Cheat Sheet for College Algebra, Trigonometry and Precalculus, Cheat Sheet of Algebra

Download Cheat Sheet for College Algebra, Trigonometry and Precalculus and more Cheat Sheet Algebra in PDF only on Docsity! 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 Distance Formula If 𝑃1 = (𝑥1, 𝑦1) and 𝑃2 = (𝑥2, 𝑦2), the distance from 𝑃1 to 𝑃2 is 𝑑(𝑃1, 𝑃2) = √(𝑥2 − 𝑥1) 2 + (𝑦2 − 𝑦1) 2 Standard Equation Of a Circle The standard equation of a circle of radius 𝑟 with center at (ℎ, 𝑘) is (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2 Slope Formula The slope 𝑚 of the line containing the points 𝑃1 = (𝑥1, 𝑦1) and 𝑃2 = (𝑥2, 𝑦2) is 𝑚 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1 if 𝑥1 ≠ 𝑥2 𝑚 is undefined if 𝑥1 = 𝑥2 Point-slope Equation of a Line The equation of a line with slope 𝑚 containing the points (𝑥1, 𝑦1) is 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) Slope-Intercept Equation of a Line The equation of a line with slope 𝑚 and 𝑦-intercept 𝑏 is 𝑦 = 𝑚𝑥 + 𝑏 Quadratic Formula The solutions of the equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, 𝑎 ≠ 0, are 𝑥 = −𝑏 ± √𝑏2 − 4𝑎𝑐 2𝑎 LIBRARY OF FUNCTIONS Constant Function 𝑓(𝑥) = 𝑏 Identity Function 𝑓(𝑥) = 𝑥 Square Function 𝑓(𝑥) = 𝑥2 Cube Function 𝑓(𝑥) = 𝑥3 Reciprocal Function 𝑓(𝑥) = 1 𝑥 Squared Reciprocal Function 𝑓(𝑥) = 1 𝑥2 Square Root Function 𝑓(𝑥) = √𝑥 Cube Root Function 𝑓(𝑥) = √𝑥 3 Absolute Function 𝑓(𝑥) = |𝑥| Exponential Function 𝑓(𝑥) = 𝑒𝑥 Natural Logarithm Function 𝑓(𝑥) = ln 𝑥 Greatest Integer Function 𝑓(𝑥) = ⟦𝑥⟧ (𝟏, 𝒆) (𝟎, 𝟏) ൬−𝟏, 𝟏 𝒆 ൰ 𝒚 𝒙 𝒇(𝒙) = 𝒆𝒙 𝒇(𝒙) = 𝐥𝐧𝒙 (𝒆, 𝟏) ൬ 𝟏 𝒆 , −𝟏൰ 𝒙 𝒚 (𝟏, 𝟎) 𝒙 𝒚 𝒇(𝒙) = ⟦𝒙⟧ TRIGONOMETRIC FUNCTIONS Of an Acute Angle sin 𝜃 = 𝑏 𝑐 = Opposite Hypotenuse csc 𝜃 = 𝑐 𝑏 = Hypotenuse Opposite cos 𝜃 = 𝑎 𝑐 = Adjacent Hypotenuse sec 𝜃 = 𝑐 𝑎 = Hypotenuse Adjacent tan 𝜃 = 𝑏 𝑎 = Opposite Adjacent cot 𝜃 = 𝑎 𝑏 = Adjacent Opposite Of a General Angle sin 𝜃 = 𝑏 𝑟 csc 𝜃 = 𝑟 𝑏 , 𝑏 ≠ 0 cos 𝜃 = 𝑎 𝑟 sec 𝜃 = 𝑟 𝑎 , 𝑎 ≠ 0 tan 𝜃 = 𝑏 𝑎 , 𝑎 ≠ 0 cot 𝜃 = 𝑎 𝑏 , 𝑏 ≠ 0 APPLICATIONS Arc Length: 𝑠 = 𝑟𝜃, 𝜃 in radians Area of Sector: 𝐴 = 1 2 𝑟2𝜃, 𝜃 in radians Angular Speed: 𝜔 = 𝜃 𝑡 , 𝜃 in radians Linear Speed: 𝑣 = 𝑠 𝑡 , 𝑣 = 𝜔𝑟 SOLVING TRIANGLES Law of Sine: sin 𝐴 𝑎 = sin 𝐵 𝑏 = sin 𝐶 𝑐 Law of Cosine: 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴 𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑐 cos 𝐴 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐴 𝒓 = √𝒂𝟐 + 𝒃𝟐 (𝒂, 𝒃) 𝜽 𝒙 𝒚 𝑨 𝑩 𝑪 b 𝐜 𝒂 𝒔 TRIGONOMETRIC IDENTITIES Fundamental Identities tan𝜃 = sin 𝜃 cos 𝜃 cot 𝜃 = cos 𝜃 sin 𝜃 csc 𝜃 = 1 sin 𝜃 sec 𝜃 = 1 cos 𝜃 cot 𝜃 = 1 tan 𝜃 sin2 𝜃 + cos2 𝜃 = 1 1 + tan2 𝜃 = sec2 𝜃 1 + cot2 𝜃 = csc2 𝜃 Even-Odd Identities Cofunction Identities sin(−𝜃) = − sin 𝜃 csc( − 𝜃) = − csc 𝜃 cos(−𝜃) = cos 𝜃 sec( − 𝜃) = sec 𝜃 tan(−𝜃) = − tan 𝜃 cot( − 𝜃) = − cot 𝜃 cos(90° − 𝜃) = sin 𝜃 sin(90° − 𝜃) = cos 𝜃 tan(90° − 𝜃) = tan 𝜃 Sum and Difference Formulas Double-Angle Formulas sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽 sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽 cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽 cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽 tan(𝛼 + 𝛽) = tan 𝛼 + tan 𝛽 1 − tan 𝛼 tan 𝛽 tan(𝛼 − 𝛽) = tan 𝛼 − tan 𝛽 1 + tan 𝛼 tan 𝛽 sin(2𝜃) = 2 sin 𝜃 cos 𝜃 cos(2𝜃) = cos2 𝜃 − sin2 𝜃 = 2 cos2 𝜃 − 1 = 1 − 2sin2 𝜃 tan(2𝜃) = 2 tan 𝜃 1 − tan2 𝜃 LIBRARY OF TRIGONOMETRIC FUNCTIONS Sine Function 𝑓(𝑥) = sin 𝑥 Cosine Function 𝑓(𝑥) = cos 𝑥 Tangent Function 𝑓(𝑥) = tan 𝑥 Secant Function 𝑓(𝑥) = sec 𝑥 Cosecant Function 𝑓(𝑥) = csc 𝑥 Cotangent Function 𝑓(𝑥) = cot 𝑥 POLAR EQUATIONS OF CONICS(Focus at the Pole, Eccentricity 𝒆) Equation Description 𝑟 = 𝑒𝑝 1 − 𝑒 cos 𝜃 Directrix is perpendicular to the polar axis at a distance 𝑝 units to the left of the pole: 𝑥 = −𝑝 𝑟 = 𝑒𝑝 1 + 𝑒 cos 𝜃 Directrix is perpendicular to the polar axis at a distance 𝑝 units to the right of the pole: 𝑥 = 𝑝 𝑟 = 𝑒𝑝 1 + 𝑒 sin 𝜃 Directrix is parallel to the polar axis at a distance 𝑝 units above the pole: 𝑦 = 𝑝 𝑟 = 𝑒𝑝 1 − 𝑒 sin 𝜃 Directrix is parallel to the polar axis at a distance 𝑝 units below the pole: 𝑦 = −𝑝 Eccentricity If 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 SEQUENCE 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 𝑆𝑛 = 𝑎1 + (𝑎1 + 𝑑) + (𝑎1 + 2𝑑) + ⋯ + [𝑎1 + (𝑛 − 1)𝑑] = 𝑛 2 [2𝑎1 + (𝑛 − 1)𝑑] = 𝑛 2 [𝑎1 + 𝑎𝑛] GEOMETRIC SEQUENCE 𝑎𝑛 = 𝑎1𝑟 𝑛−1 𝑆𝑛 = 𝑎1 + 𝑎1𝑟 + 𝑎1𝑟 2 + ⋯ + 𝑎1𝑟 𝑛−1 = 𝑎1(1−𝑟 𝑛) 1−𝑟 GEOMETRIC SERIES If |𝑟| < 1, 𝑎1 + 𝑎1𝑟 + 𝑎1𝑟 2 + ⋯ = ∑ 𝑎1𝑟 𝑘−1 ∞ 𝑘=1 = 𝑎1 1 − 𝑟 If |𝑟| ≥ 1, the infinite geometric series does not have a sum. PERMUTATIONS/COMBINATIONS 0! = 1 1! = 1 𝑛! = 𝑛(𝑛 − 1) ∙ ⋯ ∙ (3)(2)(1) 𝑃(𝑛, 𝑟) = 𝑛! (𝑛 − 𝑟)! C(n, r) = ( 𝑛 𝑟 ) = 𝑛! (𝑛 − 𝑟)! 𝑟! BINOMIAL THEOREM (𝑎 + 𝑏)𝑛 = 𝑎𝑛 + ( 𝑛 1 ) 𝑏𝑎𝑛−1 + ( 𝑛 2 ) 𝑏2𝑎𝑛−2 + ⋯ + ( 𝑛 𝑛 − 1 ) 𝑏𝑛−1𝑎 + 𝑏𝑛