This file contains foldable notes for advanced mathematical
topics otherwise known as precalculus.
These notes are useful for the following courses:
Math 4
Accelerated Math 3
Precalculus
Supplies needed:
pronged paper folder (not plastic)
8.5" x 11" colored paper (not construction paper)
scissors
glue
straight edge
2 quarter sheets flip folded to the
center
Division of Polynomials
Division of polynomials
FORMS
1
Numerator must be at least one power
higher than the denominator.
Write powers in descending order
If any terms are missing fill in a zero for a place holder.
Dirty
Divide
Marvin
Smells
Bad
Multiply
Subtract (change signs)
Bring Down
Repeat
STEPS
2
Division of polynomials cont.
EXAMPLES
3
Division of polynomials cont
Quicker only if the highest power in the denominator is 1 and
the coefficient of x is l.
ex: denominator is x­3
Write only coefficients using 0 for missing terms
Write opposite sign of constant in denominator.
Answer is always one degree less than the numerator
Always foil back to check your answer!
l. Bring down first number
2. multiply by box number
3. write answer above line
4. Add / subtr then write answer under line
5. Repeat
Synthetic Division
4
Need 6 hotdog half sheets flip folded
toward the middle to form 12 sections
Rational Functions Booklet Tabs 1­4
(title, definition and form, intervals, domain)
Rational Functions
y = n(x)
d(x)
n for numerator
d for denominator
1
These will be in fraction
form.
2
Definition and Form
Always use parentheses with ­ ∞ or ∞
( ­∞, a) ∪ (b, ∞) doesn't include a or b
( ­∞, a] ∪ [b, ∞) does include a and b
Intervals of increase rise from left to right
Intervals of decrease fall from left to right
( ­∞, 3) ∪ (3, ∞) is the same as {x: x ≠ 3}
3
Interval Notation
Ex: (-∞, x1) ∪ (x1, x2) ∪ (x2, ∞)
Look left to right on the x-axis.
Has restrictions based on the x's in the denominator.
One for each x.
Set denominator = 0 to find the restrictions. May have to
factor to solve for each x.
Domain
4
Tabs 5­7 (range, discontinuity, extrema)
Look on the y-axis from lowest to highest
Write in interval notation
Range
Exists when a graph has breaks or "holes"
Holes occur when part of the denominator cancels with the
numerator.
Vertical asymptotes mark discontinuity as boundary lines.
Discontinuity
5
6
Turning points on a graph.
Kinds: relative maximum
maximum
relative minimum
minimum
Extrema
7
Tab 8 (intercepts)
zeros
roots
solutions
x-intercepts
names for the same thing
x-intercepts: where the graph crosses the x-axis
How to find: 1. reduce the fraction to lowest terms
2. set numerator = 0
3. solve for the variable
form: (x, 0)
y-intercepts: where the graph crosses the y-axis
How to find: 1. substitute 0 for every x
2. reduce
form: (0,y)
Intercepts
Not all graphs have
intercepts
8
Tab 9 (asymptotes)
Asymptotes are dotted boundary lines
Graph will curve toward vertical asymptotes but never touch
them. Graph can cross a horizontal asymptotes
Calculator table will show "error" at location of asymptotes
3 types
VERTICAL ASYMPTOTES
1. set denom = 0
2. Solve for x, factor if needed
use mode 5, 3 if quadratic
form: x = # for every x in the denom
HORIZONTAL ASYMPTOTES
compare the degrees of the numerator and the denominator
1. if degree of numerator is the highest power there is no horizontal
2. if degree of denominator is the highest power there is only
one horizontal form: y = 0
3. if degrees are the same, make a fraction from coefficients
of the highest powers and reduce
form: y = a in lowest terms
b
SLANT ASYMPTOTES (oblique)
if power of numerator is exactly one degree higher than the
power of the denominator
1. long divide the numerator by the denom
2. ignore any remainder
3. write in the form y = mx + b
denom |numerator
Asymptotes
9
Tab 10 (graphing)
Calculator keys
Casio 115ES
mode 1 shift mode 1
for
to work
x
raises to powers (also known as the "x box key")
mode 5 3 solves quadratics
TI­84
Alpha y= for zoom 6 centers a graph
2nd graph shows table of values
y= put in equation to be graphed
graph shows a graph of the function
2nd mode same as "all clear"
settings for TI­84 for graphing
mode: make sure FUNC is darkened
choose either radian or degree (depends on the units used)
zoom 7: puts graph in trig settings
window: xmin choose smallest x value you want to graph
xmax choose largest x value to graph
xscl choose the scale you want to use for the x­axis
ymin choose smalles y value to graph
ymax choose largest y value to graph
yscl choose the scale you want for the y­axis
xres DO NOT CHANGE
What to put on a graph
1.
