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 If 푃 = (푥 , 푦 ) and 푃 = (푥 , 푦 ), the distance from 푃 to 푃 is Distance Formula 1 1 1 2 2 2 1 2 2 2 푑(푃1, 푃2) = √(푥2 − 푥1) + (푦2 − 푦1) Standard Equation The standard equation of a circle of radius 푟 with center at (ℎ, 푘) is Of a Circle (푥 − ℎ)2 + (푦 − 푘)2 = 푟2 The slope 푚 of the line containing the points 푃1 = (푥1, 푦1) and 푃2 = (푥2, 푦2) is 푦2 − 푦1 Slope Formula 푚 = if 푥1 ≠ 푥2 푥2 − 푥1 푚 is undefined if 푥1 = 푥2 Point-slope The equation of a line with slope 푚 containing the points (푥1, 푦1) is Equation of a Line 푦 − 푦1 = 푚(푥 − 푥1) Slope-Intercept Equation of The equation of a line with slope 푚 and 푦-intercept 푏 is a Line 푦 = 푚푥 + 푏 The solutions of the equation 푎푥2 + 푏푥 + 푐 = 0, 푎 ≠ 0, are Quadratic Formula −푏 ± √푏2 − 4푎푐 푥 = 2푎 LIBRARY OF FUNCTIONS Constant Function Identity Function Square Function Cube Function 푓(푥) = 푏 푓(푥) = 푥 푓(푥) = 푥2 푓(푥) = 푥3 Reciprocal Function Squared Reciprocal Square Root Function Cube Root Function 1 Function 푓(푥) = 푥 푓(푥) = 3 푥 푓(푥) = √ √ 푥 1 푓(푥) = 푥2 Absolute Function Exponential Function Natural Logarithm Greatest Integer 푓(푥) = |푥| 푓(푥) = 푒푥 Function Function 푓(푥) = ln 푥 푓(푥) = ⟦푥⟧ 풚 풚 풇(풙) = 풆풙 풚 풇(풙) = 퐥퐧풙 풇(풙) = ⟦풙⟧ (ퟏ, 풆) ퟏ ൬−ퟏ, ൰ (풆, ퟏ) 풆 (ퟎ, ퟏ) (ퟏ, ퟎ) 풙 풙 ퟏ 풙 ൬ , −ퟏ൰ 풆 GEOMETRY FOMULAS r 푟 = Radius, 퐴 = Area, 퐶 = Circumference Circle ퟐ 푨 = 흅풓 푪 = ퟐ흅풓 푏 = Base, ℎ = Altitude(Height), 퐴 = Area Triangle h ퟏ 푨 = 풃풉 ퟐ b 푤 푙 = Length, 푤 = Width, 퐴 = Area, 푃 = Perimeter Rectangle 푨 = 풍풘 푷 = ퟐ풍 + ퟐ풘 푙 푙 = Length, 푤 = Width, ℎ = Height, Rectangular 푉 = Volume, 푆 = Surface Area Box 푽 = 풍풘풉 푺 = ퟐ풍풘 + ퟐ풍풉 + ퟐ풘풉 PROPERTIES OF LOGARITHMS • log푎(푀푁) = log푎 푀 + log푎 푁 푀 • log ( ) = log 푀 − log 푁 푎 푁 푎 푎 푟 • log푎 푀 = 푟 log푎 푀 log 푀 ln 푀 • log 푀 = = 푎 log 푎 ln 푎 • 푎푥 = 푒푥 ln 푎 Math 1316 Trigonometry Students in Trigonometry should know all the formulas from Math 1314 College Algebra plus the following. Unit Circle 풙ퟐ + 풚ퟐ = ퟏ TRIGONOMETRIC FUNCTIONS Of an Acute Angle 푏 Opposite 푐 Hypotenuse sin 휃 = = csc 휃 = = 푐 Hypotenuse 푏 Opposite 푎 Adjacent 푐 Hypotenuse cos 휃 = = sec 휃 = = 푐 Hypotenuse 푎 Adjacent 푏 Opposite 푎 Adjacent tan 휃 = = cot 휃 = = 푎 Adjacent 