013 014 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?... II. CRAMMING FOR SUCCESS III. GENERAL MATH... 3 IV. ALGEBRA.... 5 V. GEOMETRY........15 VI. TRIGONOMETRY... VII. CRUNCH KIT... 6 VIII. ABOUT THE AUTHOR AND EDITOR 8 BY STEVEN ZHU HARVARD UNIVERSITY FRISCO HIGH SCHOOL EDITED BY SOPHY LEE HARVARD UNIVERSITY PEARLAND HIGH SCHOOL DEDICATED TO PYTHAGORAS, FOR BEING SUCH A HOMIE. 013 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 WHAT IS A CRAM KIT? A Word from the Editor COMPETITION IS NEARING 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. 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. 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. STRUCTURE OF A CRAM KIT 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. 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. 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% 60% Geometry 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 Fractions must have a common denominator before 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): 1 4 4 1 Step. Multiply the first fraction by the reciprocal of the second: 1 4 1 4 4 = = 1 1 Step 3. Simplify the fraction: 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 5 0 $40.00 x (1- ) x (1- ) $4 o a percentage 100 100 x Formula for sale prices: (1- )(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 5%. Joe uses a 0%-off coupon to purchase the shirt. How much does he have to pay for the shirt before tax? MULTIPLICATION PRINCIPLE Helps us find the total number of possibilities when we 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 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)! A club of 1 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 1! 1! 1P3 1 x 11x 10 130 (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 ncr (r!)(n-r)! A club of 1 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 1! 1! 1 x 11x 10 1C3 0 3!(1-3)! 3!9! 3 x x 1 TRY THIS MNEMONIC! Permutations = Prizes (order matters) Combinations = Committees (order doesn t matter)
Math Cram Kit 4 BASIC COUNTING TECHNIQUES (PT. ) GENERAL MATH More Counting; Winning in Vegas PROBABILITY OF EQUALLY LIKELY EVENTS ARRANGEMENT RULES 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 Number of possible circular arrangements = h 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 4 When arranging keys on a keychain, we must divide the result by 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 PROBABILITY The chance that an event will happen RULES The probability that event A happens is P(A) csc 6x 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 5 cards Such a deck has 4 suits ( red and black) of 13 cards each A deck s face cards are the Jack, Queen, and King of each suit (1 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, 4-5, 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 5-card deck has four Queens, two of which are red 1 P(Q r) 5 6 What is the probability of a coin landing heads four tosses in a row? 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! For each toss, the chance of landing heads is 1 The tosses are independent events, since each toss does not affect the result of any other toss 1 1 P(4H) P(H) P(H) P(H) P(H) 16 4
Math Cram Kit 5 SOLVING POLYNOMIAL EQUATIONS (THE BASICS) ALGEBRA Separate but Equal SOLVING POLYNOMIAL EQUATIONS (LINEAR) EQUATION A mathematical statement that two expressions are equal Examples 3 + 7 = 14 -- 4 4x + 5 = y POLYNOMIAL An expression containing variables 4 5x x 3 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 1x y 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 + 3x 4 y -- 81z 4xyz has a degree of 1 + 1 + 1 = 3 3x 4 y has a degree of 4 + = 6-81z has a degree of 1 Thus, 4xyz + 3x 4 y -- 81z is a 6 th order polynomial The leading coefficient of a polynomial is the coefficient of the term with the highest degree The leading coefficient of 7x -- 9x 3 + 15x - 64 is -9 LINEAR POLYNOMIALS Equations that have a degree of 1 and straight-line graphs SLOPE-INTERCEPT FORM y = mx + b m is slope y y1 m, given points (x1, y1) and (x, y) x x 1 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 y1 m(x x 1) m is slope (x1, y1) is a given point 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 The roots of a quadratic equation are