Midterms Flashcards
two-valued logic
every statement is either True or False
truth table
used to determine the truth or falsity of a complicated statement based on the truth or falsity of its simple components
conjunction
“and”; true when both statements are true; ^
disjunction
“or”; true when @ least 1 statement is true, V
negation
“not”, ~
inclusive “or”
doing 1/other/both
when is p → q not true?
when p is true and q is false
tautology
rule of logic
a formula which is “always true” - all the end results are true
p ⇔ q
p iff q
both p & q r equivalent. true if p & q r both true/both false
contradiction
opposite of a tautology, a formula which is “always false”
what r p &q called in p ⇒ q?
p = hypothesis
q = conclusion
p ⇒q
if p, then q
p implies q
p if q
4 ways to rewrite a statement
1) if p, then q
2) Every p has q.
3) The fact that p, implies that q
4) p iff/if/only if q
converse
q ⇒ p
inverse
~p ⇒ ~q
contrapositive
~q ⇒ ~p
Direct Argument
p ⇒ q
p
…q
premise
a statement that is assumed to be true
a given statement in an argument. the resulting statement is called the conclusion
Indirect Argument
p ⇒ q
~q
… ~p
Chain Rule
p ⇒ q
q ⇒ r
…p ⇒ r
Or Rule
p V q
~p
…q
p V q
~q
…p
good definition
built from a true conditional with a true converse
invalid argument
argument that doesn’t use rules of logic
4 rules of biconditionals
p ⇔ q
p
… q
p ⇔ q
q
… p
p ⇔ q
~p
… ~q
p ⇔ q
~q
… ~p
Venn diagram placement for conditionals/implications
two-column proof
a proof written in 2 columns
statements r listed in 1 column & justifications r listed in the other column
paragraph proof
a proof whose statements & justifications r written in paragraph form
flow proof
’s written over the arrows refer to a #-ed list of the justifications 4 the statements
a proof written as a diagram using arrows to show the connections b/w statements
postulate
a statement assumed to be true w/out proof
Addition Property of Equality
If the same # is added to equal #’s, the sums r equal
a = b → a + c = b + c
Subtraction Property of Equality
If the same # is subtracted from equal #’s the diff’s r equal
a = b → a - c = b - c
multiplication property of equality
If equal #’s r multiplied by the same #, the products r equal
a = b → ac = bc
division property of equality
if equal #’s r divided by the same nonzero #, the quotients r equal
a = b and c =/ 0 → a/c = b/c
reflexive prop of equality
a # is equal to itself
a = a
substitution property
if values r equal, 1 value may be substituted 4 the other
a = b → a may be substituted for b
distributive prop
An expression of the form a(b + c) is equivalent to ab + ac
a(b + c) = ab + ac
square root
one of 2 equal factors of a #
straight angle postulate
If the sides of an angle form a straight line, then the angle is a straight angle with a measure of 180
angle or segment addition postulate
or
whole and parts postulate
for any segment/angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts
Supplements of Angles Theorem
If 2 angles r supplementary to the same angle, then they r equal in measure
complements of angles theorem
If 2 angles r complements of the same angle, then they r equal in measure
vertical angle theorem
All vertical angles r equal in measure
corresponding angles postulate
if 2 parallel lines r intersected by a transversal, then corresponding angles r equal in measure
alternate interior angles theorem
if 2 parallel lines r intersected by a transversal, then alternate inteiror angles r equal in measure
what kinds of angles r these?
angles 3 & 6 r alternate interior angles
angles 1 & 8 r alternate exterior angles
angles 3 & 5 r cointerior angles
angles 2 & 6 r corresponding angles
co-interior angles theorem
If 2 parallel lines r intersected by a transversal, then co-interior angles r supplementary
what kinds of angles r shown below?
Angles A & D r consecutive angles
Angles A & C r opposite angles
Consecutive Angles th
If a quadrilateral is a parallelogram, then consecutive angles r supplementary
opposite angle theorem
If a quadrilateral is a parallelgoram, then opposite angles r equal in measure
Venn diagram placement for biconditionals
hypothesis
the if part of an if-then statement
conclusion
the then part of an in-then statement
implication/conditional
a statement with an if part & a then part
biconditional
the conjunction of a true conditional & its true converse, usually written using the phrase if and only if (iff)
converse of corresponding angles postulate
If 2 lines r intersected by a transversal, & corresponding angles r ocngruent, then the lines r parallel
converse of cointerior angle th
if 2 lines r cut by a transversal & cointerior angles r supplementary, then the lines r parallel
converse of alt interior angle theorem
if 2 lines r cut by a transversal & alternate interior angles r congruent, then the lines r parallel
perpendicular & parallel th
if 2 lines r perp 2 the same transversal, then they r parallel
perp & transversal theorem
if a transversal is perp to one of 2 parallel lines, then it is also perp to the other line
trapezoid
quadrilateral w/ only 1 pair of parallel sides
triangle sum theorem
the sum of the measures of the angles of a triangle is 180
quadrilateral sum theorem
the sum of the measures of the angles of a quadrilateral is 360
quadrilateral is parallelogram theorem
if both pairs of opp angles of a quadrilateral are equal, then the quadrilateral is a parallelogram
exterior angle theorem
an exterior angle of a triangle is equal in measure to the sum of its 2 remote interior angles
definition of similar
2 triangles are similar iff their vertices can be matched up so that corresponding angles r equal & corresponding sides r equal in proportion
definition of congruent
2 triangles r congruent iff their vertices can be matched up so that the corresponding parts (angles & sides) of the triangles r equal in measure
triangle similarity postulate (AA Post.)
