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Math Help - Derivatives

  1. #1
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    Derivatives

    I need to find the equation of the tangent line to the curve at the given value of t:
    x=2cost
    y=2sint
    t=pi/3
    Thanks for the help.
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  2. #2
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    \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}
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  3. #3
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    Or, since the original curve is given in terms of the parameter t, write the tangent line in the same way:
    \frac{dx}{dt}= -2sin(t)
    \frac{dy}{dt}=  2cos(t)
    so, at t= t_0, i.e. the point (2cos(t_0), 2sin(t_0)) the tangent line is given by x= 2cos(t_0)- 2sin(t_0)(t- t_0), y= 2sin(t_0)+ 2 cos(t_0)(t- t_0).

    In particular, with t= \pi/3, sin(\pi/3)= \sqrt{3}/2 and cos(\pi/3)= 1/2 so the tangent line there is x= 1- \sqrt{3}(t- \pi/3), y= \sqrt{3}+ (t- \pi/3).
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