algebra final
how to know whether or not if it's a reflection
the sign in front of the A value
how to find the average of sinusoidal equations
the sinusoidal axis
how to find determinant of 2 by 2 matrix
|a b | |c d | = ad -bc
converges
|r| < 1
diverges
|r|≥ 1, no sum
sigma notation (what all the numbers mean)
∑ the n on top would be the number of terms the k is what you start with stuff on the side
recursive formula for geometric
a₁ = a, An= r(An-1)
factorial symbol
! 0!=1 1!= 1 factorial means to multiply by all the terms after it
intersection of sets A and B
"AND" elements shared by set A and set B
signs for quadrants
"all students take calculus" Quadrant 1: all are positive Quadrant 2: sin and csc are positive Quadrant 3: tan and cot are positive Quadrant 4: cos and sec are positive
probability of an event
# of ways event can occur _________________________________________ total number of outcomes
how to make altering signs for rules
(-1)^n-1
coordinate points 180 degrees
(-1,0)
range inverse tan function
(-pi/2, pi/2)
inverse csc domain
(-∞, -1] U [1,∞) value
inverse sec domain
(-∞, -1] U [1,∞) value
inverse cot domain
(-∞, ∞) value
domain inverse tan function
(-∞,∞)
inverse cot range
(0, pi)
coordinate points 270 degrees
(0,-1)
coordinate points 90 degrees
(0,1)
coordinate points 0/360 degrees
(1,0)
unit circle 60 coordinate points
(1/2, root 3/2)
how to find trig functions from the unit circle
(cos, sin)
vertex for horizontal major axis ellipse
(h + a, k) (h - a, k)
covertices for vertical major axis ellipse
(h + b, k) (h-b, k)
foci for horizontal major axis ellipse
(h+c, k) (h-c, k)
foci for vertical major axis ellipse
(h, k + c) (h, k-c)
(y-k)^2= 4a(x-h) vertex
(h, k)
center for horizontal major axis ellipse
(h, k)
hyperbola center
(h, k)
vertex for vertical major axis ellipse
(h, k+a) (h, k-a)
covertices for horizontal major axis ellipse
(h, k+b) (h, k-b)
(x-h)^2=4a(y-k) vertex
(h,k)
center for vertical major axis ellipse
(h,k)
unit circle 45 coordinate points
(root 2/2, root 2/2)
unit circle 30 coordinate points
(root 3/2, 1/2)
horizontal transverse axis equation hyperbola (branches opening left and right)
(x-h)^2 (y-k)^2 __________ - ____________ = 1 a^2 b^2
horizontal major axis ellipse equation
(x-h)^2 (y-k)^2 ___________ + ___________ = 1 a^2 b^2
vertical major axis ellipse equation
(x-h)^2 (y-k)^2 ___________ + ___________ = 1 b^2 a^2
equation for parabola opening up and down
(x-h)^2=4a(y-k)
vertical transverse axis equation hyperbola (branches opening up and down)
(y-k)^2 (x-h)^2 ___________ - __________ = 1 a^2 b^2
opening right and left equation parabola
(y-k)^2= 4a(x-h)
transformations for tan and cot
-A value is a vertical stretch. it changes the magnitude. (you multiply A by the y value) -period is pi/B. the new period decides where your asymptotes are
transformations sec, csc
-amplitude sets the range (-infinity, -A] U [A, infinity), basically you move the max/min of the parabola the number of the amplitude -period 2pi/B (moves the asymptotes around)
if given the characteristics for sin and cos
-amplitude= A value -the period = 2pi/B so you take the period and set it equal to 2pi/B and cross multiply to find the B -the vertical shift is the D value, down is negative, up is positive
basic look for cot graph
-asymptote on y axis, next asymptote one period away -then halfway between the two asymptotes is one the axis, and halfway in between pt and axis (to the left) is up, and halfway in between pt and axis (to the right) is down
basic look of csc graph
-bunch of parabolas -draw the sin graph first, the asymptotes are where sin is 0 -the first asymptote is on the y-axis, then at half the period, then at one period away -the first parabola to the right opens up (where the first sin pt would be), then the first parabola to the left opens down
basic look of sec graph
-bunch of parabolas -straddles x axis -draw the cos graph first, the asymptotes are where cos is 0 -asymptotes make up 1/2 a period -first parabola straddle opens up, the both next to the left and right are opening down
phase shift
-c/b (x-coordinate where the cycle begins, horizontal shifts)
how to expand binomial
-coefficient of pascal triangle -first term descending -second term ascending
if it gives you two GEOMETRIC terms and their values how do you find the first term and ratio?
