Question

find a parametrization for the circle of radius 8, centered on the origin, in the clockwise direction from the point (0, -8) to (-8, 0). use t as the parameter for your answers.
x(t) =
y(t) =
for ≤ t ≤
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Explanation:

Step1: Recall the general parametric form of a circle

The general parametric equations for a circle of radius $r$ centered at the origin are $x = r\cos\theta$ and $y = r\sin\theta$. For a clock - wise orientation, we use negative angles. Here $r = 8$, so initially $x=8\cos(-\theta)$ and $y = 8\sin(-\theta)$. Let $\theta=t$.

Step2: Find the starting and ending values of the parameter

The point $(0, - 8)$ corresponds to $\theta=-\frac{\pi}{2}$ and the point $(-8,0)$ corresponds to $\theta =-\pi$ in the clock - wise direction. So the range of $t$ is $-\frac{\pi}{2}\leq t\leq-\pi$.

Step3: Write the parametric equations

Since $x = 8\cos(-t)=8\cos t$ and $y = 8\sin(-t)=-8\sin t$.

Answer:

$x(t)=8\cos t$
$y(t)=-8\sin t$
for $-\frac{\pi}{2}\leq t\leq-\pi$