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This file contains foldable notes for advanced mathematicaltopics otherwise known as precalculus.
These notes are useful for the following courses:Math 4Accelerated Math 3PrecalculusSupplies needed:pronged paper folder (not plastic)8.5" x 11" colored paper (not construction paper)scissorsgluestraight edge
Division of polynomials
FORMS
Numerator must be at least one power
higher than the denominator.Write powers in descending orderIf any terms are missing fill in a zero for a place holder.
Dirty DivideMarvin MultiplySmells Subtract (change signs)Bad Bring Down
Repeat
STEPS
1
2
Division of Polynomials 2 quarter sheets flip folded to the center
Division of polynomials cont.
3EXAMPLES
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 termsWrite opposite sign of constant in denominator.
Answer is always one degree less than the numerator
Always foil back to check your answer!
Synthetic Division 4
l. Bring down first number
2. multiply by box number3. write answer above line4. Add / subtr then write answer under line
5. Repeat
Rational Functions Booklet (title, definition and form, intervals, domain)
Need 6 hotdog half sheets flip folded toward the middle to form 12 sections
Rational Functions 1
Definition and Form 2
y = n(x) d(x)
n for numeratord for denominator
Interval Notation
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 rightIntervals of decrease fall from left to right
( ∞, 3) ∪ (3, ∞) is the same as {x: x ≠ 3}
3
These will be in fractionform.
Ex: (-∞, x1) ∪ (x1, x2) ∪ (x2, ∞)
Domain 4
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 tofactor to solve for each x.
Tabs 14
Tabs 57 (range, discontinuity, extrema)
Range 5
Look on the y-axis from lowest to highestWrite in interval notation
Discontinuity 6
Exists when a graph has breaks or "holes"Holes occur when part of the denominator cancels with thenumerator.Vertical asymptotes mark discontinuity as boundary lines.
Extrema 7
Turning points on a graph.
Kinds: relative maximum maximum relative minimum minimum
Tab 8 (intercepts)
zerosrootssolutionsx-intercepts
names for the same thing
x-intercepts: where the graph crosses the x-axisHow to find: 1. reduce the fraction to lowest terms 2. set numerator = 0 3. solve for the variableform: (x, 0)
y-intercepts: where the graph crosses the y-axis How to find: 1. substitute 0 for every x 2. reduceform: (0,y)
Not all graphs haveintercepts
Intercepts 8
Tab 9 (asymptotes)
Asymptotes are dotted boundary linesGraph will curve toward vertical asymptotes but never touch them. Graph can cross a horizontal asymptotesCalculator table will show "error" at location of asymptotes
3 typesVERTICAL ASYMPTOTES 1. set denom = 0 2. Solve for x, factor if needed use mode 5, 3 if quadraticform: 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 keysCasio 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
TI84Alpha y= for zoom 6 centers a graph2nd graph shows table of valuesy= put in equation to be graphedgraph shows a graph of the function2nd mode same as "all clear"
What to put on a graph
1. draw in all asymptotes with dotted lines2. plot all intercepts3. locate any holes in the graph4. check table for graphable points and plot them5. sketch graph in between the asymptotes
Graphing 10
settings for TI84 for graphingmode: make sure FUNC is darkened choose either radian or degree (depends on the units used)zoom 7: puts graph in trig settingswindow: 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
Tab 11 (solve rational equations)
Always check answers for extraneous roots
If only two parts : a = c b d1. cross multiply ad=bc2. isolate the variable3. check answer against restrictions
If more than 2 parts:1. find LCD2. multiply each piece of the equation by the LCD3. cancel where possible to get rid of the denom in each piece4. solve the new equation5. 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 LCD3. 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 interval6. for ≥ or > look for the sections that have +'s for ≤ or < look for the sections that have 's7. write in interval notation
Solve Rational Inequalities 12
Rational Expressions Need 2 quarter sheets tabbed, folded to center.
1
Rational Expressions
Canceling: (a+b) is a unit powers are subtracted numbers are divided
Simplifying
2Mult or Div Rational Expressions
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.
3Add or Subtr Rational Expressions
Denominators must be alike to add or subtract!1. Find a common denominator2. Add or subtract only the numerators
To find a common denominator:Multiply each fraction by the least common multiple.
4Complex Fractions
Treat as a division of two fractions problem.fractionfraction
both numerator and denominator are fractions
rewrite, keep it, change it, flip it.
Balance method or Kriss Kross Boom!
DMS <=>DD
If doesn't work use fractions
DMS <=> DD
ownreferenceangle
θ
180 - θθ
θ
θ-180
θ
360-θ
If θ is negative: convert to positive then find using the quadrants.
Q II Q I
Reference AnglesQ III Q IV
o ' " => Use after each number
+ +60 3600
Degrees, Minutes, Seconds and Reference Angles 1/4 sheet folded in half top down with a tab at the bottom
1
2
hypotenuseleg
leg SOHCAHTOA1. locate right angle2. locate θ3. label triangle sides4. Choose function5. Cross multiply and solve
tangent cosecantcotangent secant
Solving Right Triangles need 1/2 hamburger sheet folded into a square, trimmed, folded into 4 triangles
inside
Angle Measures
initial side
terminalside
Standard position
Degrees <=> Radians
Coterminal Anglesadd or subtract 360 for degreesadd or subtract 2π for radians
Finding Angle Measure
Radians to degrees
Degrees to radians
radians x 180 = degreesπ
(π should cancel--if it doesn't change it to 3.14 and divide)
degrees x π = radians180
Keep π in answer
θ
.
line of sigh
t
line of sight
angle of elevation
angle of depression
Angles of depressionand elevation are equal.
verticalheight
horizontallength
Angles are alternate interior angles.
line o
f sigh
t
Angles of elevation and depression
Use SOHCAHTOA
depression
elevation
))
inside
1/4 sheet folded to the middle
The angle of depression is not usually inside the triangle, that's why you put the anglemeasure on the angle of elevation inside thetriangle at the ground level!
1
Solving Non-right (Oblique) Triangles
A =B =C =
a =b =c =
Graphic Organizer
Soving nonright triangles flip chart Need seven1/4 sheets folded into flip book
2Law of Sines Formula
Longest side is always across from largest angle, smallest side is across from smallest angle.
Angles in triangle add up to 180 degrees.
Sin A Sin B Sin C a b c
= =
Must have an angle and an opp side to use LS
Use two fractions at a timeCross multiply to solve for missing values.Use shift key if solving for an angle.
3Law of Sine Setups AAS and ASA
AAS
ASA
A B
C
b ? Find side a first because it matches <A
A B
C
c
l. Find C first (180AB)2. Set up fractions to findother sides.
SSA
SSA setup 4
If opp side is greater than the adj only 1 solution
adj oppθIf angle θ is acute:
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.
Make two graphic organizers. label them Triangle 1 and Triangle 2
Law of Cosines on next page of file**********
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 isright
If opp side < h
No solution⇒sidetoo short
If opp side > h
2 Δ's formed
adj oppθ
hadj opp
θ h
adj
θ
opp
h
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.
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
A B
C
c
a
Angle Formulas:
a
b
c
The two smallest sides added together must be greater than the third side or no triangle exists.
1. solve for smallest angle first2. Use cos-1 to find the angle.3. Use Law of Sines to find the next smallest angle.
SSS 7
Law of Cosines sections of flip book
negative sign
negative sign
negative sign
Triangle AreaFormulas
A = 1/2 bh
SAS setupArea = 1/2 abSinCArea = 1/2 bcSinAArea = 1/2 acSinB
Answers arein square units.
Heron
's Fo
rmula
SSS setup Area = K = s(sa)(sb)(sc)
where s = 1/2 (a+b+c)
s is the semiperimeter
Als
o ca
lled
Her
o's
Form
ula
Triangle Area Formulas
Need 1/4 sheet then fold into asquare. Cut off excess.
