If $x = a (t - sin t), y = a (1 - cos t),$ find $d y / d x$ . at $t = π$

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Solution

$x = a (t - sin t), y = a (1 - cos t),$
$∴ \frac{d y}{d t} = a (1 - cos t), \frac{d y}{d t} = a sin t$
$∴ \frac{d y}{d x} = \frac{˙ y}{˙ x} = \frac{a sin t}{a (1 - cos t)} = cot \frac{t}{2}$
$∴ y^{'} (π) = cot (\frac{π}{2}) = 0$

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