Question

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

Hint:

For parametric differentiation, use the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}. \] Applying trigonometric identities simplifies the process.

Answer 1

Step 1: Differentiate \( x \) with respect to \( \theta \). Given: \[ x = a(1 - \cos\theta) \] Differentiate both sides with respect to \( \theta \): \[ \frac{dx}{d\theta} = a \sin\theta. \] Step 2: Differentiate \( y \) with respect to \( \theta \). Given: \[ y = a(\theta + \sin\theta) \] Differentiate both sides with respect to \( \theta \): \[ \frac{dy}{d\theta} = a(1 + \cos\theta). \] Step 3: Compute \( \frac{dy}{dx} \). Using the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a(1 + \cos\theta)}{a \sin\theta}. \] Simplify: \[ \frac{dy}{dx} = \frac{1 + \cos\theta}{\sin\theta}. \] Using the identity: \[ 1 + \cos\theta = 2\cos^2 \frac{\theta}{2}, \] and \[ \sin\theta = 2\sin\frac{\theta}{2} \cos\frac{\theta}{2}, \] we get: \[ \frac{dy}{dx} = \frac{2\cos^2\frac{\theta}{2}}{2\sin\frac{\theta}{2} \cos\frac{\theta}{2}}. \] Canceling common terms: \[ \frac{dy}{dx} = \frac{\cos\frac{\theta}{2}}{\sin\frac{\theta}{2}} = \cot\frac{\theta}{2}. \] Conclusion: The correct answer is \( \mathbf{\frac{\cot\theta}{2}} \).