Step-by-Step Solution
Key Concept: Express linear combinations of sine and cosine in the form $R\sin(\theta + \phi)$ or $R\cos(\theta + \phi)$
Rewrite $\sqrt{3}\cos z - \sin z$ as $2(\frac{\sqrt{3}}{2}\cos z - \frac{1}{2}\sin z) = 2\cos(z + \frac{\pi}{6})$. For the equation $2\cos(z + \frac{\pi}{6}) = \pm 2$, we need $\cos(z + \frac{\pi}{6}) = \pm 1$. This gives $z + \frac{\pi}{6} = n\pi$, so $z = n\pi - \frac{\pi}{6}$. The sum of all roots is found by summing the individual solutions in the given interval, yielding a total of $10\pi - \frac{4\pi}{3} = \frac{26\pi}{3}$.
Correct Answer: 6