Find the equation of the tangent line at the given t if x = 5cost, y =3sint at t=pi/4

To find the equation of the tangent line at the given t if
x = 5cost, y =3sint at t=pi/4.

x = 5cost and y =
3sint.

To find the tangent at t =
pi/4. 

Therefore at t = pi/4, the
coordintes :

 x1 = 5cospi/4 = 5/qsrt2 and y1 = 5sinpi/4 =
5/sqrt2.

dy/dx = (sint)' = cost. dx/dt = (cost)' =
-sint.

So the slope of the tangent at at x= pi, is dy/dx =
(dy/dt)/(dx/dt) = {(cost/sint) at t = pi/4} = (1/sqrt2)/(-1/sqrt2) =
-1.

Therefore the tangent at t = pi is given
by:

y-y1 = {slope at t =
pi/4}(x-x1).

y-5/sqrt2 =
(-1)(x-5/sqrt2).

x+y =10/sqrt2 = 10sqrt2/2 =
5sqrt2.

x+y = 5sqrt2 is the equation of the tangent at to t
=pi/4.