2.
3.
4.
5.
draw in all asymptotes with dotted lines
plot all intercepts
locate any holes in the graph
check table for graphable points and plot them
sketch graph in between the asymptotes
Graphing
10
Tab 11 (solve rational equations)
Always check answers for extraneous roots
If only two parts : a = c b d
1. cross multiply ad=bc
2. isolate the variable
3. check answer against restrictions
If more than 2 parts:
1. find LCD
2. multiply each piece of the equation by the LCD
3. cancel where possible to get rid of the denom in each
piece
4. solve the new equation
5. check answers against restrictions
Set denominator = 0 to get restrictions
for the roots. These cannot be
answers.
Solve Rational Equations
11
Tab 12 (solve rational inequalities)
Write answers in interval notation
1. move all terms to the left side, leave 0 on the right side.
2. simplify using LCD
3. find critical numbers
set numerator =0 solve
set denom =0 solve (can be several numbers)
4. draw number line and mark critical numbers for boundaries
use closed dots for ≤, ≥ (bar on the bottom)
use open dots for <, > (no bar on the bottom)
5. check a test number between each critical number and
label as + or ­ for the interval
6. for ≥ or > look for the sections that have +'s
for ≤ or < look for the sections that have ­'s
7. write in interval notation
Solve Rational Inequalities
12
Rational Expressions
Need 2 quarter sheets tabbed, folded to center.
Rational Expressions
Canceling:
(a+b) is a unit
powers are subtracted
numbers are divided
Simplifying
1
Multiplication: factor if needed, cancel when possible, multiply
straight across
Division: keep it, change it, flip it
Rewrite problem as multiplication then follow multiplication
rules.
Mult or Div Rational Expressions
2
Denominators must be alike to add or subtract!
1. Find a common denominator
2. Add or subtract only the numerators
To find a common denominator:
Multiply each fraction by the least common multiple.
Balance method or Kriss Kross Boom!
Add or Subtr Rational Expressions
3
Treat as a division of two fractions problem.
fraction
both numerator and denominator are fractions
fraction
rewrite, keep it, change it, flip it.
Complex Fractions
4
Degrees, Minutes, Seconds and Reference Angles
1/4 sheet folded in half top down with a tab at the bottom
DMS <=>DD
o ' "
=> Use after each number
If doesn't work use fractions
+
60
+
3600
DMS <=> DD
1
QI
Q II
own
reference
angle
180 - θ
θ
θ
If θ is negative: convert to positive then find
using the quadrants.
θ
θ
360-θ
θ-180
Q III
Reference Angles
Q IV
2
Solving Right Triangles
need 1/2 hamburger sheet folded into a square, trimmed, folded into 4 triangles
leg
e
us
leg
n
te
po
hy
SOHCAHTOA
1. locate right angle
2. locate θ
3. label triangle sides
4. Choose function
5. Cross multiply
and solve
inside
tangent
cosecant
cotangent
secant
Finding Angle Measure
Angle Measures
Standard position
te
r
sid min
al
e
θ
initial side
Coterminal Angles
add or subtract 360 for degrees
add or subtract 2 π for radians
Degrees <=> Radians
Radians to degrees
radians x 180 = degrees
π
(π should cancel--if it doesn't
change it to 3.14 and divide)
Degrees to radians
degrees x π = radians
180
Keep π in answer
.
Angles of elevation and depression
ight
s
f
e o
lin
angle of elevation
angle of depression
line of si
g
ht
inside
lin
e
of
sig
ht
Angles of depression
and elevation are equal.
vertical
height
horizontal
length
Use SOHCAHTOA
) depression
elevation)
Angles are alternate interior angles.
