(a) Note that, the position of the particle is given by the parametric equations .
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 this set of equations denotes which direction the particle is traveling based on different times t.
For example, at in a clockwise direction
As the period of the parametric equations is , to find for the particle to travel a full rotation around the circle.
It will take the time 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
Thus, the time that will be taken by the particle to go once around the circle is
(b) Note that, the particle travels clockwise.
For example, at in a clockwise direction.
The parametric equations when the particle travels in the opposite direction, the parametric equations will be exchanged.
Thus, the particle travels clockwise and if the particle travels in opposite direction around the circle, the parametric equations are
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