Answer
\[f(\theta)=\sin\theta\;,\; a=\frac{\pi}{6}\]
and \[\lim_{\theta\rightarrow \pi/5}\frac{\sin\theta-\frac{1}{2}}{\theta-\frac{\pi}{6}}=\frac{\sqrt 3}{2}\]
Work Step by Step
Let \[L=\lim_{\theta\rightarrow \pi/5}\frac{\sin\theta-\frac{1}{2}}{\theta-\frac{\pi}{6}}\]
Compare $L$ with definition of derivative:
\[f'(a)=\lim_{\theta\rightarrow a}\frac{f(\theta)-f(a)}{\theta-a}\]
\[\Rightarrow f(\theta)=\sin\theta\;,\; a=\frac{\pi}{6}\]
and \[L=f'(\frac{\pi}{6})\;\;\;\;\;\;\ldots (1)\]
\[f(\theta)=\sin\theta\Rightarrow f'(\theta)=\cos\theta\]
\[\Rightarrow f'(\frac{\pi}{6})=\frac{\sqrt{3}}{2}\]
From (1)
\[L=\frac{\sqrt{3}}{2}\]
