Question

Hello, struggling with finding the 1st and 2nd time the tangent line will be vertical of

An object is moving in the plane according to these parametric equations:
x(t) = πt + cos(4πt + π/2)
y(t) = sin(4πt + π/2)
A portion of the "cycloidal" path (on the time interval [0,1]) is shown in this figure:

Answer 1

find dx/dt
equate it to zero and find t

Answer 2

this is the reasoning:
the velocity vector V=< dx/dt , dy/dt > is tangent to the path
thus we need to look for points where V has only a 'Y' component
so just set dx/dt=0 and solve for t

Answer 3

@kkzhao99: Done, or do you still need help? Please let the rest of us know your status.