Chapter 8.4, Problem 57E

(a)

To determine

To find: that parametric for the curate cycloid .

Answer to Problem 57E

Therefore, the parametric for the curate cycloid is

  $\begin{array}{l}x=a\theta -b\sin \theta \\ y=a-b\cos \theta \end{array}$

Explanation of Solution

Given:

  $x=a\theta -b\sin \theta$ , $y=a-b\cos \theta$

Calculation:

Consider the following sketch of a curate cycloid.

  

Create a two equations for the curate cycloid for the point $P$ .Recall the distance from the origin to point C is $a\theta$ . Then the distance from the origin to point $P$ for the $x$ component will be $x=a\theta -b\sin \theta$ .The distance to point $P$ from the origin for the $y$ component will be $y=a-b\cos \theta$ .

  $\begin{array}{l}x=a\theta -b\sin \theta \\ y=a-b\cos \theta \end{array}$

Conclusion:

Therefore, the parametric for the curate cycloid is $\begin{array}{l}x=a\theta -b\sin \theta \\ y=a-b\cos \theta \end{array}$

(b)

To determine

To find: the graph using $a=3$and $b=2$

Answer to Problem 57E

  

Explanation of Solution

Given: $a=3$ and $b=2$

Calculation:

Plug in for $a=3$ , $b=2$

  $\begin{array}{l}x=3\theta -2\sin \theta \\ y=3-2\cos \theta \end{array}$

Now plug in values and create a table.

$\theta$	$x$	$y$
$0$	$0$	$1$
$\frac{\pi }{2}$	$\frac{3\pi }{2}-2$	$3$
$\pi$	$3\pi$	$5$
$\frac{3\pi }{2}$	$\frac{9\pi }{2}+2$	$3$
$2\pi$	$6\pi$	$1$

Plot the points on a coordinate plane.

  

Conclusion: