Download Math Cram Kit File

2013
2014
19
YE
AR
S
EDITION
DO
ING
OU
RB
EST
, SO
MATHEMATICS
MATH
Algebra &
Trigonometry
CRAM KIT
EDITOR
Sophy Lee
®
the World
Scholar’s Cup®
ALPACA-IN-CHIEF
Daniel Berdichevsky
YO
U
CA
N
DO
YO
U
RS
MATH
®
CRAM KIT
I.
WHAT IS A CRAM KIT?................................................................. 2
II.
CRAMMING FOR SUCCESS……………………………………………… 2
III.
GENERAL MATH……………………………………................................. 3
IV. ALGEBRA……………….……………....................................................... 5
V.
GEOMETRY……………...…..………......................................................15
VI. TRIGONOMETRY……………………................................................... 22
VII. CRUNCH KIT……………..................................................................... 26
VIII. ABOUT THE AUTHOR AND EDITOR………………………………… 28
BY
EDITED BY
STEVEN ZHU
SOPHY LEE
HARVARD UNIVERSITY
FRISCO HIGH SCHOOL
HARVARD UNIVERSITY
PEARLAND HIGH SCHOOL
DEDICATED TO PYTHAGORAS,
FOR BEING SUCH A HOMIE.
© 2013 DEMIDEC
DemiDec, Scholar’s Cup, Power Guide, and Cram Kit are registered trademarks of the DemiDec Corporation.
Learn more about DemiDec and the World Scholar’s Cup at www.demidec.com and www.scholarscup.org, respectively.
Math Cram Kit | 2
WHAT IS A CRAM KIT?
A Word from the Editor
COMPETITION IS NEARING…
STRUCTURE OF A CRAM KIT
The handful of days before competition can be the most
overwhelming. You don’t have enough time to review
everything, so a strategic allocation of your resources is
crucial. Cram Kits are designed with one goal in mind-----to
provide you with the most testable and most easily
forgotten facts.
The main body of every Cram Kit is filled with charts and
diagrams for efficient studying. You’ll also find helpful
quizzes to reinforce the information as you review.
This Crunch Kit presents the most important formulas that
you need to know for the math test. Realize, however, that
knowing when to apply each formula is half the battle.
Plugging in the numbers is often the easiest step.
As for math-----the very word strikes fear into the hearts of
many, and that’s before we start doing funky things with
letters like m, a, t, and h. But don’t be discouraged-----like
any other event, it can be mastered through studying, and
perhaps more than any other, through practice tests.
It also includes material for geometry that you don’t need
to know this year-----at least, not specifically. Geometry can
be useful to understanding trigonometry, so we’re leaving
that segment in here. Plus, next year’s curriculum might
cover geometry in more depth. Same with general math.
This Cram Kit is not meant to teach math-----even our
resource can hardly claim to accomplish that. It is meant
as a quick review tool to remind you of needed formulas
and to correct minor misconceptions that may cost you
points at competition. I advise you to go through this
guide with tests nearby-----perhaps our focused quizzes,
organized by topic. Doing example problems is the best
way to reinforce the concepts that you learn.
Last, remember to relax. In the final moments before you
open your test booklet, confidence is your most important
asset. Good luck and happy cramming!
Sophy Lee
CRAMMING FOR SUCCESS
A Word from the Author
PIECES OF THE MATH PIE
0%
Algebra
40%
Geometry
60%
Everything Else
TIME IS TICKING!
If you have one day left, read the whole guide.
*
If you have one hour left, read the Crunch Kit.
*
If you have one minute left, scan the List of Lists
*
If you have one second left, good luck. You should already have put me away.
Math Cram Kit | 3
GENERAL MATH
The Deceptively Simple and the Utterly Confusing
INTEGERS, FRACTIONS, DECIMALS,
AND PERCENTS
BASIC COUNTING TECHNIQUES
FRACTIONS
MULTIPLICATION PRINCIPLE
 Fractions must have a common denominator before
 Helps us find the total number of possibilities when we


we can add or subtract them
When multiplying fractions, try to cancel out
common factors
When dividing fractions, flip over the second
fraction and multiply it by the first one
Step 1. Turn the second fraction upside-down
(the reciprocal ):
4
1
4
1
Step 2. Multiply the first fraction by the reciprocal of the
second:
1×4
1
4
4
×
=
=
2×1
2
1
2
Step 3. Simplify the fraction: 2

are choosing one item from each of several groups
Multiply the number of choices from each group
 If Sally can choose an outfit from 4 pairs of jeans, 5
shirts, and 3 pairs of shoes, she has 4 x 5 x 3 = 60
outfit choices
FACTORIALS, PERMUTATIONS, AND COMBINATIONS
FACTORIALS
 ‘‘!’’ denotes a factorial
 (50!  50 x 49 x 48... x 2 x 1)
PERMUTATIONS
 Arrangements of a set of objects in which order
matters
 When arranging r objects out of a set of n total objects,
n!
the number of permutations is nPr 
(n-r)!

PERCENTAGES
 1% represents one in 100
 Divide a percentage by 100 to convert it to a

decimal
Multiply a decimal by 100 to convert it t
25
20
$40.00 x (1) x (1)  $24 o a percentage
100
100
x
)(originalprice)
100
Successive discounts do NOT have the same effect
as a cumulative discount
If more than one discount applies to an item, keep
multiplying the right side of the above formula by
discount
(1)
100
 A shirt originally priced $40.00 is marked down
by 25%. Joe uses a 20%-off coupon to
purchase the shirt. How much does he have to
pay for the shirt before tax?
 Formula for sale prices: (1



A club of 12 people wants to elect a president, a
vice-president, and a treasurer. How many different
results can this election have?
The three positions are different, so order matters
12!
12!

 12 x 11 x 10  1320
12 P3 
(n-3)! 9!
COMBINATIONS
 Arrangements of a set of objects in which order does
NOT matter
 When arranging r objects out of a set of n total objects,
n!
the number of combinations is n Cr 
(r!)(n-r)!



