Chapter 1, Problem 45RE

(a)

To determine

To find: The graph of the parametric equations, the initial and terminal points. Also, indicate the direction in which the curve is traced.

Answer to Problem 45RE

The graph of the parametric equations with the direction is shown in figure (1), the initial point is $\left(5,0\right)$ and terminal point is $\left(5,0\right)$ .

Explanation of Solution

Given information:The given equations are $x=5\cos t$ and $y=2\sin t$ such that $0\le t\le 2\pi$ .

Calculation:

Use the following step to graph the parametric equations by graphing calculator.

Step 1: First press the “ON” button graphical calculator.

Step 2: Press the $\boxed{\text{MODE}}$ button, under FUNC and choose PAR option.

Step 3: Press $\boxed{\text{Y=}}$ , and set the parametric equations under ${\text{X}}_{\text{1T}}$ and ${\text{Y}}_{\text{1T}}$ .

Step 4: Press $\boxed{\text{GRAPH}}$ to obtain the graph of the parametric equations and the window can be set to $\left[-5,5\right]$ by $\left[-2,2\right]$ as shown below.

  

Figure (1)

Substitute 0 for $t$ in the equation $x=5\cos t$ .

  $\begin{array}{c}x=5\cos \left(0\right)\\ =5\end{array}$

Substitute 0 for $t$ in the equation $y=2\sin t$ .

  $\begin{array}{c}y=2\sin \left(0\right)\\ =0\end{array}$

The initial points is $\left(5,0\right)$ .

Substitute $2\pi$ for $t$ in the equation $x=5\cos t$ .

  $\begin{array}{c}x=5\cos \left(2\pi \right)\\ =5\end{array}$

Substitute $2\pi$ for $t$ in the equation $y=2\sin t$ .

  $\begin{array}{c}y=2\sin \left(2\pi \right)\\ =0\end{array}$

The terminal point is $\left(5,0\right)$ .

Therefore,the graph of the parametric equations with the direction is shown in figure (1), the initial point is $\left(5,0\right)$ and terminal point is $\left(5,0\right)$ .

(b)

To determine

To find: The Cartesian equation and the portion of the graph of the Cartesian equation that is traced by the parameterized curve.

Answer to Problem 45RE

The Cartesian equation is $\frac{y^2}{4}+\frac{x^2}{25}=1$ and the parameterized curve traces the entire ellipse in counterclockwise direction.

Explanation of Solution

Given information:The given equations are $x=5\cos t$ and $y=2\sin t$ such that $0\le t\le 2\pi$ .

Calculation:

Square both the given equations and add them.

  $\begin{array}{c}{x}^2+y^2={\left(5\sin t\right)}^2+{\left(2\cos t\right)}^2\\ {\left(\frac{x}{5}\right)}^2+{\left(\frac{y}{2}\right)}^2={\sin }^{2}t+{\cos }^{2}t\\ \frac{x^2}{25}+\frac{y^2}{4}=1\end{array}$

This is an equation of a horizontal ellipse centered at the origin. The parameterized curve traces the complete ellipse in counterclockwise direction.

Therefore, the Cartesian equation is $\frac{y^2}{4}+\frac{x^2}{25}=1$ and the parameterized curve traces the entire ellipse in counterclockwise direction.