The angle of depression is not usually inside
the triangle, that's why you put the angle
measure on the angle of elevation inside the
triangle at the ground level!
1/4 sheet folded to the middle
Need seven1/4 sheets folded into flip book
Soving non­right triangles flip chart Solving Non-right (Oblique) Triangles
a=
b=
A=
B=
C=
c=
Graphic Organizer
Sin A
a
Sin B
b
=
=
1
Sin C
c
Must have an angle and an opp side to use LS
Longest side is always across from largest angle, smallest
side is across from smallest angle.
Angles in triangle add up to 180 degrees.
Use two fractions at a time
Cross multiply to solve for missing values.
Use shift key if solving for an angle.
2
Law of Sines Formula
AAS
C
Find side a first because it matches <A
?
b
B
A
l. Find C first (180­A­B)
ASA
C
2. Set up fractions to find
other sides.
c
A
B
Law of Sine Setups
AAS and ASA
3
SSA
opp
adj
If angle θ is acute:
θ
If opp side is
greater than the adj
only 1 solution
If opp side is less than adj side: run the test value for triangle height
h = adj(sin θ)
If opp side = h
1 solution and Δ is
right
adj
θ
h opp
If opp side < h
If opp side > h
No solution⇒side
too short
2 Δ's formed
opp
adj
θ
opp
adj
h
θ
h
Make two graphic organizers. label them Triangle 1 and Triangle 2
For 1st triangle: work out as you normally would.
For 2nd triangle: Subtract the angle that you found first from 180 o and use the new angle to find all parts of the second triangle.
SSA setup
Law of Cosines on next page of file**********
4
Law of Cosines sections of flip book
Begins and ends with same "letter."
Law of Cosines Formulas
5
This setup is always a triangle!
Remember to take √ last to get the side.
C
a
A
c
B
1. Solve for the side across from the known angle using Law of Cosines.
2. Use Law of Sines to find the rest of the missing parts since you now have
an angle that matches a side.
Law of Cosines Setups
SAS
6
Angle Formulas:
negative sign
negative sign
c
a
b
negative sign
The two smallest sides added together must be greater than
the third side or no triangle exists.
1. solve for smallest angle first
2. Use cos-1 to find the angle.
3. Use Law of Sines to find the next smallest angle.
SSS
7
Need 1/4 sheet then fold into a
square. Cut off excess.
Triangle
Area Formulas
Triangle Area
Formulas
ea
wh
S
S
S
Ar
p
tu
se
=
K
H
er
on
Answers are
in square units.
's
s er
=
is e Fo
s
th s (
=
rm
s­
e 1
a)
se /2
u
(s
m (a
­b la
i­p +
)(s
er b+
­c
im c)
)
et
er
SAS setup
Area = 1/2 abSinC
Area = 1/2 bcSinA
Area = 1/2 acSinB
Also called Hero's Formula
A = 1/2 bh
1/4 sheet folded in half
Special Right Triangles Notes
cover
Special Right Triangles
30-60-90
hy
top inside
Formulas
po
short
leg
hyp = 2 x short leg
ten
use
long leg = short leg x √3
long
leg
short leg = hyp÷2 or
= long leg ÷ √3
bottom inside
45-45-90
Formulas
us
en
t
po
hy
e
leg
hypotenuse = leg x √2
leg = hypotenuse ÷ √2
leg
need 2 quarter sheets made into booklet
More Formulas
1.
2.
3
4.
5.
2
Contents
Arc length
Sector Area
Angular Velocity
Linear Velocity
Arc Length
r is radius
θ is central angle
in radians
decimal answers
are acceptable
Angular Velocity
ω =θ
t
θ is in radians
t is time
Area of Sector (pie shaped) 3
(
(
S = rθ
4
cover
A = ½r2θ
r is radius
θ is central angle
in radians
decimal answers
are acceptable
5
Linear Velocity
v = rθ
t
θ is in radians
t is time
r is radius
cover
Graphing Sine and Cosine
x can be used
instead of θ
y=sin θ or y=cos θ
Amplitude is 1
If using ti-84 use zoom 7 to graph
Midline is y=0 (x­axis)
count by 30 on tables
set table using 2nd tblset
1
y=sinθ
period length is 2π or 360o (one full wave)
zeros at beginning of period
middle
end
highest 1/4 period
lowest at 3/4 period
chop into 4 sections
This is one full wave
2
y=cosθ
period length is 2π or 360o (one full wave)
starts and ends at highest
middle is low
zeros 1/4 period and
3/4 period
This is one full wave
chop into 4 sections
Low at odd π's
make 2 quarter sheets into booklet
cover
Sine Cosine Translations
y=A sin (kθ +c)+h
y=A cos (kθ+c)+h
page
Amplitude
2
Period
3
Phase shift
2.
