1i 12 iGiven.

12 i

ii 1

iiiii 123

111224 iii

iiiii 145

111246 iii

iiiii 1347

111448 iii

Notice a pattern!

i to an odd power is always equal to i, just need to

determine the sign.

i to an even power is always equal to 1, just need to

determine the sign.

iiodd 1eveniTime to determine the sign.

Odd number of negatives multiplied together is negative and an Even number of negatives is a positive.

Find the number of i2 are in in.

2nEVEN Whole number = +

ODD Whole number = –

75i5.37275 i 89i

5.44289 i 102i

512102 1 124i

622124 1

MAT 182 Chapter 8Section 8.1 Complex Numbers.

Simplify radicals with negatives.

3 4 11 50

205 4916

31

31

3i

i

4i

2ii2

i

11i

i

225 i5

25i

205ii

i i

1002i

10

i i

4916 ii

ii 74

i11

bia Real

NumberimaginaryNumber

Definition of Complex Numbers.

Complex Numbers.

Real Numbers. Imaginary Numbers.

12 i

When the directions read, “Leave the answers in a + bi form.” The answer will have to include a zero if there is no real number or imaginary number.

For example.If the answer is 2, then we write the answer as 2 + 0i.

If the answer is -5i, then we write the answer as 0 – 5i.

Combine Like Terms. Treat i like a variable.

ii 73212 i414

ii 8956

i1315

Distribute the minus sign.

ii 8596

iii 3252 2610 ii

1610 i

i106

iiii 25352434 iiii 22322333 21015812 iii

110712 i

10712 i

i722

24669 iii

149 49

13

Complex Conjugate Product Rule. 22 babiabia

ii 3535

Rationalize the Denominator above rule!

i

i

i

i

25

25

25

32

425

615410 2

iii

6

ii

29

11

29

16

29

1116

i

i

i

i

5

47

The denominator is a single term, just multiply

by i top and bottom.

2

2

5

47

i

ii 5

47

i

5

74

i

5

7

5

4 i

22 35 34925

Solving Quadratic equations that create complex solutions.Solve for x.

094 2 x 0123 2 x

4

92 x

ix2

3

123 2 x

323 ix

323 ix

Solving Quadratic equations that create complex solutions.Solve for x.

01062 xx 0683 2 xx

____10____62 xx 23

13 2 x

9

ix 3

ix 3

a

acb

a

bx

2

4

2

2

6

6348

6

8 2 x

6

7264

3

4 x

6

12

3

4 x

6

32

3

4 ix

3

3

3

4 ix

SECTION 8.2 Complex Numbers in Polar Form A complex number a + bi is represented as a point (a, b) in a coordinate plane. The horizontal axis of the coordinate plane is called the real axis. The vertical axis is called the imaginary axis. The coordinate system is called the complex plane.When we represent a complex number as a point in the complex plane, we say that we are plotting the complex number.

We plot (a, b) as if it were (x, y).

ii 3426 i52

bia

A complex number in the form a + bi is said to be in rectangular form.

The expression is called the polar form of a complex number. sincos ir

a

b

22 bar

r

Polar notations.

cosra sinrb

a

btan

The number r is the modulus of a + bi, and is called an argument of a + bi.

22 bar

A shortcut notation for cisrir sincos

Writing a Complex number (Rect.) into Polar Form.

Convert – 2 – 2i into Polar Form.

1st Plot the point to determine the angle.

2,2

2

2tan

a

b

real

imaginary

451tan 12

2

4

5225

or

2nd Find r.

2222 22 bar

228 r

sincos ir

225sin225cos22 i

45

45 sincos22 i

Writing a Complex number (Rect.) into Polar Form.

Convert into Polar Form.

1st Plot the point to determine the angle. 1,3

3

1tan

a

b

real

imaginary

30tan3

111

3

6

5150

or

2nd Find r.

2222 13 bar

24 r

sincos ir

150sin150cos2 i

65

65 sincos2 i

i 3

Writing a Complex number (Polar) into Rectangular Form.

Convert into Rectangular Form.

1st Find the exact values for the cosine and sine.

3

60sin60cos2 i

60sin60cos2 i 60

301

2

2

3

2

12 i

1 3i

Convert into Rectangular Form.

1st Find the values for the cosine and sine with the calculator.

115sin115cos6 i

i906307787.04226182617.06

i437846722.553570957.2

Section 8.3 Multiplication and Division of Complex Numbers.Given two complex numbers in trigonometric form.

The product is

AiAr sincos1 BiBr sincos2 and

BiBAiArr sincossincos21

BAiABiBAiBArr sinsinsincossincoscoscos 221

BAABiBAiBArr sinsinsincossincoscoscos21

ABBAiBABArr sincossincossinsincoscos21

Use the sum formulas for sine and cosine,

BAiBArr sincos21

F O I L

Find the product of the complex numbers.

50sin50cos3 i 130sin130cos4 i

212121 sincos irr

13050sin13050cos43 i

180sin180cos12 i

Find the product of the complex numbers.

0112 i 12

8.497.22.1253.9 ciscis 2.125sin2.125cos3.9 i 8.49sin8.49cos7.2 i

8.492.1257.23.9 cis

17511.25 cis 175sin175cos11.25 i i1885.20144.25

and

Section 8.3 Multiplication and Division of Complex Numbers.Given two complex numbers in trigonometric form.

The quotient is

AiAr sincos1 BiBr sincos2 and BiBr

AiAr

sincos

sincos

2

1

Multiply top and bottom by the conjugate of

the denominator.

