Chapter 3: Derivatives - Section 3.5 - Derivatives of Trigonometric Functions - Exercises 3.5 - Page 142: 49

$\frac{\sqrt 3}{2}$

$\lim_{\theta\to\pi/6}\frac{sin\theta-1/2}{\theta-\pi/6}=\lim_{\theta\to\pi/6}\frac{sin\theta-sin(\pi/6)}{\theta-\pi/6}=(sin\theta)'|_{\theta=\pi/6}=(cos\theta)|_{\theta=\pi/6}=cos(\pi/6)=\frac{\sqrt 3}{2}$