Lone Star College-CyFair Formula Sheet

Lone Star College-CyFair Formula SheetThe following formulas are critical for success in the indicated course. Student CANNOTbring these formulas on a formula sheet or card to tests and instructors MUST NOTprovide them during the test either on the board or on a handout. They MUST bememorized.Math 1314 College AlgebraFORMULAS/EQUATIONSDistance FormulaIf 𝑃1 (𝑥1 , 𝑦1 ) and 𝑃2 (𝑥2 , 𝑦2 ), the distance from 𝑃1 to 𝑃2 is𝑑(𝑃1 , 𝑃2 ) (𝑥2 𝑥1 )2 (𝑦2 𝑦1 )2Standard EquationOf a CircleThe standard equation of a circle of radius 𝑟 with center at (ℎ, 𝑘) is(𝑥 ℎ)2 (𝑦 𝑘)2 𝑟 2Slope FormulaThe slope 𝑚 of the line containing the points 𝑃1 (𝑥1 , 𝑦1 ) and 𝑃2 (𝑥2 , 𝑦2 ) is𝑦2 𝑦1𝑚 if 𝑥1 𝑥2𝑥2 𝑥1𝑚 is undefined if 𝑥1 𝑥2Point-slopeEquation of a LineThe equation of a line with slope 𝑚 containing the points (𝑥1 , 𝑦1 ) is𝑦 𝑦1 𝑚(𝑥 𝑥1 )Slope-Intercept Equation of The equation of a line with slope 𝑚 and 𝑦-intercept 𝑏 isa Line𝑦 𝑚𝑥 𝑏Quadratic FormulaThe solutions of the equation 𝑎𝑥 2 𝑏𝑥 𝑐 0, 𝑎 0, are 𝑏 𝑏 2 4𝑎𝑐𝑥 2𝑎

LIBRARY OF FUNCTIONSConstant FunctionIdentity FunctionSquare FunctionCube Function2𝑓(𝑥) 𝑥 3𝑓(𝑥) 𝑏𝑓(𝑥) 𝑥𝑓(𝑥) 𝑥Reciprocal FunctionSquared ReciprocalFunctionSquare Root FunctionCube Root Function𝑓(𝑥) 𝑥𝑓(𝑥) 𝑥Natural LogarithmFunctionGreatest IntegerFunction𝑓(𝑥) ln 𝑥𝑓(𝑥) ⟦𝑥⟧1𝑓(𝑥) 𝑥Absolute Function𝑓(𝑥) 𝑥 𝑓(𝑥) 1𝑥2Exponential Function𝑓(𝑥) 𝑒𝑥𝒚 𝒇(𝒙) 𝒆𝒙3𝒚𝒚𝒇(𝒙) 𝐥𝐧𝒙𝒇(𝒙) ⟦𝒙⟧(𝟏, 𝒆)𝟏൬ 𝟏, ൰𝒆(𝒆, 𝟏)(𝟎, 𝟏)𝒙(𝟏, 𝟎)𝒙𝟏൬ , 𝟏൰𝒆𝒙

GEOMETRY FOMULAS𝑟 Radius, 𝐴 Area, 𝐶 Circumference𝑨 𝝅𝒓𝟐rCircle𝑪 𝟐𝝅𝒓𝑏 Base, ℎ Altitude(Height), 𝐴 AreahTriangle𝑨 b𝑤Rectangle𝑙𝟏𝒃𝒉𝟐𝑙 Length, 𝑤 Width, 𝐴 Area, 𝑃 Perimeter𝑨 𝒍𝒘𝑷 𝟐𝒍 𝟐𝒘𝑙 Length, 𝑤 Width, ℎ Height,𝑉 Volume, 𝑆 Surface AreaRectangularBox𝑽 𝒍𝒘𝒉𝑺 𝟐𝒍𝒘 𝟐𝒍𝒉 𝟐𝒘𝒉PROPERTIES OF LOGARITHMS log 𝑎 (𝑀𝑁) log 𝑎 𝑀 log 𝑎 𝑁 log 𝑎 ( 𝑁 ) log 𝑎 𝑀 log 𝑎 𝑁 log 𝑎 𝑀𝑟 𝑟 log 𝑎 𝑀 log 𝑎 𝑀 𝑎 𝑥 𝑒 𝑥 ln 𝑎𝑀log 𝑀log 𝑎 ln 𝑀ln 𝑎