Special Right Triangles Notes
Special Right Triangles
30-60-90
45-45-90
longleg
shortleg
hypotenuse
hyp = 2 x short leg
long leg = short leg x √3
short leg = hyp÷2 or = long leg ÷ √3
Formulas
Formulas
leg
leg
hypotenusehypotenuse = leg x √2
leg = hypotenuse ÷ √2
cover
top inside
bottom inside
1/4 sheet folded in half
More Formulas
need 2 quarter sheets made into booklet
cover
2. Arc length3 Sector Area4. Angular Velocity5. Linear Velocity
1. Contents
2 Arc Length
S = rθ r is radiusθ is central anglein radians
3Area of Sector (pie shaped)
A = ½r2θ
4 5Angular Velocity Linear Velocity
ω =θ t
θ is in radianst is time
v = rθ t
θ is in radianst is timer is radius
((decimal answersare acceptable
decimal answersare acceptable
r is radiusθ is central anglein radians
Graphing Sine and Cosine
Amplitude is 1Midline is y=0 (xaxis)
y=sin θ or y=cos θ
count by 30 on tablesset table using 2nd tblset
cover
1 y=sinθ zeros at beginning of period middle end
period length is 2π or 360o (one full wave)
highest 1/4 periodlowest at 3/4 period
2 y=cosθ period length is 2π or 360o (one full wave) starts and ends at highestmiddle is low
zeros 1/4 period and 3/4 period
chop into 4 sections
chop into 4 sections
If using ti-84 use zoom 7 to graph
This is one full wave
This is one full wave
x can be usedinstead of θ
Low at odd π's
Sine Cosine Translationscover
y=A sin (kθ +c)+hy=A cos (kθ+c)+h
Amplitude 2
Period 3Phase shift 4Vertical shift 5
How to graph 6-7
page
2. Amplitudeheight
Amplitude=|A|
y=A Sinθ
y=A cosθ
3. Period
y= ASin kθ
y= ACos kθ
periodk is pos
4.Phase Shift
horizontal translationy=Asin (kθ + c)y=A cos(kθ+c)
If c is positive: shift left
if c is negative: shift right
5.Vertical Shift
y=A sin(kθ+c)+h
if h is positive: upif h is negative: down
Graphing Sine and Cosine Translations1. Find the vertical shift and graph the midline (h)2. find the amplitude (max and min)
3. find the period
4. find the phase shift
5. translate and graph sections (chop into sections)
k is pos
y=A cos(kθ+c)+h
(A)
(K)
(C)
make 2 quarter sheets into booklet
Phase shift = c k>0 (change sign on c) k
2π k
c k
p 6-7
If there are no parentheses, there is no phase shift!
Other Trig Graphs
Contents
Tangent 12
Cotangent 34
Cosecant 56
Secant 78
Other graphs booklet: need three 1/4 sheets
cover
These fns do not have amplitudes because their range is all reals.
1
Tangent y =tan x or sin xcos x
Two full waves in 2π, one wave in π
2
Properties of Tangent graph
always increasesamplitude: noneperiod formula:
Cosine is the denominator sowhere cosine is zero you will locate the 2 asymptotes
Cotangent
Two waves in 2π: onewave in π.3.
4.
Properties of Cotangent graphalways decreases
period formula: p= π k
Asymptotes are atbeginning, middle and endof two periods because sineis in the denominator.
y = cot x or cos xsin x
Where sine is zero you will locatethe 3 asymptotes!
Tan and Cot notes this filepage. Csc andSec notes next file page.
0, π, 2π
number pages110 after the content page
graph 2 Waves
One full wave contains pos and neg values.Graph can be split.
Cosecant and Secant notes for other trig graphs
5
Cosecant y = csc x or 1sin x
One wave in 2π: one wave is twopieces
Cosecant properties: (flip the sin graph)
sine in denominator means 3 asymptotes
6
formula for period is same as sine.
asymptotes located where sineis zero
Notice the 1's!! Graph flips up at+1, down at -1
Secant y = sec x or 1cos x
7one wave in 2π: has twopieces like the csc.
8
Secant properties: (flip the cos graph)
cos in denominator meanstwo 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
Definitions of inverse fns Make a square from hamburger half.Fold like an envelope (points to the center).
folded closed
Definitions ofInverse TrigFunctions
y = Cos x y = Sin x
y = Tan x
0 ≤ x ≤ πdomain
range
[1,1]QuadrantsI and II
range
[1,1]QuadrantsI and IV
domain
-π/2 ≤ x ≤ π/2
-π/2 ≤ x ≤ π/2domain
rangeall reals
Quadrants I and IV
Inverses must pass verticalline test whengraphed.