-draw blanks w/ a₂, a₃, etc. on bottom -write in the values -divide the last value from the first THATS GIVEN -the number of blanks in between is how many xs are being multiplied, so thats the kind of root you root it by, which gives you the ratio. then use the values and the ratio to plug in and then find the equation
if it gives you two ARITHMETIC terms and their values how do you find the nth term and common difference?
-find the slope so rewrite the terms as A(whatever term # they give you) = An (whatever term value they give you) -the common difference would be the slope, which would be the difference of the term values/ over the difference of the term #s -use this to plug into formula, along with values, to find the first term and then the equation
systems of equations substitution method
-ideal to use if coefficient for x or y is 1 -isolate the variable with coefficient of 1 -substitute into other equation -back substitute for other variable
matrix addition/subtraction
-may do it only if the matrices have the same dimensions -add/subtract the corresponding ones in the matrix
how to rewrite with a common denominator
-multiply each part by the other denominator, not the whole thing -remember the numerator of the multiplied fraction is only multiplied by the other numerator, same thing applies for the denominators -make sure to multiply top and bottom by the other denominator!!
systems of equations elimination method
-multiply one (or both) equations to get opposites for coefficients of one variable -add equations together so one variable drops out and solve -back substitute for other variable
what to do with the even odd property when simplifying
-must apply first -if it's something like sin^2 (-theta), then it's really like saying (sin(-theta)^2 and you'd have to apply within the squared
3 x3 system with inverse matrix
-must have zero placeholders -do the same thing but use the calc
when can you use (cos, sin)
-on the unit circle -quadrantal angles -when the triangle's hyp = 1 -Q1: special 30, 60, and 45s
formula for arithmetic sequence
An= a₁+ (n-1)d
recursive arithmetic sequence formula
A₁ =a, An = An-1 + d
quadrant order
I- both positive II- x negative, y positive III- both negative IV- x positive, y negative
computing probabilities of complementary events
P (E') = 1- P(E)
conditional probability
P(A⌒B) ____________ P(B)
multiplication rule
P(A⌒B)= P(A) x P(B|A)
what if it gives you that the second event occurs and you have to work backwards
P(intersection) ______________________ probability of second event
infinite geometric sum formula
S= a₁ / 1-r
cases for law of cosines
SAS and SSS
sum of first n terms of a geometric sequence formula
Sn= a₁ (1-rⁿ /1-r)
arithmetic sum formula
Sn= n/2 (a₁ +An) (An is the last term)
how to find the regression equation of a sinusoidal equation
TURN CALC TO RADIANS 1) turn your scatterplot on 2) stat edit, then enter your data 3) then go to stat calc, scroll down to sin regression and it'll give you the values
domain inverse cos function
[-1, 1] value
domain inverse sin function
[-1,1] value
inverse csc range
[-pi/2, pi/2] angle
range inverse sin function
[-pi/2, pi/2] angle
range inverse cos function
[0, pi]
inverse sec range
[0, pi] y≠pi/2 angle
how to find range
[axis-amplitude, axis+amplitude]
geometric sequence
a common ratio "r" is multiplied to get the next term
arithmetic sequence
a constant d is added to get the next term
sequence
a function whose domain (input) is a set of positive integers
determinant
a real number associated with a square matrix
hyperbola definition
a set of all points in a plane such the difference of the distances from two fixed points, called the foci is constant
vertex of hyperbola
a units from center -for horizontal, you count left and right -for vertical, you count up and down
(y-k)^2= 4a(x-h) directrix
a units from the vertex, opposite direction from focus
(y-k)^2= 4a(x-h) focus
a units from vertex count the way it's opening
(x-h)^2=4a(y-k) focus
a units from vertex (if opening down, you count down, if opening up, count up)
(x-h)^2=4a(y-k) directrix
a units from vertex, opposite direction of focus
cos
adj/hyp
union of sets A and B
all elements in A, B, both "OR"
standard position
an angle with the vertex at the origin and initial side on positive x axis
permutation
an ordered arrangement of objects
coterminal angles
