How do I convert this parametric expression to an implicit one I have:
$$x=5+8 \cos \theta$$
$$y=4+8 \sin \theta$$
With $ -\frac {3\pi}4 \le \theta \le 0$
If I wanted to write that implicitly, how would I do it? I get that it's a circle, and I can easily write the circle implicitly, but I'm not sure how to convert the domain from $ -\frac {3\pi}4 \le \theta \le 0$ to a suitable domain in x and y.
I can think of a way to do it that would involve using the $Min$ function, but is that allowed in an implicit representation?

Use that $\exists \theta : (v,w) = (\cos\theta, \sin\theta) \iff v^2 + w^2 = 1$.
You immediately get that the parametric curve is a part of the curve defined by
the implicit equation
$$
\left(\frac{x-5}8\right)^2 + \left(\frac{y-4}8\right)^2 = 1$$
To get only the part that you want, you must also make sure that $-\frac{3\pi}4\le \theta\le  0$, which is equivalent to
$$
\cos \theta \ge -\frac 12, \sin\theta< 0
$$
and then you get the equation
\begin{cases}
\left(\frac{x-5}8\right)^2 + \left(\frac{y-4}8\right)^2 = 1\\
\frac{x-5}8 \ge -\frac 12\\
\frac{y-4}8 \le 0
\end{cases}
