SAT Subject Math Level 2 Facts amp FormulasNumbers Sequences FactorsIntegers -3 -2 -1 0 1 2 3 Reals integers plus fractions decimals and irrationals (radic2 radic3 _ etc)Order Of Operations PEMDAS(Parentheses Exponents Multiply Divide Add Subtract)Arithmetic Sequences each term is equal to the previous term plus dSequence t1 t1 + d t1 + 2d The nth term is tn = t1 + (n minus 1)dNumber of integers from in to im = im minus in + 1Sum of n terms Sn = (n2) (t1 + tn)Geometric Sequences each term is equal to the previous term times rSequence t1 t1 r t1 r2 The nth term is tn = t1 rnminus1

Sum of n terms Sn = t1 (rn minus 1)(r minus 1)Sum of infinite sequence (r lt 1) is Sinfin = t1(1 minus r)Prime Factorization break up a number into prime factors (2 3 5 7 11 )200 = 4 times 50 = 2 times 2 times 2 times 5 times 552 = 2 times 26 = 2 times 2 times 13Greatest Common Factor multiply common prime factors200 = 2 times 2 times 2 times 5 times 560 = 2 times 2 times 3 times 5GCF(200 60) = 2 times 2 times 5 = 20Least Common Multiple check multiples of the largest numberLCM(200 60) 200 (no) 400 (no) 600 (yes)Percentages use the following formula to find part whole or percentpart =percent100 times wholehttpwwweriktheredcomtutor pg 1SAT Subject Math Level 2 Facts amp FormulasAverages Counting Statistics Probabilityaverage =sum of termsnumber of termsaverage speed =total distancetotal timesum = average times (number of terms)mode = value in the list that appears most oftenmedian = middle value in the list (which must be sorted)Example median of 3 10 9 27 50 = 10Example median of 3 9 10 27 = (9 + 10)2 = 95Fundamental Counting PrincipleIf an event can happen in N ways and another independent eventcan happen in M ways then both events together can happen inN timesM ways (Extend this for three or more N1 times N2 times N3 )Permutations and Combinations

The number of permutations of n things is nPn = nThe number of permutations of n things taken r at a time is nPr = n(n minus r)The number of permutations of n things a of which are indistinguishable b ofwhich are indistinguishable etc is nPn(a b ) = n(a b )The number of combinations of n things taken r at a time is nCr = n1048576(n minus r) r_Probabilityprobability =number of desired outcomesnumber of total outcomesThe probability of two different events A and B both happening isP(A and B) = P(A) P(B) as long as the events are independent(not mutually exclusive)If the probability of event A happening is P(A) then the probability of event A nothappening is P(not A) = 1 minus P(A)httpwwweriktheredcomtutor pg 2SAT Subject Math Level 2 Facts amp FormulasLogic (Optional)The statement ldquoevent A implies event Brdquo is logically the same as ldquonot event B implies notevent Ardquo However ldquoevent A implies event Brdquo is not logically the same as ldquoevent B impliesevent Ardquo To see this try an example such as A = it rains and B = the road is wetIf it rains then the road gets wet (A rArr B) alternatively if the road is not wet it didnrsquotrain (not B rArr not A) However if the road is wet it didnrsquot necessarily rain (B 6rArr A)Powers Exponents Rootsxa xb = xa+b

(xa)b = xab

x0 = 1xaxb = xaminusb

(xy)a = xa ya

radicxy = radicx radic y1xb = xminusb

(minus1)n = _+1 if n is evenminus1 if n is oddIf 0 lt x lt 1 then 0 lt x3 lt x2 lt x lt radicx lt radic3 x lt 1Factoring Solving(x + a)(x + b) = x2 + (b + a)x + ab ldquoFOILrdquoa2 minus b2 = (a + b)(a minus b) ldquoDifference Of Squaresrdquoa2 + 2ab + b2 = (a + b)(a + b)a2 minus 2ab + b2 = (a minus b)(a minus b)x2 + (b + a)x + ab = (x + a)(x + b) ldquoReverse FOILrdquo

You can use Reverse FOIL to factor a polynomial by thinking about two numbers a and bwhich add to the number in front of the x and which multiply to give the constant Forexample to factor x2 + 5x + 6 the numbers add to 5 and multiply to 6 ie a = 2 andb = 3 so that x2 + 5x + 6 = (x + 2)(x + 3)To solve a quadratic such as x2+bx+c = 0 first factor the left side to get (x+a)(x+b) = 0then set each part in parentheses equal to zero Eg x2 + 4x + 3 = (x + 3)(x + 1) = 0 sothat x = minus3 or x = minus1The solution to the quadratic equation ax2 + bx + c = 0 can always be found (if it exists)using the quadratic formulax = minusb radicb2 minus 4ac2aNote that if b2 minus 4ac lt 0 then there is no solution to the equation If b2 minus 4ac = 0 thereis exactly one solution namely x = minusb2a If b2 minus 4ac gt 0 there are two solutions to theequationhttpwwweriktheredcomtutor pg 3SAT Subject Math Level 2 Facts amp FormulasTo solve two linear equations in x and y use the first equation to substitute for a variablein the second Eg suppose x+y = 3 and 4xminusy = 2 The first equation gives y = 3minusxso the second equation becomes 4x minus (3 minus x) = 2 rArr 5x minus 3 = 2 rArr x = 1 y = 2Solving two linear equations in x and y is geometrically the same as finding where two linesintersect In the example above the lines intersect at the point (1 2) Two parallel lineswill have no solution and two overlapping lines will have an infinite number of solutionsFunctionsA function is a rule to go from one number (x) to another number (y) usually writteny = f(x)The set of possible values of x is called the domain of f() and the corresponding set ofpossible values of y is called the range of f() For any given value of x there can only beone corresponding value yTranslationsThe graph of y = f(x minus h) + k is the translation of the graph ofy = f(x) by (h k) units in the plane

Absolute value|x| = _+x if x ge 0minusx if x lt 0|x| lt n rArr minusn lt x lt n|x| gt n rArr x lt minusn or x gt nParabolasA parabola parallel to the y-axis is given byy = ax2 + bx + cIf a gt 0 the parabola opens up If a lt 0 the parabola opens down The y-intercept is cand the x-coordinate of the vertex is x = minusb2aNote that when x = minusb2a the y-value of the parabola is either a minimum (a gt 0) or amaximum (a lt 0)EllipsesAn ellipse is essentially a squashed circle The equation of an ellipse centered on the originwhich intersects the x-axis at ( a 0) and the y-axis at (0b) isx2

a2 +y2

b2 = 1httpwwweriktheredcomtutor pg 4SAT Subject Math Level 2 Facts amp FormulasHyperbolas (Optional)A hyperbola looks like two elongated parabolas pointed away from one another Theequation of a hyperbola centered on the origin pointing down the positive and negativex-axes and which intersects the x-axis at ( a 0) isx2

a2 minusy2

b2 = 1Compound FunctionsA function can be applied directly to the y-value of another function This is usuallywritten with one function inside the parentheses of another function For examplef(g(x)) means apply g to x first then apply f to the resultg(f(x)) means apply f to x first then apply g to the resultf(x)g(x) means apply f to x first then apply g to x then multiply the resultsFor example if f(x) = 3xminus2 and g(x) = x2 then f(g(3)) = f(32) = f(9) = 3 9minus2 = 25Inverse FunctionsSince a function f() is a rule to go from one number (x) to another number (y) an inverse

