Question

(a) If \( x = a(\theta + \sin \theta) \), \( y = a(1 - \cos \theta) \), find \( \frac{dy}{dx} \):

Hint:

For parametric equations, use \( \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \) and simplify carefully.

Answer 1

We are given: \[ x = a(\theta + \sin \theta), \quad y = a(1 - \cos \theta). \] Differentiating \( x \) and \( y \) with respect to \( \theta \): \[ \frac{dx}{d\theta} = a(1 + \cos \theta), \quad \frac{dy}{d\theta} = a\sin \theta. \] Using the chain rule, \( \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \): \[ \frac{dy}{dx} = \frac{a\sin \theta}{a(1 + \cos \theta)} = \frac{\sin \theta}{1 + \cos \theta}. \]