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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.


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

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This is the alternate segment theorem.


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