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Math Help - find the unit tangent to curve

  1. #1
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    find the unit tangent to curve

    find the unit tangent to the curve r(t) = t cos(t) i - tsin(t) j at time t = pi/2

    thanks for any help.
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  2. #2
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    Quote Originally Posted by rcmango View Post
    find the unit tangent to the curve r(t) = t cos(t) i - tsin(t) j at time t = pi/2

    thanks for any help.
    Hi,

    first I've transformed your equation into:
    x(t) = t*cos(t)
    y(t) = -t*sin(t)

    Then the point T where the tangent touches the curve has the coordinates:
    T(0, -(π/2))

    To calculate the tangent you need the gradient of the curve in T. I use the formula:

    dy/dx = (dy/dt) / (dx/dt). Use product rule:

    dy/dx = (sin(t)*(-1) + (-t)*cos(t)) / (cos(t) + t*(-sin(t)))

    Now plug in t = π/2

    therefore the slope of the tangent is 2/π.

    Use the point-slope-formula to get the equation of the straight line:

    t: y = (2/π)*x-(π/2)

    I've attached an image of your curve and the tangent.

    EB
    Attached Thumbnails Attached Thumbnails find the unit tangent to curve-cycloidtangente.gif  
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