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8/6/2019 High School Math Notes
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Geometry Axioms
Point: an undefined term in geometry, a point can be thought of as a dot that represents a location on a plane or
in space; just a location, no thickness or size
Line: an undefined term in geometry, a line is understood to be straight, containing an infinite number of points
extending infiting in two directions, and having no thickness
Line Segment: a portion of a line with two end points
Ray: consists of an intial point on a line and all of the points on that line on one side of it
Plane: an undefined term in geometry, a plane is understood to be a flat surface that extends infinitely in al
directions
Angle: a figure formed by two rayss that have the same endpoint
Collinear: three or more points not all of which lie on the same line
Noncolinear: points that lie on the same line
Perpendicular Lines: lines that cross at a 90 angleParallel Lines: Lines in the same plane that never meet and hav ethe same slope
Skewed Lines: lines that are not parallel and do not cross; lines in different planes
one and only one line passes through two points
one and only one plane passes through two lines
two points define an unique line
three non-collinear points define an unique plane
Logic
Postulate: a statement that is accepted as true without proof
Theorem: a statement that must be proved to be true
Inductive Reasoning: forming conjectures on the basis of an observed pattern
Deductive Reasoning: the process of drawing conclusion by using logical reasoning
Conjecture: an educated guess based on an observation
Hypothesis: the phrase in a conditional statement following the word if
Conditional Statement: a statement that can be written in the form: if P then QConverse Statement: the statement formed by interchanging the hypothesis and conclusion of a conditiona
statement
Counter Example: an example which proves that a conditional statement is false in that the hypothesis is true but
the conclusion is false
Conclusion: the phrase in a conditional statement following the word then
Lines
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Ruler Postulate: The points on a line can be matched one to one with the set of real numbers. The real numbe
that corresponds with a point is the coordinate of the point. The distance, AB between two points, A and B on a
line is equal to the absolute value of the difference between the coordinates of A and B.
Segment Addition Postulate: If B is between A and C then AB +BC= AC
Parallel Postulate: IF there is a line and a point not on the line, THEN there is exactly on a line through the point
parallel to the given line
Perpendicular Postulate: IF there is a line and a point not on the line, THEN there is exactly on a line through the
point perpendicular to the given line
Transitive Property of Parallel Lines: IF two lines are parallel to the same line THEN they are parallel to each
other Property of Perpendicular Lines: IF two coplanar lines are perpendicular to the same line THEN they are
parallel to each other
Perpendicular Bisector Theorem: IF a point is on the perpendicular bisector of a segment THEN it is equidistan
from the endpoints of the segment
Perpendicular Bisector Converse: IF a point is equidistant from the endpoints of a segment THEN it lies on the
perpendicular bisector of the segment
Angle Bisector Theorem: IF a point is on the bisector of an angle THEN it is equidistant from the two sides of the
angle
Angle Bisector Converse: IF a point is in the interior of an angle and equidistant from the sides of the angle
THEN it lies on the bisector of the angle
IF two distinct planes intersect, THEN their intersection is a line
IF two distinct points lie in a plane, THEN the line containing them lies in the plane
IF two distinct lines intersect, THEN their intersection is exactly one point
IF three parallel lines intersect two transversals, THEN they divide the transversals proportionally
IF two parallel lines intersect two other parallel lines and IF the distance between the first two lines is equa
to the distance between the second two lines THEN their intersection forms four congruent segments
Angles
Protractor Postulate: Let OA be a ray and consider on of the half-plane P determined by the line OA. The rays
of the form ray DO, where D is in a half-plane P can be put in one-to-one correspondence with the real numbers
between 0 and 180, including 180. If C and D are in the half-plane P, then the measure of angle COD is equal to
the absolute value of the difference between the real numbers for ray OCand ray OD.
