Question

If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \] 

Hint:

When differentiating parametric equations, use the chain rule \( \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \).

Answer 1

Step 1: Differentiate both \( x \) and \( y \) with respect to \( \theta \): \[ \frac{dx}{d\theta} = a \cdot (- \cos \theta), \] \[ \frac{dy}{d\theta} = a \cdot (- \sin \theta). \] Step 2: Use the chain rule to find \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{-a \sin \theta}{-a \cos \theta} = \frac{\sin \theta}{\cos \theta} = \tan \theta. \] Thus, \( \frac{dy}{dx} = \tan \theta \).