function fminus1() can be defined as a rule to go from the number y back to the number x Inother words if y = f(x) then x = fminus1(y)To get the inverse function substitute y for f(x) solve for x in terms of y and substitutefminus1(y) for x For example if f(x) = 2x+ 6 then x = (y minus 6)2 so that fminus1(y) = y2minus 3Note that the function f() given x = 1 returns y = 8 and that fminus1(y) given y = 8returns x = 1Usually even inverse functions are written in terms of x so the final step is to substitutex for y In the above example this gives fminus1(x) = x2 minus 3 A quick recipe to find theinverse of f(x) is substitute y for f(x) interchange y and x solve for y and replace ywith fminus1(x)Two facts about inverse functions 1) their graphs are symmetric about the line y = xand 2) if one of the functions is a line with slope m the other is a line with slope 1mLogarithms (Optional)Logarithms are basically the inverse functions of exponentials The function logb x answersthe question b to what power gives x Here b is called the logarithmic ldquobaserdquo So ify = logb x then the logarithm function gives the number y such that by = x For examplelog3 radic27 = log3 radic33 = log3 332 = 32 = 15 Similarly logb bn = nhttpwwweriktheredcomtutor pg 5SAT Subject Math Level 2 Facts amp FormulasThe natural logarithm ln x is just the usual logarithm function but with base equal to thespecial number e (approximately 2718)A useful rule to know is logb xy = logb x + logb yComplex NumbersA complex number is of the form a + bi where i2 = minus1 When multiplying complexnumbers treat i just like any other variable (letter) except remember to replace powersof i with minus1 or 1 as follows (the pattern repeats after the first four)i0 = 1i4 = 1i1 = ii5 = ii2 = minus1i6 = minus1i3 = minusii7 = minusi

For example using ldquoFOILrdquo and i2 = minus1 (1 + 3i)(5 minus 2i) = 5 minus 2i + 15i minus 6i2 = 11 + 13iLines (Linear Functions)Consider the line that goes through points A(x1 y1) and B(x2 y2)Distance from A to B p(x2 minus x1)2 + (y2 minus y1)2

Mid-point of the segment AB _x1 + x2

2y1 + y2

2 _Slope of the liney2 minus y1

x2 minus x1

=riserunPoint-slope form given the slope m and a point (x1 y1) on the line the equation of theline is (y minus y1) = m(x minus x1)Slope-intercept form given the slope m and the y-intercept b then the equation of theline is y = mx + bTo find the equation of the line given two points A(x1 y1) and B(x2 y2) calculate theslope m = (y2 minus y1)(x2 minus x1) and use the point-slope formParallel lines have equal slopes Perpendicular lines (ie those that make a 90 anglewhere they intersect) have negative reciprocal slopes m1 m2 = minus1httpwwweriktheredcomtutor pg 6SAT Subject Math Level 2 Facts amp Formulasa

b

a

b

mla

b

b a

a

b

ab

Intersecting Lines Parallel Lines (l k m)Intersecting lines opposite angles are equal Also each pair of angles along the same lineadd to 180 In the figure above a + b = 180Parallel lines eight angles are formed when a line crosses two parallel lines The four bigangles (a) are equal and the four small angles (b) are equal

TrianglesRight trianglesabcxradic3x 2x30

60

xxxradic245

45

a2 + b2 = c2 Special Right TrianglesA good example of a right triangle is one with a = 3 b = 4 and c = 5 also called a 3ndash4ndash5right triangle Note that multiples of these numbers are also right triangles For exampleif you multiply these numbers by 2 you get a = 6 b = 8 and c = 10 (6ndash8ndash10) which isalso a right triangleAll triangleshbArea =12 b hhttpwwweriktheredcomtutor pg 7SAT Subject Math Level 2 Facts amp FormulasAngles on the inside of any triangle add up to 180The length of one side of any triangle is always less than the sum and more than thedifference of the lengths of the other two sidesAn exterior angle of any triangle is equal to the sum of the two remote interior anglesOther important trianglesEquilateral These triangles have three equal sides and all three angles are 60The area of an equilateral triangle is A = (side)2 radic34Isosceles An isosceles triangle has two equal sides The ldquobaserdquo angles(the ones opposite the two sides) are equal (see the 45 triangle above)Similar Two or more triangles are similar if they have the same shape Thecorresponding angles are equal and the corresponding sides are inproportion The ratio of their areas equals the ratio of the correspondingsides squared For example the 3ndash4ndash5 triangle and the 6ndash8ndash10triangle from before are similar since their sides are in a ratio of 2 to 1Trigonometry

Referring to the figure below there are three important functions which are defined forangles in a right triangleadjacentoppositehypotenuse_sin _ =oppositehypotenuseldquoSOHrdquocos _ =adjacenthypotenuseldquoCAHrdquotan _ =oppositeadjacentldquoTOArdquo(the last line above shows a mnemonic to remember these functions ldquoSOH-CAH-TOArdquo)An important relationship to remember which works for any angle _ issin2 _ + cos2 _ = 1For example if _ = 30 then (refer to the Special Right Triangles figure) we have sin 30 =12 cos 30 = radic32 so that sin2 30 + cos2 30 = 14 + 34 = 1httpwwweriktheredcomtutor pg 8SAT Subject Math Level 2 Facts amp FormulasThe functions y = a sin bx and y = a cos bx both have periods T = 360b ie they repeatevery 360b degrees The function y = a tan bx has a period of T = 180bThere are three other trig functions (cosecant secant and cotangent) that are related tothe usual three (sine cosine and tangent) as followscsc _ =hypotenuseopposite= 1 sin _sec _ =hypotenuseadjacent= 1 cos _cot _ =adjacentopposite= 1 tan _It is very useful to know the following trigonometric identitiessin(A + B) = sinAcosB + cosAsinBcos(A + B) = cosAcosB minus sinAsinB

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

The number of permutations of n things is nPn = nThe number of permutations of n things taken r at a time is nPr = n(n minus r)The number of permutations of n things a of which are indistinguishable b ofwhich are indistinguishable etc is nPn(a b ) = n(a b )The number of combinations of n things taken r at a time is nCr = n1048576(n minus r) r_Probabilityprobability =number of desired outcomesnumber of total outcomesThe probability of two different events A and B both happening isP(A and B) = P(A) P(B) as long as the events are independent(not mutually exclusive)If the probability of event A happening is P(A) then the probability of event A nothappening is P(not A) = 1 minus P(A)httpwwweriktheredcomtutor pg 2SAT Subject Math Level 2 Facts amp FormulasLogic (Optional)The statement ldquoevent A implies event Brdquo is logically the same as ldquonot event B implies notevent Ardquo However ldquoevent A implies event Brdquo is not logically the same as ldquoevent B impliesevent Ardquo To see this try an example such as A = it rains and B = the road is wetIf it rains then the road gets wet (A rArr B) alternatively if the road is not wet it didnrsquotrain (not B rArr not A) However if the road is wet it didnrsquot necessarily rain (B 6rArr A)Powers Exponents Rootsxa xb = xa+b