angles in standard position with the same terminal side (infinite amount)
combination
arrangement of objects where order does not matter, nCr
how to express answers to systems
as ordered pairs
equal sets
both sets have exactly the same elements
how to find coterminal angles
by adding and subtracting multiples of 360 or 2pi. might have to do it a couple times to get it positive or negative. you gotta make sure its in the parameters of what it wants it in. also to state what quadrant it is in, you can draw it to figure out its location. if it lies on an axis, you gotta say positive or negative as well
foci of hyperbola
c units from center on transverse axis where c^2= a^2+ b^2 count in same direction of vertex
how to know common difference from eq (arithmetic)
coefficient of n
complementary angle theorem
cofunctions of complementary angles are equal. basically if you have a function of an angle, take the cofunction (ex: sin goes to cosine) and then its complement in either degrees or radian
periodic property: cos/sec
cos (θ+2π) = cos θ same w/ sec -also works w/ multiples of 2π and 360 degrees
even trigonometric functions
cos, sec
cos(-θ) / sec (-θ)
cosθ secθ
cotangent quotient identity
cot=cos/sin
reciprocal identities
csc= 1/sin sec= 1/cos cot= 1/tan
representing a sequence
define the rule in terms of n
venn diagrams
diagrams using overlapping circles in a rectangle -intersections: shade the overlap -unions shade everything
multiply matrix by scalar
distribute to every term, fractions must be reduced
how to do 30-60-90s and 45-45-90
draw the triangle and find the trig functions for each one
what to do if not in unit circle
draw the triangle. but if the hyp. is less than your legs, it is undefined. also if it's ones with different trig functions, you won't need to find the angle because you can figure it out through the triangle.
subset
each element in A is also in B
writing a sequence
each number in the sequence is a term
angle of depression
falls from the horizontal NOT ALWAYS IN TRIANGLE ITS FROM THE HORIZONTAL!!! ex: building higher up
how to find trig functions through definition
find all the sides and then use soh-cah-toa
directrix parabola
fixed line
focus parabola
fixed point inside parabola
initial side
fixed ray the angle is generated from (on the x axis)
periodic function definition
functions that repeat at regular intervals
negative angles
go clockwise
positive angles
go counterclockwise
periodic function
graph has a repeating pattern that continues indefinitely
how to find a for parabolas
have to divide by 4
sinusoidal axis
horizontal line through the middle of the graph always y= something
common ratio
the number that is multiplied to each term to get the next term
reference angle quadrant 1
same as theta
latus rectum pts parabola
segment joining the two points on the parabola through the focus and parallel to the directrix (endpoints equidistant from focus and directrix. count distance from focus to directrix, and then go that distance away from focus)
parabola
set of all points in a plane equidistant from a fixed point and a fixed line
ellipse
set of all points in a plane such that the sum of the distances from two fixed points is constant
summation notation
sigma notation -notation to express the sum of n terms of a sequence ∑
periodic property: sin/csc
sin (θ+2π)= sinθ same w/ csc -also works w/ multiples of 2π and 360 degrees
law of sines
sin A/a = sin B/b = sinC/ c (calc mode degrees)
odd trigonometric functions
sin, csc, tan, cot
cofunctions for complementary angle theorem
sin/cos csc/sec tan/cot
pythagorean identities
sin^2 theta + cos^2 theta =1 tan^2 theta + 1 = sec^2 theta cot^2 theta + 1 = csc^2 theta
how to find the five key points on the x axis
start with the phase shift (or just 0 if no phase shift), the period is the 5th point. then you divide the period by four to find out what you add each time. OR you could just divide the period by two and that's the middle point, and then you divide the middle point by two to find the other 2 points NEED TO BE EVEN INTERVALS
5 key pts with phase shift
start with the phase shift (take the C value in equation and do the -c/B thing). then add the period to it to find the last pt. the divide the period by 4 to find how much you add each time
basic of cos curve
