Question

The curve given parametrically by x = 2t[cubed], y = t[sixth power], intersects the graph given parametrically by x = 1 + cost, y = 1 + sin t at the point (2; 1). What is the angle between the tangent lines to the two graphs at that point?

Answer 1

Just find slope of tangents at the two points by differnentitaion. Here it is parametric differentiation. So dy/dx= (dy/dt)/(dx/dt). So for the first it becomes 6t^5/6t^2=t^3. (at x=2,t=1)=1.
Now the second becomes -tan t=0. (As at x=2 cos t=1 tan t=0).
so m1=1, m2=0. So m= (m2-m1)/(1+m1m2)=1.
So the angle is tan inverse 1= pi/4.