Angle Addition Postulate: If B is the interior of angle
AOC, then
AOB +
BOC =
AOCLinear Pair Postulate: IF two angles form a linear pair, THEN they are supplementary; that is, the sum of their
measures is 180
Corresponding Angles Postulate: IF two parallel lines are cut by a transversal, THEN the pairs of corresponding
angles are congruent
Corresponding Angles Postulate: IF two parallel lines are cut by a transversal so that corresponding angles are
congruent, THEN the lines are parallel
Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the
two arcs.
Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal to ???????????
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Interior Angle Sum: 180 (n2) n = # of sidesExterior Angle Sum: always = 360
# of Diagonals:n (n3)
2n = # of sides
Slope:rise
run=y
x=
y2y1x2x1 = m
Area Congruence Postulate: IF two polygons are congruent THEN they have the same area
Area Addition Postulate: the area of a region is the sum of the areas of all its non-overlapping partsCavalieris Principle: IF two solids have the same height and the same cross-sectional area at every level, THEN
they have the same volume
IF two polygons are similar THEN the ratio of their perimeters is equal to the ratio of their corresponding
sides
IF two polygons are similar with corresponding sides in the ratio of a : b THEN the ratio of their areas is
a2 : b2
Areas
A =1
2BH (triangle)
A = s2 (square)
A = BH (rectangle)
A = BH (parallelogram)
A =
1
2 (B1 +B2)H (trapezoid)
A =1
2d1d2 = BH (rhombus)
A = r2 (circle)
A = r1r2 (ellipse)
A =2
3BH (under symmetrical parabola)
Surface Areas
SA = 6s2 (cube)
SA = 2 (rectangular prism)
SA = (rectangle parallelpiped)
Pyramid:
Cone:
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Sphere: SA = 4r2 (sphere)
Ellipsoid:
Volumes
V = s3 (cube)
V = HL2 (rectangular prism)
V = LWH (rectangular parallelpiped) V = BH (irregular prism)
V =1
3BH (pyramid)
V =1
3r2h (cone)
V = r2h (cylinder)
V =4
3r3 (sphere)
V =4
3r1r2r3 (ellipsoid)
Triangles
Similar:
Congruent: AAS, HL
Triangle Sum Theorem: the sum of the measures of the angles of a triangle is 180
Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of
the two remote (nonadjacent) interior angles
Exterior Angle Inequality: the measure of an exterior angle of a triangle is greater than the measure of either of
the two nonadjacent interior angles
Triangle Proportionality Theorem: IF a line parallel to one side of a triangle intersects the other two sides THEN
it divides the two sides proportionally
Triangle Proportionally Theorem Converse: IF a line divides tow sides of a triangle proportionally THEN it isparallel to the third side
Midsegment Theorem: The segment connection the midpoints of two dies of a triangle is parallel to the third side
and is half its length
Triangle Inequality: the sum of the lengths of any two sides of a triangle is greater than the length of the third side
Third Angles Theorem: IF two angles of one triangle are congruent to two angles of a second triangle THEN the
third angles are also congruent
Base Angles Theorem: IF two sides of triangles are congruent THEN the angles opposite them are congruent
Corollary: IF a triangle is equilateral THEN it is also equiangular
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Hinge Theorem: IF two sides of one triangle are congruent to two sides of another triangle, and the included angle
of the first is larger than the includes angle of the second THEN the third side of the first is longer than the third
side of the second
Converse of Hinge Theorem: IF two sides of one triangle are congruent to two sides of another triangle, and the
third side of the first is longer than the third side of the second THEN the included angle of the first is larger than
the included angle of the second
