• p 58 – 60

• #13, 21, 27, 37, 41, 49, 51, 55, 57, 61, 65

In this unit we will answer…9.1: graph in the polar coordinate system and use

the corresponding distance formula (9-1, 9-2)

9.2: convert between polar and rectangular coordinates and equations (9-3)

9.3: simplify complex numbers (9-5)

9.4: perform operations on complex numbers in polar form (9-6, 9-7, 9-8)

9.1: graph in the polar coordinate system and use the corresponding distance formula (9-1, 9-2)

In this section we will answer…

• What is the polar coordinate system?• How do I write and graph points in polar form?• How do I write and graph simple equations in

polar form?• Is there a way to find the distance between two

points in polar form?

• Coordinate (r,θ)• r = the distance from

the pole to the point.• θ = the angle.

• Plot some.3,4

55,

6

A

B

2, 15

4,75

( 2, 30 )

C

D

E

( 5,45 )F

3(4, )

2G

• First, let’s look at one variable equations in rectangular.

• Graph x = 4 and y = -3

• r = 6

• θ = -60˚

1

2

( 2,210 )

(4,135 )

P

P

2 21 2 1 2 1 2 2 12 cos( )PP r r r r

3 32, and 4,

4 2

• Surveying• You are standing in the parking lot of a

historical site reading the map of the area. You notice there is a monument 700 feet away and 40˚ to the left of your position and a gift shop 350 feet away and 35˚ to the right.

• How far is the monument from the gift shop?

• P 558 #17 – 49 every other odd

9.1: graph in the polar coordinate system and use the corresponding

distance formula (9-1, 9-2)

In this lesson we will answer…• How are equations graphed in polar form?

• What are the basic families of graphs possible in polar form?

• How can I solve a system of polar equations?

• Graph r = sin θ

• Use a T-chart.

• Connect points as you go so that you don’t mix them up.

• How does this differ from rectangular?

• What do you expect it to look like?

• How do you think it will differ from r = sin θ?

• Graph it on your calculator.

• You do NOT need to memorize these!

23

32

y x

y x

1 cos

1 cos

r

r

• P 565 #11 – 27 odd – you may graph them on your calculators then sketch the result.

• Choose one polar equation from p 565 #11 – 22 to present on large polar graph paper.

• Must show a full, completed T-chart.

• Quiz Grade!

Warm-up:

•p 197 - 201

• #1 – 9 all, 17 - add respect to origin,

• 19, 21, 23,

• 33 – graph both function and inverse,

9.2: convert between polar and rectangular coordinates and

equations (9-3)

In this section we will answer…• Can we convert from rectangular form to

polar form and back again?• How do I rename a polar point in rectangular

form? A rectangular point in polar?• How can I convert rectangular equations into

polar form and visa versa?

Can we convert from rectangular form to polar form and back again?

How do I rename a polar point in rectangular form?

cos

sin

x r

y r

6,3

Do another.

cos

sin

x r

y r

5,45

How about this?

cos

sin

x r

y r

32,

2

Now, name a rectangular point in polar.

3, 4

2 2 r x y

1tanyx

Now, name a rectangular point in polar.

5,6

2 2 r x y

1tanyx

How can I convert rectangular equations into polar form and visa

versa?

3r

A little harder…

cscr

One more…

2 sin2 8r

Okay, now rectangular to polar…

7x

Again…

2 2 25x y

Oooo…what about this?

2 2 1x y

Have fun!

Homework:

•p 572 #15 – 39 odd

•Quiz tomorrow!!!

Warm-up:

• p 269 #15, 17, 21, 25,

• 35 – find # possible pos and neg roots, list all possible rational roots, then find the actual rational roots.

• 43 – use your calculators

• 45, 47, 51, 53, 57

Homework:

9.3: simplify complex numbers (9-5)

In this section we will answer…

• Do I remember how to work with complex numbers?

• How do I rationalize with complex rational numbers?

What is a complex number?

Written in the f orm:

a bi

where 1i

Let’s review the powers of “i”:

Operations on Complex Numbers

• Addition and Subtraction

• Multiplication

• Division

Write an equation which has the solutions –2,

3+i, 3-i.

Homework:

• p 583 #13 – 35 odd

9.4: perform operations on complex numbers in polar form

(9-6, 9-7, 9-8)In these sections we will answer…

• Can complex numbers be graphed?• Is it possible to change a complex number into

polar form?• How do I get back to rectangular form fom polar?• How do I multiply and divide complex numbers in

polar form? • Why on earth would anyone work in polar form?

Is it possible to change a complex number into polar form?

6 8i

Polar Form of Complex Numbers

(cos sin )r i

You do a couple…

3 3i

2i

How do I get back to rectangular form from polar?

4 cos sin6 6

i

You try one…

5 52 cos sin

6 6i

How do I multiply and divide complex numbers in polar form?

• First, let’s review the rules for multiplying bases with exponents.

The Product of Complex Numbers in Polar Form

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

Find the product then express the product in rectangular form.

7 77 cos sin 3 cos sin

12 12 12 12i i

Find the product then express the product in rectangular form.

2 cos240 sin240 3 cos60 sin60i i

Let’s review the rules for with dividing exponents.

Division of Complex Numbers in Polar Form

1 1 1 2 2 2

1 1 12 2 2

(cos sin ) (cos sin )

(cos( ) sin( )

r i r i

r r i

Find the quotient then express the quotient in rectangular form.

3 310 cos sin 2 cos sin

10 10 20 20i i

How do I raise complex numbers in polar form to a power or take a

root?• Review the rules for raising exponents to a

power.

Powers and Roots of Complex Numbers in Polar Form

[ (cos sin )]

(cos sin )

n

n

r i

r n i n

Find the power then express the result in rectangular form.

3

3 cos sin6 6

i

Why on earth would anyone work in polar form?

How about this? Doesn’t that look like fun?

81 i

Taking Roots of Complex Numbers.

• Don’t bother.

Change the root to a power and follow the power rule!

1

(cos sin )

[ (cos sin )]

n

n

r i

r i

4 1 2i

Homework: do one a day!

• P 590 #27 – 41 odd

• P 597 #11 – 25 odd

• P 605 #13 – 25 odd