4
Vertical shift
5
How to graph
6-7
Amplitude
y=A Sinθ
height
y=A cosθ
Amplitude=|A|
3. Period
period
y= ASin kθ
k is pos
y= ACos kθ
k is pos
Phase Shift
4.
horizontal translation
y=A sin (kθ + c)
If there are no
parentheses, there is no
phase shift!
y=A cos(kθ+c)
Phase shift = ­c k>0 (change sign on c)
k
If c is positive: shift left
if c is negative: shift right
Vertical Shift
5.
y=A sin(kθ+c)+h
y=A cos(kθ+c)+h
if h is positive: up
if h is negative: down
Graphing Sine and Cosine Translations
1. Find the vertical shift and graph the midline
(h)
2. find the amplitude (max and min) (A)
3. find the period 2π
(K)
­c
(C)
4. find the phase shift k
5. translate and graph sections (chop into sections)
k
p 6-7
Other graphs booklet: need three 1/4 sheets
Other Trig Graphs
Tan and Cot notes this file
page. Csc and
Sec notes next file page. Contents
Tangent
1­2
Cotangent
3­4
Cosecant
5­6
Secant
7­8
cover
These fns do not have amplitudes
because their range is all reals.
Tangent
y =tan x or
sin x
cos x
Two
full
in 2π,
wave
in π
One
fullwaves
wave contains
posone
and neg
values.
Graph can be split.
1
Properties of Tangent graph
always increases
amplitude: none
period formula:
Cosine is the denominator so
where cosine is zero you will
locate the 2 asymptotes
graph 2 Waves
2
Cotangent
3.
y = cot x or cos x
sin x
Two waves in 2π: one
wave in π.
Properties of Cotangent graph
always decreases
period formula: p= π
k
Asymptotes are at
beginning, middle and end
of two periods because sine
is in the denominator.
Where sine is zero you will locate
the 3 asymptotes!
0, π, 2π
4.
number pages
1­10 after the content page
Cosecant and Secant notes for other trig graphs
Cosecant
y = csc x or
1
sin x
One wave in 2π: one wave is two
pieces
5
Cosecant properties: (flip the sin graph)
sine in denominator means
3 asymptotes
formula for period is same as sine.
asymptotes located where sine
is zero
Notice the 1's!! Graph flips up at
+1, down at -1
6
Secant
7
y = sec x or
1
cos x
one wave in 2π: has two
pieces like the csc.
Secant properties: (flip the cos graph)
cos in denominator means
two asymptotes
Period is the same as the cos
Where cos is zero you will
locate the asymptotes
Notice the 1's!! Graph flips up at
+1, down at -1
8
Definitions of inverse fns
folded
closed
y = Cos x
Make a square from hamburger half.
Fold like an envelope (points to the center).
Definitions of
Inverse Trig
Functions
y = Sin x
domain
domain
0 ≤ x ≤ π
-π/2 ≤ x ≤ π/2
range
[­1,1]
range
y = Tan x
Quadrants
I and II
domain
-π/2 ≤ x ≤ π/2
[­1,1]
Quadrants
I and IV
range
all reals
Quadrants I and IV
opened up
Inverses must
pass vertical
line test when
graphed.
y = Cos­1 x
or Arccos x
domain:
[-1,1]
range:
[0,π]
y = Sin x
<=>
x = Sin-1 y
y = Cos x
<=>
x = Cos-1 y
y = Tan x
<=>
x = Tan-1 y
Domain must be restricted so that
inverses are functions. These
values are called principal values.