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 xor Arctan x
y = Cos1 xor Arccos x
y = Sin1 xor Arcsin x
domain:All Reals
[-π/2 ,π/2]range:
range:[0,π]
[-π/2 ,π/2]Range
domain:[-1,1]
domain:[-1,1]
opened up
Graphs of Inverse Trig Functions
Graphs of Inverse Trig Functions
1/4 sheet folded in half
cover
How to write sinusoidal functions
cover
1 quarter sheet hamburger folded
How to Write Sinusoidal Functionsfrom Real World Data
y = A (kθ + c) + hsinorcos
Use sine if function begins near 0Use cosine if functions begins at a maximum or minimum value
inside topTo find A
A = greatest value - least value 2
To find h
h = greatest value + least value 2
period = 2π k
Look for "every" to get the period length then solve for K
t is used for θ in the time functions
To find k
inside bottom
You will have a chart of values or a dot plot.
Basic Trig Identities4 quarter sheet flip book fold back instead of to center
Basic Trig Identities
tan θ = sin θ cos θ
Cot θ = cos θ sin θ
Quotient Identities
top flap
x can b
e used
instea
d of θ.
Reciprocal Identities
Pythagorean Identities
Other Names for sin, cos, and 1
other forms
factored forms
other forms
factored forms
other forms
factored forms
Note: both fns have "co"!
Signs depend on Quadrant!
2
3
1
2
3
4
4
Conversions One 1/4 sheet flip book
Conversions
How to change sin <=> cos
1. square the value2. subtract from 13. √ the result4. assign + or - according to the quadrant
Method 1: use identities
1. identify the quadrant2. substitute into an identity3. assign + or - according to quadrant
Method 2: use Pythagorean Theorem and SOHCAHTOA
1. identify the quadrant2. draw a right triangle in quad with θ at origin3. use pythagorean theorem to find needed sides4. set up function needed using SOHCAHTOA5. Assign + or - according to quadrant
∎ Find values in a specified quadrant ∎
top
Verifying Identities one layered 1/4 sheet
Overview
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
top flap
Strategies
⇒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
inside
Other Identities Use 2 quarter sheetsto make a booklet that opensupwards
1Other Identities
Pages
2-3 Sum and Difference (α ± β)4-5 Double Angles (2α)6-7 Half Angle (α/2)
2
α is alpha β is beta
Sum and Difference
cos (α ± β) = cosαcosβ sinαsinβ
sin (α ± β) = sinαcosβ ± cosαsinβ
tan (α ± β) = tanα ± tanβ 1 tanαtanβ
You will need sin and cos of both angles for cos ( α±β) or sin(α±β)
Find exact values by substituting into the formulasx 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!!!≫
3
4Double Angle
sin (2θ) = 2sinθcosθ
cos (2θ) = cos2θ - sin2θ = 2cos2θ - 1 = 1 - 2sin2θ
tan (2θ) = 2tanθ 1-tan 2θ
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) 2 x ½ sin 60 1 ≠ √3/2
5
folded edge
Other Identities cont.
6Half-Angle
sin(α/2) = ± 1-cosα 2√cos(α/2) = ± 1+cosα 2√tan(α/2) = ± 1-cosα 1+cosα√ cos α ≠ -1
Determine the quadrant of the angle first to choose pos or neg for answerYou 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
7
Solving trig identities use a quarter sheet foldedinto an upside down flip book
top view closed
Solving Trig Equations
Types of Solutions
Principal Value Solutions
sin and tancos
Quads1 and 41 and 2
-90o≤x≤90o
0o≤x≤180o
Infinite Solutions
use x+360ok or x+2πk for sinx and cosx
use x+180ok or x+πk for tanx
flips down
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 sheetsmade into a tabbed flip book
FORMULA
d = Ax1 + By1 + C ±√A2 + B2
Point => (x1,y1)Line => Ax + By + C = 0
Distance from a point to a line
cover
How to Compute
1. put the equation in Ax+By+C=0 form2. 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
2
3Ways 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=Cx 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
Polar Coordinates
Pole:the origin
Polar Axis:Horizontal ray to right from the pole at 0o
Point:
(r, θ)
closed
First ring is the Unit Circle!
|r| is thedistance frompole to thepoint
To plot points:From 0 go out to r then go tothe angle
Distan
ce
Ways to name the same angle:(r,θ) = (r, θ + 2πk) = (-r, θ + (2k+1)π)(r,θ) = (r, θ + 360ok) = (-r, θ + (2k+1)180o)
P(r,θ)
P(r,θ)
rrθ
θ
r > 0r is poscount up
r < 0r is negcount down
P1(r1,θ1)P2(r2,θ2)
P 1P 2 =
√r 12 +r 2
2 - 2r 1r 2
cos(θ
2-θ 1)
Polar Graphing Full sheet folded to center from sides and ends into 16 equal sections
Polar Graphing
(r,θ)
TI-84 settings
mode: radians POL
To set Window useZOOM 7 ZTRIG θ min 0θ max 2π
graphs with sinθ center on the y axisgraphs with cosθ center on x axis
USUALLY!
