Chapter 8.4, Problem 68E

(a)

To determine

To find: parametric equations and the time taken by the particle to revolve around the circle

Answer to Problem 68E

The parametric equations are

  $\begin{array}{l}x=\sin 2t\\ y=\cos 2t\end{array}$

The time taken by the particle to revolve around the circle is $2\pi$

Explanation of Solution

Given: $x=sint,y=cost$

Calculation:

Parametric equations contain more than just the shape of the curve. They also denote which direction the curve is traveling. Therefore if a position of a particle is determined by the following equations $x=sint,y=cost$ this set of equations shows which direction the particle is traveling based of different times t .

For example:

At $t=0$ the particle is at the point $\left(0,1\right)$ but at time $t=\pi /2$ the particle has moved to the point $\left(1,0\right)$ in a clockwise direction. So in order for the particle to travel a full rotation around the circle since the period of the parametric equations is $2\pi$ it will take the particle $t=2\pi$ to traverse the circle in a clockwise direction. In order to travel the circle twice as fast simply double the coefficient inside each trigonometric function and the result will be

  $\begin{array}{l}x=\sin 2t\\ y=\cos 2t\end{array}$

Conclusion:

Therefore, the parametric equations are

  $\begin{array}{l}x=\sin 2t\\ y=\cos 2t\end{array}$

The time taken by the particle to revolve around the circle is $2\pi$

(b)

To determine

To find: parametric equations and the justify whether the particle travel clockwise or counter clockwise

Answer to Problem 68E

Parametric equations is

  $\begin{array}{l}x=\cos t\\ y=\sin t\end{array}$

Explanation of Solution

Calculation:

The particle travels clockwise.

For example:

At $t=0$ the particle is at the point $\left(0,1\right)$ but at time $t=\pi /2$ the particle has moved to the point $\left(1,0\right)$ in a clockwise direction. In order to travel the opposite direction simply exchange the trigonometric functions.

  $\begin{array}{l}x=\cos t\\ y=\sin t\end{array}$

Conclusion:

Therefore, parametric equations is

  $\begin{array}{l}x=\cos t\\ y=\sin t\end{array}$