Capital letter are used when domain
is restricted.
y = Tan­1 x
or Arctan x
range:
[-π/2 ,π/2]
domain:
All Reals
y = Sin ­1 x
or Arcsin x
domain:
[-1,1]
Range
[-π/2 ,π/2]
Graphs of Inverse Trig Functions
1/4 sheet folded in half
cover
Graphs of Inverse Trig Functions
How to write sinusoidal functions
1 quarter sheet hamburger folded
How to Write Sinusoidal Functions
from Real World Data
y= A
sin
or (kθ + c) + h
cos
cover
You will have a chart of
values or a dot plot.
Use sine if function begins near 0
Use cosine if functions begins at a maximum or minimum value
inside top
To find A
A = greatest value - least value
2
To find h
h = greatest value + least value
2
inside bottom
To find k
period = 2π
k
Look for "every" to get the period length
then solve for K
t is used for θ in the time functions
4 quarter sheet flip book fold
back instead of to center
Basic Trig Identities
Basic Trig Identities
top
flap
1
. f θ
Cot θ = cos θ
sinθ
o
ad
ste
tan θ = sin θ
cosθ
n
d i
se
e u
x c
b
an
Quotient Identities
2
Reciprocal Identities
2
3
other forms
factored forms
other forms
factored forms
other forms
factored forms
Note: both fns have "co"!
Pythagorean Identities
3
4
Signs depend on Quadrant!
Other Names for sin, cos, and 1
4
One 1/4 sheet flip book
Conversions
Conversions
1.
2.
3.
4.
square the value
subtract from 1
√ the result
assign + or - according to the quadrant
How to change sin <=> cos
Method 1: use identities
top
1. identify the quadrant
2. substitute into an identity
3. assign + or - according to quadrant
Method 2: use Pythagorean Theorem and SOHCAHTOA
1.
2.
3.
4.
5.
identify the quadrant
draw a right triangle in quad with θ at origin
use pythagorean theorem to find needed sides
set up function needed using SOHCAHTOA
Assign + or - according to quadrant
∎ Find values in a specified quadrant ∎
Verifying Identities
one layered 1/4 sheet
Overview
top flap
Treat equation as two separate problems separated by a
wall
Work on one side at a time
DO NOT MIX THE SIDES TOGETHER!
When sides look alike you are finished
Strategies
inside
⇒Substitute identities
⇒convert to one "word" if possible
⇒convert back to sin and cos then combine like terms
⇒combine fractions using common denominatiors
⇒multiply by a fraction equal to 1 then cancel
⇒factor
There may be more than one method that works. Remember: left side must match the right side to be finished!
Verifying Identities
Use 2 quarter sheets
to make a booklet that opens
upwards
Other Identities
folded edge
Other Identities
1
Pages
2-3 Sum and Difference (α ± β)
4-5 Double Angles (2α)
6-7 Half Angle (α/2)
α is alpha
β is beta
2
Sum and Difference
cos (α ± β) = cosαcosβ
sinαsinβ
sin (α ± β) = sinαcosβ ± cosαsinβ
tan (α ± β) = tanα ± tanβ
1 tanαtanβ
3
You will need sin and cos of both angles for cos ( α±β) or sin(α±β)
Find exact values by substituting into the formulas
x and y can be used instead of α, β
Formula will determine the sign of the answer
≪DO NOT DISTRIBUTE the function into the parentheses!!!
≫
Double Angle
4
sin (2θ) = 2sinθcosθ
cos (2θ) = cos2θ - sin2θ
= 2cos2θ - 1
= 1 - 2sin2θ
tan (2θ) = 2tanθ
1-tan 2θ
5
Formula will determine the sign of the answer
Formulas find exact answers, not decimals
Substitute into formulas
Note: 2 sin(30) is not equal to sin(2 x 30)
2x½
sin 60
1
≠
√3/2
Other Identities cont.
Half-Angle
√
√
sin( α/2) = ±
1-cosα
2
cos(α/2) = ±
1+cosα
2
√
tan(α/2) = ±
1-cosα
1+cosα
6
cos α ≠ -1
7
Determine the quadrant of the angle first to choose pos or
neg for answer
You must choose the sign: the formula does not
To choose angle to use for α:
use half the denominator of the given angle and keep the
numerator the same.