A club of 12 people wants to elect three people to a
committee. How many different results can this
election have?
The three seats on the committee are the same, so
order does not matter
12!
12! 12 x 11 x 10


 220
12 C3 
3!(12-3)! 3!9!
3x2x1
TRY THIS MNEMONIC!
 Permutations = Prizes (order matters)
 C ombinations = C ommittees (order doesn’t matter)
Math Cram Kit | 4
GENERAL MATH
More Counting; Winning in Vegas
BASIC COUNTING TECHNIQUES (PT. 2)
PROBABILITY OF EQUALLY LIKELY EVENTS
ARRANGEMENT RULES
PROBABILITY
ARRANGEMENT PRINCIPLE
 When a set has two or more identical objects, we
need to take away the redundant arrangements
caused by the identical objects
 To arrange the letters in CALIFORNIA, we need
to find the number of permutations and divide
by the factorials of the identical letters
 CALIFORNIA has two A’s and two I’s
 possible arrangements
ARRANGING OBJECTS IN CIRCLES
 When we arrange objects in circles, we need to
make sure that each arrangement represents a
distinct ordering of objects, not a mere rotation of
another arrangement
h
 Number of possible circular arrangements =
k
 We have to keep one object in place to mark the
‘‘beginning’’ of the arrangements
 How many different ways can four people sit
around a circular table?
 Keep one person in place and rearrange the
other three
 24
 When arranging keys on a keychain, we must divide
the result by 2 since we can flip the keychain over,
which makes arrangements that are mirror images
of each other identical
 In how many different ways can 4 keys be
arranged on a keychain?
5

3
CALCULATOR USE
When dealing with permutations and combinations,
use the built-in functions on your scientific or graphing
calculator to avoid typing in the formulas. Master these
(and other calculator techniques) before the test!
The chance that an event will happen
RULES
 The probability that event A happens is P(A)

csc  6x  2   8
 The probability that independent, unrelated events A

and B will occur is P(A+B) = P(A) x P(B)
If events A and B are not mutually exclusive, the
probability of one or the other occurring is
P(A or B) = P(A) + P(B) --- P(A+B)
USEFUL FACTS
 A standard poker deck has 52 cards
 Such a deck has 4 suits (2 red and 2 black) of 13 cards
each
 A deck’s face cards are the Jack, Queen, and King of
each suit (12 face cards total in a standard poker deck)
 A standard die has 6 faces
EXAMPLES
 What is the probability of rolling a sum of 9 with two
dice?
 We have 4 outcomes with a sum of 9 (3-6, 6-3, 45, 5-4)
 The total possible number of outcomes is 6 x 6 = 36
 The probability of rolling a sum of 9 is 4/36 = 1/9
 What is the probability of drawing a red Queen from a
standard deck of cards?
 A 52-card deck has four Queens, two of which are
red
2
1

 P(Qr ) 
52 26
 What is the probability of a coin landing heads four
tosses in a row?
1
 For each toss, the chance of landing heads is
2
 The tosses are independent events, since each toss
does not affect the result of any other toss
4

1
 1
P(4H)  P(H)  P(H)  P(H)  P(H)    
 2  16
Math Cram Kit | 5
ALGEBRA
Separate but Equal
SOLVING POLYNOMIAL EQUATIONS
(THE BASICS)
SOLVING POLYNOMIAL EQUATIONS
(LINEAR)
EQUATION
LINEAR POLYNOMIALS
A mathematical statement that two expressions are
equal
Equations that have a degree of 1 and straight-line graphs
 Examples


3 + 7 = 14 --- 4
4x + 5 = 2y
POLYNOMIAL
 An expression containing variables
2
 5x4  x  23
9
 The variables cannot be contained in fraction
denominators
 The variables also cannot be contained in
exponents
 Polynomials with only one term are called
monomials




SLOPE-INTERCEPT FORM
 y = mx + b
 m is slope

m
y2  y1
, given points (x1, y1) and (x2, y2)
x2  x1
 b is the y-intercept

b is the value of y when the line crosses the y-axis,
when x = 0
POINT-SLOPE FORM
 y  y 1  m(x  x 1 )
 m is slope
 (x1, y1) is a given point
12x2y is an example of a monomial
Even though the expression has two variables, x
and y, the variables are contained in one term
The degree or order of a polynomial is the same as
the degree of the term with the highest sum of
exponents
 Consider 4xyz + 3x4y2 --- 81z
 4xyz has a degree of 1 + 1 + 1 = 3
 3x4y2 has a degree of 4 + 2 = 6
 -81z has a degree of 1
 Thus, 4xyz + 3x4y2 --- 81z is a 6th order
polynomial
The leading coefficient of a polynomial is the
coefficient of the term with the highest degree
 The leading coefficient of 7x --- 9x3 + 15x2 - 64 is
-9
STANDARD FORM
 Ax  By  C


A
is slope
B
C
(0, ) is the y-intercept
B

Math Cram Kit | 6
ALGEBRA
The Root of the Problem
SOLVING POLYNOMIAL EQUATIONS
QUADRATIC EQUATIONS
Equations that have a degree of 2
REMAINDER AND FACTOR THEOREMS
 Remainder Theorem: To find the remainder when a
 The roots of a quadratic equation are the values of






x for which y = 0 (where the graph intersects the xaxis)
Roots are also called zeroes or x-intercepts
If the equation is in the form y = Ax2 + Bx + C, we
can use the quadratic formula to find the roots
B  B2  4AC
Quadratic formula: x 
2A
The part of the quadratic formula under the radical
sign, B2 --- 4AC, is called the discriminant
 If the discriminant is positive, then the equation
has two real roots (graph crosses the x-axis
twice)
 If the discriminant is 0, then the equation has
one real root (graph touches the x-axis once)
 If the discriminant is negative, then the
equation has no real roots (graph does not
intersect the x-axis)
Sometimes we can solve quadratic polynomials by
factoring
Think of factoring as reverse distribution

4x2  4x  3  0

(2x  3)(2x  1)  0

If either factor equals 0, the whole expression
equals 0
Thus, we will set both factors equal to 0 to find
the roots
3
2x  3  0  x 
2



2x  1  0  x  
1
2
HIGHER ORDER EQUATIONS
Equations that have a degree higher than 2
 Some cubic polynomials are factorable
 Sum of cubes formula:

x3 + y3 = (x + y)(x2 --- xy + y2)
Difference of cubes formula:
x3 --- y3 = (x --- y)(x2 + xy + y2)

polynomial is divided by (x --- c), plug c into the
polynomial
 What is the remainder when x4 --- 5x + 27 is divided
by x + 3?
 In this example, c = ---3, as x + 3 = x --- (---3)
 The remainder is (---3)4 --- 5(---3) + 27 = 123
Factor Theorem: If dividing a polynomial by (x --- c)
yields a remainder of 0, then (x --- c) is a factor of the
polynomial
 The remainder when x3 --- 5x2 --- x + 5 is divided by
(x --- 5) is (5)3 --- 5(5)2 + 5 = 0
 Thus, (x --- 5) is a factor of x3 --- 5x2 --- x + 5
ROOT THEOREMS
 Rational Roots Theorem: To find all of the possible

