Parametric Equations and Curves — Question 5

Question 5

Problem

Two students parametrize the unit circle by $(\cos t,\sin t)$ and $(\cos2t,-\sin2t)$ on $0\le t\le2\pi$. Compare orientation and number of traversals.

Question 5 – Solution

Solution

1. For the first parametrization, $$(x,y)=(\cos t,\sin t),$$ the polar angle is $t$. As $t$ increases from $0$ to $2\pi$, the angle increases through one full revolution.

2. It begins at $(1,0)$, then moves through $(0,1)$, $(-1,0)$, and $(0,-1)$ before returning to $(1,0)$. Therefore, it traces the circle counterclockwise exactly once.

3. Rewrite the second parametrization as $$(x,y)=(\cos(2t),-\sin(2t)) =(\cos(-2t),\sin(-2t)).$$ Its polar angle is therefore $-2t$.

4. As $t$ increases from $0$ to $2\pi$, the angle decreases from $0$ to $-4\pi$. A decreasing angle produces clockwise motion, and a total angular change of $4\pi$ represents two complete revolutions.

5. Thus $$\boxed{(\cos t,\sin t)\text{ travels counterclockwise once,}}$$ whereas $$\boxed{(\cos2t,-\sin2t)\text{ travels clockwise twice.}}$$