# Tangent line equation.Given x = 5cost, y =3sint, what is the equation of the tangent line if t=pi/4.

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It is given that x = 5*cos t and y = 3*sin t. We have to find the equation of the tangent if t = pi/4

When t = pi/4 , x = 5*(1/sqrt 2) and y = 3/sqrt 2

dx/dt = -5*sin t and dy/dt = 3*cos t

dy/dx = (dy/dt)/(dx/dt)

=> 3*cos t/ -5*sin t

=> (-3/5)/tan t

at t = pi/4

=> -3/5

The equation of the tangent is (y - 3/sqrt 2)/(x - 5/sqrt 2) = -3/5

=> 5y - 15/sqrt 2 = -3x + 15/sqrt 2

=> 3x + 5y - 30/sqrt 2 = 0

**The equation of the tangent is 3x + 5y - 30/sqrt 2 = 0**

We'll use the chain rule to differentiate:

dy/dx = (dy/dt)/(dx/dt)

dx/dt = (d/dt)(5cost)

dx/dt = -5 sin t

dy/dt = (d/dt)(3sint)

dy/dt = 3 cos t

dy/dx = 3 cos t/-5 sin t

dy/dx = 3 cos(pi/4)/-5 sin (pi/4)

dy/dx = 3/-5

At t = pi/4, we'll get the point (x(t),y(t))= (5cos(pi/4),3sin(pi/4))

(x(t),y(t))= (5sqrt2/2,3sqrt2/2)

The equation of the tangent line is:

y - 3sqrt2/2 = m(x - 5sqrt2/2)

y - 3sqrt2/2 = (-3/5)(x - 5sqrt2/2)

**y = (-3/5)(x - 5sqrt2/2) + 3sqrt2/2**