rational roots of a polynomial, divide all the factors of
the constant by all the factors of the leading coefficient
 Find all possible rational roots of 3x2 --- 6 + 5x3 + 2x
 The constant is ---6, and the leading coefficient is 5
because the third term has the highest degree
 Now we list all the positive and negative factors of 6 over all of the positive and negative factors of 5
1 2 3 6 1 2 3 6
  , , , , , , ,
1 1 1 1 5 5 5 5
 The list includes all possible rational roots, but
none of them has to be a root of the polynomial
Given a polynomial in the form Ax2 + Bx + C, two
formulas exist for finding the sum and the product of
the roots
B
1. Sum of roots formula: 
A
2. Product of roots formula:
C
 for odd  numberedpolynomials
A
C
and for even  numberedpolynomials
A
Math Cram Kit | 7
ALGEBRA
More or Less
SOLVING INEQUALITIES
INEQUALITY
A mathematical statement that two expressions
are not equal
 As with solving an equation, solve an inequality by

isolating the variable
When multiplying or dividing by a negative term,
flip the sign of the inequality
LINEAR INEQUALITY
 An inequality with a degree of 1
 18 < ---5x --- 7
 25 < ---5x
 5>x
ABSOLUTE VALUE INEQUALITIES
 A number’s absolute value is its distance from 0 on a
number line
 Absolute value is always non-negative (by definition)
 When an inequality contains an absolute value, we
have to solve two inequalities based on the original
 Consider 2x --- 3 < 5
 The first inequality is the same as the original, but
without the absolute value signs





2x  3x  8  43

2x  3x  35  0

(2x  7)(x  5)  0

At this point, we will plot the roots on a number
line, dividing it into three regions
2
2
-5




7
2
We will pick a value in each of the three regions
to test the inequality in each region
We will use -6, 0, and 4
Plugging -6 and 4 into the polynomial satisfy
the inequality, so we will place checks in those
regions
Plugging 0 into the polynomial makes the
inequality false, so we will place an ‘‘x’’ in that
region
–5


7
2
The inequality is true when x < ---5 or x >
7
2
2x  8
x4
For the second inequality, we multiply the right side
by -1 and flip the sign of the inequality



QUADRATIC INEQUALITY
 An inequality with a degree of 2
2x  3  5
2x  3  5
2x  2
x  1
Thus, 2x --- 3 < 5 holds true when x < 4 and x > ---1
Math Cram Kit | 8
ALGEBRA
Putting the Fun in Function!
FUNCTIONS (BASICS)
FUNCTIONS (COMPOSITE AND INVERSE)
WHAT IS A FUNCTION?
TYPES OF FUNCTIONS (PART 1)
A relationship between an independent variable x and a
dependent variable y
 f(x) denotes a function
 Functions can only have one value of y for each

value of x
Vertical-line test: If you can place a vertical line at
every x-value of an equation’s graph, and the line
crosses the graph at no more than one point, then
the equation is a function
 The following graph is not a function because a
vertical line would cross the graph at two points
whenever x > 0
DOMAIN AND RANGE
 The domain of a function consists of all the x-values
that have corresponding y-values
1
 Find the domain of f(x) 
x
 At x = 0, the function is undefined (no
corresponding y-value), so the domain is all real
numbers except 0
 The range of a function consists of all its possible yvalues
 The following graph has a range of -1 to 1
COMPOSITE FUNCTION
 Combines two or more functions together
 For two functions f(x) and g(x), a possible composite
function is f(g(x)) or, written in another form, (f  g)(x)
 In function (f  g)(x) , plug x into g(x) and plug that
result into f(x)
 Find a(b(x)) if a(x) = 3x2, b(x) = 5x + 7, and x = 2
 b(2) = 5(2) + 7 = 17
 a(b(2)) = a(17) = 3(17)2 = 867
INVERSE FUNCTIONS
 To find the inverse function f-1(x) of a function f(x),
replace f(x) with y and switch the positions of x and y
 The inverse of y = 3x +2 is x = 3y + 2
 Because we switch the x’s and the y’s, the graphs of
inverse functions are mirror images of the original
graphs across the line y = x
Math Cram Kit | 9
ALGEBRA
Functions: The Logarithm Strikes Back (With Rational Exponential Force)
FUNCTIONS
(RATIONAL, EXPONENTIAL, LOGARITHMIC)
FUNCTIONS
(OPERATIONS ON LOGARITHMIC FUNCTIONS)
TYPES OF FUNCTIONS (PART 2)
WORKING WITH LOGS
RATIONAL FUNCTIONS
 Functions in which variables are in the
denominators of fractions
 Fractions are ratios, hence, rational functions

5x4  41
is a rational function
2x3
EXPONENTIAL FUNCTIONS
 Functions in which the independent variable x is in
an exponent
 3x is an exponential function
 A common exponential function is ex
 e is a constant like  and can be found on a
scientific or graphing calculator
 e = 2.71828…
LOGARITHMIC FUNCTIONS
 Functions in which the independent variable x is in
the argument of a logarithm
 Logarithms are the reverse of exponents
 Logarithms follow the form logbase (argument) =
exponent, such that baseexponent = argument
 Log7(49) = 2 because 49 is 7 to the 2nd power
 When the logarithm does not have a base written,
assume that the base is 10
 Log(1000) = 3, since 103 = 1000
 Logs with a base of e are called natural logarithms
 Natural logarithms are denoted ln(x)
 Logarithms and exponential expressions cancel
each other out to yield the exponent when the
bases are the same

Ln(e13 )  13

Log 4 (4 x )  x
ADDITION
 When adding two logarithms of the same base, we can
combine them into one logarithm with the arguments
multiplied together
 log 12 (x  1)  log 12 (x  3)  log 12 ((x  1)(x  3))
SUBTRACTION
 When subtracting two logarithms of the same base, we
can combine them into one logarithm with the first
argument divided by the second

 x1 
log 12 (x  1)  log12 (x  3)  log12 

 x3 
OTHER CASES
 When the entire argument of a logarithm has an
exponent, we can turn the exponent into a coefficient
of the logarithm

log((5x2  9)3 )  3log(5x2  9)

We can pull the 3 out because it applies to the
whole argument
We cannot pull the 2 out because it only applies to
one term in the argument

