Derivatives

**nystudent2729** · January 3rd 2009, 02:15 PM

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.

**skeeter** · January 3rd 2009, 02:18 PM

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

**HallsofIvy** · January 3rd 2009, 03:42 PM

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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