The Pythagorean Theorem: in a right triangle, the square of the length of the hypotenuse is equal to the sum of
the squares of the lengths of the legsThe Converse of The Pythagorean Theorem: IF the square of the length of the longest side of a triangle is equa
to the sum of the squares of the lengths of the two shorter sides, THEN the triangle is a right triangle
Concurrency Properties:
the lines containing the perpendicular bisectors of a triangle are concurrent
their common point is the circumcenter of the triangle
the circumcenter is equidistant from the three vertices of the triangle
the angle bisectors of a triangle are concurrent their common point is the incenter of the triangle
the incenter is equidistant from the three sides of the triangle
the medians of a triangle are concurrent
their common points is the centroid of the triangle
the centroid is two thirds of the distance from each vertex to the midpoint of the opposite side
the lines containing the altitudes of a triangle are concurrent
their common point is the orthocenter
IF two angles of a triangle are congruent THEN the sides opposite them are congruent
the acute angles of a right triangle are complementary
every triangle is congruent to itself
IF one side of a triangle is longer than another side THEN the angle opposite the longer side is larger than
the angle opposite the shorter side
IF one angle of a triangle is larger than another angle THEN the side opposite the larger angle is longer than
the side opposite the smaller angle
IF a ray bisects an angle of a triangle THEN it divides the opposite side into segments whose lengths are
proportional to the lengths of the other two sides
IF the altitude is drawn to the hypotenuse of a right triangle THEN the two triangles formed are similar to the
original triangle and to each other
in a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of
the lengths of the two segments of the hypotenuse
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Right Triangles
hc =ab
c(altitude of c)
ta = 2bccos
A2
(b + c)
(angle bisector of a)
tb = 2accos
B2
(a + c)
(angle bisector of b)
tc = ab
2
(a + b)(angle bisector of c)
ma =
4b2 + a2
2(median of a)
mb =
4a2 + b2
2(median of b)
mc =c2
(median of c)
r=ab
a + b + c(inscribed circle radius)
R =c
2(circumscribed circle radius)
Equilateral Triangles
K= a2
3
4(area)
ha = hb = hc = a
3
2(altitude)
ma = mb = mc = a
3
2(median)
ta = tb = tc = a
3
2(angle bisector)
R = a
3
3(circumscribed circle radius)
r= a
3
6(inscribed circle radius)
Quadrilaterals
IF both pairs of opposite sides of a quadrilateral are congruent THEN the quadrilateral is a parallelogram
IF both pairs of opposite angles of a quadrilateral are congruent THEN the quadrilateral is a parallelogram
IF an angle of a quadrilateral is supplementary to both of its consecutive angles THEN the quadrilateral is a
parallelogram
IF the diagonals of a quadrilateral bisect each other THEN the quadrilateral is a parallelogram
IF the diagonals of a quadrilateral are perpendicular THEN the area of the quadrilateral is half the sum of the
lengths of the diagonals or A = 0.5 (d1 + d2)
IF one pair of opposite sides of a quadrilateral are congruent and parallel THEN the quadrilateral is a paral-
lelogram
a parallelogram is a rhombus IF AND ONLY IF its diagonals are perpendicular
a parallelogram is a rhombus IF AND ONLY IF each diagonal bisects a pair of opposite angles
a parallelogram is a rectangle IF AND ONLY IF its diagonals are congruent
a quadrilateral is a rhombus IF AND ONLY IF it has four congruent sides
a quadrilateral is a rectangle IF AND ONLY IF it has four right angles
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SASAS Congruence Theorem: IF three sides and the included angles of one quadrilateral are congruent to the