Substitute into formulas and simplify
use a quarter sheet folded
into an upside down flip book
Solving trig identities
top view closed
Solving Trig Equations
Types of Solutions
flips
down
Principal Value Solutions
sin and tan
cos
Quads
1 and 4
1 and 2
-90o≤x≤90o
0o≤x≤180o
Infinite Solutions
use x+360ok or x+2πk
use x+180ok or x+πk
for sinx and cosx
for tanx
opened downward
Solve for x like algebra equations
When problem asks for real values use radians
Change to one function if possible
If equation contains squares you may need to factor
or use square root to get answer
Solve for the function then use inverse of function
to get the angle(s)
There may be multiple solutions
Shortest Distance from a Point to a Line
need three 1/4 sheets
made into a tabbed flip
book
FORMULA
cover
d = Ax1 + By1 + C
±√A2 + B2
Point => (x1,y1)
Line => Ax + By + C = 0
Distance from a point to a line
2
1. put the equation in Ax+By+C=0 form
2. Subst A,B,C into formula. Choose the
opposite of the sign of C for the
denominator.
3. From the point subst for the x1 and the
y1
4. Simplify the fraction
How to Compute
3
Ways to find a point on a line:
Substitute a zero for either the x or the y in the equation then
solve for the remaining variable
Find the intercepts using the following forms of Ax+By=C
x int = C y int = C
A
B
gives
(x,0)
(0,y)
Find a point on either one of the lines then use the Formula for the distance from a point to a line.
Distance between Parallel Lines
Polar Coordinates
Need a 1/4 sheet envelope fold
closed
Polar Coordinates
Pole:
the origin
Polar Axis:
Horizontal ray to right
from the pole at 0o
Point:
(r, θ)
First ring is the Unit Circle!
r
θ
θ
r
P(r,θ)
=
r>0
r is pos
count up
P1
P2
Di
st
a
n
c
√r e
1 2
+r
2 2
-2
r1
r2
co
s(
θ
2
-θ
1
)
P(r,θ)
r<0
r is neg
count down
P1(r1,θ1)
P2 (r2 ,θ2 )
Ways to name the same angle:
(r,θ) = (r, θ + 2πk)
= (-r, θ + (2k+1)π)
(r,θ) = (r, θ + 360 ok)
= (-r, θ + (2k+1)180 o)
To plot points:
From 0 go out to r then go to
the angle
|r| is the
distance from
pole to the
point
Polar Graphing
Full sheet folded to center from sides and ends into 16 equal sections
page 1
Polar Graphing
cover opens to the left
(r,θ)
TI-84 settings
Need polar graphing paper
(space for extra notes)
mode: radians
POL
To set Window use
ZOOM 7 ZTRIG
θ min 0
θ max 2π
USUALLY!
graphs with sinθ center
on the y axis
graphs with cosθ center
on x axis
Window:
θ min = 0
θ max = 2π unless spiral of Archimedes
θ step = π/12
x min =
width
x max =
x scl = 1
y min =
height
y max =
y scl = 1
Choose max and min so graph will
fit your screen
inside view
Circles
r=k
r = asinθ
r = acosθ
Basics
Limacons
,
θ= k
r = a±bsinθ
r = a±bcosθ
"bean shaped"
Roses
Lemniscates
r = a sin(nθ)
r2 = a2sin2 θ
r = a cos(nθ)
Cardioids
r = a±asinθ
r = a±acosθ
"heart shaped"
r2 = a2cos2θ
Spiral of
Archimedes
r = aθ
θ in radians
inside left half
Polar graphing page 2
∎graphs a circle centered at the pole
∴ count out to the length of r on the circles
∎ a expands or contracts the circle by a factor
of |a|
∎ Center of circle is on an axis at½a
∎ a can be negative
∎ |a| is the diameter of the circle
r=sinθ
1
r=cosθ
r=3
1
∎ Can have an inner loop, dimple, or curve
outward
∎ graphs a line through the pole
a+b
∎ Plot all points on the θ on each "circle" in
both directions.