starts at an amplitude away from the axis, goes down to the axis, one amplitude down from the axis, axis, one amplitude above the axis *remember translations vary this*
basic look of sin curve
starts at origin, goes up the amplitude away from the axis, back down to the axis, goes down one amplitude from axis, then back at the axis *remember translations vary this*
pascal's triangle
starts out with one. each row you add the two terms above it (on each side) to find the term
periodic property: tan/cot
tan (θ+π)= tanθ same w/ cot -also works w/ multiples of π and 180 degrees
ones that straddle the x axis
tan, sec
tangent quotient identity
tan= sin/cos
An
term
n
term #
An-1
term before it
how to find the amplitude from cos/sin equations
the absolute value of A
reference angle definition
the acute, positive angle formed by the terminal side of theta and the x axis
denominators of cramer's rule
the determinant of the coefficient matrix
numerators of cramer's rule
the determinant of the matrix formed by using the column of constants as replacements for the coefficients of the variable you are solving for
recursive formula
the first terms are given, then the rule is defined using the term before it
transverse axis
the line containing the foci
major axis ellipse
the line that contains the foci
minor axis ellipse
the line through the center that is perpendicular to the major axis
center ellipse
the midpoint of the segment joining the foci
binomial expansion patterns
-powers of first term descend from highest power -powers of 2nd term ascend -coefficients are derived from pascal's triangle
sin (-θ) / csc (-θ)
-sinθ -cscθ
basic look for tan graph
-straddles y-axis (on origin) -the asymptotes combine for one period away from origin (half the amplitude to right, the other half to the left) -you go right one, up one and left one down one (unless vertical stretch from different amplitude)
tan (-θ) / cot (-θ)
-tan θ -cot θ
how to find the term of binomial expansion
-the choose number on bottom is one less than the term -so you put the choose number on top from exponent, one less than the term for bottom choose number -bottom choose number is the exponent of the last term -first term exponent is top choose number minus bottom choose number -multiply all out to find term
binomial theorem
-top number of the combo is the sum of the exponent -bottom number is the second term exponent
unit circle: quadrantal angle measures and their radian measures
0, 90, 180, 270, 360 0, pi/2, pi, 3pi/2, 2pi
how to find the inverse matrix
1 over the determinant of the original matrix is distributed out in front. the first position is switched with the position diagonal from it. the other two positions become opposite sides
how to find the trig functions of an angle on a coordinate plane (not necessarily on the unit circle)
1) draw the terminal side/plot the point it gives you 2) drop an altitude onto the x axis 3)use the x and y values to find the legs, then either use triples or the pytag theorem 4) then use the reference angles for the values and ASTC for signs
how to find equation from grapgh
1) first determine where the sinusoidal axis is (middle of graph) 2) figure out if it is sin or cos based on if its on or above the SA 3) figure out what the period is (when it gets back to originating y value) 4) find out the amplitude 5) the equation is in either the form y= A sin (Bx) + D or y= A cos (Bx) + D 6) the amplitude is the A value 7) the SA is the D value 8) the period = 2pi/B so you take the period and set it equal to 2pi/B and cross multiply to find the B
how to graph a hyperbola
1) plot the center 2) find a, then count a units away from the center to find the vertex 3) find b, then count in the opposite direction and draw a slash mark (then draw your box using this) 4) the asymptotes are the diagonals of the box 5) find c, c units away from the center in the direction of the vertex are your foci 6) draw the branches from your vertex approaching the asymptotes
how to draw angles on coordinate plane
1) put the initial side on the positive x axis 2) note that the axises are 90, 180, 270, 360/0 degrees and pi/2, pi, 3pi/2, 0/2pi radians 3) put the terminal side in the quadrant it belongs 4) draw arrow to show which way the angle is rotating
determinants in calculator
1. 2nd x^-1 (matrix) -> edit (input entries in matrix) 2. quit 3. matrix -> math 1: det( 4. matrix names (to select the matrix u want the determinant of)