(xa)b = xab

x0 = 1xaxb = xaminusb

(xy)a = xa ya

radicxy = radicx radic y1xb = xminusb

(minus1)n = _+1 if n is evenminus1 if n is oddIf 0 lt x lt 1 then 0 lt x3 lt x2 lt x lt radicx lt radic3 x lt 1Factoring Solving(x + a)(x + b) = x2 + (b + a)x + ab ldquoFOILrdquoa2 minus b2 = (a + b)(a minus b) ldquoDifference Of Squaresrdquoa2 + 2ab + b2 = (a + b)(a + b)a2 minus 2ab + b2 = (a minus b)(a minus b)x2 + (b + a)x + ab = (x + a)(x + b) ldquoReverse FOILrdquo

You can use Reverse FOIL to factor a polynomial by thinking about two numbers a and bwhich add to the number in front of the x and which multiply to give the constant Forexample to factor x2 + 5x + 6 the numbers add to 5 and multiply to 6 ie a = 2 andb = 3 so that x2 + 5x + 6 = (x + 2)(x + 3)To solve a quadratic such as x2+bx+c = 0 first factor the left side to get (x+a)(x+b) = 0then set each part in parentheses equal to zero Eg x2 + 4x + 3 = (x + 3)(x + 1) = 0 sothat x = minus3 or x = minus1The solution to the quadratic equation ax2 + bx + c = 0 can always be found (if it exists)using the quadratic formulax = minusb radicb2 minus 4ac2aNote that if b2 minus 4ac lt 0 then there is no solution to the equation If b2 minus 4ac = 0 thereis exactly one solution namely x = minusb2a If b2 minus 4ac gt 0 there are two solutions to theequationhttpwwweriktheredcomtutor pg 3SAT Subject Math Level 2 Facts amp FormulasTo solve two linear equations in x and y use the first equation to substitute for a variablein the second Eg suppose x+y = 3 and 4xminusy = 2 The first equation gives y = 3minusxso the second equation becomes 4x minus (3 minus x) = 2 rArr 5x minus 3 = 2 rArr x = 1 y = 2Solving two linear equations in x and y is geometrically the same as finding where two linesintersect In the example above the lines intersect at the point (1 2) Two parallel lineswill have no solution and two overlapping lines will have an infinite number of solutionsFunctionsA function is a rule to go from one number (x) to another number (y) usually writteny = f(x)The set of possible values of x is called the domain of f() and the corresponding set ofpossible values of y is called the range of f() For any given value of x there can only beone corresponding value yTranslationsThe graph of y = f(x minus h) + k is the translation of the graph ofy = f(x) by (h k) units in the plane

Absolute value|x| = _+x if x ge 0minusx if x lt 0|x| lt n rArr minusn lt x lt n|x| gt n rArr x lt minusn or x gt nParabolasA parabola parallel to the y-axis is given byy = ax2 + bx + cIf a gt 0 the parabola opens up If a lt 0 the parabola opens down The y-intercept is cand the x-coordinate of the vertex is x = minusb2aNote that when x = minusb2a the y-value of the parabola is either a minimum (a gt 0) or amaximum (a lt 0)EllipsesAn ellipse is essentially a squashed circle The equation of an ellipse centered on the originwhich intersects the x-axis at ( a 0) and the y-axis at (0b) isx2

a2 +y2

b2 = 1httpwwweriktheredcomtutor pg 4SAT Subject Math Level 2 Facts amp FormulasHyperbolas (Optional)A hyperbola looks like two elongated parabolas pointed away from one another Theequation of a hyperbola centered on the origin pointing down the positive and negativex-axes and which intersects the x-axis at ( a 0) isx2

a2 minusy2

b2 = 1Compound FunctionsA function can be applied directly to the y-value of another function This is usuallywritten with one function inside the parentheses of another function For examplef(g(x)) means apply g to x first then apply f to the resultg(f(x)) means apply f to x first then apply g to the resultf(x)g(x) means apply f to x first then apply g to x then multiply the resultsFor example if f(x) = 3xminus2 and g(x) = x2 then f(g(3)) = f(32) = f(9) = 3 9minus2 = 25Inverse FunctionsSince a function f() is a rule to go from one number (x) to another number (y) an inverse

function fminus1() can be defined as a rule to go from the number y back to the number x Inother words if y = f(x) then x = fminus1(y)To get the inverse function substitute y for f(x) solve for x in terms of y and substitutefminus1(y) for x For example if f(x) = 2x+ 6 then x = (y minus 6)2 so that fminus1(y) = y2minus 3Note that the function f() given x = 1 returns y = 8 and that fminus1(y) given y = 8returns x = 1Usually even inverse functions are written in terms of x so the final step is to substitutex for y In the above example this gives fminus1(x) = x2 minus 3 A quick recipe to find theinverse of f(x) is substitute y for f(x) interchange y and x solve for y and replace ywith fminus1(x)Two facts about inverse functions 1) their graphs are symmetric about the line y = xand 2) if one of the functions is a line with slope m the other is a line with slope 1mLogarithms (Optional)Logarithms are basically the inverse functions of exponentials The function logb x answersthe question b to what power gives x Here b is called the logarithmic ldquobaserdquo So ify = logb x then the logarithm function gives the number y such that by = x For examplelog3 radic27 = log3 radic33 = log3 332 = 32 = 15 Similarly logb bn = nhttpwwweriktheredcomtutor pg 5SAT Subject Math Level 2 Facts amp FormulasThe natural logarithm ln x is just the usual logarithm function but with base equal to thespecial number e (approximately 2718)A useful rule to know is logb xy = logb x + logb yComplex NumbersA complex number is of the form a + bi where i2 = minus1 When multiplying complexnumbers treat i just like any other variable (letter) except remember to replace powersof i with minus1 or 1 as follows (the pattern repeats after the first four)i0 = 1i4 = 1i1 = ii5 = ii2 = minus1i6 = minus1i3 = minusii7 = minusi

For example using ldquoFOILrdquo and i2 = minus1 (1 + 3i)(5 minus 2i) = 5 minus 2i + 15i minus 6i2 = 11 + 13iLines (Linear Functions)Consider the line that goes through points A(x1 y1) and B(x2 y2)Distance from A to B p(x2 minus x1)2 + (y2 minus y1)2

Mid-point of the segment AB _x1 + x2

2y1 + y2

2 _Slope of the liney2 minus y1

x2 minus x1

=riserunPoint-slope form given the slope m and a point (x1 y1) on the line the equation of theline is (y minus y1) = m(x minus x1)Slope-intercept form given the slope m and the y-intercept b then the equation of theline is y = mx + bTo find the equation of the line given two points A(x1 y1) and B(x2 y2) calculate theslope m = (y2 minus y1)(x2 minus x1) and use the point-slope formParallel lines have equal slopes Perpendicular lines (ie those that make a 90 anglewhere they intersect) have negative reciprocal slopes m1 m2 = minus1httpwwweriktheredcomtutor pg 6SAT Subject Math Level 2 Facts amp Formulasa

b

a

b

mla

b

b a

a

b

ab

Intersecting Lines Parallel Lines (l k m)Intersecting lines opposite angles are equal Also each pair of angles along the same lineadd to 180 In the figure above a + b = 180Parallel lines eight angles are formed when a line crosses two parallel lines The four bigangles (a) are equal and the four small angles (b) are equal

