Chapter 10.1, Problem 55ES

In the and-1850s the French physicist Jules Antoine Lissajous (1822-1880) became interested in parametric equations of the form $x=\sin at,y=\sin bt$ in the course of studying vibrations that combine two perpendicular sinusoidal motions. If $a/b$ is a rational number, then the combined effect of the oscillations is a periodic motion along a path called a Lissajous curve.

(a) Use a graphing utility to generate the complete graph of the Lissajous curves corresponding to $a=1,b=2;a=2,b=3;a=3,b=4;\text{and }a=4,b=5.$

(b) The Lissajous curve $x=\sin t,y=\sin 2t\left(0\le t\le 2\pi \right)$ crosses itself at the origin (see Figure Ex-55 on the next page). Find equations for the two tangent lines at the origin.