Question

Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?

Answer 1

\[\text{ We have, x } = a\left( \theta + \sin\theta \right) \text{ and } y = a\left( 1 - \cos\theta \right)\]

\[\Rightarrow \frac{dx}{d\theta} = a\left( 1 + \cos\theta \right) \text{ and } \frac{dy}{d\theta} = a \sin\theta \]

\[\therefore \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a \sin\theta}{a\left( 1 + \cos\theta \right)} = \frac{2 \sin\frac{\theta}{2}\cos\frac{\theta}{2}}{2 \cos^2 \theta} = \tan\frac{\theta}{2}\]