푏 Opposite Of a General Angle 풚 푏 푟 sin 휃 = csc 휃 = , 푏 ≠ 0 ퟐ ퟐ 푟 푏 풓 = √풂 + 풃 푎 푟 cos 휃 = sec 휃 = , 푎 ≠ 0 푟 푎 (풂, 풃) 휽 풙 푏 푎 tan 휃 = , 푎 ≠ 0 cot 휃 = , 푏 ≠ 0 푎 푏 APPLICATIONS 풔 Arc Length: 푠 = 푟휃, 휃 in radians 1 Area of Sector: 퐴 = 푟2휃, 휃 in radians 2 휃 Angular Speed: 휔 = , 휃 in radians 푡 푠 Linear Speed: 푣 = , 푣 = 휔푟 푡 SOLVING TRIANGLES sin 퐴 sin 퐵 sin 퐶 푩 Law of Sine: = = 푎 푏 푐 퐜 풂 Law of Cosine: 푎2 = 푏2 + 푐2 − 2푏푐 cos 퐴 2 2 2 푨 푪 푏 = 푎 + 푐 − 2푎푐 cos 퐴 b 푐2 = 푎2 + 푏2 − 2푎푏 cos 퐴 TRIGONOMETRIC IDENTITIES Fundamental Identities sin 휃 cos 휃 tan휃 = cot 휃 = cos 휃 sin 휃 1 1 1 csc 휃 = sec 휃 = cot 휃 = sin 휃 cos 휃 tan 휃 sin2 휃 + cos2 휃 = 1 1 + tan2 휃 = sec2 휃 1 + cot2 휃 = csc2 휃 Even-Odd Identities Cofunction Identities sin(−휃) = − sin 휃 csc( − 휃) = − csc 휃 cos(90° − 휃) = sin 휃 cos(−휃) = cos 휃 sec( − 휃) = sec 휃 sin(90° − 휃) = cos 휃 tan(−휃) = − tan 휃 cot( − 휃) = − cot 휃 tan(90° − 휃) = tan 휃 Sum and Difference Formulas Double-Angle Formulas sin(훼 + 훽) = sin 훼 cos 훽 + cos 훼 sin 훽 sin(2휃) = 2 sin 휃 cos 휃 sin(훼 − 훽) = sin 훼 cos 훽 − cos 훼 sin 훽 cos(2휃) = cos2 휃 − sin2 휃 cos(훼 + 훽) = cos 훼 cos 훽 − sin 훼 sin 훽 = 2 cos2 휃 − 1 cos(훼 − 훽) = cos 훼 cos 훽 + sin 훼 sin 훽 = 1 − 2sin2 휃 tan 훼 + tan 훽 2 tan 휃 tan(훼 + 훽) = tan(2휃) = 1 − tan 훼 tan 훽 1 − tan2 휃 tan 훼 − tan 훽 tan(훼 − 훽) = 1 + tan 훼 tan 훽 LIBRARY OF TRIGONOMETRIC FUNCTIONS Sine Function 푓(푥) = sin 푥 Cosine Function 푓(푥) = cos 푥 Tangent Function 푓(푥) = tan 푥 Secant Function 푓(푥) = sec 푥 Cosecant Function 푓(푥) = csc 푥 Cotangent Function 푓(푥) = cot 푥 Math 2412 Precalculus Students in Precalculus should know all the formulas from Math 1314 College Algebra and Math 1316 Trigonometry plus the following. Half Angle Formulas 1−cos 2휃 1+cos 2휃 1−cos 2휃 sin 휃 = ±√ cos 휃 = ±√ tan 휃 = 2 2 sin 2휃 Products and Quotients of Complex Numbers in Polar Form Let 푧1 = 푟1(cos 휃1 + sin 휃1) and 푧2 = 푟2(cos 휃2 + sin 휃2). Then 푧1푧2 = 푟1푟2[cos(휃1 + 휃2) + sin(휃1 + 휃2)] and, if 푧2 ≠ 0, 푧1 푟1 = [cos(휃1 − 휃2) + sin(휃1 − 휃2)]. 푧2 푟2 DeMoivre’s Theorem If 푧 = 푟(cos 휃 + sin 휃) and n is a positive integer, 푧푛 = 푟푛[cos(푛휃) + sin(푛휃)]. Complex Roots Let 푤 = 푟(cos 휃0 + sin 휃0) be a complex number and let n≥2be an integer. If 푤 ≠ 0, there are n distinct complex nth roots of 푤, given by the formula 휃0 2푘휋 휃0 2푘휋 푧 = 푛√푟 [cos ൬ + ൰ + sin ൬ + ൰]. 