REVERSAL
 These three rules can also be used in reverse
 A logarithm whose argument is a product can be split
into the sum of two logarithms whose arguments are
that product’s factors
 Log12((x --- 5)(x + 9)) = Log12(x --- 5) + Log12(x + 9)
 A logarithm with one argument divided by another can
be split into the difference of two logarithms, such that
the divisor becomes the argument of the subtracted
logarithm

 x 5 
log 
  log 12 (x  5)  log 12 (x  9)
 x9
 A coefficient of a logarithm can become the exponent
of the logarithm’s entire argument
 3(Log(5x2 + 9)) = Log((5x2 + 9)3)
Math Cram Kit | 10
ALGEBRA
Use Your Imagination; Walk the Line
COMPLEX NUMBERS
READING GRAPHS OF FUNCTIONS (LINEAR)
WHAT IS A COMPLEX NUMBER?
LINEAR FUNCTIONS
Any number in the form a + bi
 a and b are real numbers
 i is an imaginary number such that
i  1
OPERATIONS WITH COMPLEX NUMBERS
 We can simplify higher powers of i
 Find i47
 We know that i2 = ---1
 i47 is the same as (i46)(i)
 (i46)(i) = (i2)23(i)
 Thus, i47 = (---1)23(i)
 i47 = ---1
COMPLEX CONJUGATES
 Pairs of complex numbers in forms a + bi and a --bi
 A fraction with an imaginary number in the
denominator is simplified by multiplying its
numerator and denominator by the complex
conjugate of the denominator
1 i
 Simplify
2  3i


1  i 2  3i 2  3i  2i  3 1  5i



2  3i 2  3i 4  6i  6i  9
13
Notice that multiplying by the complex
conjugate removes i from the denominator
COMPLEX QUADRATIC ROOTS
 In a quadratic equation whose discriminant (b2 --4ac) is negative, the roots are complex numbers
 If the roots are complex numbers, they will be
complex conjugates
 A polynomial with the root 35 + 9i must also
have the root 35 --- 9i
 Linear functions are always straight lines
 First, we find the y-intercept of the function



The line above crosses the y-axis at y = 3
In slope-intercept form, which is y = mx + b, the yintercept is b, so b = 3
To find m, the slope, we need two points from the graph
 We already know that the y-intercept is (0,3)
 We can also read the x-intercept from the graph,
which is (---6,0)

Using the formula for slope, m 
the slope is m 
y2  y1
, we find that
x2  x1
0  3 3 1


6  0 6 2
1
2
 Therefore, the graph above represents y  x  3
Math Cram Kit | 11
ALGEBRA
Read Between the Curves
READING GRAPHS OF FUNCTIONS (QUADRATIC)
READING GRAPHS OF FUNCTIONS (HIGHER ORDER)
QUADRATIC FUNCTIONS
HIGHER ORDER EQUATIONS
 If the degree of the equation is even, the graph will start
and end on the same side of the y-axis

1 6 3
x  x , which
4
starts and ends on the positive side of the y-axis
The following graph represents y 
 Quadratic functions are always U-shaped or n



shaped
The graphs of quadratic equations are called
parabolas
The standard form for the equation of a parabola is
y = A(x --- h)2 + k
The point (h,k) is the vertex-----the turning point of
the curve
 In the graph above, the vertex is (---2,1)
We can plug points into the standard form for the
equation of a parabola to obtain the equation of the
graph
 We can plug the vertex of the graph above to
get
y = A(x --- (---2))2 + 1, which becomes
y = A(x + 2))2 + 1
 We still need to find A by plugging in a point for
(x,y)
 We can read from the graph the point (0,---1)

2
A
4
1
 A
2
Thus, the equation of the graph above is
1
y   (x  2)2  1
2


1  A(0  2)2  1
 If the degree of the equation is odd, the graph will start
and end on opposite sides of the y-axis
 The following graph represents y = ---x7 + x4, which
starts on the positive side of the y-axis and ends on
the negative side
Math Cram Kit | 12
ALGEBRA
Flipped Functions and Arithmetic Arrangements
READING GRAPHS OF FUNCTIONS
(EXPONENTIAL AND LOGARITHMIC)
SEQUENCES, SERIES, AND MEANS
(ARITHMETIC)
EXPONENTIAL FUNCTIONS
ARITHMETIC SEQUENCE
 Exponential functions create graphs with horizontal
Pattern of numbers that has a common difference d


asymptotes
Asymptotes are lines at which the x or y value of a
function approaches infinity or negative infinity (but
never reaches it)
The following graph represents y = ex, which has a
horizontal asymptote at y = 0
 As x approaches negative infinity, y will
approach 0 but will never reach it



1, 8, 15, 22, 29
Common difference is 7 because each term is 7
more than the previous one
Formula to find the nth term of an arithmetic sequence:
nth term = first term + d(n --- 1)
 Find the 9th term of the sequence: 68, 64, 60, 56…
 n = 9 and d = ---4 (---4 = 64 --- 68 = 60 --- 64, and so
on)
 9th term = 68 + (---4)(9 --- 1) = 68 --- 32 = 36
ARITHMETIC SERIES
The sum of an arithmetic sequence
 Formula to find the sum of the first n terms:
(first term  last term)
2
Formula to find n, the number of terms in the series:
(last term  first term)
n
1
d
 Find the sum of the arithmetic progression: 17, 20,
23…44, 47, 50
 d = 3, the last term is 50, and the first term is 17
(50  17)
 1  12
 n
3
(17  50)
 402
 Now we can find the sum  12 
2
Summation problems may use sigma () notation
n

LOGARITHMIC FUNCTIONS
 Logarithmic functions create graphs with vertical

asymptotes
The following graph represents y = ln(x), which has
a vertical asymptote at x = 0
As x approaches 0, y approaches negative
infinity

5

 k = the sum of the numbers 1 through 5
k 1



The index k starts at 1, the lower bound, and
increases by 1 for each term until it reaches 5
The expression on the right side of the sigma sign
(here, k) represents an element of the series
The expression above is the same as 1+2+3+4+5
ARITHMETIC MEAN
The average of two or more numbers

The arithmetic mean of 1, 4, 7, 10, and 13 is
1  4  7  10  13
7
5
Math Cram Kit | 13
ALGEBRA
Rational Commonists
SEQUENCES, SERIES, AND MEANS
(GEOMETRIC)
SEQUENCES, SERIES, AND MEANS
(GEOMETRIC AND INFINITE)
GEOMETRIC SEQUENCE
GEOMETRIC MEAN
Pattern of numbers with a common ratio r
The square root of the product of two terms



2, 6, 18, 54…
The common ratio is 3 because each term is 3
times the previous one
Formula to find the nth term of a geometric
sequence: nth term  (first term)rn1




What is the geometric mean of 4 and 64?


th
What is the 8 term of the sequence that
begins: 625, 125, 25, 5…?
1
The common ratio is
5
k 1
3
Find
(4)  
2
k 1
10

4  64  16
4, 16, and 64 form a geometric series with a
common ratio of 4
7

1
 1
 1 
8th term  (625)    (625) 


5
 78125  125
INFINITE SERIES
The sum of a sequence with an infinite number of terms
 For an infinite series to be solvable, r has to be less
GEOMETRIC SERIES
The sum of a geometric sequence
 Formula to find the sum of the first n terms of a
geometric sequence:
(first term)(1  rn )
1r
k 1
3
(4)  
2
k 1
10


