Chapter 1.4, Problem 9E

(a)

To determine

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

Answer to Problem 9E

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

Explanation of Solution

Given information: The equations are $x=\cos t$ and $y=\sin t$ , $0\le t\le \pi$.

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

Step 3: Press $\boxed{\text{GRAPH}}$ to obtain the graph of the parametric equations and the window can be set to $\left[-3,3\right]$ by $\left[-2,2\right]$ as shown below.

  

Figure (1)

As $t$ increases from 0 to 1, the graph starts at the point $\left(1,0\right)$ and moves in a anticlockwise direction ending at the point $\left(-1,0\right)$.

So, the initial point is $\left(1,0\right)$ and terminal point is $\left(-1,0\right)$.

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

(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 9E

The Cartesian equation is $x^2+y^2=1$ and parameterized curve traces the upper half of the circle.

Explanation of Solution

Given information: The equations are $x=\cos t$ and $y=\sin t$ , $0\le t\le \pi$.

Calculation:

Square both the given equations and add them.

  $\begin{array}{c}{x}^2+y^2={\sin }^{2}t+{\cos }^{2}t\\ =1\end{array}$

The above equation is form of a circle. From the graph shown in part (a), it can be seen that the parameterized curve traces the upper half of the circle.

Therefore, the Cartesian equation is $x^2+y^2=1$ and parameterized curve traces the upper half of the circle.