have the same value for n.

3. Divide theta by n to find the first angle.

4. Divide a full circle by n to find out how much you add to theta to get to each subsequent answer.

The formula

cos sin

360 360 2 2cos sin cos sinn n n

z r i

k k k kz r i or r i

n n n n

k starts at 0 and goes up to n-1

This is easier than it looks.

Example Find the 4th root of 81 cos80 sin80z i

1. Find the 4th root of 81 4 81 3r

2. Divide theta by 4 to get the first angle.

8020

4

3. Divide a full circle (360) by 4 to find out how far apart

the answers are.

36090 between answers

4

4. List the 4 answers.

• The only thing that changes is the angle.

• The number of answers equals the number of roots.

1

2

3

4

3 cos20 sin 20

3 cos 20 90 sin 20 90 3 cos110 sin110

3 cos 110 90 sin 110 90 3 cos200 sin 200

3 cos 200 90 sin 200 90 3 cos290 sin 290

z i

z i i

z i i

z i i