corresponding three sides and included angles of another quadrilateral THEN the quadrilateral are congruent
ASASA Congruence Theorem: IF three angles and the included side of one quadrilateral are congruent to the
corresponding three angles and the included sides of another quadrilateral THEN the quadrilateral are congruent
Parallelograms
opposite sides are congruent
opposite angles are congruent
consecutive angles are supplementary
diagonals bisect each other
Trapezoids
Trapezoid Base Angles Theorem: IF a trapezoid is isosceles THEN each pair of base angles is congruent
Trapezoid Diagonals Theorem: IF a trapezoid is isosceles THEN its diagonals are congruent
Midsegment Theorem for Trapezoids: the midsegment of a trapezoid is parallel to each base and its length is
half the sum of the lengths of its bases
IF a trapezoid has one pair of congruent base angles THEN it is an isosceles trapezoid
IF a trapezoid has congruent diagonals THEN it is an isosceles trapezoid
Kites
diagonals are perpendicular
exactly one pair of opposite angles are congruent
Circles
Standard Form: (xh)2 + (y k)2 = r2radius = r center = (h, k)
L = (arc length)
A = (arc sector area)
Major Arc: an arc of more than 180; represented by three lettersMinor Arc: an arc of less than 180; represented by two lettersInscribed Angle of a Circle: an angle whose vertex is on the circle and its sides are chords of the circle
Intercepted Arc: the arc that lies in the interior of an inscribed angle
Tangent of a Circle: a line that intersects the circle at exactly one point
Secant of a Circle: a line that intersects the circle in two points
IF a line is tangent to a circle THEN it is perpendicular to the radius drawn to the point of tangency in a plane
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IF a line is perpendicular to a radius of a circle at its endpoint on the circle THEN the line is tangent to the
circle
IF two segments from the same exterior point are tangent to a circle THEN the segments are congruent
in the same circle or in congruent circles, two arcs are congruent IF AND ONLY IF their central angles are
congruent
in the same circle or in congruent circles, two minor arcs are congruent IF AND ONLY IF their corresponding
chords are congruent
IF a diameter of a circle is perpendicular to a chord THEN the diameter bisects the chord and its arc
IF a chord AB is perpendicular bisector of another chord THEN AB is a diameter
in the same circle or in congruent circles, two chords are congruent IF AND ONLY IF they are equidistant
from the center
IF an angle is inscribed in a circle THEN its measure is half the measure of its intercepted arc
IF two inscribed angles of a circle intercept the same arc THEN the angles are congruent
an angle that is inscribed in a circle is a right angle IF AND ONLY IF its corresponding arc is a semicircle
a quadrilateral can be inscribed in a circle IF AND ONLY IF its opposite angles are congruent
IF a tangent and a chord intersect at a point on a circle THEN the measure of each angle from it is half the
measure of its intercepted arc
IF two chord intersect in the interior of a circle THEN the measure of each angle is half the sum of the
measure of the arcs intercepted by the angle and its vertical angle
IF a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle THEN the measure
of the angle from is half the difference of the measure of the intercepted arcs
Trigonometry
A2 +B2 = C2 (Pythagorian theorem)
A sin () +B cos () =
A2 +B2 sin+
4
=
A2 +B2 cos
4
(sum of sin and cos)
Right Angle Trigonometry:
Sine = oppositehypotenuse
Cosine = adjacenthypotenuse
Tangent = oppositeadjacent
Secant = hypotenuseopposite
Cosecant = hypotenuseadjacent
Cotangent = oppositeadjacent
sinA
a=
sinB
b=
sinC
c(law of sines)
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Law of Cosines:
A2 = B2 +C22BCcosAB2 = A2 +C22ACcosBC2 = A2 +B22AB cosCLaw of Tangents: cosA =cosB cosC+ sinB sinCcosA
Linear Equations
Standard Form: Ax +By = C A,B,Care integers and A is postive
Slope-Intercept Form: y = mx + b m = slope, b = y-intercept
Point Slope Form: yy0 = m (xx0) (y0,x0) = a point on the line m = slope
m = tan =y2y1
x2x1 (slope of a line)
= tan1 m2m11 + m1m2
(angle between intersecting lines)
(x2x1)2 + (y2y1)2 (distance between two points)
x1 +x2
2,
y1 +y22
(midpoint of a line segment P1P2)
m1x2 + m2x1
m1 + m2,
m1y2 + m2y1m1 + m2
(point dividing a line segment P1P2 in a ratio of m1 : m2)
Quadratic Equations
y = Ax2 +Bx +C (quadratic equation; standard form)
y = a (xh)2 + k (quadratic equation; vertex form)
x =b2a
or x = h (axis of symmetry)
b2a
, f
22a
or (h, k) (vertex)
(0,C) or (0, k) (Y intercept)
x = b
b2
4ac
2a (quadratic formula)
Completing The Square:
ax2 + bx + c = 0 (standard form)
h = b2a
k= c b2
4a
a (xh)2 + k= 0 (vertex form)
x = hk
a(roots)
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Polynomials
Factoring:
x2y2 = (x +y)(xy) (difference of squares) x2 + 2xy +y2 = (x +y)2 (trinomial square sums)
x22xy +y2 = (xy)2 (trinomial square difference) (x +y)3 = x3 + 3x2y + 3xy2 +y3 (cubed binomial sum)
(xy)3 = x33x2y + 3xy2y3 (cubed binomial difference) x3 +y3 = (x +y)
x2xy +y2 (sum of cubes)
x3y3 = (xy)x2 +xy +y2 (difference of cubes)Fundamental Theorem of Algebra: every non-zero single-variable polynomial with complex coefficients has ex-
actly as many complex roots as its degree, if each root is counted up to its multiplicityIntegral Root Theorem: all possible rational roots of the polynomial xn + an1xn1 + + a0 = 0 are of the form
x =p where:
p is an integer factor of the constant term a0
Rational Root Theorem: all possible rational roots of the polynomial anxn + an1xn1 + + a0 = 0 are of the
form: x =pq
where:
p is an integer factor of the constant term a0
q is an integer factor of the leading coefficient an
Radical Conjugate Root Theorem:
if a polynomial P (x) with rational coefficients has a +
b as a zero, where a, b are rational and
b is
irrational, then a
b is also a zero
Complex Conjugate Root Theorem:
if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbersthen its complex conjugate a
bi is also a root of P
Descartes Rule of Signs:
the number of positive roots of the polynomial anxn + an1xn1 + + a0 = 0 is either equal to the numbe
of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2
multiple roots of the same value are counted separately
the number of negative roots is the number of sign changes after negating the coefficients of odd-power
terms (otherwise seen as substituting the negation of the variable for the variable itself), or fewer than it by a
multiple of 2
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Exponents
the variables m and n are integers
no denominater equals zero
a
0
= 1(a = 0
)
am an = am+n
(ab)n = anbn
(am)n = amn
an =1
an
am
an = am
n
ab
n=
an
bn
a1n = n
a
amn =
n
am =
n
am
Radicals
even roots require if you divid by a variable in an equation, you mus
check for zero as a root
n
an
= |a| (nis even)
n
a n
b =n
ab
n
an
b= n
a
b(b = 0)
a1n = n
a
amn =
n
am =
n
am
x mn nm = |x|
n
m
x = mn
x
a n
x + b n
x = ab n
x (like radicals)
a +
b
a
b
= a2b2 (conjugates)
conjugates can be used to rationalize a binomia
denominator
Simplest Form
Cant Have: negative exponent
Fix: make fraction and rationalize denominator or multiply by another negative
Cant Have: rational exponent
Fix: turn base into number raised to a power equal to the denominator
Cant Have: radical in the denominator
Fix: use conjugate for binomials multiply radical by itself number of times equal to the index
Rationalize denominator
make radical a fraction and make denominator perfect square, etc.