TrianglesRight trianglesabcxradic3x 2x30

60

xxxradic245

45

a2 + b2 = c2 Special Right TrianglesA good example of a right triangle is one with a = 3 b = 4 and c = 5 also called a 3ndash4ndash5right triangle Note that multiples of these numbers are also right triangles For exampleif you multiply these numbers by 2 you get a = 6 b = 8 and c = 10 (6ndash8ndash10) which isalso a right triangleAll triangleshbArea =12 b hhttpwwweriktheredcomtutor pg 7SAT Subject Math Level 2 Facts amp FormulasAngles on the inside of any triangle add up to 180The length of one side of any triangle is always less than the sum and more than thedifference of the lengths of the other two sidesAn exterior angle of any triangle is equal to the sum of the two remote interior anglesOther important trianglesEquilateral These triangles have three equal sides and all three angles are 60The area of an equilateral triangle is A = (side)2 radic34Isosceles An isosceles triangle has two equal sides The ldquobaserdquo angles(the ones opposite the two sides) are equal (see the 45 triangle above)Similar Two or more triangles are similar if they have the same shape Thecorresponding angles are equal and the corresponding sides are inproportion The ratio of their areas equals the ratio of the correspondingsides squared For example the 3ndash4ndash5 triangle and the 6ndash8ndash10triangle from before are similar since their sides are in a ratio of 2 to 1Trigonometry

Referring to the figure below there are three important functions which are defined forangles in a right triangleadjacentoppositehypotenuse_sin _ =oppositehypotenuseldquoSOHrdquocos _ =adjacenthypotenuseldquoCAHrdquotan _ =oppositeadjacentldquoTOArdquo(the last line above shows a mnemonic to remember these functions ldquoSOH-CAH-TOArdquo)An important relationship to remember which works for any angle _ issin2 _ + cos2 _ = 1For example if _ = 30 then (refer to the Special Right Triangles figure) we have sin 30 =12 cos 30 = radic32 so that sin2 30 + cos2 30 = 14 + 34 = 1httpwwweriktheredcomtutor pg 8SAT Subject Math Level 2 Facts amp FormulasThe functions y = a sin bx and y = a cos bx both have periods T = 360b ie they repeatevery 360b degrees The function y = a tan bx has a period of T = 180bThere are three other trig functions (cosecant secant and cotangent) that are related tothe usual three (sine cosine and tangent) as followscsc _ =hypotenuseopposite= 1 sin _sec _ =hypotenuseadjacent= 1 cos _cot _ =adjacentopposite= 1 tan _It is very useful to know the following trigonometric identitiessin(A + B) = sinAcosB + cosAsinBcos(A + B) = cosAcosB minus sinAsinB

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

You can use Reverse FOIL to factor a polynomial by thinking about two numbers a and bwhich add to the number in front of the x and which multiply to give the constant Forexample to factor x2 + 5x + 6 the numbers add to 5 and multiply to 6 ie a = 2 andb = 3 so that x2 + 5x + 6 = (x + 2)(x + 3)To solve a quadratic such as x2+bx+c = 0 first factor the left side to get (x+a)(x+b) = 0then set each part in parentheses equal to zero Eg x2 + 4x + 3 = (x + 3)(x + 1) = 0 sothat x = minus3 or x = minus1The solution to the quadratic equation ax2 + bx + c = 0 can always be found (if it exists)using the quadratic formulax = minusb radicb2 minus 4ac2aNote that if b2 minus 4ac lt 0 then there is no solution to the equation If b2 minus 4ac = 0 thereis exactly one solution namely x = minusb2a If b2 minus 4ac gt 0 there are two solutions to theequationhttpwwweriktheredcomtutor pg 3SAT Subject Math Level 2 Facts amp FormulasTo solve two linear equations in x and y use the first equation to substitute for a variablein the second Eg suppose x+y = 3 and 4xminusy = 2 The first equation gives y = 3minusxso the second equation becomes 4x minus (3 minus x) = 2 rArr 5x minus 3 = 2 rArr x = 1 y = 2Solving two linear equations in x and y is geometrically the same as finding where two linesintersect In the example above the lines intersect at the point (1 2) Two parallel lineswill have no solution and two overlapping lines will have an infinite number of solutionsFunctionsA function is a rule to go from one number (x) to another number (y) usually writteny = f(x)The set of possible values of x is called the domain of f() and the corresponding set ofpossible values of y is called the range of f() For any given value of x there can only beone corresponding value yTranslationsThe graph of y = f(x minus h) + k is the translation of the graph ofy = f(x) by (h k) units in the plane

Absolute value|x| = _+x if x ge 0minusx if x lt 0|x| lt n rArr minusn lt x lt n|x| gt n rArr x lt minusn or x gt nParabolasA parabola parallel to the y-axis is given byy = ax2 + bx + cIf a gt 0 the parabola opens up If a lt 0 the parabola opens down The y-intercept is cand the x-coordinate of the vertex is x = minusb2aNote that when x = minusb2a the y-value of the parabola is either a minimum (a gt 0) or amaximum (a lt 0)EllipsesAn ellipse is essentially a squashed circle The equation of an ellipse centered on the originwhich intersects the x-axis at ( a 0) and the y-axis at (0b) isx2

a2 +y2

b2 = 1httpwwweriktheredcomtutor pg 4SAT Subject Math Level 2 Facts amp FormulasHyperbolas (Optional)A hyperbola looks like two elongated parabolas pointed away from one another Theequation of a hyperbola centered on the origin pointing down the positive and negativex-axes and which intersects the x-axis at ( a 0) isx2

a2 minusy2

b2 = 1Compound FunctionsA function can be applied directly to the y-value of another function This is usuallywritten with one function inside the parentheses of another function For examplef(g(x)) means apply g to x first then apply f to the resultg(f(x)) means apply f to x first then apply g to the resultf(x)g(x) means apply f to x first then apply g to x then multiply the resultsFor example if f(x) = 3xminus2 and g(x) = x2 then f(g(3)) = f(32) = f(9) = 3 9minus2 = 25Inverse FunctionsSince a function f() is a rule to go from one number (x) to another number (y) an inverse

function fminus1() can be defined as a rule to go from the number y back to the number x Inother words if y = f(x) then x = fminus1(y)To get the inverse function substitute y for f(x) solve for x in terms of y and substitutefminus1(y) for x For example if f(x) = 2x+ 6 then x = (y minus 6)2 so that fminus1(y) = y2minus 3Note that the function f() given x = 1 returns y = 8 and that fminus1(y) given y = 8returns x = 1Usually even inverse functions are written in terms of x so the final step is to substitutex for y In the above example this gives fminus1(x) = x2 minus 3 A quick recipe to find theinverse of f(x) is substitute y for f(x) interchange y and x solve for y and replace ywith fminus1(x)Two facts about inverse functions 1) their graphs are symmetric about the line y = xand 2) if one of the functions is a line with slope m the other is a line with slope 1mLogarithms (Optional)Logarithms are basically the inverse functions of exponentials The function logb x answersthe question b to what power gives x Here b is called the logarithmic ldquobaserdquo So ify = logb x then the logarithm function gives the number y such that by = x For examplelog3 radic27 = log3 radic33 = log3 332 = 32 = 15 Similarly logb bn = nhttpwwweriktheredcomtutor pg 5SAT Subject Math Level 2 Facts amp FormulasThe natural logarithm ln x is just the usual logarithm function but with base equal to thespecial number e (approximately 2718)A useful rule to know is logb xy = logb x + logb yComplex NumbersA complex number is of the form a + bi where i2 = minus1 When multiplying complexnumbers treat i just like any other variable (letter) except remember to replace powersof i with minus1 or 1 as follows (the pattern repeats after the first four)i0 = 1i4 = 1i1 = ii5 = ii2 = minus1i6 = minus1i3 = minusii7 = minusi

