Given the parametric curve x= sin(2t) and y=2 sin(t)+ sin(2t).

Question

1) Calculate the slope of the tangent line at any value of t.

2) Find the tangent line at t= (1/2)(Pi).

Answer 1

To find dy/dx , you need to find ( dy/dt) / (dx/dt), which will be (2cost + 2cos2t) /( 2cos 2t)

This is the "slope function" you were looking for.

Find the slope at t=(1/2)(Pi):

m = (2cos(Pi/2) + 2cos(Pi)) / 2cos(Pi) = (0 - 2) / (-1) = 2

The point is (sinPi , 2sin(Pi/2)i+sin Pi) = (0, 2)

The equation of the tangent line is: y - 2 = 2*(x - 0)

y = 2x + 2

Answer 2

You want to find dy/dt but you do not have a function for y in terms of x.

Rather you have x(t) and y(t).

So dy/dt = (dy/dt)*(dx/dt). So compute dy/dt and dx/dt.