Step 1
(a) Note that, the position of the particle is given by the parametric equations \(x =\ \sin t,\ and\ y =\ \cos\ t\).
The parametric equations contain more than just shape of the curve. They also represent the direction of curve as traveling. If a position of a particle is determined by the equation \(x =\ \sin\ t,\ y =\ \cos\ t,\) this set of equations denotes which direction the particle is traveling based on different times t.
For example, at \(t = 0,\ \text{the particle is at the point}\ (0,\ 1)\ \text{but at time}\ t=\ \frac{\pi}{2}\ \text{the particle has moved to the point}\ (1,\ 0)\) in a clockwise direction
As the period of the parametric equations is \(2\ \pi\), to find for the particle to travel a full rotation around the circle.
It will take the time \(t = 2\ \pi\) to traverse the circle in a clockwise direction.
To travel the circle twice as fast simply double the coefficient inside each trigonometric function and the parametric equations are \(x =\ \sin\ 2t,\ y =\ \cos\ 2t.\)
Thus, the time that will be taken by the particle to go once around the circle is \(t = 2\ \pi\ \text{and the parametric equations, the particle moves twice as fast around the circle are}\ x =\ \sin\ 2t,\ y =\ \cos\ 2t.\)
Step 2
(b) Note that, the particle travels clockwise.
For example, at \(t = 0,\ \text{the particle is at the point}\ (0,\ 1),\ \text{but at the time}\ t =\ \frac{\pi}{2}\ \text{the particle has moved to the point}\ (1,\ 0)\) in a clockwise direction.
The parametric equations when the particle travels in the opposite direction, the parametric equations will be exchanged.
That are, \(x =\ \cos\ t,\ y =\ \sin\ t.\)
Thus, the particle travels clockwise and if the particle travels in opposite direction around the circle, the parametric equations are \(x =\ \cos\ t,\ y =\ \sin\ t.\)