Chapter 10.6, Problem 6E

a.

To determine

To find: The table of $x$ - axis and $y$ -axis using $\theta =-\pi /2,and-\pi /4,0,and-\pi /2$ .

Answer to Problem 6E

The table for the equations is shown in table (1).

Explanation of Solution

Given information:

The given parametric equations are $x=4{\cos }^2\theta$ and $y=2\sin \theta$ .

Calculation:

Using theparametric equations

  $\begin{array}{l}x=4{\cos }^2\theta \\ y=2\sin \theta \end{array}$

The table for the above equation is shown in table (1).

$\theta$	$-\frac{\pi }{2}$	$-\frac{\pi }{4}$	$0$	$\frac{\pi }{4}$	$\frac{\pi }{2}$
$x$	$0$	$2$	$1$	$2$	$0$
$y$	$-2$	$-\sqrt{2}$	$0$	$\sqrt{2}$	$2$

Table (1)

Therefore, the table for the equations is shown in table (1).

b.

To determine

To find: The Plot for the points $\left(x,y\right)$ from table (1), and sketch a graph of the parametric equations.

Answer to Problem 6E

The graph of the parametric equations $f\left(x\right)=x^2$ and $g\left(x\right)=x-1$ is shown in figure (1).

Explanation of Solution

Given information:

The given parametric equations are $x=4{\cos }^2\theta$ and $y=2\sin \theta$ .

Calculation:

The graph is shown in graph (1).

  

Figure (1)

Therefore, the graph of the parametric equations $x=4{\cos }^2\theta$ and $y=2\sin \theta$ is shown in figure (1).

c.

To determine

To find: The rectangular equation by eliminating the parameter and plot the graph.

Answer to Problem 6E

The graph for the rectangular equation $x+y^2=4$ is shown in figure (2).

Explanation of Solution

Given information:

The given parametric equations are $x=4{\cos }^2\theta$ and $y=2\sin \theta$ .

Calculation:

The parametric equations are.

  $\begin{array}{l}x=4{\cos }^2\theta \\ {\cos }^2\theta =\frac{x}{4}\end{array}$

And,

  $y=2\sin \theta$

Square on both sides of the above equation.

  ${\sin }^2\theta =\frac{y^2}{4}$

Using the trigonometric identity.

  $\begin{array}{l}{\sin }^2\theta +{\cos }^2\theta =1\\ \frac{x}{4}+\frac{y^2}{4}=1\\ x+y^2=4\end{array}$

The graph for the rectangular equations is shown in figure (2).

  

Figure (2)

The difference between two graphs is that the rectangular equation shows the complete parabola rather than just the right half

Therefore, the graph for the rectangular equation $x+y^2=4$ is shown in figure (2).