Math 1316 TrigonometryStudents in Trigonometry should know all the formulas from Math 1314 College Algebraplus the following.Unit Circle 𝒙𝟐 𝒚𝟐 𝟏

TRIGONOMETRIC FUNCTIONSOf an Acute Anglesin 𝜃 cos 𝜃 𝑏𝑐𝑎Opposite𝑐 HypotenuseAdjacent𝑐 Hypotenuse𝑏Oppositetan 𝜃 𝑎 csc 𝜃 𝑏 𝑐sec 𝜃 𝑎 t 𝜃 𝑏 OppositeAdjacentOf a General Anglesin 𝜃 𝒚𝑏𝑟csc 𝜃 𝑏 , 𝑏 0𝑟cos 𝜃 𝑎𝒓 𝒂𝟐 𝒃𝟐𝑟sec 𝜃 𝑎 , 𝑎 0𝑟𝑏(𝒂, 𝒃)𝒙𝑎tan 𝜃 𝑎 , 𝑎 0cot 𝜃 𝑏 , 𝑏 0APPLICATIONS𝒔Arc Length: 𝑠 𝑟𝜃, 𝜃 in radians1Area of Sector: 𝐴 𝑟 2 𝜃, 𝜃 in radians2𝜃Angular Speed: 𝜔 , 𝜃 in radians𝑡𝑠Linear Speed: 𝑣 , 𝑣 𝜔𝑟𝑡SOLVING TRIANGLES𝑩Law of Sine:𝐜sin 𝐴𝑎 sin 𝐵𝑏 sin 𝐶𝑐𝒂Law of Cosine: 𝑎2 𝑏 2 𝑐 2 2𝑏𝑐 cos 𝐴𝑨𝑪b𝜽𝑏 2 𝑎2 𝑐 2 2𝑎𝑐 cos 𝐴𝑐 2 𝑎2 𝑏 2 2𝑎𝑏 cos 𝐴

TRIGONOMETRIC IDENTITIESFundamental Identitiessin 𝜃tan𝜃 cos 𝜃cot 𝜃 1cos 𝜃sin 𝜃1csc 𝜃 sin 𝜃sec 𝜃 cos 𝜃1cot 𝜃 tan 𝜃sin2 𝜃 cos 2 𝜃 11 tan2 𝜃 sec 2 𝜃1 cot 2 𝜃 csc 2 𝜃Even-Odd IdentitiesCofunction Identitiessin( 𝜃) sin 𝜃csc( 𝜃) csc 𝜃cos(90 𝜃) sin 𝜃cos( 𝜃) cos 𝜃sec( 𝜃) sec 𝜃sin(90 𝜃) cos 𝜃tan( 𝜃) tan 𝜃cot( 𝜃) cot 𝜃tan(90 𝜃) tan 𝜃Sum and Difference FormulasDouble-Angle Formulassin(𝛼 𝛽) sin 𝛼 cos 𝛽 cos 𝛼 sin 𝛽sin(2𝜃) 2 sin 𝜃 cos 𝜃sin(𝛼 𝛽) sin 𝛼 cos 𝛽 cos 𝛼 sin 𝛽cos(2𝜃) cos2 𝜃 sin2 𝜃cos(𝛼 𝛽) cos 𝛼 cos 𝛽 sin 𝛼 sin 𝛽 2 cos2 𝜃 1cos(𝛼 𝛽) cos 𝛼 cos 𝛽 sin 𝛼 sin 𝛽 1 2sin2 𝜃tan(𝛼 𝛽) tan 𝛼 tan 𝛽1 tan 𝛼 tan 𝛽tan(𝛼 𝛽) tan 𝛼 tan 𝛽1 tan 𝛼 tan 𝛽tan(2𝜃) 2 tan 𝜃1 tan2 𝜃

