Question
The curves with equations $x=a \sin n t, y=b \cos t$ are called Lissajous figures. Investigate how these curves vary when $a, b$, and $n$ vary. (Take $n$ to be a positive integer.)
Step 1
The Lissajous figures are parametric curves defined by the equations \( x = a \sin(nt) \) and \( y = b \cos(t) \), where \( a \) and \( b \) are amplitudes, and \( n \) is a positive integer that affects the frequency of the sine function in the x-component. Show more…
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The curves with equations $ x = \alpha \sin nt $, $ y = b \cos t $ are called Lissajous figures. Investigate how these curves vary when $ a $, $ b $, and $ n $ vary. (Take $ n $ to be a positive integer.)
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$$ \begin{array}{l}{\text { The curves with equations } x=a \sin n t, y=b \cos t \text { are }} \\ {\text { called Lissajous figures. Investigate how these curves vary }} \\ {\text { when } a, b, \text { and } n \text { vary. (Take } n \text { to be a positive integer.) }}\end{array} $$
The parametric curve given by $x=\cos a t$ and $y=\sin b t$ is known as a Lissajous figure. The $x$ -coordinate oscillates $a$ times between 1 and -1 as $t$ goes from 0 to 2 $\pi$, while the $y$ -coordinate oscillates $b$ times over the same $t$ interval. This behavior is repeated over every interval of length $2 \pi$. The entire motion takes place in a unit square. Plot the following Lissajous figures for a range of $t$ that ensures that the resulting figure is a closed curve. In each case, count the number of times that the curve touches the horizontal and vertical borders of the unit square. (a) $x=\sin t, y=\cos t$ (b) $x=\sin 3 t, y=\cos 5 t$ (c) $x=\cos 5 t, y=\sin 15 t$ (d) $x=\sin 2 t, y=\cos 9 t$
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