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