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.
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 $\displaystyle x= 1- \sqrt{3}(t- \pi/3)$, $\displaystyle y= \sqrt{3}+ (t- \pi/3)$.