Anonymous
Anonymous asked in Science & MathematicsMathematics · 10 months ago

x =  4 cos t, y =  4 sin t,   π  ≤  t ≤  2 π?

Parametric equations for the motion of a particle in the xy -plane are given. Find a Cartesian equation for it. Indicate the portion of the graph traced by the particle.

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  • MyRank
    Lv 6
    10 months ago

    x = 4 cos t, y = 4 sin t,

    x = 4cost

    x² = 4²cos²t

    x² = 16cos²t…(1

    y = 4sint

    y² = 4²sin²t

    y² = 16 sin²t…(2)

    from equation (1) and (2)

    x² + y² = 16cos²t + 16 sin²t

    x² + y² = 16(cos²t + sin²t)

    x² + y² = 16

  • Ash
    Lv 7
    10 months ago

    x = 4 cos t

    cos t = x/4 ...(1)

    y = 4 sin t

    sin t = y/4 ...(2)

    We know sin²t + cos²t = 1

    plug from (1) and (2)

    (y/4)²+(x/4)²=1

    y²/16  + x²/16  = 1

    y² + x² = 16

    y² = 16 - x²  ....(3)

    Since π ≤ t ≤ 2 π, we have  -1≤cos t ≤1

    so the condition for x is -1≤ x/4 ≤1 or -4≤x≤4

    Also since π ≤ t ≤ 2 π, we have  -1≤sin t ≤0

    so the condition for y is -1≤ y/4 ≤0 or -4≤y≤0

    Cartesian equation is y² = 16 - x²

    -4≤x≤4 ; -4≤y≤0

    The portion traced by the graph is in III and IV quadrant

    Attachment image
  • 10 months ago

    Ask your math teacher.

    Do you seriously think the brain dead idiots on Yahoo Answers who can't even write a complete sentence in English could possibly help you with your higher level mathematics equations?

  • Pope
    Lv 7
    10 months ago

    The parametric equations define a semicircle, centered on the origin, with radius 4, on or below the x-axis. Here is the Cartesian equation:

    x² + y² = 16, y ≤ 0

    or

    y = -√(16 - x²)

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