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Mathematics Dictionary
Dr. K. G. Shih
Parametric equations
Subjects


  • AN 15 00 | - Outlines
  • AN 15 01 | - Parametric equations including trigonometric functions
  • AN 15 02 | - Properties of the graph of x = tan(t) and y = sec(t)
  • AN 15 03 | - Properties of the graph of x = sec(t) and y = tan(t)
  • AN 15 04 | - Ellipse : x = h + a*cos(t) and y = k + b*sin(t)
  • AN 15 05 | - x = sin(t) and y = sin(t)
  • AN 15 06 | -
  • AN 15 07 | -
  • AN 15 08 | -
  • AN 15 09 | -
  • AN 15 10 | -
    • Answers


    AN 15 01. Parametric equations including trigonometric functions

    Diagram of equations
    • Study Subjects | diagram of parametric equation
    • Equations : There are 36 equations, e.g.
      • 1. x = cos(t) and y = sin(t)
      • 2. x = tan(t) and y = sec(t)
      • 3. x = sec(t) and y = tan(t)

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    AN 15 02. Properties of the graph of x = tan(t) and y = sec(t)

    Properties from rectangular form
    • Change it to rectangular form using tan(t)^2 + 1 = sec(t)^2
      • x^2 - y^2 = tan(t)^2 - sec(t)^2
      • Hence x^2 - y^2 = -1
      • This is a hyperbola
    • The principal axis is y = 0
    • The semi-axese : a = 1 and b = 1
    • The focal length is Sqr(a^2 + b^2) = Sqr(2)
    • The center is (0, 0)
    • The Foci are (0, -Sqr(2)) and (0, Sqr(2))
    • The vertices are (0, -1) and (0, 1)
    • The asymptotes are y = x and y = -x
    Properties from polar form
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    AN 15 03. Properties of the graph of x = sec(t) and y = tan(t)

    Properties from rectangular form
    • Change it to rectangular form using tan(t)^2 + 1 = sec(t)^2
      • x^2 - y^2 = sec(t)^2 - tan(t)^2
      • Hence x^2 - y^2 = +1
      • This is a hyperbola
    • The principal axis is x = 0
    • The semi-axese : a = 1 and b = 1
    • The focal length is Sqr(a^2 + b^2) = Sqr(2)
    • The center is (0, 0)
    • The Foci are (-Sqr(2), 0) and (Sqr(2), 0)
    • The vertices are (-1, 0) and (1, 0)
    • The asymptotes are y = x and y = -x
    Properties from polar form
  • The polar form R = (D*E)/(1 - e*cos(A))
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    AN 15 04. The graph of x = h + a*cos(t) and y = h + b*sin(t)

    Properties from rectangular form
    • Change it to rectangular form using cos(t)^2 + sin(t)^2 = 1
      • ((x-h)/a)^2 - ((y-k)/b)^2 = cos(t)^2 + sin(t)^2
      • Hence ((x-h)/a)^2 - ((y-k)/b)^2 = +1
      • This is an ellipse
    • The principal axis is y = 0 if a > b
    • The semi-axese : a and b
    • The focal length is Sqr(f^2 - a^2)
    • The center is (h, k)
    • The Foci are (h - f, 0) and (h + f, 0)
    • The vertices are (h - a, 0) and (h + a, 0)
    Properties from polar form
  • The polar form R = (D*E)/(1 - e*cos(A))
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    AN 15 05. x = sin(t) and y=sin(t)
    Properties of sketch
    • t = 000 to 090 : Plot from (0,0) to (1,1)
    • t = 090 to 180 : Plot from (1,1) to (0,0)
    • t = 180 to 270 : Plot from (0,0) to (-1,-1)
    • t = 270 to 360 : Plot from (-1,-1) to (0,0)
    Properties of graph in oxy coordinate
    • Domain : x = -1 to 1
    • Range : y = -1 to 1
    • The line section is [-1, -1] to [1,1]
    • The slope is 1
    Diagrams
    • Study Subjects | Program 15 01
    • Note : If t from 0 to n*pi, the slow motion will be in S.H.M.

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    AN 15 06. x=cos(t) and y=sin(t)

    Properties of sketch
    • t = 000 to 090 : Plot arc of circle from (1,0) to (0,1)
    • t = 090 to 180 : Plot arc of circle from (0,1) to (-1,0)
    • t = 180 to 270 : Plot arc of circle from (-1,0) to (0,-1)
    • t = 270 to 360 : Plot arc of circle from (0,-1) to (1,0)
    Properties of graph in oxy coordinate
    • It is a unit circle
    • Equation : x^2 + y^2 = 1
    • Center (0,0) and radius = 1
    Diagrams
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    AN 15 07. Answer


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    AN 15 08. Answer

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    AN 15 09. Answer

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    AN 15 10. Answer


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    AN 15 00. Answer

    Pythagorean relation
    • x = cos(t) and y = sin(t) is unit circle
    • x = sec(t) and y = tan(t) is unit hyperbola
    • x = tan(t) and y = sec(t) is unit hyperbola

    Conic sections
    • Circle : x = h + r*cos(t) and y = k + r*sin(t)
    • Ellipse : x = h + a*cos(t) and y = k + b*sin(t)
    • Hyperbola : x = h + a*sec(t) and y = k + b*tan(t)
    • Hyperbola : x = h + a*tan(t) and y = k + b*sec(t)

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    Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
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