Find

We plug in k  1 to find the first term:
3
(4)  
2

11
4

We’re trying to find the sum of the terms from
k =1 to k = 10, so n = 10
The ratio that we multiply to find each
3
3
consecutive term is , so r =
2
2

  3 10 
(4)  1    
 2 

  453.32
Thus, the sum is
3
1
2



than 1
 The infinite series of the sequence that begins with
1 1 1
2,1,  , ,  ... will have a value because each
2 4 8
1
term is  times the previous one
2
 The terms will eventually be so close to 0 that
adding them to the series does not change the sum
These types of series are said to converge, or reach a
definite sum
If r is 1 or higher, the sequence will keep generating
larger numbers, and the series will have an indefinite
value
 The series of the sequence that begins with ---2, 4, --8, 16, ---32… does not have a value because every
term is -2 times the previous one
 The terms will keep increasing, and the sum will
never stay at a definite number
These types of series are said to diverge, or not reach a
definite sum
Math Cram Kit | 14
ALGEBRA
Can You See the Pattern?
SEQUENCES, SERIES, AND MEANS (GRAPHING)
SEQUENCES, SERIES, AND MEANS (GRAPHING)
Arithmetic Sequence
Geometric Sequence
20
300
250
15
200
150
10
100
5
50
0
0
0
5
0
10
2
4
6
8
10
 In the above geometric sequence, each term is twice as
 In arithmetic sequences, the terms have equal
large the previous one
vertical distances between them because the
common difference d never changes
Arithmetic Series
Geometric Series
80
70
60
50
40
30
20
10
0
5
4
3
2
1
0
0
5
10
 In an arithmetic series, the sums do not have equal
vertical distances between them because each term
added is larger than the previous term
0
50
100
150
 In the above geometric series, the sum approaches 4 as

n extends to infinity, meaning that the series converges
In a diverging series, the sum would approach infinity
Math Cram Kit | 15
GEOMETRY
Triangles with Little Squares in the Corner
RIGHT TRIANGLES
SPECIAL RIGHT TRIANGLES
PYTHAGOREAN THEOREM
45-45-90 TRIANGLES
c
45 
a
s 2
s
b
45 
 A right triangle contains a right angle (90°)
 The two sides adjacent to the right angle are called



legs
 In the above diagram, a and b are legs
The hypotenuse is the side opposite the right angle
 In the above diagram, c is the hypotenuse
The Pythagorean theorem states a relationship
between the three sides
s
 45-45-90 triangles are right triangles with legs of equal
length
They are also called right isosceles triangles

 The hypotenuse is equal to
 a2  b2  c2
The theorem can also give us information about
other types of triangles in which c is the longest
side

If a2  b2  c2 , then the triangle is acute (all
angles are less than 90°)
If a2  b2  c2 , then the triangle is obtuse (one
angle is greater than 90°)
A Pythagorean triple is a set of three integers that
fit the theorem
 3, 4, 5
 5, 12, 13
 7, 24, 25
 8, 15, 17
 9, 40, 41
Any multiple of a Pythagorean triple will also be a
Pythagorean triple
 6, 8, 10
 10, 24, 26
2 times a side
30-60-90 TRIANGLES
60 
2s
s



30 
s 3
 The shorter leg is opposite the 30° angle
 The hypotenuse is twice the length of the shorter leg

 The longer leg, which is opposite the 60 angle, is 3
times the shorter leg
Math Cram Kit | 16
GEOMETRY
Point-Line Coordination
COORDINATE GEOMETRY (POINTS)
COORDINATE GEOMETRY (LINES)
MIDPOINT
PARALLEL AND PERPENDICULAR LINES
The point that is exactly in the middle of two other
points
 Two lines are parallel if they have the same slope
 A line that crosses two parallel lines is called a
transversal
 Given two points (x1, y1) and (x2, y2), their midpoint
is the average of their coordinates:
 x 1  x2 y 1  y2 
,


2 
 2

Find the midpoint of (-2, 3) and (5, -6)

 2  5 3  (6)   3 3 
 2 , 2    2 , 2 

 

SLOPE
The rate of change of a line
 In other words, slope is a ratio of how fast the line is

changing vertically over how fast the line is
changing horizontally
Given two points (x1,y1) and (x2,y2) that lie on the
y y
same line, the slope of the line is m  2 1
x2  x 1
 Note that slope is change in y (vertical) over

change in x (horizontal)
 Thus, slope can be remembered as ‘‘rise over
run’’
In equations, slope is usually denoted as m
DISTANCE FORMULA
 The distance between two points (x1, y1) and
(x2, y2) is:
(x 1  x2 )2  (y 1  y2 )2
8
5
7
6
4
1
3
2
 Two angles that add up to 180 degrees are called






supplementary angles (1 & 2, 4 & 3, 1 & 4, etc.)
Two angles that add up to 90 degrees are called
complementary angles
All of the larger angles (1, 3, 5, 7) are equal to each
other
All of the smaller angles (2, 4, 6, 8) are equal to each
other
The sum of any larger angle and any smaller angle is
180°
Two lines are perpendicular if they intersect and form
right angles
The slopes of perpendicular lines are negative
reciprocals of each other (the product of their slopes is
-1)
 Find the slope of a line perpendicular to the line
4
y  x 3
7

The slope of the given line is
perpendicular line is 
7
4
4
, so the slope of the
7
Math Cram Kit | 17
GEOMETRY
Four-sided Shapes That Are Almost, but Not Entirely, Unlike Triangles
COORDINATE GEOMETRY (QUADRILATERALS)
QUADRILATERAL
A four-sided polygon
RECTANGLE
 A parallelogram with four right angles
TRAPEZOID
 A quadrilateral with one pair of parallel sides
 Area = (base)(height)
 In a coordinate system, opposite sides have the same
 The parallel sides are called bases
 The non-parallel sides are called legs
 The height is the distance from one base to the
other
1
(base1 + base2)(height)
2
In a coordinate system, the two parallel bases have
the same slope, and the two legs have different
slopes
slope and length, and adjacent sides must be
perpendicular
RHOMBUS
 A parallelogram with four congruent sides
 Area =

PARALLELOGRAM
 A quadrilateral with two pairs of parallel sides
 The diagonals form right angles
 The diagonals bisect each other and bisect the angles,


 Opposite sides are congruent (equal in magnitude)
 Opposite angles are congruent
 Consecutive angles are supplementary (add up to



