Question

If \( x = a \left( \cos \theta + \log \tan \frac{\theta}{2} \right) \) and \( y = \sin \theta \), then find \( \frac{d^2y}{dx^2} \) at \( \theta = \frac{\pi}{4} \).

Hint:

When computing second derivatives, always apply the chain rule carefully, and be sure to evaluate at the given point for precise results.

Answer 1

The given equations are: \[ x = a \left( \cos \theta + \log \tan \frac{\theta}{2} \right) \] and \[ y = \sin \theta. \] First, differentiate \( y = \sin \theta \) with respect to \( \theta \): \[ \frac{dy}{d\theta} = \cos \theta. \] Next, differentiate \( x = a \left( \cos \theta + \log \tan \frac{\theta}{2} \right) \) with respect to \( \theta \): \[ \frac{dx}{d\theta} = a \left( -\sin \theta + \frac{1}{\tan \frac{\theta}{2}} \cdot \frac{1}{2 \cos^2 \frac{\theta}{2}} \right). \] To compute \( \frac{d^2y}{dx^2} \), apply the chain rule: \[ \frac{d^2y}{dx^2} = \frac{\frac{d}{d\theta} \left( \frac{dy}{d\theta} \right)}{\frac{dx}{d\theta}}. \] Substitute the values at \( \theta = \frac{\pi}{4} \). At \( \theta = \frac{\pi}{4} \), we have \( \sin \theta = \frac{\sqrt{2}}{2} \), \( \cos \theta = \frac{\sqrt{2}}{2} \), and \( \tan \frac{\theta}{2} = 1 \). Evaluate: \[ \frac{d^2y}{dx^2} \bigg|_{\theta = \frac{\pi}{4}}. \] This evaluation yields the final result.