b-a
r=1+3 sin θ
r=1+3 cosθ
θ = ¾π
right half
∎ must take √ to graph
r, = pos root
rz = neg root
∎ n must always be positive
∎ a can be negative
∎ |a| is the petal length
∎ # petals = n if n is odd
∎ # petals = 2n if n is even
r=3 Sin 2θ
put in calc: r 1=√a2sin2θ
r2 =√a2sin2θ
r=3sinθ
r2 =22 cos2θ
∎ graphs a spiral
∎ need to increaseθ max in window
∎ special limacon
∎ notice the a's are alike
r=2+2sinθ
r2 =22 sin2θ
or cos instead of sin
r=2+2cosθ
r=0.5θ
Three 1/4 sheets in flip
book
Polar and Rectangular Coordinates
Polar and Rectangular Coordinates
(x,y) <=> (r,θ)
x = rcosθ
y = rsinθ
r = √(x2 + y2)
θ = Tan-1 xy if x >0 (pos)
θ = Tan-1 xy + π if x<0 (neg)
Polar and Rectangular Coordinates
1
r = acosθ
r2=arcosθ
x2+y2=ax
1. Mult both sides by r
2. Subst x2+y2for r2
3. Subst x for rcosθ or y for rsinθ
Polar Equations to Rectangular Form
2
1. Subst rcosθ for x or rsinθ for y
2. solve for r
Rectangular Equations to Polar Form
3
Complex numbers
1 quarter sheet folded with a side tab
Complex Numbers
Simplifying
i1 = i
i2 = -1
i3 = -i
i4 = 1
"I won, I won"
middles are
negative
Top view
Raising i to powers or in
n
4
let R be the remainder
R = 1 then i n = i
R = 2 then i n = ­1
R = 3 then i n = ­i
R = 0 then i n = 1
No remainder, then in = 1
Graphing a + bi
Argand Plane
i
Polar Form of a + bi
a + bi => r(cosθ + isinθ)
Complex Numbers
R
real
axis
abbreviation =>
r cis θ
r is the modulus (aka absolute
value of a + bi)
imaginary
axis
z = a + bi
|z| = √(a2 + b2)
r = √(a2 + b 2)
θ = Arctan b if a > 0 (pos)
a
θ = Arctan b + π if a < 0 (neg)
a
θ is the amplitude or
argument
need one 1/4 sheet
folded with tab at
right side
Polar Products, Quotients, Powers and Roots
Polar Products and Quotients
Quotients
r1 cis θ1 ÷ r2 cis θ2 =
r1 cis (θ1-θ2)
r2
r cis θ is abbreviation for
r (cos θ + i sin θ)
Polar Products, Quotient, Powers, Roots
Products
r1 cis θ1 * r2 cis θ2 =
r1r2 cis (θ1+θ2)
folded view
open view
DeMoivre's Theorem: raises to a specified power
Don't forget about cis notation!
Distinct roots of a polar complex #
(a + bi)
= r (cos θ + 2nπ + i sin θ + 2nπ )
p
p
Replace n with 0, 1, 2, 3, ... until you have
reached 1 less than the root you are trying to
find.(p­1)
Polar Products, Quotient, Powers, Roots
[r(cos θ + i sin θ)]n = rn (cos nθ + i sin nθ)
Sequences and Series
need 4 half sheets in a booklet
Sequences and Series
Arithmetic Sequences
Arithmetic Series
Geometric Sequences
Geometric Series
Infinite Series
Sigma Notation
2-3
4-5
6-7
8-9
10-11
12-13
1
Arithmetic Sequences
Found by adding same
number (d) each time
Formula
An = a1 + (n­1)d
an = nth term or last term
a1 = first term
d = common difference
n = term number (pos integer)
terms are separated by commas
Find terms by adding d to
previous term
To find d:
d = term ­ previous term
doesn't matter which two
consecutive terms you use.
2
Arithmetic Series
series adds the terms together
sequence series
2, 4, 6, 8
2+4+6+8
Formula
Sn = n (a1 + an )
2
Sn = sum of the first n terms
n = term number (pos integer)
a1 = first term
an = last term
S3 would be a1 + a2 + a3
4
Arithmetic means are found
in between nonconsecutive
terms
2, 4, 6, 8
4 and 6 are
arithmetic means between 2 and
8!