BiBr

AiAr

sincos

sincos

2

1

BiB

BiB

sincos

sincos

BiB

BAiBAiBAiBA

r

r222

2

2

1

sincos

sinsincossinsincoscoscos

BB

BABAiBABA

r

r22

2

1

sincos

sincoscossinsinsincoscos

1

1

BAiBAr

r sincos

2

1

Find the quotient of the complex numbers.

70sin70cos8 i 130sin130cos4 i

Find the quotient of the complex numbers.

507.2

251.8

cis

cis

and

BAiBAr

r sincos

2

1 13070sin13070cos4

8i

60sin60cos2 i

2

3

2

12 i 31 i

5025sin5025cos7.2

1.8i

25sin25cos3 i

i2679.17189.2

Power of Complex Numbers.Given a complex number in trigonometric form.

AiAr sincos

The pattern leads to DeMoivre’s Theorem

212121 sincos irr

AiArAiArAiAr sincossincossincos 2

AiArAiArAiAr sincos2sin2cossincos 23

AiAr 4sin4cos4

AiAr 5sin5cos5

ninrir nn sincossincos , where n is a positive integer.

AiAr 2sin2cos2

AiAr 3sin3cos3

4sincos AiAr

5sincos AiAr

Find and write the result in rectangular form.

60

1

3

360

or 24312222 bar

8860sin60cos231 ii

608sin608cos2 8 i

480sin480cos256 i

23

21256 i 3128128 i

831 i

120sin120cos256 i60

1

3

2

2

Working De Moivre’s Theorem backwards.

Find the 3 cube roots of .

135sin135cos8 i

135sin135cos83sin3cos3 iir

This implies that r = 2, and .135sin3sin135cos3cos and

3 must represent a coterminal angle with .135

k 3601353

3

360135 k k is any integer.

453

03601350 k 45sin45cos2 i

3

1360451

k 165sin165cos2 i

120165

3

2360452

k 285sin285cos2 i

120285

Finding nth Roots of a Complex Numbers.Given a complex number in trigonometric form and n is a positive integer, has exactly n distinct roots given by sincos ir

n

ki

n

krn 2

sin2

cos , where k = 0, 1, 2, 3, … n – 1.

Find all complex fourth roots of .

388 i

; k = 0, 1, 2, 3

iii

i

32

1

2

3230sin30cos2

09030sin4

0360120cos164

388 i

318 i 360

30

1

2

120sin120cos16 i 36464

38822

22

bar

1628464

2Q

n

ki

n

krn 2

sin2

cos

real

imaginary

38

81

1682

0k

ii

i

31120sin120cos2

19030sin19030cos164

ii

i

13210sin210cos2

29030sin29030cos164

ii

i

31300sin300cos2

39030sin39030cos164

1k

2k

3k

Find all complex roots of x5 – 1 = 0. There is one real solution, x = 1, but there are 5 complex solutions. The first one is rewriting 1 in trigonometric form, where r = 1.

0sin0cos1011 ii

5

3600 k for k = 0, 1, 2, 3, 4.

007200 k

115 k 720

0sin0cos1 i

7217201 k 72sin72cos1 i

14427202 k 144sin144cos1 i

21637203 k 216sin216cos1 i

28847204 k 288sin288cos1 i

SECTION 8.5The foundation of the polar coordinate system is a horizontal ray that extends to the right. The ray is called the polar axis. The endpoint of the ray is called the pole. A point P in the polar coordinate system is represented by an ordered pair of numbers . We refer to the ordered pair as the polar coordinates of P. r is a directed distance from the pole to P. is an angle from the polar axis to the line segment from the pole to P. This angle can be measured in degrees or radians. Positive angles are measured counterclockwise from the polar axis. Negative angles are measured clockwise from the polar axis.

,r ,rP

60o

90o

120o

150o

210o

240o

270o

300o

330o

30o

2 4 6180o 0o Plot the following points.

Find 3 different ways to rewrite the coordinates of point A.

135,4,300,6

,120,3,195,5

DC

BAA

15,5A

B

C

D

165,5A 375,5A 555,5A

Relations between Polar and Rectangular Coordinates

Find the rectangular coordinates of the points with the following polar coordinates:

2

3,2

6,10

2

3,2,

r

cosrx 2

3cos2 002

sinry 2

3sin2 212

2,0

6,10,r

cosrx 6cos10 3510 2

3

sinry 6sin10

5,35

510 21

Find the polar coordinates of the points with the following rectangular coordinates:

3,122222 yxrryx

13tan 24

603tan 1

32,2

2Q

x

ytan

22 31 r

60

3

2120

or

Convert each rectangular equation to a polar equation.

5 yx 11 22 yx

Replace x with r cos and y with r sin . Simplify and solve for r.

5sincos rr 5sincos r

sincos

5

r

1sin1cos 22 rr

1sin1cos2cos 2222 rrr

11cos2sincos 2222 rrr

11cos2sincos 222 rr

0cos22 rr

0cos2 rr

0cos2 r0r

cos2r

Convert the polar equation to rectangular equations.

222 ryx x

ytan

5r

cosrx sinry We will need the following equations.

252 r

2522 yx

A.4

B.

4tantan

1tan

1x

y

xy 1

Convert the polar equation to rectangular equations.

222 ryx x

ytan

csc3r

cosrx sinry We will need the following equations.

C. D.

sin

13r

sinsin

13sin r

3sin r

3y

cos6r

cos6 rrr

cos62 rr

xyx 622

996 22 yxx

93 22 yx

Convert the polar equation to rectangular equations.

222 ryx x

ytan cosrx sinry

We will need the following equations.

sin4cos4 r

sin4cos4 rrrr

sin4cos42 rrr yxyx 4422

044 22 yyxx

22 ____0__4__4 22 yyxx 22 4 4

822 22 yx