Question

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

Answer 1

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

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

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

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

\[ \Rightarrow \frac{dx}{d\theta} = a\theta \cos\theta \text{ and} \frac{dy}{d\theta} = a\theta \sin\theta\]

\[ \therefore \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a\theta \sin\theta}{a\theta \cos\theta} = \tan\theta\]