For example using ldquoFOILrdquo and i2 = minus1 (1 + 3i)(5 minus 2i) = 5 minus 2i + 15i minus 6i2 = 11 + 13iLines (Linear Functions)Consider the line that goes through points A(x1 y1) and B(x2 y2)Distance from A to B p(x2 minus x1)2 + (y2 minus y1)2

Mid-point of the segment AB _x1 + x2

2y1 + y2

2 _Slope of the liney2 minus y1

x2 minus x1

=riserunPoint-slope form given the slope m and a point (x1 y1) on the line the equation of theline is (y minus y1) = m(x minus x1)Slope-intercept form given the slope m and the y-intercept b then the equation of theline is y = mx + bTo find the equation of the line given two points A(x1 y1) and B(x2 y2) calculate theslope m = (y2 minus y1)(x2 minus x1) and use the point-slope formParallel lines have equal slopes Perpendicular lines (ie those that make a 90 anglewhere they intersect) have negative reciprocal slopes m1 m2 = minus1httpwwweriktheredcomtutor pg 6SAT Subject Math Level 2 Facts amp Formulasa

b

a

b

mla

b

b a

a

b

ab

Intersecting Lines Parallel Lines (l k m)Intersecting lines opposite angles are equal Also each pair of angles along the same lineadd to 180 In the figure above a + b = 180Parallel lines eight angles are formed when a line crosses two parallel lines The four bigangles (a) are equal and the four small angles (b) are equal

TrianglesRight trianglesabcxradic3x 2x30

60

xxxradic245

45

a2 + b2 = c2 Special Right TrianglesA good example of a right triangle is one with a = 3 b = 4 and c = 5 also called a 3ndash4ndash5right triangle Note that multiples of these numbers are also right triangles For exampleif you multiply these numbers by 2 you get a = 6 b = 8 and c = 10 (6ndash8ndash10) which isalso a right triangleAll triangleshbArea =12 b hhttpwwweriktheredcomtutor pg 7SAT Subject Math Level 2 Facts amp FormulasAngles on the inside of any triangle add up to 180The length of one side of any triangle is always less than the sum and more than thedifference of the lengths of the other two sidesAn exterior angle of any triangle is equal to the sum of the two remote interior anglesOther important trianglesEquilateral These triangles have three equal sides and all three angles are 60The area of an equilateral triangle is A = (side)2 radic34Isosceles An isosceles triangle has two equal sides The ldquobaserdquo angles(the ones opposite the two sides) are equal (see the 45 triangle above)Similar Two or more triangles are similar if they have the same shape Thecorresponding angles are equal and the corresponding sides are inproportion The ratio of their areas equals the ratio of the correspondingsides squared For example the 3ndash4ndash5 triangle and the 6ndash8ndash10triangle from before are similar since their sides are in a ratio of 2 to 1Trigonometry

Referring to the figure below there are three important functions which are defined forangles in a right triangleadjacentoppositehypotenuse_sin _ =oppositehypotenuseldquoSOHrdquocos _ =adjacenthypotenuseldquoCAHrdquotan _ =oppositeadjacentldquoTOArdquo(the last line above shows a mnemonic to remember these functions ldquoSOH-CAH-TOArdquo)An important relationship to remember which works for any angle _ issin2 _ + cos2 _ = 1For example if _ = 30 then (refer to the Special Right Triangles figure) we have sin 30 =12 cos 30 = radic32 so that sin2 30 + cos2 30 = 14 + 34 = 1httpwwweriktheredcomtutor pg 8SAT Subject Math Level 2 Facts amp FormulasThe functions y = a sin bx and y = a cos bx both have periods T = 360b ie they repeatevery 360b degrees The function y = a tan bx has a period of T = 180bThere are three other trig functions (cosecant secant and cotangent) that are related tothe usual three (sine cosine and tangent) as followscsc _ =hypotenuseopposite= 1 sin _sec _ =hypotenuseadjacent= 1 cos _cot _ =adjacentopposite= 1 tan _It is very useful to know the following trigonometric identitiessin(A + B) = sinAcosB + cosAsinBcos(A + B) = cosAcosB minus sinAsinB

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

Absolute value|x| = _+x if x ge 0minusx if x lt 0|x| lt n rArr minusn lt x lt n|x| gt n rArr x lt minusn or x gt nParabolasA parabola parallel to the y-axis is given byy = ax2 + bx + cIf a gt 0 the parabola opens up If a lt 0 the parabola opens down The y-intercept is cand the x-coordinate of the vertex is x = minusb2aNote that when x = minusb2a the y-value of the parabola is either a minimum (a gt 0) or amaximum (a lt 0)EllipsesAn ellipse is essentially a squashed circle The equation of an ellipse centered on the originwhich intersects the x-axis at ( a 0) and the y-axis at (0b) isx2

a2 +y2

b2 = 1httpwwweriktheredcomtutor pg 4SAT Subject Math Level 2 Facts amp FormulasHyperbolas (Optional)A hyperbola looks like two elongated parabolas pointed away from one another Theequation of a hyperbola centered on the origin pointing down the positive and negativex-axes and which intersects the x-axis at ( a 0) isx2

a2 minusy2

b2 = 1Compound FunctionsA function can be applied directly to the y-value of another function This is usuallywritten with one function inside the parentheses of another function For examplef(g(x)) means apply g to x first then apply f to the resultg(f(x)) means apply f to x first then apply g to the resultf(x)g(x) means apply f to x first then apply g to x then multiply the resultsFor example if f(x) = 3xminus2 and g(x) = x2 then f(g(3)) = f(32) = f(9) = 3 9minus2 = 25Inverse FunctionsSince a function f() is a rule to go from one number (x) to another number (y) an inverse

function fminus1() can be defined as a rule to go from the number y back to the number x Inother words if y = f(x) then x = fminus1(y)To get the inverse function substitute y for f(x) solve for x in terms of y and substitutefminus1(y) for x For example if f(x) = 2x+ 6 then x = (y minus 6)2 so that fminus1(y) = y2minus 3Note that the function f() given x = 1 returns y = 8 and that fminus1(y) given y = 8returns x = 1Usually even inverse functions are written in terms of x so the final step is to substitutex for y In the above example this gives fminus1(x) = x2 minus 3 A quick recipe to find theinverse of f(x) is substitute y for f(x) interchange y and x solve for y and replace ywith fminus1(x)Two facts about inverse functions 1) their graphs are symmetric about the line y = xand 2) if one of the functions is a line with slope m the other is a line with slope 1mLogarithms (Optional)Logarithms are basically the inverse functions of exponentials The function logb x answersthe question b to what power gives x Here b is called the logarithmic ldquobaserdquo So ify = logb x then the logarithm function gives the number y such that by = x For examplelog3 radic27 = log3 radic33 = log3 332 = 32 = 15 Similarly logb bn = nhttpwwweriktheredcomtutor pg 5SAT Subject Math Level 2 Facts amp FormulasThe natural logarithm ln x is just the usual logarithm function but with base equal to thespecial number e (approximately 2718)A useful rule to know is logb xy = logb x + logb yComplex NumbersA complex number is of the form a + bi where i2 = minus1 When multiplying complexnumbers treat i just like any other variable (letter) except remember to replace powersof i with minus1 or 1 as follows (the pattern repeats after the first four)i0 = 1i4 = 1i1 = ii5 = ii2 = minus1i6 = minus1i3 = minusii7 = minusi

