Question
Graph several members of the family of curves$x=\sin t+\sin n t, y=\cos t+\cos n t,$ where $n$ is a positive integer. What features do the curves have in common?What happens as $n$ increases?
Step 1
In this case, the parametric equations become $x=\sin t + \sin t = 2\sin t$ and $y=\cos t + \cos t = 2\cos t$. This represents a circle of radius 2. Show more…
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$$ \begin{array}{l}{\text { Graph several members of the family of curves }} \\ {x=\sin t+\sin n t, y=\cos t+\cos n t \text { where } n \text { is a positive }} \\ {\text { integer. What features do the curves have in common? What }} \\ {\text { happens as } n \text { increases? }}\end{array} $$
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Graph several members of the family of curves $ x = \sin t + \sin nt $, $ y = \cos t + \cos nt $, where $ n $ is a positive integer. What features do the curves have in common? What happens as $ n $ increases?
Graph several members of the family of curves with parametric equations $x=t+a \cos t, y=t+a \sin t,$ where $a > 0 .$ How does the shape change as $a$ increases? For what values of $a$ does the curve have a loop?
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