multiply fraction by the denominator to make denominator which is a radical into an integer
Logarithms
the variables M, N, and b are positive numbers
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the variable b = 1 IF y = bx TH E N logby = x
loga ax = x
loga 1 = 0
loga a = 1
alogax = x
logb (MN) = logbM+ logbN
logb
M
N
= logbM logbN
logb (Mx) = x logbM
logbM=logcM
logc b(change of base; b = 1, c = 1, M > 0)
Complex Fractions
Complex Numbers
x = ix, x > 0
in =
r= 0 1
r= 1 i
r= 2 1r= 3 i
(value of i after dividing exponent by 4)
Re [Z] = A = rcos (real part)
Im [Z] = B = rsin (imaginary part)
r= |z| = |A +Bi| =
A2 +B2 (modulus; absolute value)
= tan
1BA
(argument)
arguments have 2 periodicity
z = r(cos+ i sin) (polar form)
z = rei (alternate polar form)
ei = cos+ i sin (Euler relation)
z1z2 = r1r2[cos (1 +2) + i sin (1 +2)] (complex product; polar form)
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z1
z2=
r1
r2[cos (12) + i sin (12)] , z2 = 0 (complex quotient; polar form)
zn = rn (cos n+ i sin n) , n = integer (complex to power; polar form)
wk = r1n
cos
+ 2k
n
+ i sin
+ 2k
n
, k= {0, 1, 2, . . . , n1} (root of complex; polar form)
d
dtei(t) = iei(t) d
dt(complex derivative)
d
dt(Re [Z]) = Re
dZ
dt
(commutativity)
Inequalities
Trichotomy Property: given any two real numbers a and b, then only one of the following statements must hold
true
a < b a = b a > b
for a, b, and c are real numbers:
all of the following properties hold for and in addition to < and > remember to change the direction of the inequality when multiplying or dividing by a negative number
IF: a < b THEN: a + c < b + c (inequality addition property)
IF: a > b THEN: a + c > b + c (inequality addition property)
IF: a < b THEN: a
c < b
c (inequality subtraction property)
IF: a > b THEN: ac > b c (inequality subtraction property) IF: a < b THEN: ac < bc (inequality multiplication property; c > 0)
IF: a > b THEN: ac > bc (inequality multiplication property; c > 0)
IF: a > b THEN: ac < bc (inequality multiplication property; c < 0)
IF: a < b THEN: ac > bc (inequality multiplication property; c < 0)
IF: a < b THEN:a
c 0)
IF: a > b THEN:a
c>
b
c(inequality division property; c > 0)
IF: a > b THEN:a
c
b
c(inequality division property; c < 0)
Absolute Values
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|a| 0 |a| = 0 a = 0 |ab| = |a| |b| |a + b| |a|+ |b| |a| = |a|
|ab| = 0 a = b
|ab| |a c|+ |cb|
ab
= |a||b| if b = 0
|a
b| ||
a| |
b||
Number Theory
Number Divisibility Condition
1 automatic
2 last digit is even
3 sum of the digits is divisible by 3
4 the last two digits are divisible by 4
5 the last digit is a 0 or a 5
6 the number is divisible by 2 AND 3
7 the sum of [three times the first digit] and [the second digit] is divisible by 7
8 the last three digits are divisible by 8
9 the sum of the digits is divisible by 9
10 the last digit is zero
limk
n
A = xk+1 =1
n
(n1)xk+ A
xn1k
(nth root numerical approximation algorithm)
Greatest Common Divisor (Greatest Common Denominator) (GCD) Properties:
every common divisor of a and b is a divisor of gcd (a, b)
if a divides the product bc, and gcd (a, b) = d, then a/d divides c if m is a non-negative integer, then gcd (m a, m b) = m gcd (a, b) if m is any integer, then gcd (a + m b, b) = gcd (a, b)
if m is a nonzero common divisor of a and b, then gcdam
,bm=
gcd(a, b)m
the gcd is a multiplicative function in the following sense: if a1 and a2 are relatively prime, then gcd (a1 a2, b) gcd (a1, b) gcd (a2, b)
the gcd is a commutative function: gcd (a, b) = gcd (b, a)
the gcd is an associative function: gcd (a, gcd (b, c)) = gcd (gcd (a, b) , c)
Fundamental Theorem of Artihmetic (Unique Prime Factorization Theorem): any integer greater than 1 can
be written as a unique product (up to ordering of the factors) of prime numbers
there are infinitely many prime numbers
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Mihailescus Theorem (Catalans Conjecture): the only solution in the natural numbers ofxayb = 1 forx, a, y,is x = 3, a = 2, y = 2, b = 3
Combinatorics
Artihmetic Series: constant adding or subtracting
Geometric Series: constant multiplying or dividing
an = a1 + r(n1) (nth term of arithmetic series)
Sn =n
2(a1 + an) (sum of first n terms of arithmetic series)
an = a1rn1 (nth term of geometric series)
Sn =a1 (1 rn)
1 r (sum of first n terms of geometric series)
S =a1
1
r, |r| < 1 (infinite sum of geometric series)
Factorials:
n! =n
1
(n > 0)
0! = 1
Permutation: n things ordered in the permutation number of ways taking r at a time
nPr =n!
