Question

find a parametrization for the circle of radius 4, centered on the origin, counterclockwise from the point (0, 4) to (-4, 0). use t as the parameter for your answers.

$x(t)=$

$y(t)=$

for $leq tleq$

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Explanation:

Step1: Recall circle - parametric form

The general parametric equations for a circle of radius $r$ centered at the origin are $x = r\cos(t)$ and $y = r\sin(t)$. Here $r = 4$, so $x(t)=4\cos(t)$ and $y(t)=4\sin(t)$.

Step2: Determine the range of $t$

We start at the point $(0,4)$. When $x = 0$ and $y = 4$, we have $4\cos(t)=0$ and $4\sin(t)=4$. Solving $\sin(t) = 1$, we get $t=\frac{\pi}{2}$.
We end at the point $(-4,0)$. When $x=-4$ and $y = 0$, we have $4\cos(t)=-4$ and $4\sin(t)=0$. Solving $\cos(t)=-1$, we get $t=\pi$.

Answer:

$x(t)=4\cos(t)$
$y(t)=4\sin(t)$
for $\frac{\pi}{2}\leq t\leq\pi$