the values of x for which y = 0 (where the graph intersects the x- axis) Roots are also called zeroes or x-intercepts If the equation is in the form y = Ax + Bx + C, we can use the quadratic formula to find the roots B B 4AC Quadratic formula: x A The part of the quadratic formula under the radical sign, B -- 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 4x 4x 3 0 (x 3)(x 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 x 3 0 x 3 x 1 0 x 1 HIGHER ORDER EQUATIONS Equations that have a degree higher than Some cubic polynomials are factorable Sum of cubes formula: x 3 + y 3 = (x + y)(x -- xy + y ) Difference of cubes formula: x 3 -- y 3 = (x -- y)(x + xy + y ) REMAINDER AND FACTOR THEOREMS Remainder Theorem: To find the remainder when a polynomial is divided by (x -- c), plug c into the polynomial What is the remainder when x 4 -- 5x + 7 is divided by x + 3? In this example, c = --3, as x + 3 = x -- (--3) The remainder is (--3) 4 -- 5(--3) + 7 = 13 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 x 3 -- 5x -- x + 5 is divided by (x -- 5) is (5) 3 -- 5(5) + 5 = 0 Thus, (x -- 5) is a factor of x 3 -- 5x -- 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 3x -- 6 + 5x 3 + x 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 3 6 1 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 Ax + Bx + C, two formulas exist for finding the sum and the product of the roots 1. Sum of roots formula: B A. 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 5 < --5x 5 > x QUADRATIC INEQUALITY An inequality with a degree of x 3x 8 43 x 3x 35 0 (x 7)(x 5) 0 At this point, we will plot the roots on a number line, dividing it into three regions 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 x -- 3 < 5 The first inequality is the same as the original, but without the absolute value signs x 3 5 x 8 x 4 For the second inequality, we multiply the right side by -1 and flip the sign of the inequality x 3 5 x x 1 Thus, x -- 3 < 5 holds true when x < 4 and x > --1-5 7 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 The inequality is true when x < --5 or x > 7
Math Cram Kit 8 FUNCTIONS (BASICS) ALGEBRA Putting the Fun in Function! FUNCTIONS (COMPOSITE AND INVERSE) WHAT IS A FUNCTION? 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 TYPES OF FUNCTIONS (PART 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) = 3x, b(x) = 5x + 7, and x = b() = 5() + 7 = 17 a(b()) = a(17) = 3(17) = 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 + is x = 3y + 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 DOMAIN AND RANGE The domain of a function consists of all the x-values that have corresponding y-values Find the domain of f(x) 1 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 y- values The following graph has a range of -1 to 1
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 ) RATIONAL FUNCTIONS Functions in which variables are in the denominators of fractions Fractions are ratios, hence, rational functions 4 5x 41 is a rational function 3 x EXPONENTIAL FUNCTIONS Functions in which the independent variable x is in an exponent 3 x is an exponential function A common exponential function is e x e is a constant like and can be found on a scientific or graphing calculator e =.7188 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 base exponent = argument Log7(49) = because 49 is 7 to the nd power When the logarithm does not have a base written, assume that the base is 10 Log(1000) = 3, since 10 3 = 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 13 Ln(e ) 13 Log (4 x ) x 4 WORKING WITH LOGS ADDITION When adding two logarithms of the same base, we can combine them into one logarithm with the arguments multiplied together log 1(x 1) log 1(x 3) log 1((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 1(x 1) log 1(x 3) log1 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 3 log((5x 9) ) 3log(5x 9) We can pull the 3 out because it applies to the whole argument We cannot pull the 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 Log1((x -- 5)(x + 9)) = Log1(x -- 5) + Log1(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 1(x 5) log 1(x 9) x 9 A coefficient of a logarithm can become the exponent of the logarithm s entire argument 3(Log(5x + 9)) = Log((5x + 9) 3 )