For example using ldquoFOILrdquo and i2 = minus1 (1 + 3i)(5 minus 2i) = 5 minus 2i + 15i minus 6i2 = 11 + 13iLines (Linear Functions)Consider the line that goes through points A(x1 y1) and B(x2 y2)Distance from A to B p(x2 minus x1)2 + (y2 minus y1)2

Mid-point of the segment AB _x1 + x2

2y1 + y2

2 _Slope of the liney2 minus y1

x2 minus x1

=riserunPoint-slope form given the slope m and a point (x1 y1) on the line the equation of theline is (y minus y1) = m(x minus x1)Slope-intercept form given the slope m and the y-intercept b then the equation of theline is y = mx + bTo find the equation of the line given two points A(x1 y1) and B(x2 y2) calculate theslope m = (y2 minus y1)(x2 minus x1) and use the point-slope formParallel lines have equal slopes Perpendicular lines (ie those that make a 90 anglewhere they intersect) have negative reciprocal slopes m1 m2 = minus1httpwwweriktheredcomtutor pg 6SAT Subject Math Level 2 Facts amp Formulasa

b

a

b

mla

b

b a

a

b

ab

Intersecting Lines Parallel Lines (l k m)Intersecting lines opposite angles are equal Also each pair of angles along the same lineadd to 180 In the figure above a + b = 180Parallel lines eight angles are formed when a line crosses two parallel lines The four bigangles (a) are equal and the four small angles (b) are equal

TrianglesRight trianglesabcxradic3x 2x30

60

xxxradic245

45

a2 + b2 = c2 Special Right TrianglesA good example of a right triangle is one with a = 3 b = 4 and c = 5 also called a 3ndash4ndash5right triangle Note that multiples of these numbers are also right triangles For exampleif you multiply these numbers by 2 you get a = 6 b = 8 and c = 10 (6ndash8ndash10) which isalso a right triangleAll triangleshbArea =12 b hhttpwwweriktheredcomtutor pg 7SAT Subject Math Level 2 Facts amp FormulasAngles on the inside of any triangle add up to 180The length of one side of any triangle is always less than the sum and more than thedifference of the lengths of the other two sidesAn exterior angle of any triangle is equal to the sum of the two remote interior anglesOther important trianglesEquilateral These triangles have three equal sides and all three angles are 60The area of an equilateral triangle is A = (side)2 radic34Isosceles An isosceles triangle has two equal sides The ldquobaserdquo angles(the ones opposite the two sides) are equal (see the 45 triangle above)Similar Two or more triangles are similar if they have the same shape Thecorresponding angles are equal and the corresponding sides are inproportion The ratio of their areas equals the ratio of the correspondingsides squared For example the 3ndash4ndash5 triangle and the 6ndash8ndash10triangle from before are similar since their sides are in a ratio of 2 to 1Trigonometry

Referring to the figure below there are three important functions which are defined forangles in a right triangleadjacentoppositehypotenuse_sin _ =oppositehypotenuseldquoSOHrdquocos _ =adjacenthypotenuseldquoCAHrdquotan _ =oppositeadjacentldquoTOArdquo(the last line above shows a mnemonic to remember these functions ldquoSOH-CAH-TOArdquo)An important relationship to remember which works for any angle _ issin2 _ + cos2 _ = 1For example if _ = 30 then (refer to the Special Right Triangles figure) we have sin 30 =12 cos 30 = radic32 so that sin2 30 + cos2 30 = 14 + 34 = 1httpwwweriktheredcomtutor pg 8SAT Subject Math Level 2 Facts amp FormulasThe functions y = a sin bx and y = a cos bx both have periods T = 360b ie they repeatevery 360b degrees The function y = a tan bx has a period of T = 180bThere are three other trig functions (cosecant secant and cotangent) that are related tothe usual three (sine cosine and tangent) as followscsc _ =hypotenuseopposite= 1 sin _sec _ =hypotenuseadjacent= 1 cos _cot _ =adjacentopposite= 1 tan _It is very useful to know the following trigonometric identitiessin(A + B) = sinAcosB + cosAsinBcos(A + B) = cosAcosB minus sinAsinB

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

function fminus1() can be defined as a rule to go from the number y back to the number x Inother words if y = f(x) then x = fminus1(y)To get the inverse function substitute y for f(x) solve for x in terms of y and substitutefminus1(y) for x For example if f(x) = 2x+ 6 then x = (y minus 6)2 so that fminus1(y) = y2minus 3Note that the function f() given x = 1 returns y = 8 and that fminus1(y) given y = 8returns x = 1Usually even inverse functions are written in terms of x so the final step is to substitutex for y In the above example this gives fminus1(x) = x2 minus 3 A quick recipe to find theinverse of f(x) is substitute y for f(x) interchange y and x solve for y and replace ywith fminus1(x)Two facts about inverse functions 1) their graphs are symmetric about the line y = xand 2) if one of the functions is a line with slope m the other is a line with slope 1mLogarithms (Optional)Logarithms are basically the inverse functions of exponentials The function logb x answersthe question b to what power gives x Here b is called the logarithmic ldquobaserdquo So ify = logb x then the logarithm function gives the number y such that by = x For examplelog3 radic27 = log3 radic33 = log3 332 = 32 = 15 Similarly logb bn = nhttpwwweriktheredcomtutor pg 5SAT Subject Math Level 2 Facts amp FormulasThe natural logarithm ln x is just the usual logarithm function but with base equal to thespecial number e (approximately 2718)A useful rule to know is logb xy = logb x + logb yComplex NumbersA complex number is of the form a + bi where i2 = minus1 When multiplying complexnumbers treat i just like any other variable (letter) except remember to replace powersof i with minus1 or 1 as follows (the pattern repeats after the first four)i0 = 1i4 = 1i1 = ii5 = ii2 = minus1i6 = minus1i3 = minusii7 = minusi

For example using ldquoFOILrdquo and i2 = minus1 (1 + 3i)(5 minus 2i) = 5 minus 2i + 15i minus 6i2 = 11 + 13iLines (Linear Functions)Consider the line that goes through points A(x1 y1) and B(x2 y2)Distance from A to B p(x2 minus x1)2 + (y2 minus y1)2

