Chapter 4.2, Problem 59E

a.

To determine

Graph the entered equations and describe the graph.

Answer to Problem 59E

  

Explanation of Solution

Given information:

With a graphing utility in radian and parametric modes, enter the equations $X_{IT}=\cos Tand{Y}_{IT}=\sin T$ and use the setting below.

  $\begin{array}{l}{T}_{\min }=0,T_{\max }=6.3,T_{Step}=0.1\\ X_{\min }=-1.5,X_{\max }=1.5,X_{Scl}=1\\ Y_{\min }=-1,Y_{\max }=1,Y_{Scl}=1\end{array}$

Graph the entered equations and describe the graph.

Calculation:

Consider the given equation $X_{IT}=\cos Tand{Y}_{IT}=\sin T$ and,

  $\begin{array}{l}{T}_{\min }=0,T_{\max }=6.3,T_{Step}=0.1\\ X_{\min }=-1.5,X_{\max }=1.5,X_{Scl}=1\\ Y_{\min }=-1,Y_{\max }=1,Y_{Scl}=1\end{array}$

We get the circle with centre $\left(0,0\right)$ and radius $1$ as the equation $X=\cos TandY=\sin T$ with

  $T_{\min }=0,T_{\max }=6.3,T_{Step}=0.1$

Given conditions,

  $\begin{array}{l}{X}_{\min }=-1.5,X_{\max }=1.5,X_{Scl}=1\\ Y_{\min }=-1,Y_{\max }=1,Y_{Scl}=1\end{array}$

Represents the parametric form of equation of circle, $x^2+y^2=1$

Hence, the plot is.

  

b.

To determine

What do the $x$ - and $y$ -values represent?

Answer to Problem 59E

  $\boxed{coordinate}$

Explanation of Solution

Given information:

With a graphing utility in radian and parametric modes, enter the equations $X_{IT}=\cos Tand{Y}_{IT}=\sin T$ and use the setting below.

  $\begin{array}{l}{T}_{\min }=0,T_{\max }=6.3,T_{Step}=0.1\\ X_{\min }=-1.5,X_{\max }=1.5,X_{Scl}=1\\ Y_{\min }=-1,Y_{\max }=1,Y_{Scl}=1\end{array}$

Use the trace feature to move the cursor around the graph. What do the $t$ -values represent? What do the $x$ - and $y$ -values represent?

Calculation:

Use the trace feature to move the cursor around the graph we get the vales $t$ represents the central angle in radians in the anti-clock wise direction.

  $x$ - and $y$ -values represent the coordinate in the plane corresponding to the value of $t$ on joining these points we get the circle.

Hence, $x$ - and $y$ -values represent the $\boxed{coordinate}$ .

c.

To determine

What are the least and greatest values of $x$ - and $y$ ?

Answer to Problem 59E

  $\boxed{-1\le x\le 1,-1\le y\le 1}$

Explanation of Solution

Given information:

With a graphing utility in radian and parametric modes, enter the equations $X_{IT}=\cos Tand{Y}_{IT}=\sin T$ and use the setting below.

  $\begin{array}{l}{T}_{\min }=0,T_{\max }=6.3,T_{Step}=0.1\\ X_{\min }=-1.5,X_{\max }=1.5,X_{Scl}=1\\ Y_{\min }=-1,Y_{\max }=1,Y_{Scl}=1\end{array}$

What are the least and greatest values of $x$ - and $y$ ?

Calculation:

  $x$ - and $y$ -values lies between $1$ and $-1$

Hence the values of $x$ - and $y$ is $\boxed{-1\le x\le 1,-1\le y\le 1}$