<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
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\newcommand{\pl}{\partial}
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{\bf Question}

For the curve $$r = a \cos (\theta - \alpha),$$ show that $$ \phi
= \frac{1}{2} \pi + \theta - \alpha$$

Illustrate this result geometrically.


\vspace{.25in}

{\bf Answer}

$\ds r = a\cos (\theta-\alpha)$

$\cot \phi = \frac{1}{r} \frac{dr}{d\theta} =
\frac{-a\sin(\theta-\alpha)}{r} = -\tan(\theta-\alpha)$

So $\phi = \frac{\pi}{2} + \theta-\alpha$

\begin{center}
\epsfig{file=cg-3-1.eps, width=50mm}
\end{center}

This is the alternate segment theorem.


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