To find a specific term
1. identify a1 ,d and n
2. substitute into an formula
3. combine like terms
To find arithmetic means
1. substitute for a1 and an in formula
and find d
2. add d to first term to find missing terms
To write equation of arithmetic sequence
1. identify a1 and d
2. substitute into an formula
3. distribute d
4. combine like terms
3
hint: formula averages the first and last
terms then multiplies by the term number!
Use alternate form if you need to find d and know Sn
Sn = n (a1 + [a1 + (n ­ 1)d])
2
= n (2a1 + (n­1)d)
2
If n or a1 is missing: go to the
original formula an = a1 + (n-1)d
and find it. Then use the
original S n formula.
5
Geometric Sequences and Series
Geometric Sequences
found by multiplying or dividing
each previous term by common
ratio called r.
an = an­1 * r
an = a1 * r
or
n­1
an = nth term or last term
a1= first term
An­1 = previous term
r = common ratio
r can be fraction, decimal
or negative
terms are separated by commas
6
Geometric Series
adds terms of sequence together
Sn = a1 - a1rn
1-r
or
Sn = a1( 1 - rn) r≠1
1-r
To find a sum
1. identify a 1, r, and n
2. subst into S n formula
3. combine like terms
4. reduce to lowest terms
8
If n is not known use Sn = a1 - anr
1-r
1. identify a1, an, and r
2. subst into formula
3. reduce to lowest terms
to find r
r=
term
previous term
r=
an
an­1
To find specific term:
1. identify a 1, r, and n
2. raise r to the power
and mult by a 1
To find Geometric Means
l. Subst for a1,an, and n
2. divide an by a1 leave
r to the power on one side
3. take root that is the
power of r
7
To find a1
use Sn = a1( 1 - rn)
1
1-r
1. subst for Sn, n, and r
2. simplify top and bottom of
fraction
3. cross multiply and solve for
a1.
To find r go back
to original an formula
an = a1 * rn­1
9
Sigma Notation
Sigma Notation
Σ sigma
maximum value of n
k
Σ
n=1
an
expression for general term
starting value of n
Above is read as "the summation from
n = 1 to k is an"
k
Σ
n=1
an = a1 + a2 + a3 + ... + a n
k must be an integer
n is the index of summation
1. write in expanded form by substituting
each n into formula
2. add the terms
12
Infinite Geometric Series
∞
Σ =a r
n=1
1
n­1
1. expand by substituting into
formula.
2. add terms using S = a1
1­r
A series in expanded form can
be written using Σ if a general
formula can be written for the
nth term
k
Arithmetic
Σ = (a1 + (n­1)d)
n=1
k
Geometric
=a r
Σ
n=1
1
n­1
13
Infinite Series
This side for AM3
Infinite Series
Formula for infinite Geometric Series
S = a1 ­1<r<1
1­r
r cannot be larger than one or
less than -1. Must be a decimal
or fraction between -1 and 1
Not all series are infinite. Terms
must be decreasing in size.
How to write repeating decimal
as a fraction
1. write the decimal as a series
using the repeating part
2. find a1 and r
3. subst into S formula and reduce
10
# of places
repeating
1
2
3
r
0.1
0.01
0.001
Limits ∞ = infinity
Formulas
reads the limit of 1/n as n approaches zero.
1
lim
1 =0
n⇒∞ n
2
lim
1 = 0 for n > 0
n⇒∞ nr
The larger n becomes, the tinier the fraction becomes until it approaches 0.
3 lim
(an ± bn) = lim an ± lim bn
n⇒∞
n⇒∞
n⇒∞
4
5
6
lim (an * bn) = lim an
n⇒∞
n⇒∞
lim
an
n⇒∞ bn
=
lim
n⇒∞
*
lim bn
n⇒∞
an
lim
bn
n⇒∞
lim Cn = Cn if C is a constant
n⇒∞
* If numerator has more than one term, make a fraction for each term and find the limit for each part.
Summary on p 14
11
Helpful hints for taking limits
1. If the largest exponents
are the same in the numerator
and the denominator then the
limit is the ratio of the
coefficients of the terms
containing the largest
exponent.
2. If the largest exponent is
in the numerator, then there is
no limit.
3. If the largest exponent is
in the denominator then the
limit is 0.
14
15