Given $x = a(\theta - \sin\theta)$, $y = a(1 - \cos\theta)$ (cycloid). Then $\dfrac{dy}{dx}$ equals:
A$\cos\theta$
B$\tan(\theta/2)$
C$\sin\theta$
D$\cot(\theta/2)$
Answer & Solution
Correct answer: D. $\cot(\theta/2)$
$dx/d\theta = a(1 - \cos\theta) = 2a\sin^2(\theta/2)$. $dy/d\theta = a\sin\theta = 2a\sin(\theta/2)\cos(\theta/2)$. Ratio = $\cos(\theta/2)/\sin(\theta/2) = \cot(\theta/2)$.
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