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Consider the curve C parametrized by

(x, y) = (cos (5t), sin(−4t)), for −π ≤ t ≤ 2pi

How would I go about finding how many times the curve is traversed? Also, how would I find out the radius and center of the circle is? I don't need answers I just need explanations on how to go about doing this.

Thanks for your help!

1 answer

x has a period of 2pi/5
y has a period of pi/2
The LCM is 2pi, during which time x has gone around 5 times and y has gone around 4 times. The entire Lissajou figure will have been completed once.

so, adding another pi means y goes around twice more, but x only 2.5 times more. You can get a feel for this by looking at the plot at

https://www.wolframalpha.com/input/?i=plot+x%3Dcos%285t%29%2C+y%3Dsin%28-4t%29+for+-pi+%3C%3D+t+%3C%3D+9pi%2F10

You can see that increasing the interval to [-pi,pi] will complete the curve one time
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