Rectangular Polar Parametric

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1 Harold s Precalculus Rectangular Polar Parametric Cheat Sheet 15 October 2017 Point Line Rectangular Polar Parametric f(x) = y (x, y) (a, b) Slope-Intercept Form: y = mx + b Point-Slope Form: y y 0 = m (x x 0 ) General Form: Ax + By + C = 0 Calculus Form: f(x) = f (a) x + f(0) (r, θ) or r θ Polar Rect. Rect. Polar x = r cos θ y = r sin θ tan θ = y x r 2 = x 2 + y 2 r = ± x 2 + y 2 θ = tan 1 ( y x ) Point (a,b) in Rectangular: x(t) = a y(t) = b < a, b > t = 3 rd variable, usually time, with 1 degree of freedom (df) < x, y > = < x 0, y 0 > + t < a, b > < x, y > = < x 0 + at, y 0 + bt > where < a, b > = < x 2 x 1, y 2 y 1 > x(t) = x 0 + ta y(t) = y 0 + tb m = y x = y 2 y 1 x 2 x 1 = b a Plane n x (x x 0 ) +n y (y y 0 ) + n z (z z 0 ) = 0 Vector Form: n (r r 0 ) = 0 r = r 0 + sv + tw Copyright by Harold Toomey, WyzAnt Tutor 1

2 General Equation for All Conics: General Equation for All Conics: Conics Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 where Line: A = B = C = 0 Circle: A = C and B = 0 Ellipse: AC > 0 or B 2 4AC < 0 Parabola: AC = 0 or B 2 4AC = 0 Hyperbola: AC < 0 or B 2 4AC > 0 Note: If A + C = 0, square hyperbola Rotation: If B 0, then rotate coordinate system: A C cot 2θ = B x = x cos θ y sin θ y = y cos θ + x sin θ New = (x, y ), Old = (x, y) rotates through angle θ from x-axis r = p 1 e cos θ a (1 e 2 ) 0 e < 1 where p = { 2d for { e = 1 a (e 2 1) e > 1 p = semi-latus rectum or the line segment running from the focus to the curve in a direction parallel to the directrix Eccentricity: Circle e = 0 Ellipse 0 e < 1 Parabola e = 1 Hyperbola e > 1 Copyright by Harold Toomey, WyzAnt Tutor 2

3 x 2 + y 2 = r 2 (x h) 2 + (y k) 2 = r 2 Center: (h, k) Vertices: NA Focus: (h, k) Circle Centered at Origin: r = a (constant) θ = θ [0, 2π] or [0, 360 ] x(t) = r cos(t) + h y(t) = r sin(t) + k [t min, t max ] = [0, 2π) (h, k) = center of circle (h, k) Copyright by Harold Toomey, WyzAnt Tutor 3

4 (x h) 2 a 2 (y k)2 + b 2 = 1 Center: (h, k) Vertices: (h ± a, k) and (h, k ± b) Foci: (h ± c, k) Focus length, c, from center: c = a 2 b 2 Ellipse: r = a (1 e2 ) for 0 e < e cos θ where e = c a = a2 b 2 a relative to center (h,k) x(t) = a cos(t) + h y(t) = b sin(t) + k [t min, t max ] = [0, 2π] (h, k) = center of ellipse Rotated Ellipse: x(t) = a cos t cos θ b sin t sin θ + h y(t) = a cos t sin θ + b sin t cos θ + k Ellipse θ = the angle between the x-axis and the major axis of the ellipse Interesting Note: The sum of the distances from each focus to a point on the curve is constant. d 1 + d 2 = k Copyright by Harold Toomey, WyzAnt Tutor 4