180°)
Area = (base)(height)
In the above diagram, the base is the side on the
bottom, and the height is the vertical dotted line
In a coordinate system, opposite sides have the
same slope and length
forming four congruent right triangles
1
Area =
(diagonal1)(diagonal2)
2
In a coordinate system, the diagonals are
perpendicular, and the side lengths are all equal
SQUARE
 A quadrilateral with four congruent sides and four right
angles, making it both a type of rectangle and rhombus
 Area = (side)2
 In a coordinate system, all sides have the same length,
and adjacent sides are perpendicular
Math Cram Kit | 18
GEOMETRY
Movin’ On Up, Dimensionally

PLANE AND SOLID FIGURES (AREA)
PLANE AND SOLID FIGURES (VOLUME)
AREA OF A TRIANGLE
VOLUME OF SOLID FIGURES
1
Area  (base)(height)
2
 Works best for right triangles and triangles
whose base and height are known
 Prism: V = (area of base)(height)

 Heron’s Formula: Area  (s)(s  a)(s  b)(s  c)


a, b, and c are the sides of the triangle, and
ab  c
s
2
 When using this formula, find s first and store it
as a variable in your calculator
 Be careful to calculate the formula correctly
 This formula works for any triangle, but you
need to know the lengths of all three sides
1
Area = ab(sinC)
2
 a and b are two sides, and C is the angle
between them
SURFACE AREA OF SOLID FIGURES
 Prism: SA = Area of 2 bases + area of lateral faces
 Pyramid: SA = Area of the base + area of lateral


triangles
Cylinder: SA = 2  r2 + 2  rh
 r is the radius of the base, and h is the height of
the cylinder
Sphere: SA = 4  r2
 r is the radius of the sphere
 Cone: SA =  r2 +  r r2  h2

r is the radius of the base, and h is the height of
the cone

r2  h2 is the lateral height, the distance from
the edge of the base to the apex of the cone
If the lateral height is given, substitute it for

r2  h2
1
3
Cylinder: V =  r2h
 r is the radius of the base
 h is the height of the cylinder
4
Sphere: V =  r3
3
 r is the radius of the sphere
1
Cone: V =  r2h
3
 r is the radius of the base
 h is the height of the cone
 Pyramid: V  (area of thebase)(height)


Math Cram Kit | 19
GEOMETRY
Circle Time
PLANE AND SOLID FIGURES (CIRCLES)
MEASURING CIRCLES
LINES AND CIRCLES (PART 1)
CIRCUMFERENCE OF A CIRCLE
 Circumference = 2  r
 Circumference is the perimeter of a circle
 Tangents are lines that intersect a circle at one point

A tangent will be perpendicular to the radius of the
circle at the point where it touches the circle
AREA OF A CIRCLE
 Area =  r2
 r is the radius of the circle
LOOKING INSIDE
ANGLES IN A CIRCLE
 A circle has 360° or 2  radians
 180° =  radians
 A central angle has the same measure as its
intercepted arc
 Secants are lines that intersect a circle at two points
 Chords are line segments that have endpoints on the
rim of a circle
 An inscribed angle has half the measure of its
intercepted arc
90 
45 


The longest chord is the diameter
If two chords are the same distance from the
center of a circle, they have the same length and
intercept the same-sized arc
Math Cram Kit | 20
GEOMETRY
Circle Time: Part Deux
PLANE AND SOLID FIGURES (CIRCLES) (CONT’D)
LINES AND CIRCLES (PART 2)
TWO CHORDS
 In the above diagram, two chords intersect at a
LINES AND CIRCLES (PART 3)
A TANGENT AND A SECANT
 In the above diagram, AB is a tangent and AC is a
secant that intersects the circle at point D
point E


AB  DC
AEB  CED 
and
2
AD  BC
AEB  BEC 
2
A 

(AB)2  AD  AC
AE EC  BE ED
TWO TANGENTS
 In the above diagram, two tangents have a common
endpoint at A and intersect circle O at B and C
The lengths of the two tangents are the same

 The two radii OB and CO are perpendicular to
TWO SECANTS
 In the above diagram, two secants originating from
point A intersect a circle at points D and E
A 
major arc BC  minor arc BC
2
BC  DE
2

A 

AD  AB  AE  AC
their respective tangents

BC  BD
2

Math Cram Kit | 21
GEOMETRY
A Striking Resemblance
CONGRUENCE
SIMILARITY
PROPERTIES OF CONGRUENT FIGURES
PROPERTIES OF SIMILAR FIGURES
 Two figures are congruent if their corresponding


sides have the same length and the sides form the
same angles
The figures may be flipped or rotated
The following figures are all congruent
CONGRUENT TRIANGLES
 SSS (Side-Side-Side): If the corresponding sides of
two triangles are congruent, the triangles are
congruent
 A triangle with side lengths 3, 4, and 5 is
congruent to a triangle with side lengths 3, 4,
and 5
 SAS (Side-Angle-Side): If two triangles have the
same angle, and the corresponding sides adjacent
to the angle are congruent, then the triangles are
congruent
 A triangle with side lengths of 2 and 6
separated by an angle of 54 degrees is
congruent to another triangle with side lengths
of 2 and 6 separated by 54 degrees
 ASA (Angle-Side-Angle): If two triangles have two
matching angles, and the sides between both
angles are congruent, then the triangles are
congruent
 A triangle with angles of 34 and 89 degrees
separated by a side of length 7 is congruent to
another triangle with angles of 34 and 89
degrees separated by a side of length 7
 Two figures are similar if corresponding sides form


equal ratios and the sides form the same angles
The figures may be flipped or rotated
The following figures are all similar
SIMILAR TRIANGLES
 SSS: If the corresponding sides of two triangles form
equal ratios, then the triangles are similar
 A triangle with side lengths 4, 7, and 9 is similar to
a triangle with side lengths 8, 14, and 18
 SAS: If two triangles have the same angle, and the
corresponding sides adjacent to the angle form equal
ratios, then the triangles are similar
 A triangle with side lengths of 3 and 5 separated by
an angle of 80 degrees is similar to a triangle with
side lengths of 12 and 20 separated by 80 degrees
 AA (Angle-Angle): Triangles with two corresponding
angles are similar
 Since a triangle only has three angles, the third one
can be found if two of them are known
Math Cram Kit | 22
TRIGONOMETRY
Sine Here
RIGHT TRIANGLE RELATIONSHIPS
TRIGONOMETRIC FUNCTIONS
SIDES AND ANGLES
TRIG FUNCTIONS AND QUADRANTS
 To remember what the trig functions mean, use the

mnemonic SOHCAHTOA (soak-a-toe-a)
Opposite
Sine(angle) 
Hypotenuse

Cosine(angle) 

