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Question
Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?
Solution
\[\text{ We have, x } = a\left( \theta + \sin\theta \right) \text{ and } y = a\left( 1 - \cos\theta \right)\]
\[\Rightarrow \frac{dx}{d\theta} = a\left( 1 + \cos\theta \right) \text{ and } \frac{dy}{d\theta} = a \sin\theta \]
\[\therefore \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a \sin\theta}{a\left( 1 + \cos\theta \right)} = \frac{2 \sin\frac{\theta}{2}\cos\frac{\theta}{2}}{2 \cos^2 \theta} = \tan\frac{\theta}{2}\]
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