Mid-point of the segment AB _x1 + x2

2y1 + y2

2 _Slope of the liney2 minus y1

x2 minus x1

=riserunPoint-slope form given the slope m and a point (x1 y1) on the line the equation of theline is (y minus y1) = m(x minus x1)Slope-intercept form given the slope m and the y-intercept b then the equation of theline is y = mx + bTo find the equation of the line given two points A(x1 y1) and B(x2 y2) calculate theslope m = (y2 minus y1)(x2 minus x1) and use the point-slope formParallel lines have equal slopes Perpendicular lines (ie those that make a 90 anglewhere they intersect) have negative reciprocal slopes m1 m2 = minus1httpwwweriktheredcomtutor pg 6SAT Subject Math Level 2 Facts amp Formulasa

b

a

b

mla

b

b a

a

b

ab

Intersecting Lines Parallel Lines (l k m)Intersecting lines opposite angles are equal Also each pair of angles along the same lineadd to 180 In the figure above a + b = 180Parallel lines eight angles are formed when a line crosses two parallel lines The four bigangles (a) are equal and the four small angles (b) are equal

TrianglesRight trianglesabcxradic3x 2x30

60

xxxradic245

45

a2 + b2 = c2 Special Right TrianglesA good example of a right triangle is one with a = 3 b = 4 and c = 5 also called a 3ndash4ndash5right triangle Note that multiples of these numbers are also right triangles For exampleif you multiply these numbers by 2 you get a = 6 b = 8 and c = 10 (6ndash8ndash10) which isalso a right triangleAll triangleshbArea =12 b hhttpwwweriktheredcomtutor pg 7SAT Subject Math Level 2 Facts amp FormulasAngles on the inside of any triangle add up to 180The length of one side of any triangle is always less than the sum and more than thedifference of the lengths of the other two sidesAn exterior angle of any triangle is equal to the sum of the two remote interior anglesOther important trianglesEquilateral These triangles have three equal sides and all three angles are 60The area of an equilateral triangle is A = (side)2 radic34Isosceles An isosceles triangle has two equal sides The ldquobaserdquo angles(the ones opposite the two sides) are equal (see the 45 triangle above)Similar Two or more triangles are similar if they have the same shape Thecorresponding angles are equal and the corresponding sides are inproportion The ratio of their areas equals the ratio of the correspondingsides squared For example the 3ndash4ndash5 triangle and the 6ndash8ndash10triangle from before are similar since their sides are in a ratio of 2 to 1Trigonometry

Referring to the figure below there are three important functions which are defined forangles in a right triangleadjacentoppositehypotenuse_sin _ =oppositehypotenuseldquoSOHrdquocos _ =adjacenthypotenuseldquoCAHrdquotan _ =oppositeadjacentldquoTOArdquo(the last line above shows a mnemonic to remember these functions ldquoSOH-CAH-TOArdquo)An important relationship to remember which works for any angle _ issin2 _ + cos2 _ = 1For example if _ = 30 then (refer to the Special Right Triangles figure) we have sin 30 =12 cos 30 = radic32 so that sin2 30 + cos2 30 = 14 + 34 = 1httpwwweriktheredcomtutor pg 8SAT Subject Math Level 2 Facts amp FormulasThe functions y = a sin bx and y = a cos bx both have periods T = 360b ie they repeatevery 360b degrees The function y = a tan bx has a period of T = 180bThere are three other trig functions (cosecant secant and cotangent) that are related tothe usual three (sine cosine and tangent) as followscsc _ =hypotenuseopposite= 1 sin _sec _ =hypotenuseadjacent= 1 cos _cot _ =adjacentopposite= 1 tan _It is very useful to know the following trigonometric identitiessin(A + B) = sinAcosB + cosAsinBcos(A + B) = cosAcosB minus sinAsinB

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

For example using ldquoFOILrdquo and i2 = minus1 (1 + 3i)(5 minus 2i) = 5 minus 2i + 15i minus 6i2 = 11 + 13iLines (Linear Functions)Consider the line that goes through points A(x1 y1) and B(x2 y2)Distance from A to B p(x2 minus x1)2 + (y2 minus y1)2

Mid-point of the segment AB _x1 + x2

2y1 + y2

2 _Slope of the liney2 minus y1

x2 minus x1

=riserunPoint-slope form given the slope m and a point (x1 y1) on the line the equation of theline is (y minus y1) = m(x minus x1)Slope-intercept form given the slope m and the y-intercept b then the equation of theline is y = mx + bTo find the equation of the line given two points A(x1 y1) and B(x2 y2) calculate theslope m = (y2 minus y1)(x2 minus x1) and use the point-slope formParallel lines have equal slopes Perpendicular lines (ie those that make a 90 anglewhere they intersect) have negative reciprocal slopes m1 m2 = minus1httpwwweriktheredcomtutor pg 6SAT Subject Math Level 2 Facts amp Formulasa

b

a

b

mla

b

b a

a

b

ab

Intersecting Lines Parallel Lines (l k m)Intersecting lines opposite angles are equal Also each pair of angles along the same lineadd to 180 In the figure above a + b = 180Parallel lines eight angles are formed when a line crosses two parallel lines The four bigangles (a) are equal and the four small angles (b) are equal

TrianglesRight trianglesabcxradic3x 2x30

60

xxxradic245

45

a2 + b2 = c2 Special Right TrianglesA good example of a right triangle is one with a = 3 b = 4 and c = 5 also called a 3ndash4ndash5right triangle Note that multiples of these numbers are also right triangles For exampleif you multiply these numbers by 2 you get a = 6 b = 8 and c = 10 (6ndash8ndash10) which isalso a right triangleAll triangleshbArea =12 b hhttpwwweriktheredcomtutor pg 7SAT Subject Math Level 2 Facts amp FormulasAngles on the inside of any triangle add up to 180The length of one side of any triangle is always less than the sum and more than thedifference of the lengths of the other two sidesAn exterior angle of any triangle is equal to the sum of the two remote interior anglesOther important trianglesEquilateral These triangles have three equal sides and all three angles are 60The area of an equilateral triangle is A = (side)2 radic34Isosceles An isosceles triangle has two equal sides The ldquobaserdquo angles(the ones opposite the two sides) are equal (see the 45 triangle above)Similar Two or more triangles are similar if they have the same shape Thecorresponding angles are equal and the corresponding sides are inproportion The ratio of their areas equals the ratio of the correspondingsides squared For example the 3ndash4ndash5 triangle and the 6ndash8ndash10triangle from before are similar since their sides are in a ratio of 2 to 1Trigonometry

Referring to the figure below there are three important functions which are defined forangles in a right triangleadjacentoppositehypotenuse_sin _ =oppositehypotenuseldquoSOHrdquocos _ =adjacenthypotenuseldquoCAHrdquotan _ =oppositeadjacentldquoTOArdquo(the last line above shows a mnemonic to remember these functions ldquoSOH-CAH-TOArdquo)An important relationship to remember which works for any angle _ issin2 _ + cos2 _ = 1For example if _ = 30 then (refer to the Special Right Triangles figure) we have sin 30 =12 cos 30 = radic32 so that sin2 30 + cos2 30 = 14 + 34 = 1httpwwweriktheredcomtutor pg 8SAT Subject Math Level 2 Facts amp FormulasThe functions y = a sin bx and y = a cos bx both have periods T = 360b ie they repeatevery 360b degrees The function y = a tan bx has a period of T = 180bThere are three other trig functions (cosecant secant and cotangent) that are related tothe usual three (sine cosine and tangent) as followscsc _ =hypotenuseopposite= 1 sin _sec _ =hypotenuseadjacent= 1 cos _cot _ =adjacentopposite= 1 tan _It is very useful to know the following trigonometric identitiessin(A + B) = sinAcosB + cosAsinBcos(A + B) = cosAcosB minus sinAsinB

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

TrianglesRight trianglesabcxradic3x 2x30

60

xxxradic245

45

a2 + b2 = c2 Special Right TrianglesA good example of a right triangle is one with a = 3 b = 4 and c = 5 also called a 3ndash4ndash5right triangle Note that multiples of these numbers are also right triangles For exampleif you multiply these numbers by 2 you get a = 6 b = 8 and c = 10 (6ndash8ndash10) which isalso a right triangleAll triangleshbArea =12 b hhttpwwweriktheredcomtutor pg 7SAT Subject Math Level 2 Facts amp FormulasAngles on the inside of any triangle add up to 180The length of one side of any triangle is always less than the sum and more than thedifference of the lengths of the other two sidesAn exterior angle of any triangle is equal to the sum of the two remote interior anglesOther important trianglesEquilateral These triangles have three equal sides and all three angles are 60The area of an equilateral triangle is A = (side)2 radic34Isosceles An isosceles triangle has two equal sides The ldquobaserdquo angles(the ones opposite the two sides) are equal (see the 45 triangle above)Similar Two or more triangles are similar if they have the same shape Thecorresponding angles are equal and the corresponding sides are inproportion The ratio of their areas equals the ratio of the correspondingsides squared For example the 3ndash4ndash5 triangle and the 6ndash8ndash10triangle from before are similar since their sides are in a ratio of 2 to 1Trigonometry

Referring to the figure below there are three important functions which are defined forangles in a right triangleadjacentoppositehypotenuse_sin _ =oppositehypotenuseldquoSOHrdquocos _ =adjacenthypotenuseldquoCAHrdquotan _ =oppositeadjacentldquoTOArdquo(the last line above shows a mnemonic to remember these functions ldquoSOH-CAH-TOArdquo)An important relationship to remember which works for any angle _ issin2 _ + cos2 _ = 1For example if _ = 30 then (refer to the Special Right Triangles figure) we have sin 30 =12 cos 30 = radic32 so that sin2 30 + cos2 30 = 14 + 34 = 1httpwwweriktheredcomtutor pg 8SAT Subject Math Level 2 Facts amp FormulasThe functions y = a sin bx and y = a cos bx both have periods T = 360b ie they repeatevery 360b degrees The function y = a tan bx has a period of T = 180bThere are three other trig functions (cosecant secant and cotangent) that are related tothe usual three (sine cosine and tangent) as followscsc _ =hypotenuseopposite= 1 sin _sec _ =hypotenuseadjacent= 1 cos _cot _ =adjacentopposite= 1 tan _It is very useful to know the following trigonometric identitiessin(A + B) = sinAcosB + cosAsinBcos(A + B) = cosAcosB minus sinAsinB

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

Referring to the figure below there are three important functions which are defined forangles in a right triangleadjacentoppositehypotenuse_sin _ =oppositehypotenuseldquoSOHrdquocos _ =adjacenthypotenuseldquoCAHrdquotan _ =oppositeadjacentldquoTOArdquo(the last line above shows a mnemonic to remember these functions ldquoSOH-CAH-TOArdquo)An important relationship to remember which works for any angle _ issin2 _ + cos2 _ = 1For example if _ = 30 then (refer to the Special Right Triangles figure) we have sin 30 =12 cos 30 = radic32 so that sin2 30 + cos2 30 = 14 + 34 = 1httpwwweriktheredcomtutor pg 8SAT Subject Math Level 2 Facts amp FormulasThe functions y = a sin bx and y = a cos bx both have periods T = 360b ie they repeatevery 360b degrees The function y = a tan bx has a period of T = 180bThere are three other trig functions (cosecant secant and cotangent) that are related tothe usual three (sine cosine and tangent) as followscsc _ =hypotenuseopposite= 1 sin _sec _ =hypotenuseadjacent= 1 cos _cot _ =adjacentopposite= 1 tan _It is very useful to know the following trigonometric identitiessin(A + B) = sinAcosB + cosAsinBcos(A + B) = cosAcosB minus sinAsinB

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

tan(A + B) =tanA + tanB1 minus tanAtanBThe following double-angle and half-angle trigonometric identities are optional to memorizebut they can be easy shortcuts to solve problems if you are good at memorizationDouble-Angle (Optional)sin 2_ = 2 sin _ cos _cos 2_ = cos2 _ minus sin2 _tan 2_ =2 tan _1 minus tan2 _Half-Angle (Optional)sin_2= r1 minus cos _2cos_2= r1 + cos _2tan_2= r1 minus cos _1 + cos _For any triangle with angles A B and C and sides a b and c (opposite to the angles)two important laws to remember are the Law of Cosines and the Law of SinesasinA=bsinB=csinCldquoLaw of Sinesrdquoc2 = a2 + b2 minus 2ab cosC ldquoLaw of Cosinesrdquohttpwwweriktheredcomtutor pg 9SAT Subject Math Level 2 Facts amp FormulasCircles(h k)rn

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

rArcSectorArea = _r2

Circumference = 2_rFull circle = 360

Length Of Arc = (n360) 2_rArea Of Sector = (n360) _r2

Equation of the circle (above left figure) (x minus h)2 + (y minus k)2 = r2Another way to measure angles is with radians These are defined such that _ radians isequal to 180 so that the number of radians in a circle is 2_ (or 360)To convert from degrees to radians just multiply by _180 For example the number ofradians in 45 is 0785 since 45 _180 = _4 rad asymp 0785 radRectangles And FriendsRectangles and Parallelogramslw hlwRectangle Parallelogram(Square if l = w) (Rhombus if l = w)Area = lw Area = lhTrapezoidshbase1

base2

Area of trapezoid = _base1 + base2

2 _ hhttpwwweriktheredcomtutor pg 10SAT Subject Math Level 2 Facts amp FormulasPolygonsRegular polygons are n-sided figures with all sides equal and all angles equalThe sum of the inside angles of an n-sided regular polygon is (n minus 2) 180The sum of the outside angles of an n-sided regular polygon is always 360SolidsThe following five formulas for cones spheres and pyramids are given in the beginning ofthe test booklet so you donrsquot have to memorize them but you should know how to usethemVolume of right circular cone with radius r and height h V =13_r2h

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11

Lateral area of cone with base circumference c and slant height l S =12clVolume of sphere with radius r V =43_r3

Surface Area of sphere with radius r S = 4_r2

Volume of pyramid with base area B and height h V =13BhYou should know the volume formulas for the solids below The area of the rectangularsolid is just the sum of the areas of its faces The area of the cylinder is the area of thecircles on top and bottom (2_r2) plus the area of the sides (2_rh)lwhdrhRectangular Solid Right CylinderVolume = lwhArea = 2(lw + wh + lh)Volume = _r2hArea = 2_r(r + h)The distance between opposite corners of a rectangular solid is d = radicl2 + w2 + h2The volume of a uniform solid is V = (base area) heighthttpwwweriktheredcomtutor pg 11