Parametrically-Defined Curves

  1. Consider the curve defined parametrically by the coordinates x(t)=sin⁡(t)andy(t)=sin⁡(t)cos⁡(t) .x(t) = \sin(t) \quad\text{and}\quad y(t) = \sin(t)\cos(t)\,.
    1. Try to sketch this curve without the aide of technology. If you get stuck, use technology to plot the curves y=sin⁡(x){y = \sin(x)} and y=sin⁡(x)cos⁡(x){y = \sin(x)\cos(x)} to reference. Eventually use technology to plot the curve (x(t),y(t))\big(x(t), y(t)\big) to compare to your sketch.
    2. You should notice this is a closed curve in the shape of an hourglass. Find two values of the parameter t,t, call them tinitialt_\text{initial} and tterminal,t_\text{terminal}, such that (x(tinitial),y(tinitial))=(x(tterminal),y(tterminal))\big(x(t_\text{initial}), y(t_\text{initial})\big) = \big(x(t_\text{terminal}), y(t_\text{terminal})\big) and the tt between these trace out a single cycle of the curve.
  2. Consider the cycloid curve defined parametrically by x(t)=3(t−sin⁡(t))andy(t)=3(1−cos⁡(t)) .x(t) = 3\big(t-\sin(t)\big) \quad\text{and}\quad y(t) = 3\big(1-\cos(t)\big)\,.
    1. Try to sketch this curve without the aide of technology. If you get stuck, use technology to plot the curves y=3(x−sin⁡(x))y = 3\big(x-\sin(x)\big) and y=3(1−cos⁡(x))y = 3\big(1-\cos(x)\big) for reference. Eventually use technology to plot the curve (x(t),y(t))\big(x(t), y(t)\big) to compare to your sketch.
    2. Find appropriate values of tt corresponding to an initial and terminal point that trace out a single “hump” of the cycloid.