LIBRARY OF TRIGONOMETRIC FUNCTIONSSine Function𝑓(𝑥) sin 𝑥Cosine Function𝑓(𝑥) cos 𝑥Tangent Function𝑓(𝑥) tan 𝑥Secant Function𝑓(𝑥) sec 𝑥Cosecant Function𝑓(𝑥) csc 𝑥Cotangent Function𝑓(𝑥) cot 𝑥

Math 2412 PrecalculusStudents in Precalculus should know all the formulas from Math 1314 College Algebra andMath 1316 Trigonometry plus the following.Half Angle Formulassin 𝜃 1 cos 2𝜃2cos 𝜃 1 cos 2𝜃2tan 𝜃 1 cos 2𝜃sin 2𝜃Products and Quotients of Complex Numbers in Polar FormLet 𝑧1 𝑟1 (cos 𝜃1 𝑖 sin 𝜃1 ) and 𝑧2 𝑟2 (cos 𝜃2 𝑖 sin 𝜃2 ).Then 𝑧1 𝑧2 𝑟1 𝑟2 [cos(𝜃1 𝜃2 ) 𝑖 sin(𝜃1 𝜃2 )]and, if 𝑧2 0,𝑧1𝑧2 𝑟1𝑟2[cos(𝜃1 𝜃2 ) 𝑖 sin(𝜃1 𝜃2 )].DeMoivre’s TheoremIf 𝑧 𝑟(cos 𝜃 𝑖 sin 𝜃) and n is a positive integer,𝑧 𝑛 𝑟 𝑛 [cos(𝑛𝜃) 𝑖 sin(𝑛𝜃)].Complex RootsLet 𝑤 𝑟(cos 𝜃0 𝑖 sin 𝜃0 ) be a complex number and let n 2be an integer. If 𝑤 0,there are n distinct complex nth roots of 𝑤, given by the formula𝑛𝑧𝑘 𝑟 [cos ൬Where 𝑘 0, 1, 2, , 𝑛 1.𝜃0 2𝑘𝜋𝜃0 2𝑘𝜋 ൰ 𝑖 sin ൬ ൰].𝑛𝑛𝑛𝑛

CONICS𝒙𝟐 𝒚𝟐 𝟏(𝟎, 𝒓)Circle( 𝒓, 𝟎)(𝒓, 𝟎)(𝟎, 𝒓)𝒚𝟐 𝟒𝒂𝒙𝒚𝟐 𝟒𝒂𝒙𝑫: 𝒙 𝒂𝒙𝟐 𝟒𝒂𝒚𝒙𝟐 𝟒𝒂𝒚𝑫: 𝒙 𝒂𝑫: 𝒚 𝒂Parabola𝑭( 𝒂, 𝟎)𝑽𝑭(𝟎, 𝒂)𝑽𝑽𝑭(𝒂, 𝟎)𝑭(𝟎, 𝒂)𝑽𝑫: 𝒚 𝒂𝒙𝟐𝒂𝟐𝒚𝟐 𝒃𝟐 𝟏, 𝑎 𝑏, 𝑐 2 𝑎2 𝑏 2𝒙𝟐𝒃𝟐𝒚𝟐 𝒂𝟐 𝟏, 𝑎 𝑏, 𝑐 2 𝑎2 𝑏 2𝑽𝟐 (𝟎, 𝒂)(𝟎, 𝒃)𝑭𝟐 (𝟎, 𝒄)Ellipse𝑽𝟏 ( 𝒂, 𝟎) 𝑭𝟏 ( 𝒄, 𝟎)𝑭𝟐 (𝒄, 𝟎) 𝑽𝟐 (𝒂, 𝟎)( 𝒃, 𝟎)(𝒃, 𝟎)𝑭𝟏 (𝟎, 𝒄)(𝟎, 𝒃)𝑽𝟏 (𝟎, 𝒂)𝒙𝟐𝒂𝟐𝒚𝟐 𝒃𝟐 𝟏, 𝑐 2 𝑎2 𝑏 2𝑏𝑏𝑎𝑎Asymptote: 𝑦 𝑥, 𝑦 𝑥𝒚𝟐𝒂𝟐𝒙𝟐 𝒃𝟐 𝟏, 𝑐 2 𝑎2 𝑏 2𝑎𝑎𝑏𝑏Asymptote: 𝑦 𝑥, 𝑦 𝑥𝑭𝟐 (𝟎, 𝒄)Hyperbola𝐕𝟏 ( 𝒂, 𝟎)𝐕𝟐 (𝒂, 𝟎)𝐕𝟐 (𝟎, 𝒂)𝑭𝟏 ( 𝒄, 𝟎)𝑭𝟐 (𝒄, 𝟎)𝐕𝟏 (𝟎, 𝒂)𝑭𝟏 (𝟎, 𝒄)

