Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $ Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $
Can someone please explain to me how to go about doing this? 

$% Predefined Typography
 \newcommand{\paren} [1]{\left({#1}\right)}
 \newcommand{\polar}[2] {\left(#1 ~\angle~ #2\right)}
$

$$\sqrt{3}\sin(\theta) - \cos(\theta)$$

First write as $\cos() + \cos()$:
$$\sqrt{3}\sin(\theta) - \cos(\theta) = \sqrt{3}\cos\paren{\theta - \frac{\pi}2} + \cos(\theta + \pi)$$
Then add (polar coordinates) $\polar{\sqrt{3}}{-\frac{\pi}{2}} + \polar{1}{\pi}$ to get:
$$\polar{\sqrt{3}}{-\frac{\pi}{2}} + \polar{1}{\pi} = \polar{2}{-\frac{2\pi}{3}}$$
So
$$\sqrt{3}\cos\paren{\theta - \frac{\pi}2} + \cos(\theta + \pi) = 2\cos\paren{\theta - \frac{2\pi}{3}}$$