푘 푛 푛 푛 푛 Where 푘 = 0, 1, 2, ⋯ , 푛 − 1. CONICS 풙ퟐ + 풚ퟐ = ퟏ (ퟎ, 풓) Circle (−풓, ퟎ) (풓, ퟎ) (ퟎ, −풓) ퟐ ퟐ ퟐ ퟐ 풚 = ퟒ풂풙 풚 = −ퟒ풂풙 풙 = ퟒ풂풚 풙 = −ퟒ풂풚 푫: 풙 = −풂 푫: 풙 = 풂 푫: 풚 = 풂 Parabola 푭(ퟎ, 풂) 푽 푭(−풂, ퟎ) 푽 푽 푭(풂, ퟎ) 푽 푭(ퟎ, −풂) 푫: 풚 = −풂 풙ퟐ 풚ퟐ 풙ퟐ 풚ퟐ + = ퟏ, 푎 > 푏, 푐2 = 푎2 − 푏2 + = ퟏ, 푎 > 푏, 푐2 = 푎2 − 푏2 풂ퟐ 풃ퟐ 풃ퟐ 풂ퟐ 푽ퟐ(ퟎ, 풂) (ퟎ, 풃) 푭 (ퟎ, 풄) ퟐ Ellipse 푽ퟏ(−풂, ퟎ) 푭ퟏ(−풄, ퟎ) 푭ퟐ(풄, ퟎ) 푽ퟐ(풂, ퟎ) (−풃, ퟎ) (풃, ퟎ) 푭ퟏ(ퟎ, −풄) (ퟎ, −풃) 푽ퟏ(ퟎ, −풂) 풙ퟐ 풚ퟐ 풚ퟐ 풙ퟐ − = ퟏ, 푐2 = 푎2 + 푏2 − = ퟏ, 푐2 = 푎2 + 푏2 풂ퟐ 풃ퟐ 풂ퟐ 풃ퟐ 푏 푏 푎 푎 Asymptote: 푦 = 푥, 푦 = − 푥 Asymptote: 푦 = 푥, 푦 = − 푥 푎 푎 푏 푏 푭ퟐ(ퟎ, 풄) 퐕ퟏ(−풂, ퟎ) 퐕ퟐ(풂, ퟎ) Hyperbola 퐕ퟐ(ퟎ, 풂) 푭ퟏ(−풄, ퟎ) 푭ퟐ(풄, ퟎ) 퐕ퟏ(ퟎ, −풂) 푭ퟏ(ퟎ, −풄) POLAR EQUATIONS OF CONICS(Focus at the Pole, Eccentricity 풆) Equation Description 푒푝 Directrix is perpendicular to the polar axis at a distance 푝 units to the left of 푟 = 1 − 푒 cos 휃 the pole: 푥 = −푝 푒푝 Directrix is perpendicular to the polar axis at a distance 푝 units to the right of 푟 = 1 + 푒 cos 휃 the pole: 푥 = 푝 푒푝 Directrix is parallel to the polar axis at a distance 푝 units above the pole: 푟 = 1 + 푒 sin 휃 푦 = 푝 푒푝 Directrix is parallel to the polar axis at a distance 푝 units below the pole: 푟 = 1 − 푒 sin 휃 푦 = −푝 Eccentricity If 푒 = 1, the conic is a parabola; the axis of symmetry is perpendicular to the directrix. If 0 < 푒 < 1, the conic is an ellipse; the major axis is perpendicular to the directrix. If 푒 > 1, the conic is a hyperbola; the transverse axis is perpendicular to the directrix. ARITHMETIC SEQUENCE 푎푛 = 푎1 + (푛 − 1)푑 푆푛 = 푎1 + (푎1 + 푑) + (푎1 + 2푑) + ⋯ + [푎1 + (푛 − 1)푑] 푛 = [2푎 + (푛 − 1)푑] 2 1 푛 = [푎 + 푎 ] 2 1 푛 GEOMETRIC SEQUENCE 푛−1 푎푛 = 푎1푟 2 푛−1 푆푛 = 푎1 + 푎1푟 + 푎1푟 + ⋯ + 푎1푟 푛) 푎1(1−푟 = 1−푟 GEOMETRIC SERIES If |푟| < 1, ∞ 푎 푎 + 푎 푟 + 푎 푟2 + ⋯ = ∑ 푎 푟푘−1 = 1 1 1 1 1 1 − 푟 푘=1 If |푟| ≥ 1, the infinite geometric series does not have a sum. PERMUTATIONS/COMBINATIONS 0! = 1 1! = 1 푛! = 푛(푛 − 1) ∙ ⋯ ∙ (3)(2)(1) 푛! 푃(푛, 푟) = (푛 − 푟)! 푛 푛! C(n, r) = ( ) = 푟 (푛 − 푟)! 푟! BINOMIAL THEOREM 푛 푛 푛 (푎 + 푏)푛 = 푎푛 + ( ) 푏푎푛−1 + ( ) 푏2푎푛−2 + ⋯ + ( ) 푏푛−1푎 + 푏푛 1 2 푛 − 1 .