Math Cram Kit 10 COMPLEX NUMBERS ALGEBRA Use Your Imagination; Walk the Line READING GRAPHS OF FUNCTIONS (LINEAR) WHAT IS A COMPLEX NUMBER? Any number in the form a + bi a and b are real numbers i is an imaginary number such that i 1 LINEAR FUNCTIONS OPERATIONS WITH COMPLEX NUMBERS We can simplify higher powers of i Find i 47 We know that i = --1 i 47 is the same as (i 46 )(i) (i 46 )(i) = (i ) 3 (i) Thus, i 47 = (--1) 3 (i) i 47 = --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 1i Simplify 3i 1 i 3i 3i i 3 1 5i 3i 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 (b -- 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 y- intercept 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) y y1 Using the formula for slope, 1 the slope is m 0 3 3 1 6 0 6 m x x, we find that Therefore, the graph above represents 1 y x 3
Math Cram Kit 11 READING GRAPHS OF FUNCTIONS (QUADRATIC) ALGEBRA Read Between the Curves 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 The following graph represents y x 6 x 3, which 4 starts and ends on the positive side of the y-axis 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) + k The point (h,k) is the vertex----the turning point of the curve In the graph above, the vertex is (--,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 -- (--)) + 1, which becomes y = A(x + )) + 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) 1 A(0 ) 1 A 4 A 1 Thus, the equation of the graph above is 1 y (x ) 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 = --x 7 + x 4, which starts on the positive side of the y-axis and ends on the negative side
Math Cram Kit 1 READING GRAPHS OF FUNCTIONS (EXPONENTIAL AND LOGARITHMIC) ALGEBRA Flipped Functions and Arithmetic Arrangements SEQUENCES, SERIES, AND MEANS (ARITHMETIC) EXPONENTIAL FUNCTIONS Exponential functions create graphs with horizontal 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 = e x, which has a horizontal asymptote at y = 0 As x approaches negative infinity, y will approach 0 but will never reach it 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 ARITHMETIC SEQUENCE Pattern of numbers that has a common difference d 1, 8, 15,, 9 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 9 th term of the sequence: 68, 64, 60, 56 n = 9 and d = --4 (--4 = 64 -- 68 = 60 -- 64, and so on) 9 th term = 68 + (--4)(9 -- 1) = 68 -- 3 = 36 ARITHMETIC SERIES The sum of an arithmetic sequence Formula to find the sum of the first n terms: (first term last term) n 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, 0, 3 44, 47, 50 d = 3, the last term is 50, and the first term is 17 (50 17) n 1 1 3 (17 50) Now we can find the sum 1 40 Summation problems may use sigma () notation 5 k = the sum of the numbers 1 through 5 k1 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++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 SEQUENCES, SERIES, AND MEANS (GEOMETRIC) ALGEBRA Rational Commonists SEQUENCES, SERIES, AND MEANS (GEOMETRIC AND INFINITE) GEOMETRIC SEQUENCE Pattern of numbers with a common ratio r, 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 n1 sequence: nth term (first term)r What is the 8 th term of the sequence that begins: 65, 15, 5, 5? GEOMETRIC MEAN The square root of the product of two terms Find 10 k1 3 (4) k1 What is the geometric mean of 4 and 64? 464 16 4, 16, and 64 form a geometric series with a common ratio of 4 The common ratio is 1 5 1 1 1 8th term (65) (65) 5 7815 15 GEOMETRIC SERIES The sum of a geometric sequence Formula to find the sum of the first n terms of a n (first term)(1 r ) geometric sequence: 1r Find 10 k1 3 (4) k1 We plug in k 1 to find the first term: 11 3 (4) 4 7 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 consecutive term is 3, so r = 3 Thus, the sum is 3 (4) 1 3 1 10 453.3 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 than 1 The infinite series of the sequence that begins with 1 1 1,1,,,... will have a value because each 4 8 term is 1 times the previous one 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 --, 4, -- 8, 16, --3 does not have a value because every term is - 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 SEQUENCES, SERIES, AND MEANS (GRAPHING) ALGEBRA Can You See the Pattern? SEQUENCES, SERIES, AND MEANS (GRAPHING) Arithmetic Sequence Geometric