Need polar graphing paper
cover opens to the left
inside view
r = k
θ= k
Circles
Limacons
Roses
Cardioids
Lemniscates
Spiral of Archimedes
Basics
r = asinθr = acosθ
r = a±bsinθr = a±bcosθ
"bean shaped"
r = a sin(nθ)r = a cos(nθ)
r = a±asinθr = a±acosθ"heart shaped"
r2 = a2sin2θ
r2 = a2cos2θ
r = aθθ in radians
(space for extra notes)
page 1
,
Window:
θ min = 0θ max = 2π unless spiral of Archimedesθ step = π/12x min = x max = x scl = 1y min = y max = y scl = 1
width
height
Choose max and min so graph will fit your screen
Polar graphing page 2 inside left half
∎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 factorof |a|∎ Center of circle is on an axis at ½a∎ a can be negative ∎ |a| is the diameter of the circle
∎ graphs a line through the pole
∎ Plot all points on the θ on each "circle" inboth directions.
∎ Can have an inner loop, dimple, or curve outward
r = 3
θ = ¾π
right half
r=sinθ r=cosθ
a+b
r=1+3 sin θ r=1+3 cosθ
∎ 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
∎ special limacon∎ notice the a's are alike
∎ graphs a spiral∎ need to increase θ max in window
∎ must take √ to graph
r=3 Sin 2θ r=3sinθ
r=2+2sinθ
r, = pos rootrz = neg root
r=0.5θr=2+2cosθ
r2=22sin2θ r2=22cos2θ
put in calc: r1=√a2sin2θ r2=√a2sin2θ
or cos instead of sin
1
1
b-a
Polar and Rectangular Coordinates Three 1/4 sheets in flipbook
1
Polar and Rectangular Coordinates
(x,y) <=> (r,θ)
x = rcosθy = rsinθ
r = √(x2 + y2)
θ = Tan-1 y if x >0 (pos) x
θ = Tan-1 y + π if x<0 (neg) xPolar and Rectangular Coordinates
r = acosθr2=arcosθx2+y2=ax
1. Mult both sides by r2. Subst x2+y2for r2
3. Subst x for rcosθ or y for rsinθ
Polar Equations to Rectangular Form
Rectangular Equations to Polar Form
1. Subst rcosθ for x or rsinθ for y
2. solve for r
2
3
Complex numbers 1 quarter sheet folded with a side tab
Com
plex
Num
bers
Top view
Simplifyingi1 = ii2 = -1i3 = -ii4 = 1
"I won, I won"middles are negative
Raising i to powers or in
n let R be the remainder4 R = 1 then i n = i
R = 2 then i n = 1R = 3 then i n = iR = 0 then i n = 1
No remainder, then in = 1
Com
plex
Num
bers
Graphing a + bi Polar Form of a + bi
realaxis
imaginaryaxis
R
ia + bi => r(cosθ + isinθ)
r cis θabbreviation =>
r is the modulus (aka absolutevalue of a + bi)
r = √(a2 + b2)
θ = Arctan b if a > 0 (pos) aθ = Arctan b + π if a < 0 (neg) a
Argand Plane
z = a + bi|z| = √(a2 + b2)
θ is the amplitude or argument
Polar P
roducts, Quotient, P
owers, R
oots
Polar Products and Quotients
Productsr1 cis θ1 * r2 cis θ2 = r1r2 cis (θ1+θ2)
Quotientsr1 cis θ1 ÷ r2 cis θ2 = r1 cis (θ1-θ2) r2
r cis θ is abbreviation forr (cos θ + i sin θ)
Polar Products, Quotients, Powers and Rootsneed one 1/4 sheet folded with tab at right side
folded view
open view
DeMoivre's Theorem: raises to a specified power
[r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ)
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)
Don't forget about cis notation!
