#### Answer

$f(\theta)=\sin\theta$ and $a=\frac{\pi}{6}$

#### Work Step by Step

*Another way to write the derivative of a function $f$ at a number $a$ is $$f'(a)=\lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}\hspace{0.5cm}(1)$$
Here we have
$$f'(a)=\lim\limits_{\theta\to\pi/6}\frac{\sin\theta-\frac{1}{2}}{\theta-\pi/6}$$
$$f'(a)=\lim\limits_{\theta\to\pi/6}\frac{\sin\theta-\sin{\pi/6}}{\theta-\pi/6}$$
Now we match the formula found above with the formula of the derivative according to (1).
We find that $a=\frac{\pi}{6}$, $f(\frac{\pi}{6})=f(a)=\sin{\frac{\pi}{6}}$ and $f(\theta)=\sin{\theta}$