Chapter 4.2, Problem 59E

(a)

To determine

To graph: The given equations.

Explanation of Solution

Given information:

The given equations are.

  $\begin{array}{l}{\text{T}}_{\min }=0,{\text{T}}_{\max }=6\cdot 3,{\text{T}}_{\text{step}}=0\cdot 1\\ {\text{X}}_{\min }=-1\cdot 5,{\text{X}}_{\max }=1\cdot 5,{\text{X}}_{\text{sc1}}=1\\ {\text{Y}}_{\min }=-1,{\text{Y}}_{\max }=1,{\text{Y}}_{\text{sc1}}=1\end{array}$

Graph:

Using the equation $\text{X}=\cos \text{T}$ and $\text{Y}=\sin \text{T}$ .

The circle with center $\left(0,0\right)$ and radius $1$ , as the equation $\text{X}=\cos \text{T}$ and $\text{Y}=\sin \text{T}$ with given conditions.

  $\begin{array}{l}{\text{T}}_{\min }=0,{\text{T}}_{\max }=6\cdot 3,{\text{T}}_{\text{step}}=0\cdot 1\\ {\text{X}}_{\min }=-1\cdot 5,{\text{X}}_{\max }=1\cdot 5,{\text{X}}_{\text{sc1}}=1\\ {\text{Y}}_{\min }=-1,{\text{Y}}_{\max }=1,{\text{Y}}_{\text{sc1}}=1\end{array}$

Represents the parametric from of equation of curve $x^2+y^2=1$ .

The graph for the equation of curveis shown in figure (1).

  

Figure (1)

Interpretation: Graph for the equation of curve is shown in figure (1).

(b)

To determine

To find: The $t$ values representation and the $x$ and $y$ values representation.

Answer to Problem 59E

The values of $\text{t}$ represent the central angle in radian in the anti-clockwise direction and the values of $x$ and $y$ represent the coordinate in the plane.

Explanation of Solution

Given information:

The given equations are.

  $\begin{array}{l}{\text{T}}_{\min }=0,{\text{T}}_{\max }=6\cdot 3,{\text{T}}_{\text{step}}=0\cdot 1\\ {\text{X}}_{\min }=-1\cdot 5,{\text{X}}_{\max }=1\cdot 5,{\text{X}}_{\text{sc1}}=1\\ {\text{Y}}_{\min }=-1,{\text{Y}}_{\max }=1,{\text{Y}}_{\text{sc1}}=1\end{array}$

Calculation:

Using the trace feature to move the cursor around the graph we get the values of $\text{t}$ represent the central angle in radian in the anti-clockwise direction.

The values of $x$ and $y$ represent the coordinate in the plane corresponding to the values of $\text{t}$ and joining these coordinate we get a circle.

Therefore, the values of $\text{t}$ represent the central angle in radian in the anti-clockwise direction and the values of $x$ and $y$ represent the coordinate in the plane.

(c)

To determine

To find: The least and greatest value of $x$ and $y$ .

Answer to Problem 59E

The least and greatest value of $x$ and $y$ lies between $1$ and $-1$

  $-1\le x\le 1,-1\le y\le 1$ .

Explanation of Solution

Given information:

The given equations are.

  $\begin{array}{l}{\text{T}}_{\min }=0,{\text{T}}_{\max }=6\cdot 3,{\text{T}}_{\text{step}}=0\cdot 1\\ {\text{X}}_{\min }=-1\cdot 5,{\text{X}}_{\max }=1\cdot 5,{\text{X}}_{\text{sc1}}=1\\ {\text{Y}}_{\min }=-1,{\text{Y}}_{\max }=1,{\text{Y}}_{\text{sc1}}=1\end{array}$

Calculation:

From the above graph the values of $x$ and $y$ lies between $1$ and $-1$ .

  $-1\le x\le 1,-1\le y\le 1$

Therefore, the least and greatest value of $x$ and $y$ lies between $1$ and $-1$

  $-1\le x\le 1,-1\le y\le 1$ .