Polar P
roducts, Quotient, P
owers, R
oots
Sequences and Series need 4 half sheets in a booklet
Sequences and Series
Arithmetic Sequences 2-3Arithmetic Series 4-5Geometric Sequences 6-7Geometric Series 8-9Infinite Series 10-11Sigma Notation 12-13
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Arithmetic SequencesFound by adding samenumber (d) each time
FormulaAn = a1 + (n1)d
an = nth term or last terma1 = first termd = common differencen = term number (pos integer)
terms are separated by commasFind terms by adding d toprevious term
To find d:d = term previous termdoesn't matter which twoconsecutive terms you use.
Arithmetic means are foundin between nonconsecutiveterms
2, 4, 6, 8 4 and 6 are arithmetic means between 2 and 8!
To find a specific term1. identify a1 ,d and n2. substitute into an formula3. combine like terms
To find arithmetic means1. substitute for a1 and an in formula and find d2. add d to first term to find missing terms
To write equation of arithmetic sequence1. identify a1 and d2. substitute into an formula3. distribute d4. combine like terms
Arithmetic Seriesseries adds the terms together
sequence series2, 4, 6, 8 2+4+6+8
Formula
Sn = n (a1 + an) 2
Sn = sum of the first n termsn = term number (pos integer)a1 = first terman = last term
S3 would be a1 + a2 + a3
hint: formula averages the first and lastterms 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
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If n or a1 is missing: go to the original formula an = a1 + (n-1)d and find it. Then use the original Sn formula.
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Geometric Sequences and Series
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Geometric Sequencesfound by multiplying or dividing each previous term by common ratio called r.
an = nth term or last term
a1= first termAn1 = previous termr = common ratio
terms are separated by commas
to find rr= term previous term
r can be fraction, decimalor negative
To find specific term:1. identify a1, r, and n2. raise r to the power
and mult by a 1
To find Geometric Meansl. Subst for a1,an, and n2. divide an by a1 leaver to the power on one side
3. take root that is thepower of r
r = an
an1
an = an1 * r or
an = a1 * rn1
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Geometric Seriesadds terms of sequence together
Sn = a1 - a1rn
1-rorSn = a1( 1 - rn) r≠1 1-r
To find a sum1. identify a1, r, and n2. subst into S n formula3. combine like terms4. reduce to lowest terms
If n is not known use Sn = a1 - anr 1-r1. identify a1, an, and r2. subst into formula3. reduce to lowest terms
To find a1
use Sn = a1( 1 - rn) 1 1-r1. subst for Sn, n, and r2. simplify top and bottom of fraction3. cross multiply and solve for a1.
To find r go backto original an formulaan = a1 * rn1
Sigma Notation
Σ sigma
Σk
n=1an
maximum value of n
starting value of n
expression for general term
Above is read as "the summation from n = 1 to k is an"
Σ an = a1 + a2 + a3 + ... + an
k
n=1
k must be an integern is the index of summation
1. write in expanded form by substituting each n into formula2. add the terms
Sigma Notation Infinite Geometric Series
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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
Σ = a1rn1
∞
n=1
Σ = (a1 + (n1)d)k
n=1Arithmetic
Σ = a1rn1
n=1
kGeometric
Infinite Series
Infinite SeriesFormula for infinite Geometric Series
S = a1 1r
1<r<1
r cannot be larger than one orless than -1. Must be a decimalor fraction between -1 and 1Not all series are infinite. Terms must be decreasing in size.How to write repeating decimalas a fraction1. write the decimal as a series using the repeating part2. find a1 and r3. subst into S formula and reduce
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Limits ∞ = infinityreads the limit of 1/n as n approaches zero.
The larger n becomes, the tinier the fraction becomes until it approaches 0.
* If numerator has more than one term, make a fraction for each term and find the limit for each part.
lim 1 = 0 for n > 0n⇒∞ nr
lim (an ± bn) = lim an ± lim bnn⇒∞ n⇒∞ n⇒∞
lim 1 = 0n⇒∞ n
lim (an * bn) = lim an * lim bn n⇒∞ n⇒∞ n⇒∞
lim an
n⇒∞lim an = n⇒∞ bn lim bn
n⇒∞lim Cn = Cn if C is a constantn⇒∞
Formulas1
2
3
4
5
6# of places rrepeating 1 0.1 2 0.01 3 0.001
This side for AM3
Summary on p 14
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Helpful hints for taking limits1. 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.
