Lone Star College-CyFair Formula Sheet

Transcription

1 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 P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ), the distance from P 1 to P 2 is d(p 1, P 2 ) = (x 2 x 1 ) 2 + (y 2 y 1 ) 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 = r 2 Slope Formula The slope m of the line containing the points P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ) is m = y 2 y 1 x 2 x 1 if x 1 x 2 m is undefined if x 1 = x 2 Point-slope Equation of a Line The equation of a line with slope m containing the points (x 1, y 1 ) is y y 1 = m(x x 1 ) 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 ax 2 + bx + c = 0, a 0, are x = b ± b2 4ac 2a

2 LIBRARY OF FUNCTIONS Constant Function f(x) = b Identity Function f(x) = x Square Function f(x) = x 2 Cube Function f(x) = x 3 Reciprocal Function f(x) = 1 x Squared Reciprocal Function Square Root Function f(x) = x Cube Root Function 3 f(x) = x f(x) = 1 x 2 Absolute Function f(x) = x Exponential Function f(x) = e x Natural Logarithm Function f(x) = ln x Greatest Integer Function f(x) = x y f(x) = e x y y (1, e) f(x) = lnx f(x) = x ൬ 1, 1 e ൰ (0, 1) x (e, 1) (1, 0) ൬ 1 x e, 1൰ x

3 GEOMETRY FOMULAS Circle r r = Radius, A = Area, C = Circumference A = πr 2 C = 2πr Triangle h b = Base, h = Altitude(Height), A = Area A = 1 2 bh b Rectangle l w l = Length, w = Width, A = Area, P = Perimeter A = lw P = 2l + 2w Rectangular Box l = Length, w = Width, h = Height, V = Volume, S = Surface Area V = lwh S = 2lw + 2lh + 2wh PROPERTIES OF LOGARITHMS log a (MN) = log a M + log a N log a ( M N ) = log a M log a N log a M r = r log a M log a M = a x x ln a = e log M log a = ln M ln a

4 Math 1316 Trigonometry Students in Trigonometry should know all the formulas from Math 1314 College Algebra plus the following. Unit Circle x 2 + y 2 = 1

5 TRIGONOMETRIC FUNCTIONS Of an Acute Angle sin θ = b c = cos θ = a c = Opposite Hypotenuse Adjacent Hypotenuse tan θ = b a = Opposite Adjacent csc θ = c b = Hypotenuse Opposite sec θ = c a = Hypotenuse Adjacent cot θ = a b = Adjacent Opposite Of a General Angle y sin θ = b r csc θ = r b, b 0 r = a 2 + b 2 cos θ = a r sec θ = r a, a 0 tan θ = b a, a 0 cot θ = a b, b 0 (a, b) θ x APPLICATIONS s Arc Length: s = rθ, θ in radians Area of Sector: A = 1 2 r2 θ, θ in radians Angular Speed: ω = θ, θ in radians t Linear Speed: v = s, v = ωr t SOLVING TRIANGLES c B a Law of Sine: sin A a sin B sin C = = b c Law of Cosine: a 2 = b 2 + c 2 2bc cos A A b C b 2 = a 2 + c 2 2ac cos A c 2 = a 2 + b 2 2ab cos A

6 TRIGONOMETRIC IDENTITIES Fundamental Identities tanθ = sin θ cos θ csc θ = 1 sin θ cot θ = cos θ sin θ sec θ = 1 cos θ cot θ = 1 tan θ sin 2 θ + cos 2 θ = tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ Even-Odd Identities sin( θ) = sin θ cos( θ) = cos θ tan( θ) = tan θ csc( θ) = csc θ sec( θ) = sec θ cot( θ) = cot θ Cofunction Identities cos(90 θ) = sin θ sin(90 θ) = cos θ tan(90 θ) = tan θ Sum and Difference 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 Double-Angle Formulas sin(2θ) = 2 sin θ cos θ cos(2θ) = cos 2 θ sin 2 θ tan(2θ) = = 2 cos 2 θ 1 = 1 2sin 2 θ 2 tan θ 1 tan 2 θ tan(α β) = tan α tan β 1 + tan α tan β

7 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

8 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 2θ sin θ = ± 2 1+cos 2θ cos θ = ± 2 tan θ = 1 cos 2θ sin 2θ Products and Quotients of Complex Numbers in Polar Form Let z 1 = r 1 (cos θ 1 + i sin θ 1 ) and z 2 = r 2 (cos θ 2 + i sin θ 2 ). Then z 1 z 2 = r 1 r 2 [cos(θ 1 + θ 2 ) + i sin(θ 1 + θ 2 )] and, if z 2 0, z 1 z 2 = r 1 r 2 [cos(θ 1 θ 2 ) + i sin(θ 1 θ 2 )]. DeMoivre s Theorem If z = r(cos θ + i sin θ) and n is a positive integer, z n = r n [cos(nθ) + i sin(nθ)]. Complex Roots Let w = r(cos θ 0 + i sin θ 0 ) 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 formula Where k = 0, 1, 2,, n 1. n z k = r [cos ൬ θ 0 n + 2kπ n ൰ + i sin ൬θ 0 n + 2kπ n ൰].

9 CONICS x 2 + y 2 = 1 (0, r) Circle ( r, 0) (r, 0) (0, r) y 2 = 4ax y 2 = 4ax x 2 = 4ay x 2 = 4ay D: x = a D: x = a D: y = a Parabola V F(a, 0) F( a, 0) V F(0, a) V V F(0, a) D: y = a x 2 y2 a2 + = 1, a > b, b 2 c2 = a 2 b 2 x 2 y2 b2 + = 1, a > b, a 2 c2 = a 2 b 2 V 2 (0, a) (0, b) F 2 (0, c) Ellipse V 1 ( a, 0) F 1 ( c, 0) F 2 (c, 0) V 2 (a, 0) ( b, 0) (b, 0) F 1 (0, c) (0, b) V 1 (0, a) x 2 y2 a2 = 1, b 2 c2 = a 2 + b 2 Asymptote: y = b x, y = b x a a y 2 x2 a2 = 1, b 2 c2 = a 2 + b 2 Asymptote: y = a x, y = a x b b F 2 (0, c) Hyperbola V 1 ( a, 0) V 2 (a, 0) V 2 (0, a) F 1 ( c, 0) F 2 (c, 0) V 1 (0, a) F 1 (0, c)

10 POLAR EQUATIONS OF CONICS(Focus at the Pole, Eccentricity e) r = r = r = r = Equation ep 1 e cos θ ep 1 + e cos θ ep 1 + e sin θ ep 1 e sin θ Description Directrix is perpendicular to the polar axis at a distance p units to the left of the pole: x = p Directrix is perpendicular to the polar axis at a distance p units to the right of the pole: x = p Directrix is parallel to the polar axis at a distance p units above the pole: y = p 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 a n = a 1 + (n 1)d S n = a 1 + (a 1 + d) + (a 1 + 2d) + + [a 1 + (n 1)d] = n [2a (n 1)d] = n [a a n ] GEOMETRIC SEQUENCE a n = a 1 r n 1 S n = a 1 + a 1 r + a 1 r a 1 r n 1 = a 1(1 r n) 1 r

11 GEOMETRIC SERIES If r < 1, If r 1, a 1 + a 1 r + a 1 r 2 + = a 1 r k 1 = k=1 a 1 1 r 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 = a n + ( n 1 ) ban 1 + ( n 2 ) b2 a n ( n n 1 ) bn 1 a + b n