Chapter 8.4, Problem 16E

(a)

To determine

To sketch: the curve of the given parametric equation

Answer to Problem 16E

  

Explanation of Solution

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

Calculation:

Sketch the curve represented by the parametric equations

  $x={\sin }^{2}t,y=\cos t.$

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

If $t=0$ , then

  $\begin{array}{l}x={\sin }^{2}0\\ \text{ =0,}\end{array}$

and

  $\begin{array}{l}\text{y}\text{=}\cos \text{0}\\ \text{ =1,}\end{array}$

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

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

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

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 16E

Here, the given curve has the rectangular coordinate equation $x+y^2=1$ .

Explanation of Solution

Calculation:

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

  $x={\sin }^{2}t,y=\cos t,$

by eliminating the parameter.

To eliminating the parameter, use the trigonometric identify ${\sin }^{2}t+{\cos }^{2}t=1$

From the given equations,

  ${\sin }^{2}t=x,$

and

  $\cos t=y.$

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

  $x^2+{\left(y\right)}^2=1$

Thus,

  $x+y^2=1$ .

Therefore, the given curve has the rectangular coordinate equation,

  $\boxed{x+y^2=1.}$

Conclusion:

Hence, the given curve has the rectangular coordinate equation $x+y^2=1$ .