Tangent(angle) 
Adjacent
Hypotenuse
Opposite
Adjacent
 Each trig function is only positive in certain quadrants














a
c
b
sinB  cosA 
c
a
tanA  cotB 
b
b
tanB  cotA 
a
c
secA  cscB 
b
c
secB  cscA 
a
csc (cosecant) is the reciprocal of sin (sine)
sec (secant) is the reciprocal of cos (cosine)
cot (cotangent) is the reciprocal of tan (tangent)
sinA  cosB 
(mnemonic: All Students Take C lasses)
All of the trig functions have positive values in
Quadrant I
Sine is positive in Quadrant II
Tangent is positive in Quadrant III
Cosine is positive in Quadrant IV
Each reciprocal function-----cosecant, secant, and
cotangent-----has the same sign as its corresponding
function
REFERENCE ANGLES
 When drawing angles, we place the initial side at the
positive x-axis and go counter-clockwise, ending with a
terminal side
 A reference angle is the angle between the terminal
side and the x-axis
 The sine, cosine, and tangent of an angle is numerically
equivalent to its corresponding reference angle, but the
sign may need to be adjusted depending on the
quadrant in which the terminal side is located
 The above angle is 225°, and it lies in Quadrant III



Its reference angle is 225° --- 180° = 45°
sin(225°) is numerically equivalent to sin(45°), but
sine values are negative in Quadrant III
sin(225°) = ---sin(45°) = ---0.707
Math Cram Kit | 23
TRIGONOMETRY
The Arc Side and Graphic Descriptions
INVERSE TRIG FUNCTIONS
PROPERTIES OF TRIG GRAPHS
THE BASICS
PERIOD
 Inverse trig functions reverse the effects of trig

functions
If sinA = B, then arcsinB = A

The smallest interval taken for function values to repeat
 All trig functions are periodic (they repeat)
 The period of a function is always positive
 Sine, cosine, and their reciprocal functions (cosecant
 1
1
sin(30 )  , and arcsin    30
2
2
 The inverse trig functions are arcsin, arccos, arctan,

2
, where k is the
k
coefficient of x in the argument
2 

 The function sin(6x) has a period of
6 3

Tangent and cotangent have periods of , where k is
k
the coefficient of x

 The function cot(---7x) has a period of
7
and secant) have a period of
arccsc, arcsec, and arccot
The inverse trig functions can also be notated:
sin1 ,cos 1 ,tan1 ,csc 1 ,sec 1 ,cot 1
 Unlike sin2x, which means (sinx)2, sin1 x does not
mean (sinx)1
DOMAIN AND RANGE

 Inverse trig functions do not pass the vertical line

test unless we limit their domains and ranges
The following limits allow us to work with inverse
trig functions as true functions
Function
Domain
Range
arcsin
[1,1]
π π
[- , ]
2 2
arccos
[1,1]
[0, π]
AMPLITUDE
Half of the distance between the maximum and minimum
values of the function
 Sin and cos have amplitudes determined by the
coefficient of the function
 The function 3cos(5x) has an amplitude of 3
HORIZONTAL (PHASE) SHIFT
 A constant term inside the function shifts the graph
arctan
( , )
π π
( , )
2 2
arccsc
(–  ,–1]
∪[1,  )
π
π
[ ,0)  (0, ]
2
2
arcsec
(–  ,–1]
∪[1,  )
π
π
[0, )  ( , π]
2
2
arccot
(  ,  )
(0,  )
horizontally
 A function with argument (kx --- h) is shifted
h
units
k
from x = 0
 What is the phase shift of the function
tan(3x + 5)?
 First, we need to put the argument into the form
(kx --- h)
 tan(3x + 5) = tan(3x --- (---5))
 We know k = 3 and h = ---5, so the function is
h
5
shifted   units from x = 0 (in the negative
k
3
direction, or to the left)
Math Cram Kit | 24
TRIGONOMETRY
Ooh, Pretty Wave; Identity Quandary
MORE PROPERTIES OF TRIG GRAPHS
IDENTITIES
VERTICAL SHIFT
WHY DO WE USE IDENTITIES?
 A constant term outside the function shifts the
graph vertically
 What is the vertical shift of csc(6x + 2) --- 8?
 The constant term outside the function is ---8, so
the graph is shifted 8 units in the negative
direction (down)
CONSOLIDATION (SINE/COSINE)
 Asin(kx  h)  b or Acos(kx  h)  b
 Amplitude = A
 Period =
2
k
 Horizontal shift =
h
k
 Vertical shift = b
 Note that for tangent and cotangent functions,

, and amplitude is largely
k
irrelevant in graphs
period is equal to
ALL TOGETHER NOW
 The following graph represents 5sin(4x --- 8) + 2
To convert between different trigonometric
functions to solve a problem
RECIPROCAL IDENTITIES
1
1
; cscx 
 sinx 
cscx
sinx
1
1
; secx 
 cosx 
secx
cosx

tanx 
1
1
; cotx 
cotx
tanx
QUOTIENT IDENTITIES
sinx
 tanx 
cosx

cotx 
cosx
sinx
PYTHAGOREAN IDENTITIES



sin2 x  cos2 x  1
tan2 x  1  sec2 x
1  cot2 x  csc2 x
OTHER IMPORTANT IDENTITIES
 sin(x  y)  (sinx)(cosy)  (cosx)(siny)
 Amplitude (marked by the green line from the

middle to the trough of the wave) is 5
Period (marked by the bracket that covers one
2 

complete cycle) is
4 2
 Horizontal shift is
h 8
  2 units from x = 0 (to the
k 4
right)
 5sin(4x  8)  2  5sin(4x  8)  2
 Vertical shift is 2 units up because the constant
term outside the function is 2

cos(x  y)  (cosx)(cosy)  (sinx)(siny)

tan(x  y) 


sin(x  y)  (sinx)(cosy)  (cosx)(siny)

tan(x  y) 

sin(2x)  2sinx cos x
tanx  tany
1  (tanx)(tany)
cos(x  y)  (cosx)(cosy)  (sinx)(siny)
tanx  tany
1  (tanx)(tany)
cos(2x)  cos2 x  sin2 x

 1  2sin2 x
 2cos2 x  1

tan(2x) 
2tanx
1  tan2 x
Math Cram Kit | 25
TRIGONOMETRY
Triangular Relationships; Finding a Good Angle
LAW OF SINES AND COSINES
ALGEBRAIC EQUATIONS INVOLVING TRIG
FUNCTIONS
SOLUTIONS
 Unless domain and range are limited, trig functions can


LAW OF SINES
 In a triangle, the ratio of the sine of an angle to its

opposite side is the same for all three angles
sinA sinB sinC


a
b
c
LAW OF COSINES
 With a slight modification, the Pythagorean




theorem can work for any triangle, producing the
Law of Cosines
Given two sides and the angle between them, we
can find the length of the third side
c 2  a2  b 2  2ab(cosC)
a2  b 2  c 2  2bc(cos A)
b 2  a2  c 2  2ac(cosB)
have an infinite number of solutions
The answers to these functions will repeat every 360°
or 2π radians
The same reference angle in different quadrants can
produce the same result in a trig function
SOLVING
 We usually want to turn all the different types of trig
functions into just one type by substituting identities or
by canceling out common terms
 Then, we can isolate the trig expression and solve for
the angle
 1 --- cos2x + sin2x = 0
 1 --- (1 --- sin2x) + sin2x = 0





sin2x + sin2x = 0
2sin2x = 0
sin2x = 0
sinx = 0
x = 0°, 180°, 360°…
Math Cram Kit | 26
CRUNCH KIT
Formula Frenzy (Page 1)
GENERAL MATH
GEOMETRY

Pythagorean theorem: a2  b2  c2

n!
Combinations: n Cr 
(r!)(n-r)!

