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Thread: 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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    $\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
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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:
    $\displaystyle \frac{dx}{dt}= -2sin(t)$
    $\displaystyle \frac{dy}{dt}= 2cos(t)$
    so, at $\displaystyle t= t_0$, i.e. the point $\displaystyle (2cos(t_0), 2sin(t_0))$ the tangent line is given by $\displaystyle x= 2cos(t_0)- 2sin(t_0)(t- t_0)$, $\displaystyle y= 2sin(t_0)+ 2 cos(t_0)(t- t_0)$.

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