Chapter 9.4, Problem 69E

a.

To determine

To describe the orientation of the curve represented by the parametric equations $x=6\cos \theta$ and $y=6\sin \theta$ , $0\le \theta \le 6$ .

Answer to Problem 69E

Anticlockwise.

Explanation of Solution

Given: The parametric equations $x=6\cos \theta$ and $y=6\sin \theta$ , $0\le \theta \le 6$ .

Calculation:

When $\theta =0$ , $x=6,y=0$ . This means that the curve starts at the point $\left(6,0\right)$ and moves anticlockwise direction.

So, the orientation of the curve is anticlockwise.

b.

To determine

To find the range of $\theta$ that gives the graph of a circle.

Answer to Problem 69E

  $0\le \theta \le 2\pi$ .

Explanation of Solution

Given: The parametric equations $x=6\cos \theta$ and $y=6\sin \theta$ , $0\le \theta \le 6$ .

Calculation:

The parametric form of an equation of a circle is given as $x=r\cos \theta$ and $y=r\sin \theta$ , $0\le \theta \le 2\pi$ .

So, for the given curve to represent a circle, the range of $\theta$ must be $0\le \theta \le 2\pi$ .

c.

To determine

To write a set of parametric equations representing the curve so that the curve traces from the same point as the original curve but in opposite direction.

Answer to Problem 69E

  $x=6\cos \left(6-\theta \right)$ and $y=6\sin \left(6-\theta \right)$ , $0\le \theta \le 6$ .

Explanation of Solution

Given: The parametric equations $x=6\cos \theta$ and $y=6\sin \theta$ , $0\le \theta \le 6$ .

Calculation:

The given curve moves in anticlockwise direction starting from the point $\left(6,0\right)$ .

So, in order to traces the point but in opposite direction, the curve must starts at the end point of the original curve and ends at the point $\left(6,0\right)$ . This can be achieved by the equations as shown below.

  $x=6\cos \left(6-\theta \right)$ and $y=6\sin \left(6-\theta \right)$ , $0\le \theta \le 6$ .

d.

To determine

To find how the original curve changes if sine and cosine are interchanged.

Answer to Problem 69E

The curve starts at the point $\left(0,6\right)$ and move in clockwise direction.

Explanation of Solution

Given: The parametric equations $x=6\cos \theta$ and $y=6\sin \theta$ , $0\le \theta \le 6$ .

Calculation:

Interchange the sine and cosine to get the new set of equations as follows:

  $x=6\sin \theta$ and $y=6\cos \theta$ , $0\le \theta \le 6$ .

The graph of these equations can be given as shown below.

  

Now, observe that when $\theta =0$ , $x=0,y=6$ . That means, the curve starts at the point $\left(0,6\right)$ and move in clockwise direction.