Sıcak Başlangıçlar / Warm Appetizers · 57- Kapadokya Kuzu Testi Kebabı 65,00
Final Review Warm-up: p 58 – 60 #13, 21, 27, 37, 41, 49, 51, 55, 57, 61, 65.
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Transcript of Final Review Warm-up: p 58 – 60 #13, 21, 27, 37, 41, 49, 51, 55, 57, 61, 65.
• 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