Question

# Write a short paragraph explaining this statement. Use the following example and your answers How long does it take the particle to go once around the

Write a short paragraph explaining this statement. Use the following example and your answers How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. The position of a particle is given by the parametric equations $$x = \sin t, y = \cos t$$ where 1 represents time. We know that the shape of the path of the particle is a circle.

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 $$x = \sin 2t, y = \cos 2t$$.