how to establish/verify identities
1. begin with one side of the equation, the more complicated expression 2. use identities and algebraic manipulations to arrive at expression on other side YOU ARE NOT SOLVING AN EQUATION!! you can not manipulate both sides of an equation, you are just verifying that one side can be transformed to the other side
how to solve systems with three variables
1. choose two equations to eliminate one variable 2. use the third equation to eliminate the third variable 3. solve the system with two variables 4. back substitute to find the last variable
how to use reference angles to find trig functions
1. determine the quadrant of terminal side angle 2. find reference angle 3. evaluate trig function of reference angle, decide if positive or negative from "all students take calculus"
how to solve a linear system using inverse matrix
1. must write the matrix equation!! 2. find the inverse of the coefficient matrix 3. multiply inverse matrix to the left of each side 4. x= a^-1 x b
how to simplify trig expressions
1. rewrite using identities until you're down to something simple 2. multiply things out and FOIL if needed 3. you can factor! (difference of squares)
law of cosines
a²= b²+c² - 2ac cos A side²= other side² + other side squared² -2(other sides) cos of angle across from side SQUARE ROOT ANSWER
guidelines for establishing/verifying identities
1. start with the side that has the more complicated expression 2. rewrite sums and differences of quotients as a single term by getting common denominators 3. sometimes rewriting in terms of sines and cosines will help 4. always keep the end result in mind
methods for solving nonlinear systems of equations
1. substitution 2. elimination
amplitude
1/2 the distance between the max and min of the graph (or the distance from the sinusoidal axis to the max/min)
area of a sector
1/2 times r^2 times central angle IN RADIANS
reference angle quadrant 2
180- theta or pi- theta
number of subsets a set has
2^n
csc period
2pi
period of cos curve
2pi
period of sin curve
2pi
sec period
2pi
how to find the period from cos/sin equations
2pi/B
pythagorean triples
3-4-5 5-12-13 7-24-25 8-15-17 9-40-41
pi/6
30 degrees
unit circle: angles corresponding with 30 degrees and their radian measures
30, 150, 210, 330 pi/6, 5pi/6, 7pi/6, 11pi/6
reference angle quadrant 4
360 - theta or 2pi - theta
pi/4
45 degrees
unit circle: angles corresponding with 45 degrees and their radian measures
45, 135, 225, 315 pi/4, 3pi/4, 5pi/4, 7pi/4
pi/3
60 degrees
unit circle: angles corresponding with 60 degrees and their radian measures
60, 120, 240, 300 pi/3, 2pi/3, 4pi/3, 5pi/3
complement of A
A', elements not in A
cases for law of sines
ASA, AAS, SSA
formula for the nth term of geometric sequence
An= a₁ (r)^n-1
(x-h)^2=4a(y-k) how to know up or down
if a is positive it opens up
(y-k)^2= 4a(x-h) right or left?
if a is positive, opens right
how to do composite functions in general
if the inverse is on the outside, you find the value first and then the angle where that value occurs. if the inverse is on the inside, you find the angle first and then the value of that angle.
how to know if arithmetic
if you add each term
empty set
is a subset of any set, no elements in common
how to find inverse
it is looking for the angle where the value occurs, must be in restricted range
how to composite functions when it's sin(sin-1), etc.
it's asking for a value. you look at the value given on the inside. if it's not within the domain it is undefined, but otherwise that is your answer
how do u answer questions about a sinusoidal equation (like what is this when its a certain value)
just plug in
arc length
labeled as S S= radius times central angle IN RADIANS
arithmetic rule (type of equation)
linear
asymptotes hyperbola definition
lines that the branches of the hyperbola approach
how to find values w/ comp angle theorem and identities
make sure the degree measures are the same first! (WITH COFUNCTIONS sin/cos, csc/sec, tan/cot) rewrite things until you can visually cross things out or subtract the same thing SHOW ALL WORK. if its subtraction, look at pythagorean identities. if multiplication, quotient
how to find mins and maxs of sinusoidal equations
max: SA + amplitude min: SA - amplitude
cramer's rule
method where determinants are used to solve a system of equations (only linear!)