Sequence 0 300 15 50 00 10 150 5 100 50 0 0 5 10 0 0 4 6 8 10 In arithmetic sequences, the terms have equal vertical distances between them because the common difference d never changes In the above geometric sequence, each term is twice as large the previous one 80 70 60 50 40 30 0 10 0 Arithmetic Series 0 5 10 5 4 3 1 0 Geometric Series 0 50 100 150 In an arithmetic series, the sums do not have equal vertical distances between them because each term added is larger than the previous term 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 RIGHT TRIANGLES GEOMETRY Triangles with Little Squares in the Corner SPECIAL RIGHT TRIANGLES PYTHAGOREAN THEOREM 45-45-90 TRIANGLES a c s 45 s b 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 a b c The theorem can also give us information about other types of triangles in which c is the longest side If a b c, then the triangle is acute (all angles are less than 90 ) If a b c, 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, 1, 13 7, 4, 5 8, 15, 17 9, 40, 41 Any multiple of a Pythagorean triple will also be a Pythagorean triple 6, 8, 10 10, 4, 6 45-45-90 triangles are right triangles with legs of equal length They are also called right isosceles triangles The hypotenuse is equal to times a side s 60 30-60-90 TRIANGLES 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 s s 45 30
Math Cram Kit 16 COORDINATE GEOMETRY (POINTS) GEOMETRY Point-Line Coordination COORDINATE GEOMETRY (LINES) MIDPOINT The point that is exactly in the middle of two other points Given two points (x1, y1) and (x, y), their midpoint is the average of their coordinates: x1 x y1 y, Find the midpoint of (-, 3) and (5, -6) 5 3 ( 6) 3 3,, PARALLEL AND PERPENDICULAR LINES Two lines are parallel if they have the same slope A line that crosses two parallel lines is called a transversal 1 4 3 5 6 8 7 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 (x,y) that lie on the y y1 same line, the slope of the line is m x 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 (x 1, y1) and (x, y) is: (x x ) (y y ) 1 1 Two angles that add up to 180 degrees are called supplementary angles (1 &, 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 (, 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 4, so the slope of the 7 perpendicular line is 7 4
Math Cram Kit 17 GEOMETRY Four-sided Shapes That Are Almost, but Not Entirely, Unlike Triangles COORDINATE GEOMETRY (QUADRILATERALS) QUADRILATERAL A four-sided polygon TRAPEZOID A quadrilateral with one pair of parallel sides RECTANGLE A parallelogram with four right angles The parallel sides are called bases The non-parallel sides are called legs The height is the distance from one base to the other Area = 1 (base1 + base)(height) Area = (base)(height) In a coordinate system, opposite sides have the same slope and length, and adjacent sides must be perpendicular RHOMBUS A parallelogram with four congruent sides In a coordinate system, the two parallel bases have the same slope, and the two legs have different slopes PARALLELOGRAM A quadrilateral with two pairs of parallel sides The diagonals form right angles The diagonals bisect each other and bisect the angles, forming four congruent right triangles Area = 1 (diagonal1)(diagonal) In a coordinate system, the diagonals are perpendicular, and the side lengths are all equal 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 SQUARE A quadrilateral with four congruent sides and four right angles, making it both a type of rectangle and rhombus Area = (side) In a coordinate system, all sides have the same length, and adjacent sides are perpendicular
Math Cram Kit 18 PLANE AND SOLID FIGURES (AREA) GEOMETRY Movin On Up, Dimensionally PLANE AND SOLID FIGURES (VOLUME) AREA OF A TRIANGLE Area 1 (base)(height) Works best for right triangles and triangles whose base and height are known Heron s Formula: Area (s)(s a)(s b)(s c) a, b, and c are the sides of the triangle, and s a b c 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 Area = 1 ab(sinc) VOLUME OF SOLID FIGURES Prism: V = (area of base)(height) Pyramid: V 1 (area of thebase)(height) 3 Cylinder: V = r h r is the radius of the base h is the height of the cylinder Sphere: V = 4 3 r3 r is the radius of the sphere Cone: V = 1 3 r h r is the radius of the base h is the height of the cone a and b are two sides, and C is the angle between them SURFACE AREA OF SOLID FIGURES Prism: SA = Area of bases + area of lateral faces Pyramid: SA = Area of the base + area of lateral triangles Cylinder: SA = r + rh r is the radius of the base, and h is the height of the cylinder Sphere: SA = 4 r r is the radius of the sphere Cone: SA = r + r r h r is the radius of the base, and h is the height of the cone r h 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 r h