(n r)!(1
r
n)
Combination: n things grouped in the combination number of ways taking r at a time
nCr =n
r
=
n!
r! (n r)! (0 r n)
Binomial Theorem:
faster than Pascals triangle and distribution after (a + b)3
the series contains n + 1 terms
once the combinations start to repeat, just go back down the pattern
if a or b is a variable with a nonzero coefficient, both the variable and the coefficient must be raised to the
appropiate power
to find a single term, realize that the sum of the exponents of a and b is equal to n
(a + b)n =n
r=0
nCr anrbr (Compact Formal)
(a + b)n = nC0 anb0 + nC1 a
n1b1 + nC2 an2b2
+ nCn1 a1bn1 + nCn a0bn (Formal)
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(a + b)n = an + n an1b + nC2 an2b2 + n abn1 + bn (Simplified)
nCr =(nCr1) (n r)
r1 (coefficient of the rterm using the previous term)
Probability
Disjoint: cant occur togetherConditional: if A occurs then P (B)
P (A and B) = P (A) P (B) (independent) P (A or B) = P (A) + P (B) (disjoint)
P (B |A) = P (A and B)P (A)
(conditional)
P (A and B) = P (A) P (B |A) (general)
P (A or B) = P (A) + P (B)P (A and B) (general) P (x = k) =
n
k
Pk(1P)nk (binomial)
Statistics
Median: in a set with an odd number of values, the value exactly in the middle of the set ordered from smallest to
largest; with an even number of values, the average of the two in the middle
Mode: the most frequently occuring value or values in a set
Range:
Interquartile Range (IQR):
A = x =1
N
N
i=1
xi (arithmetic mean)
G =
n
i=1
ai
1n
= n
a1a2 . . .an (geometric mean)
H =1
1n ni=1 1ai
=n
1a1 + 1a2 + + 1an(harmonic mean)
H G A (relation of means) the means are only equal when every element of the data set are equal
=
1
N
N
i=1
(xi x)2 (population standard deviation)
=
1
n
1
n
i=1
(xi x)2 (sample standard deviation)
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Conics
Conic Section: a geometric figure created by the intersection of a plane and a hollow double napped cone
Locus: the set of points sharing a given property
Focus:
Directrix:
Focal Parameter: the distance between a focus and the nearest directrix
Latus Rectum:
Eccentricity (e):
Linear Eccentricity: the distance between the center and one of the foci
e =c
a=
[distance between foci]
[distance between vertices](eccentricity)
e = 0 (circle)
0 < e < 1 (ellipse)
e = 1 (parabola)
e > 1 (hyperbola)
r=ke
1 + e cos(polar conic)
Discriminant Test: Ax2 +Bxy +Cy2 +Dx +Ey + F = 0 (general quadratic curve)
B24AC< 0 (ellipse) B24AC= 0 (parabola) B24AC> 0 (hyperbola)
Degenerate Cases:
No Real Graph: circle or ellipse with negative right hand side
Point: circle or ellipse with right hand side equal to zero
Single Line: neither variable squared or only one variable present and equal to zero
Parallel Lines: one variable squared and the other absent and the right hand side positive
Intersecting Lines: hyperbola with a negative right hand side
Rotation of Axes:
x = x cosy sin y = x siny cos
Angle of Rotation:
tan (2) =B
A
C()
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Parabola: the locus of points in a plane where each point is equidistant from a focus and a directrix
(xh)2 = 4p (yk) (parabola standard form; opens up)
(h, p + k) (focus)
y = kp (directrix) (h, k) (vertex)
(xh)2 =4p (y k) (parabola standard form; opens down)
(h, kp) (focus) y = k+ p (directrix)
(h, k) (vertex)
(y k)2 = 4p (xh) (parabola standard form; opens to the right)
(p + h, k) (focus)
y = hp (directrix) (h, k) (vertex)
(y k)2 =4p (xh) (parabola standard form; opens to the left)
(hp, k) (focus) y = h + p (directrix)
(h, k) (vertex)
Circle: the locus of points in a plane that are equidistance from a focus
(xh)2 + (y k)2 = a2 (circle standard form)
(h, k) (center)
a (radius)
Ellipse: the locus of points in a plane where the sum of the distances between two focii are equal to a constant
(xh)2
a2+
(yk)2b2
= 1 (ellipse standard form; foci on the x-axis)
a = semimajor axis b = semiminor axis c =
a2b2 (center to focus distance)
(hc, k) (foci) (ha, k) (vertices) (h, k) (center)
(xh)2
b2+
(yk)2a2
= 1 (ellipse standard form; foci on the y-axis)
a = semimajor axis b = semiminor axis
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Negation:
sin (x) = sin (x)
cos (x) = cos (x)
tan (
x) =
tan (x)
csc (x) = csc (x)
sec (x) = sec (x)
cot (
x) =
cot (x)
Reciprocal:
sin (x) =1
csc (x)
cos (x) =1
sec (x)
tan (x) =1
cot (x) =sin (x)
cos (x)
Periodicity:
sin
x +
2
= cos (x)
cos
x +
2
=sin (x)
tan
x +
2
=cot (x)
csc
x + 2
= sec (x)
sec
x +
2
=csc (x)
cot
x +
2
= tan (x)
sin (x +) = sin (x)
cos (x +) = cos (x)
tan (x +) = tan (x)
csc (x +) =
csc (x)
sec (x +) = sec (x)
cot (x +) = cot (x)
sin (x + 2) = sin (x)
cos (x + 2) = cos (x)
tan (x + 2) = tan (x)
csc (x + 2) = csc (x)
sec (x + 2) = sec (x)
cot (x + 2) = cot (x)
Cofunction:
sin
2x
= cos (x)
tan2x= cot (x)
sec
2x
= csc (x)
cos
2x
= sin (x)
cot2x= tan (x)
csc
2x
= sec (x)
Addition and Subtraction:
sin (x +y) = sin (x) cos (y) + cos (x) sin (y)
sin (xy) = sin (x) cos (y)cos (x) sin (y) cos (x +y) = cos (x) cos (y)
sin (x) sin (y)
cos (xy) = cos (x) cos (y) + sin (x) sin (y)
tan (x +y) =tan (x) + tan (y)
1 tan (x) tan (y)
tan (xy) = tan (x) tan (y)1 + tan (x) tan (y)
Double Angle:
sin (2x) = 2sin (x) cos (x)
tan (2x) =2tan (x)
1 tan2 (x) cos (2x) = cos2 (x) sin2 (x) = 2cos2 (x)1 = 12sin2 (x)
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Half Angle:
sinx
2
=
1 cosx
2
cosx
2
=
1 + cosx
2
tan
x2
= 1 cosx
sinx= sinx
1 + cosx
Reducing Powers:
sin2 (x) =1cos (2x)
2
cos2 (x) =1 + cos (2x)
2
tan2 (x) =1
cos(
2x)1 + cos (2x)
Product to Sum:
sin (x) cos (y) =1
2[sin (x +y) + sin (xy)]
cos (x) sin (y) =1
2[sin (x +y) sin (xy)]
cos (x) cos (y) =1
2[cos (x +y) + cos (xy)]
sin (x) sin (y) =1
2[cos (xy) cos (x +y)]
Sum to Product:
sinx + siny = 2sin
x +y
2
cos
xy
2
sinx siny = 2cos
x +y
2
sin
xy
2
cosx + cosy = 2cosx +y2cosxy
2
cosxcosy =2sin
x +y
2
sin
xy
2