how to convert radians to degrees
multiply by 180/pi
how to convert degrees to radians
multiply by pi/180
multiplication principle of counting
multiply the number of selections
union rule for counting
n(A U B) = n(A) + n(B) - n(A⌒B)
if empty set/mutually exclusive, union rule for counting
n(A∪B)= n(A) + n(B)
permutations w/ no repetition
nPr
permutations w/ repetitition
n^r
quadrantal angles
on an axis. past 360 degrees
tan
opp/adj
sin
opp/hyp
period for cot
pi
period for tan
pi
how to know first term from eq (arithmetic)
plug in 1 for n
vertex parabola
point of intersection of the parabola with its axis of symmetry (half the distance from focus and directrix)
solve linear trig equations
pretend the trig value is an x and isolate it and solve like a linear equation
characteristics of unit circle
radius = 1 center = origin
sec
reciprocal of cos
csc
reciprocal of sin
cot
reciprocal of tan
sinusoidal graphs
represent cos as a phase shift of sin and vice versa
diagonal method for 3 x 3 matrix
rewrite the first two columns of the matrix next to the matrix. start at top, add the product of the first three diagonals together. then go to bottom, and subtract the sum of the three diagonals after
angle of elevation
rise from horizontal ex: ground
terminal side
rotated ray that determines measure of angle
dimension of a matrix
rows by columns
how to find coefficient of binomial expansion
the top of the choose will be the exponent. the first term exponent will be whatever they give you (if there is an exponent on the actual term, remember the exponents multiply to get the exponent you need). the last term exponent and bottom choose number is whatever the first choose number minus the first exponent is. in the end you must multiply that term out to figure the coefficient out
reference angle quadrant 3
theta - 180 or theta - pi
positive x axis
to the right where the numbers are positive
what to do if there's repeats within
total! ________ repeat!
ambiguous case
triangles for SSA can result in one triangle, two triangles, or no triangle at all
foci ellipse
two fixed points
vertices ellipse
two points where the ellipse intersects the major axis
co-vertices ellipse
two points where the ellipse intersects the minor axis
how to find third angle law of sines
use 180- the other sides
what to do if there's a negative within the parenthesis
use the even/odd properties before you graph. (unless its like a minus with a phase shift, this is referring to when the B value is negative). if its even get rid of the negative. if its odd pull out the negative in front
how to find an angle with trig functions
use the inverse functions in degrees
how to find trig functions on the calculator
use the sin, cos, and tan buttons. for the cofunction identities, you'll just do one divided by the cofunction. MAKE SURE YOURE IN THE RIGHT MODE!!! if there is no degree symbol, its radians.
D value
vertical shift, moves the sinusoidal axis D units away from x-axis
when can you multiply matrices
when matrices have the number of columns in the first matrix is equal to the number of rows in the second matrix
vertices in a hyperbola definition
where branches intersect the transverse axis, always on the transverse axis
asymptotes for tan
where the tan is undefined the asymptotes combine for one period away from origin (half the amplitude to right, the other half to the left)
asymptote for vertical hyperbola
y-k= ±a/b (x-h)
asymptote for horizontal hyperbola
y-k= ±b/a (x-h)
how to do composite functions when it's sin-1(sin), etc.
you are looking for an angle. you take the angle given on the inside and then manipulate it so it's within the restricted range (use reference angle!!)
how to find trig functions with identities
you can't draw a triangle and you have to use all the different identities you know to find it
reduced triangle principle
you could reduce all the sides of the triangle by a common multiple to make it easier for trig functions
how to find inverse values for sec, csc, cot
you could use the reciprocal sec and csc to make it easier BUT DONT DO FOR COT because cot is different. for cot just think of cos/sin instead
how to multiply matrices
you multiply the first term of the first row by the first term of the first column and so on. you add them all up in that position
what if the determinant of coefficient matrix equals zero
you must use a different method to solve the system
how to solve trig equations if different trig functions within
you need to rewrite any of the squared values using identities and then solve like normal
how to do composite functions when there is different trig functions
you start with the inner and perform the operation. then you take the answer and perform the next operation for your answer
what to do to find two sides in word problem
you use the angles and the side and trig functions. but when you find the second one DONT use the new side you find because its an approximation
solve trig equations quadratic in form
you would factor the equation like the trig function is x, or use the square root method BUT REMEMBER PLUS OR MINUS!
how to express all real number solutions
{ (x, y) | equation }