Math Cram Kit 19 GEOMETRY Circle Time PLANE AND SOLID FIGURES (CIRCLES) MEASURING CIRCLES CIRCUMFERENCE OF A CIRCLE Circumference = r Circumference is the perimeter of a circle AREA OF A CIRCLE Area = r r is the radius of the circle LINES AND CIRCLES (PART 1) 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 LOOKING INSIDE ANGLES IN A CIRCLE A circle has 360 or 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 0 GEOMETRY Circle Time: Part Deux PLANE AND SOLID FIGURES (CIRCLES) (CONT D) TWO CHORDS LINES AND CIRCLES (PART ) LINES AND CIRCLES (PART 3) A TANGENT AND A SECANT In the above diagram, two chords intersect at a point E AB DC AEB CED AD BC AEB BEC and In the above diagram, AB is a tangent and AC is a secant that intersects the circle at point D A BC BD (AB) AD AC AEEC BEED TWO TANGENTS TWO SECANTS 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 their respective tangents major arc BCminor arc BC A In the above diagram, two secants originating from point A intersect a circle at points D and E A BC DE ADAB AEAC
Math Cram Kit 1 CONGRUENCE GEOMETRY A Striking Resemblance SIMILARITY PROPERTIES OF CONGRUENT 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 PROPERTIES OF SIMILAR FIGURES 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 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 and 6 separated by an angle of 54 degrees is congruent to another triangle with side lengths of 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 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 1 and 0 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 TRIGONOMETRY Sine Here RIGHT TRIANGLE RELATIONSHIPS SIDES AND ANGLES To remember what the trig functions mean, use the mnemonic SOHCAHTOA (soak-a-toe-a) Sine(angle) Opposite Hypotenuse Cosine(angle) Adjacent Hypotenuse Tangent(angle) Opposite Adjacent a sina cosb c b sinb cosa c a tana cotb b b tanb cota a TRIGONOMETRIC FUNCTIONS TRIG FUNCTIONS AND QUADRANTS Each trig function is only positive in certain quadrants (mnemonic: All Students Take Classes) 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 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) The above angle is 5, and it lies in Quadrant III Its reference angle is 5 -- 180 = 45 sin(5 ) is numerically equivalent to sin(45 ), but sine values are negative in Quadrant III sin(5 ) = --sin(45 ) = --0.707
Math Cram Kit 3 INVERSE TRIG FUNCTIONS TRIGONOMETRY The Arc Side and Graphic Descriptions PROPERTIES OF TRIG GRAPHS THE BASICS Inverse trig functions reverse the effects of trig functions If sina = B, then arcsinb = A sin(30 ) 1, and 1 arcsin 30 The inverse trig functions are arcsin, arccos, arctan, arccsc, arcsec, and arccot The inverse trig functions can also be notated: 1 1 1 1 1 1 sin,cos, tan,csc,sec,cot Unlike sin x, which means (sinx) 1, sin x does not 1 mean (sinx) PERIOD 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 and secant) have a period of, where k is the k coefficient of x in the argument The function sin(6x) has a period of 6 3 Tangent and cotangent have periods of, where k is k the coefficient of x 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 arccos arctan [ 1,1] π π [-, ] [ 1,1] [0, π] (, ) arccsc (, 1] [1, ) arcsec (, 1] [1, ) π π (, ) π π [,0) (0, ] π π [0, ) (, π] arccot (, ) (0, ) The function cot(--7x) has a period of 7 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 horizontally A function with argument (kx -- h) is shifted h k units 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 4 MORE PROPERTIES OF TRIG GRAPHS TRIGONOMETRY Ooh, Pretty Wave; Identity Quandary IDENTITIES VERTICAL SHIFT A constant term outside the function shifts the graph vertically What is the vertical shift of csc(6x + ) -- 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 = k Horizontal shift = h k Vertical shift = b Note that for tangent and cotangent functions, period is equal to, and amplitude is largely k irrelevant in graphs ALL TOGETHER NOW The following graph represents 5sin(4x -- 8) + 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 complete cycle) is 4 h 8 Horizontal shift is units from x = 0 (to the k 4 right) 5sin(4x 8) 5sin(4x 8) Vertical shift is units up because the constant term outside the function is WHY DO WE USE IDENTITIES? To convert between different trigonometric functions to solve a problem RECIPROCAL IDENTITIES sinx 1 ; cscx 1 cscx sinx cosx 1 ; secx 1 secx cosx tanx 1 ; cotx 1 cotx tanx QUOTIENT IDENTITIES tanx sinx cosx cotx cosx sinx PYTHAGOREAN IDENTITIES sin x cos x 1 tan x 1 sec x 1 cot x csc x OTHER IMPORTANT IDENTITIES sin(x y) (sinx)(cosy) (cosx)(siny) cos(x y) (cosx)(cosy) (sinx)(siny) tan(x y) tanx tany 1 (tanx)(tany) sin(x y) (sinx)(cosy) (cosx)(siny) cos(x y) (cosx)(cosy) (sinx)(siny) tan(x y) tanx tany 1 (tanx)(tany) sin(x) sinx cos x cos(x) cos x sin x 1 sin x cos x 1 tanx tan(x) 1 tan x
Math Cram Kit 5 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 have an infinite number of solutions The answers to these functions will repeat every 360 or π radians The same reference angle in different quadrants can produce the same result in a trig function 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 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 -- cos x + sin x = 0 1 -- (1 -- sin x) + sin x = 0 sin x + sin x = 0 sin x = 0 sin x = 0 sinx = 0 x = 0, 180, 360 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 a b ab(cosc) a b c bc(cos A) b a c ac(cosb)
Math Cram Kit 6 CRUNCH KIT Formula Frenzy (Page 1) GENERAL MATH n! Permutations: npr (n-r)! n! Combinations: ncr (r!)(n-r)! 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) ALGEBRA y y Slope: m x x 1 1 Point-slope form: y y m(x x ) 1 1 Slope-intercept form: Standard form of a linear function: Ax By C B B 4AC Quadratic formula: x A Sum of cubes: x 3 + y 3 = (x + y)(x -- xy + y ) Difference of cubes: x 3 -- y 3 = (x -- y)(x + xy + y ) Sum of roots: B 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 nth term of a geometric sequence: nth term -- (first term)r n--1 Sum of first n terms in a geometric series: n (first term)(1 r ) 1 r Geometric mean: xy GEOMETRY Pythagorean theorem: a b c x x y y Midpoint formula:, 1 1 Distance formula: (x x ) (y y ) 1 1 Area of a trapezoid: Area = 1 (base1 + base)(height) Area of a parallelogram: Area = (base)(height) Area of a rectangle: Area = (base)(height) Area of a rhombus: Area = 1 (diagonal1)(diagonal) Area of a square: Area = (side) Area 1 (diagonal1) Area of a triangle: Area 1 (base)(height) Area (s)(s a)(s b)(s c), where s a b c Area = 1 ab(sinc) Surface area of prism: SA = Area of bases + area of lateral faces Surface area of pyramid: SA = Area of the base + area of lateral triangles Surface area of cylinder: SA = πr + π rh Surface area of sphere: SA = 4 π r Surface area of cone: SA = π r + π r r h Volume of prism: V = (area of base)(height) Volume of pyramid: V 1 (area of the base)(height) 3 Volume of cylinder: V = π r h Volume of sphere: V = 4 3 π r3 Volume of cone: V = 1 3 π r h Circumference of circle: π r Area of circle: π r 180 = π radians Central angle = intercepted arc Inscribed angle = 1 intercepted arc
Math Cram Kit 7 CRUNCH KIT Formula Frenzy (Page ) TRIGONOMETRY Sine(angle) Opposite Hypotenuse Cosine(angle) Adjacent Hypotenuse Tangent(angle) Opposite Adjacent sinx 1 ; cscx 1 csc x sinx cosx 1 ; secx 1 secx cosx tanx 1 ; cot x 1 cot x tanx tanx sinx cosx cotx cosx sinx sin x cos x 1 tan x 1 sec x 1 cot x csc 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(x) sinxcosx cos(x) cos x sin x 1 sin x cos x 1 sina sinb sinc Law of sines: a b c Law of cosines: c a b ab(cosc)
Math Cram Kit 8 FINAL TIPS AND ABOUT THE AUTHOR FINAL TIPS 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 ABOUT THE AUTHOR 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 Harvard 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-60- mile-bike-ride recovery smoothie. She hopes this non-sequitur About the Editor has distracted you from the fact that competition is tomorrow (or in 0 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.