Chapter 1.7, Problem 45E

To determine

Identify curves variation when $a,b,n$ vary.

Answer to Problem 45E

As $n$ increases, the number of oscillations increases, $a\&b$ determine the width and height.

Explanation of Solution

Given information:

The curves with equations $x=a\sin nt,y=b\cos t$ are called Lissajous figures. Investigate how these curves vary when $a,b,n$ vary. (Take $n$ to be a positive integer.)

Calculation:

Put down the equations to the Lissajous figures.

  $x=a\sin nt,y=b\cos t$ Where $n$ is a positive integer.

Put different values for $a,b,n$ in $x=a\sin nt,y=b\cos t$ and then graph them.

Put the values $a$ and $b$ , that is take $a=1;b=2$ and then vary the $n$ that means take the values of $n$ as $1,2,\mathrm{3....}$

  $\begin{array}{l}For n=1,x=\sin t,y=2\cos t\\ For n=2,x=\sin 2t,y=2\cos t\\ For n=3,x==\sin 3t,y=\cos t\\ For n=4,x=\sin 4t,y=2\cos t\end{array}$

Now graph these equations.

  

We observed that, for $n=1$ , the curve is an ellipse (observe the green curve), for $n=2$ , the curve has two loops (observe the red curve), for $n=3$ , the curve has three loops (observe the blue curve) and for $n=4$ , the curve has four loops (observe the brown curve).

And also observed that the curves are within the rectangle $\left[-1,1\right]\times \left[-2,2\right]$ .

Hence, it can be said that, the number of loops in the curve increases as $n$ increases, and the curves are within the rectangle $\left[-a,a\right]\times \left[-b,b\right]$ for the fixed values of the $a,b$ .

Put the values $a\&n$ , that is take $a=1;n=2$ and then vary the $b$ that means take the values of $b$ as $1,2,\mathrm{3...}$

  $\begin{array}{l}For b=1,x=\sin 2t,y=\cos t\\ For b=2,x=\sin 2t,y=2\cos t\\ For b=3,x=\sin 2t,y=3\cos t\\ For b=4,x=\sin 2t,y=4\cos t\end{array}$

Now graph these equation.

  

We observed that, each curve has two loops as the value of $n=2$ , the curves are within the interval $\left[-1,1\right]$ (along $x-axis$ ) as the value of $a=1$ , and the lengths of the curves are increasing vertically as the value of $b$ increases.

Hence, the curves are within the rectangle $\left[-a,a\right]\times \left[-b,b\right]$ of variable width $b$ .

Put the values $b\&n$ , that is take $b=1;n=2$ and then vary the $a$ ,that makes take the values of $a$ as $1,2,\mathrm{3...}$

  $\begin{array}{l}For a=1,x=\sin 2t,y=\cos t\\ For a=2,x=2\sin 2t,y=\cos t\\ For a=3,x=3\sin 2t,y=\cos t\\ For a=4,x=4\sin 2t,y=\cos t\end{array}$

Now graph these equations.

  

We observed that, each curve has two loops as the value of $n=2$ , the curves are within the interval $\left[-1,1\right]$ (along $y-axis$ ) as the value of $b=1$ , and the lengths of the curves are increasing horizontally as the value of $a$ increases.

the curves are within the rectangle $\left[-a,a\right]\times \left[-b,b\right]$ of variable length $a$ .

From the above discussion the following conclusions can be made.

The number of loops in the curve increases as the value of $n$ increases.

The horizontal length of the curve increases as the value of $a$ increases.

The vertical length of the curve increases as the value of $b$ increases.

Hence, as $n$ increases, the number of oscillations increases, ( $a\&b$ determine the width and height).