5 Parabola Rectangular Polar Parametric. Vertical Axis of Symmetry: x 2 = 4 py (x h) 2 = 4p(y k) Vertex: (h, k) Focus: (h, k + p) Directrix: y = k p Horizontal Axis of Symmetry: y 2 = 4 px (y k) 2 = 4p(x h) Vertex: (h, k) Focus: (h + p, k) Directrix: x = h p Parabola: 2d r = for e = e cos θ Trigonometric Form: y = x 2 r sin θ = r 2 cos 2 θ sin θ r = cos 2 = tan θ sec θ θ Vertical Axis of Symmetry: x(t) = 2pt + h y(t) = pt 2 + k (opens upwards) or y(t) = pt 2 k (opens downwards) (h, k) = vertex of parabola Horizontal Axis of Symmetry: y(t) = 2pt + k x(t) = pt 2 + h (opens to the right) or x(t) = pt 2 h (opens to the left) (h, k) = vertex of parabola Projectile Motion: x(t) = x 0 + v x t y(t) = y 0 + v y t 16t 2 feet y(t) = y 0 + v y t 4.9t 2 meters v x = v cos θ v y = v sin θ General Form: x = At 2 + Bt + C y = Dt 2 + Et + F where A and D have the same sign Copyright by Harold Toomey, WyzAnt Tutor 5

6 (x h) 2 a 2 (y k)2 b 2 = 1 Center: (h, k) Vertices: (h ± a, k) Foci: (h ± c, k) Hyperbola: r = a (e2 1) for e > e cos θ Eccentricity: where e = c a = a2 + b 2 = sec θ > 1 a relative to center (h,k) Left-Right Opening Hyperbola: x(t) = a sec( t) + h y(t) = b tan( t) + k (h, k) = vertex of hyperbola Hyperbola Focus length, c, from center: c = a 2 + b 2 Up-Down Opening Hyperbola: x(t) = a tan(t) + h y(t) = b sec(t) + k (h, k) = vertex of hyperbola Inverse Functions f(f 1 (x)) = f 1 (f(x)) = x Inverse Function Theorem: f 1 1 (b) = f (a) where b = f (a) p = semi-latus rectum or the line segment running from the focus to the curve in the directions θ = ± π 2 Interesting Note: The difference between the distances from each focus to a point on the curve is constant. d 1 d 2 = k if y = sin θ if y = cos θ if y = tan θ if y = csc θ if y = sec θ if y = cot θ then θ = sin 1 y then θ = cos 1 y then θ = tan 1 y then θ = csc 1 y then θ = sec 1 y then θ = cot 1 y General Form: x(t) = At 2 + Bt + C y(t) = Dt 2 + Et + F where A and D have different signs θ = arcsin y θ = arccos y θ = arctan y θ = arccsc y θ = arcsec y θ = arccot y Copyright by Harold Toomey, WyzAnt Tutor 6

7 Circle: L = s = rθ Arc Length Proof: L = (fraction of circumference) π (diameter) Perimeter Area Lateral Surface Area Total Surface Area Volume Square: P = 4s Rectangle: P = 2l + 2w Triangle: P = a + b + c Square: A = s² Rectangle: A = lw Rhombus: A = ½ ab Parallelogram: A = bh Trapezoid: A = (b 1+b 2 ) 2 Kite: A = d 1 d 2 2 Cylinder: S = 2πrh Cone: S = πrl Cube: S = 6s² Rectangular Box: S = 2lw + 2wh + 2hl Regular Tetrahedron: S = 2bh Cylinder: S = 2πr (r + h) Cone: S = πr² + πrl = πr (r + l) Cube: V = s³ Rectangular Prism: V = lwh Cylinder: V = πr²h Triangular Prism: V= Bh Tetrahedron: V= ⅓ bh Pyramid: V = ⅓ Bh Cone: V = ⅓ bh = ⅓ πr²h h L = ( θ ) π (2r) = rθ 2π Circle: C = πd = 2πr Ellipse: C π(a + b) Triangle: A = ½ bh Triangle: A = ½ ab sin(c) Triangle: A = s(s a)(s b)(s c), where s = a+b+c 2 Equilateral Triangle: A = ¼ 3s 2 Sphere: S = 4πr² Ellipsoid: S = (too complex) Sphere: V = 4 3 πr3 Ellipsoid: V = 4 3 πabc Frustum: A = 1 3 (b 1+b 2 2 ) h Circle: A = πr² Circular Sector: A = ½ r²θ Ellipse: A = πab Copyright by Harold Toomey, WyzAnt Tutor 7