Chapter 6.3, Problem 6E

Solution

a)

To find: The table for the parametric equations.

The complete table is as follows.

$t$	$0$	$\frac{\pi }{2}$	$\pi$	$\frac{3\pi }{2}$	$2\pi$
$x$	$1$	$0$	$-1$	$0$	$1$
$y$	$0$	$1$	$0$	$-1$	$0$

Given information:

The parametric equations are $x=\cos t$ and $y=\sin t$ .

Calculation:

Find the values of $x$ and $y$ for the different values of $t$ .

Substitute $0$ for $t$ in equation $x=\cos t$ .

  $\begin{array}{c}x=\cos 0\\ =1\end{array}$

Substitute $0$ for $t$ in equation $y=\sin t$ .

  $\begin{array}{c}y=\sin 0\\ =0\end{array}$

Substitute $\frac{\pi }{2}$ for $t$ in equation $x=\cos t$ .

  $\begin{array}{c}x=\cos \frac{\pi }{2}\\ =0\end{array}$

Substitute $\frac{\pi }{2}$ for $t$ in equation $y=\sin t$ .

  $\begin{array}{c}y=\sin \frac{\pi }{2}\\ =1\end{array}$

Substitute $\pi$ for $t$ in equation $x=\cos t$ .

  $\begin{array}{c}x=\cos \pi \\ =-1\end{array}$

Substitute $\pi$ for $t$ in equation $y=\sin t$ .

  $\begin{array}{c}y=\sin \pi \\ =0\end{array}$

Substitute $\frac{3\pi }{2}$ for $t$ in equation $x=\cos t$ .

  $\begin{array}{c}x=\cos \frac{3\pi }{2}\\ =0\end{array}$

Substitute $\frac{3\pi }{2}$ for $t$ in equation $y=\sin t$ .

  $\begin{array}{c}y=\sin \frac{3\pi }{2}\\ =-1\end{array}$

Substitute $2\pi$ for $t$ in equation $x=\cos t$ .

  $\begin{array}{c}x=\cos 2\pi \\ =1\end{array}$

Substitute $2\pi$ for $t$ in equation $y=\sin t$ .

  $\begin{array}{c}y=\sin 2\pi \\ =0\end{array}$

Complete the table as follows.

$t$	$0$	$\frac{\pi }{2}$	$\pi$	$\frac{3\pi }{2}$	$2\pi$
$x$	$1$	$0$	$-1$	$0$	$1$
$y$	$0$	$1$	$0$	$-1$	$0$

Therefore, the complete table is shown above.

b)

To graph: The calculated points in part (a).

The points are shown in figure (1).

Given Information:

From part (a), the values given in the table are as follows:

$t$	$0$	$\frac{\pi }{2}$	$\pi$	$\frac{3\pi }{2}$	$2\pi$
$x$	$1$	$0$	$-1$	$0$	$1$
$y$	$0$	$1$	$0$	$-1$	$0$

Calculation:

Plot the points as follows.

  

Figure (1)

The value of the table is shown in the figure (1).