If 2 angles of a triangle r equal to 2 angles of another triangle, then the 2 triangles r similar
how to find any angle of a triangle?
A = 1/2ab sin C
trig
SOH CAH TOA
overlapping similar triangles
if a line is drawn from a pt on 1 side of a triangle parallel to another side then it forms a triangle similar to the original triangle
ASA theorem
if 2 angles & the included side of 1 triangle r equal to the corresponding angles & side of another triangle, then the triangles r congruent
AAS theorem
if 2 triangles & a non-included side of 1 triangle r equal to corresponding angles & side of another triangle, then the triangles r equal
ASS
IZ NOT GOOD
SAS Post
if 2 sides & the included angle of 2 triangles r equal to the corresponding sides & angle of another triangle, then the triangles r congruent
angle bisector
a ray, line, or segment that divides an angle into two equal parts
segment bisector
a line, ray, / segment that divides a segment into 2 equal parts
perpendicular bisector
a line, ray, / segment that bisects the segment & is perpendicular to it
Hypotenuse-Leg Theorem
2 right triangles r equal if the hypotenuse & leg of 1 triangle r equal to the hypotenuse & leg of the other triangle
isosceles triangle
a triangle w/ 2 sides = in measure
equilateral triangle
a triangle in which all the sides r equal in measure
equiangular triangle
a triangle in which all the angles r equal in measure
Isosceles Triangle Theorem
if 2 sides of a triangle r = in measure, then the angles opposite those sides are = in measure
converse of isosceles triangle th
if 2 angles of a triangle r equal in measure, then the sides opp those angles r = in measure
equilateral triangle th
if a triangle is equilateral, then it’s also equiangular
converse of equil triangle th
if a triangle is equiangular, then it’s also equilateral
perpendicular bisector th
if a pt is the same distance from both ends of a segment, then it lies on the perp bisector of the segment
similar right triangle th
if the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed r similar to each other & the original triangle
geometric mean
if a, b, and x r pos #’s, and a/x = x/b, then x is the geometric mean b/w a and b. the GM is ALWAYS the pos root
geometric mean th
if the altitude is drawn to the hyp of a right triangle, then the measure of the altitude is the geometric mean b/w the measures of the parts of the hypotenuse
30-60-90
opp 30 - x
hypotenuse - 2x
opp 60 - x[3
45-45-90
opp 45’s - x
hypotenuse - x[2
pythagorean theorem
in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs
a2 + b2 = c2
polynomial
an expression tht can be written as a monomial/sum of monomials w/ whole # exponents
degree of a polynomial
the largest exponent
what can a polynomial NOT be?
it can’t have negative/fractional exponents
can’t have variables in denominators
BE SURE TO SOLVE FIRST SO THAT IT’S JUST AN EXPRESSION NOT AN EQUATION
polynomial equation
an equation tht can be written w/ a polynomial as 1 side & 0 as the other side
standard form of polynomials
from left to right, exponents go from largest to smallest
rational number
can be written as quotient of 2 integers
rational expression
an expression that can be written as the quotient of 2 polynomials
rational equation
an equation w/ only rational expressions on both sides
product of powers rule
am • an = am+n
quotient of powers rule
am/an = am-n
a does not equal 0
how to factor polynomials
- factor out any GCF’s
- factor normally (ax2 + bx + c)
ratio of the volume of 2 spheres
equal to the ratio of the cubes of their diameters
power of a power rule
(am)n = amn
power of a product rule
(ab)n = anbn
power of a quotient rule
(a/b)n = an/bn
provided tht b doesn’t equal 0
how to solve rational equations
- find LCD of all 3 & make the denominators all the same & cancel them out OR find LCD of left side denominators, make them the same, & then cross multiply
- simplify
extraneous solution
when a solution isn’t a solution of the original equation
test it
point of no return
the pt where it would take as much time to return to the starting point as to continue on to the destination
use t = d/r
d/r = d/r
cubic function
a polynomial function of degree 3
zero of a function
value of x that makes y = 0
same as x-intercept
double zero
cubic function w/ a squared factor
triple zero
a cubic function w/ a cubed factor
y = (x - 2)3
intersects x-axis once
y = (x - 2)2(x + 1)
intersects x-axis twice
(x+1) part goes thru at 1
(x-2)2 part touches at -2
y = (x - 2)(x + 1)(x + 4)
intersects x-axis three times
steps to solve a cubic equation w/out graphing
- put y on left side & set it to 0
- find the GCF of the right side if possible & factor it out
- if ax2 + bx + c format, use quadratic formula
how to solve cubic equations w/ calculator
- input equation
- find x-intercept (where y = 0)
square root of negative number
becomes i • pos. square root
parametric equation
equations where 2 variables r expressed in terms of a 3rd variable
parameter
3rd variable of a parametric equation
how to graph a parametric equation
- make a table of values (time, x, y)
- graph using x & y
OR
- change to PAR mode
- input equations
how to write an equation for y in terms of x
- solve for t with either of the equations (preferably x)
- substitute into the other equation not used in step 1, t
- simplify
3D coordinate plane
how to find a coordinate on a coordinate plane
3D midpt formula
(x1 + x2/2, y1 + y2/2, z1 + z2/2)
length of a diagonal of a rectangular prism
w/ edges of length x, y, and z & a diagonal length of d
d = [x2 + y2 + z2]
3D distance formula
d = [(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2]
equations of circles & spheres centered @ the origin
circle ⇒ x2 + y2 = r2
sphere ⇒x2 + y2 + z2 = r2
how to graph circle x2 + y2 = 36
- since it’s in this form, u kno the center is in (0,0) and that r2 = 36 → r = 6.
- sketch the circle w/ center at Origin & (0,6) at top, (6,0) at right, (0, -6) at bottom, and (-6,0) at left
- use a graphing calc to graph 2 semicircles: x2 + y2 = 36 → y2 = 36 - x2 → y = pos/neg [36 - x2]
\*to make circle circular, use Zoom Square
how to find the equation of a circle/sphere
- find center
- decide if it is at O or other & use the right equation
- find length of radius
- plug in radius
equations of circles not centered at origin
center (h,k) with radius r
(x - h)2 + (y - k)2 = r2
coterminal angles
angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side.
For example 30°, –330°
unit circle quadrant formulas
Quadrant I - All +, leave alone, (+,+)
Quadrant II - sin +, 180 - a, (-,+)
Quadrant III - tan +, a - 180, (-,-)
Quadrant IV - cos +, 360 - a, (+,-)
“family” unit circles
over 2 family
0: 0, (1,0)
90: pi/2, (0,1)
180: pi, (-1,0)
270: 3pi/2, (0, -1)
360: 2pi, (1,0)
over 6 family
30: pi/6, ([3]/2, 1/2)
150: 5pi/6, (-[3]/2, 1/2)
210: 7pi/6, (-[3]/2, -1/2)
330: 11pi/6, ([3]/2, -1/2)
over 4 family
45: pi/4, ([2]/2, [2]/2)
135: 3pi/4, (-[2]/2, [2]/2)
225: 5pi/4, (-[2]/2, -[2]/2)
315: 7pi/4, ([2]/2, -[2]/2)
over 3 family
60: pi/3, (1/2, [3]/2)
120: 2pi/3, (-1/2, [3]/2)
240: 4pi/3, (-1/2, -[3]/2)
300: 5pi/3, (1/2, -[3]/2)
sin, cos, tan quick sheet
quadrantal angles
Converting from Degrees to Radians & Radians to Degrees
Degree → Radians = Degree x pi/180
Radians → Degrees = Radian x 180/pi
how to find the sin cos & tan of t
- Convert to Degrees: 7pi/4 • 180/pi = 315
- 315 is in Quadrant IV
- sin315 = -sin(360 - 315) = -sin(45) cos315 = +cos(45) tan315 = -tan(45)
- -sin(45) = -[2]/2, cos(45) = [2]/2, -tan(45) = -1
degrees, radians, revolutions chart
how to find revolution:
2pi / radian = x
1/x = revolution
converting to/from DoM’S” & Decimal Form
1’ = one minute = (1/60)(1o)
1” = one second = (1/60)(1’) = (1/3600)(1o)
Ex: convert 152o15’29” to decimal degree form
152 + (15/60) + (29/3600) = 152 + .25 + .0081 = 152.26o
Ex: convert 153.66o to DMS form
153o
.66•60 = 39.6
153o39’
.6 (from 39.6) • 60 = 36
153o39’36”
how to calculate sin cos tan for degrees & radians
be sure to change mode b/w Degree & Radians
