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The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter."

$$x=\sec\theta\text{ and }y=\cos\theta$$

In the answer key, $0\le \theta < \pi/2$ and $\pi/2 < \theta \le \pi$. What about the angle on the unit circle that are in the third and fourth quadrant?

Also, in the answer key, $|x|\ge 1$ and $|y|\le 1$ Why are $x$ and $y$ restricted in such a way?

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It's the curve $y=1/x$ restricted to those values of $y$ that can be $\cos\theta$, or equivalently, those values of $x$ that can be $\sec\theta$.

Remember that $-1\le\cos\theta\le 1$, and $\sec\theta$ is always either $\ge 1$ or $\le-1$.

As $\theta$ goes from $0$ to $\pi$, $\cos\theta$ goes all the way from $1$ down to $-1$. If we went beyond $\pi$, we'd see it going through the same set of values again, from $-1$ back up to $+1$. We would not get any new values of $x$ or $y$ that we didn't get the first time.

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