Midpoint formula: 


Circular arrangements: (n --- 1)!
Probability that two independent events will occur:
P(A+B) = P(A) x P(B)
Probability that one of two mutually exclusive events will
occur: P(A or B) = P(A) + P(B) --- P(A+B)

Distance formula:

Area of a trapezoid: Area =

Permutations: nPr 

n!
(n-r)!
ALGEBRA
y2  y 1
x2  x 1

Slope: m 

Point-slope form: y  y 1  m(x  x 1 )

Slope-intercept form:

Standard form of a linear function: Ax  By  C










Quadratic formula: x 
Sum of roots: 

Area of a rhombus: Area =
xy
1

(diagonal1)(diagonal2)
2
Area of a square: Area = (side)2
1
Area  (diagonal1)2
2

1
Area of a triangle: Area  (base)(height)
2
Area  (s)(s  a)(s  b)(s  c) , where s 
ab c
2
1

Area = ab(sinC)
2
Surface area of prism:
SA = Area of 2 bases + area of lateral faces
Surface area of pyramid:
SA = Area of the base + area of lateral triangles
Surface area of cylinder: SA = 2πr2 + 2 π rh

Surface area of sphere: SA = 4 π r2

Surface area of cone: SA = π r2 + π r

Volume of prism: V = (area of base)(height)
B
Geometric mean:
1


3
A
Product of roots:
C
 for odd  numbered polynomials
A
C
and for even  numbered polynomials
A
nth term of an arithmetic sequence:
nth term = first term + d(n --- 1)
Number of terms in an arithmetic series:
(last term  first term)
n
1
d
Sum of first n terms of an arithmetic series:
(first term  last term)
n
2
nth term of a geometric sequence:
nth term --- (first term)rn---1
Sum of first n terms in a geometric series:
(first term)(1  rn )
(x1  x2 )2  (y 1  y2 )2
(base1 + base2)(height)
2
Area of a parallelogram: Area = (base)(height)
Area of a rectangle: Area = (base)(height)
2A
Sum of cubes: x + y = (x + y)(x2 --- xy + y2)
Difference of cubes: x3 --- y3 = (x --- y)(x2 + xy + y2)
3
1r

B  B2  4AC
 x 1  x2 y 1  y 2 
,

2 
 2


r 2  h2

1
Volume of pyramid: V  (area of the base)(height)
3
Volume of cylinder: V = π r2h

Volume of sphere: V =


π r2h
3
Circumference of circle: 2 π r

Area of circle: π r2

180° = π radians

Central angle = intercepted arc

Inscribed angle =

4
3
Volume of cone: V =
π r3
1
1
intercepted arc
2
Math Cram Kit | 27
CRUNCH KIT
Formula Frenzy (Page 2)
TRIGONOMETRY

Opposite
Sine(angle) 
Hypotenuse
Adjacent

Cosine(angle) 

Tangent(angle) 
Hypotenuse
Opposite
Adjacent
1
sinx 

cos x 

tanx 

tanx 

cot x 

sin2 x  cos2 x  1

tan2 x  1  sec2 x

1  cot2 x  csc2 x

sin(x  y)  (sinx)(cosy)  (cosx)(siny)

cos(x  y)  (cosx)(cosy)  (sinx)(siny)

sin(x  y)  (sinx)(cosy)  (cosx)(siny)

cos(x  y)  (cosx)(cosy)  (sinx)(siny)

sin(2x)  2sinx cos x
csc x
; csc x 
1

1
; sec x 
sec x
1
cot x
sinx
; cot x 
1
cos x
1
tanx
sinx
cos x
cos x
sinx
cos(2x)  cos2 x  sin2 x

 1  2sin2 x
 2cos2 x  1
sinA

sinB

sinC

Law of sines:

Law of cosines: c2  a2  b2  2ab(cosC)
a
b
c
Math Cram Kit | 28
FINAL TIPS AND ABOUT THE AUTHOR
FINAL TIPS
ABOUT THE AUTHOR
 Do the easy problems first; all the questions are






worth the same number of points, and the easy
problems may be at the end of the test
Use a timer in practice and at competition
Use all 30 minutes to work-----don’t give up!
When you have 5 minutes left, guess on all
remaining unanswered questions before returning
to your current problem
Be familiar with your calculator
If you don’t know how to do a problem, try plugging
in the answers, since they’re given to you
Make sure your calculator is in degree mode when
working with degrees and in radian mode when
working with radians
They say Steven Zhu shot a
man down in Reno, but that
was just a lie. Keb’ Mo’
references aside, this much is
known about Steven: he is an
economics major at H arvar d
University , he competed with
the Frisco High School
decathlon team, and he once
won a state championship in a
place called Texas. After a stint
at the Federal Reserve Bank of
Dallas this summer, Steven hopped around various cities in
China, land of Mao and slow internets. He would like to
maximize happiness instead of utility someday, but in the
meantime, he will settle for a nap.
ABOUT THE EDITOR
SOPHY LEE
Sophy Lee loves berries. In fact, she could survive on a diet composed exclusively of
strawberries, raspberries, mulberries, blackberries, and blueberries. Unfortunately,
Harvard University only offers canned blueberries and raspberries in its dining halls
during breakfast—and canned just canned cut it (bad pun; cue laughter). She
believes berries best accompany steel-cut oats and vanilla yoghurt. They also work
well with bananas and protein powder to form a scrum-diddly-umptious post-60mile-bike-ride recovery smoothie. She hopes this non-sequitur “About the Editor”
has distracted you from the fact that competition is tomorrow (or in 20 minutes)
and you haven’t read your resources yet. Remember, your gut instinct is always right
except when it’s wrong, and Sophy believes in you but only if you believe in yourself.
She is pictured here with Ryan Seacrest (or a reasonable holographic projection
thereof) after winning her first bout on NBC’s recent production, Million Second Quiz.