Chapter 8.4, Problem 23E

(a)

To determine

To sketch: the curve of the given parametric equation

Answer to Problem 23E

  

Explanation of Solution

Given: $x={\cos }^{2}t,y={\sin }^{2}t$

Calculation:

Sketch the curve represented by the parametric equations

  $x={\cos }^{2}t,y={\sin }^{2}t$.

For every value of $t$ , get a point on the curve.

If $t=0$ , then

  $\begin{array}{l}x={\cos }^2\left(0\right)\\ \text{ =1,}\end{array}$

and

  $\begin{array}{l}y={\sin }^2\left(0\right)\\ \text{ =0,}\end{array}$

So the point corresponding to $t=0$is $\left(1,0\right)$ .

The points $\left(x,y\right)$determined for different values of $t$ are shown in the following table.

$t$	$x$	$y$
$0$	$1$	$0$
$\frac{\pi }{4}$	$\frac{1}{2}$	$\frac{1}{2}$
$\frac{\pi }{2}$	$0$	$1$
$\frac{3\pi }{4}$	$\frac{1}{2}$	$\frac{1}{2}$
$\pi$	$1$	$0$

Plotting these points, get the following graph of the curve.

  

Conclusion:

Hence, the curve represented by the given parametric equation is sketched.

(b)

To determine

To find: the rectangular coordinate equation for the curve.

Answer to Problem 23E

Here, the given curve has the rectangular coordinate equation $y=-x+1$ .

Explanation of Solution

Calculation:

Find a rectangular-coordinate equation for the curve, represented by the parametric equations,

  $x={\cos }^{2}t,y={\sin }^{2}t$

by eliminating the parameter.

To eliminate the parameter, use the trigonometric identity ${\sin }^{2}t+{\cos }^{2}t=1.$

From the given equations,

  ${\sin }^{2}t=y,$

and

  ${\cos }^{2}t=x.$

Substituting the values of ${\sin }^{2}t$ and ${\cos }^{2}t$ , in ${\sin }^{2}t+{\cos }^{2}t=1$ ,

  $y+x=1.$

Thus,

  $y=-x+1.$

Therefore, the given curve has the rectangular coordinate equation,

  $\boxed{y=-x+1.}$

Conclusion:

Hence, the given curve has the rectangular coordinate equation $y=-x+1$ .