POLAR EQUATIONS OF CONICS(Focus at the Pole, Eccentricity 𝒆)EquationDescription𝑟 𝑒𝑝1 𝑒 cos 𝜃Directrix is perpendicular to the polar axis at a distance 𝑝 units to the left ofthe pole:𝑥 𝑝𝑟 𝑒𝑝1 𝑒 cos 𝜃Directrix is perpendicular to the polar axis at a distance 𝑝 units to the right ofthe pole:𝑥 𝑝𝑟 𝑒𝑝1 𝑒 sin 𝜃Directrix is parallel to the polar axis at a distance 𝑝 units above the pole:𝑟 𝑒𝑝1 𝑒 sin 𝜃Directrix is parallel to the polar axis at a distance 𝑝 units below the pole:𝑦 𝑝𝑦 𝑝EccentricityIf 𝑒 1, the conic is a parabola; the axis of symmetry is perpendicular to the directrix.If 0 𝑒 1, the conic is an ellipse; the major axis is perpendicular to the directrix.If 𝑒 1, the conic is a hyperbola; the transverse axis is perpendicular to the directrix.ARITHMETIC SEQUENCE𝑎𝑛 𝑎1 (𝑛 1)𝑑𝑆𝑛 𝑎1 (𝑎1 𝑑) (𝑎1 2𝑑) [𝑎1 (𝑛 1)𝑑] 𝑛2𝑛2[2𝑎1 (𝑛 1)𝑑][𝑎1 𝑎𝑛 ]GEOMETRIC SEQUENCE𝑎𝑛 𝑎1 𝑟 𝑛 1𝑆𝑛 𝑎1 𝑎1 𝑟 𝑎1 𝑟 2 𝑎1 𝑟 𝑛 1 𝑎1 (1 𝑟 𝑛)1 𝑟

GEOMETRIC SERIESIf 𝑟 1, 𝑎1 𝑎1 𝑟 𝑎1 𝑟 2 𝑎1 𝑟 𝑘 1 𝑘 1If 𝑟 1,𝑎11 𝑟the infinite geometric series does not have a sum.PERMUTATIONS/COMBINATIONS0! 11! 1𝑛! 𝑛(𝑛 1) (3)(2)(1)𝑃(𝑛, 𝑟) 𝑛!(𝑛 𝑟)!𝑛!𝑛C(n, r) ( ) 𝑟(𝑛 𝑟)! 𝑟!BINOMIAL THEOREM𝑛𝑛𝑛(𝑎 𝑏)𝑛 𝑎𝑛 ( ) 𝑏𝑎𝑛 1 ( ) 𝑏 2 𝑎𝑛 2 () 𝑏 𝑛 1 𝑎 𝑏 𝑛𝑛 112

Lone Star College-CyFair Formula Sheet The following formulas are critical for success in the indicated course. Student CANNOT bring these formulas on a formula sheet or card to tests and instructors MUST NOT provide them during the test either on the board or on a handout. They MUST be memorized. Math 1314 College Algebra FORMULAS/EQUATIONS