Chapter 6, Problem 83RE

Solution

a)

To find:Graph of the circle of radius 3 and the graph of two given epicycloids, that are, $x=4\cos t-\cos 4t$ and $y=4\sin t-\sin 4t$ .

  

Given information:

Two epicycloids, that are, $x=4\cos t-\cos 4t$ and $y=4\sin t-\sin 4t$ . Also, there is a circle of radius 3.

Formula used:

The parametric equations of a circle of radius $r$ is given by $x=r\cos \theta$ and $y=r\sin \theta$ .

Calculation:

The parametric equations of a circle of radius $r$ is given by $x=r\cos \theta$ and $y=r\sin \theta$ .

Consequently, the parametric equations of the circle with radius 3 and $\theta =t$ will be $x_1=3\cos t$ and $y_1=3\sin t$ .

Draw the graphs of the parametric equations $x_2=4\cos t-\cos 4t$ , $y_2=4\sin t-\sin 4t$ , $x_1=3\cos t$ , and $y_1=3\sin t$ using a graphing calculator in Parametric and Radian modes.

The obtained graph will look like:

  

b)

To find:If the radius of the large circle would be 4, then what would be change to determine defining equation? How the epicycloid would look like in this situation?

  $x=5\cos t-\cos 5t$ , $y=5\sin t-\sin 5t$ ; an epicycloid will contain 4 arcs, which will form a “flower” like structure

Given information:

Two epicycloids, that are, $x=4\cos t-\cos 4t$ and $y=4\sin t-\sin 4t$ . Also, there is a circle of radius 3.

Formula used:

The parametric equations of a circle of radius $r$ is given by $x=r\cos \theta$ and $y=r\sin \theta$ .

Calculation:

It is need to replace 4’s with 5’s into the parametric equations $x=4\cos t-\cos 4t$ and $y=4\sin t-\sin 4t$ .

Consequently, the equations will be:

  $x=5\cos t-\cos 5t$ and $y=5\sin t-\sin 5t$

Write the parametric equations of circle with radius 4 and $\theta =t$ .

  $x_1=4\cos t$ and $y_1=4\sin t$ .

From $t=0$ to $t=\frac{\pi }{2}$ , the graph travels from $\left(4,0\right)$ to $\left(0,4\right)$ , which is an arc of the epicycloid. So, the epicycloid will contain 4 arcs, which will form a flower like structure.

The resultant graph is shown below by drawing the graphs of the parametric equations $x_1=4\cos t$ , $y_1=4\sin t$ , $x_2=5\cos t-\cos 5t$ , and $y_2=5\sin t-\sin 5t$ .