Given: The position of the particle is given by the parametric equations, \(x = \sin t, y = \cos t\) Calculation: The parametric equations contain more than just shape of the curve. They also represents the direction of curve as traveling. If a position of a particle is determined by the equations \(x = \sin t\),

\(y = \cos t\), this set of equations denotes which direction the particle is traveling based on different times r. For example, \(at = 0\) the particle is at the point (0, 1) but at time \(t = \frac{\pi}{2}\) 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 \(x = \sin 2t, y = \cos 2t\). Conclusion: Hence, the time that will be taken by the particle to go once around the circle is \(t = 2\pi\) and the parametric equations, the particle moves twice as fast around the circle are \(x = \sin 2t, y = \cos 2t\).