Number of roots of the equation $\cos^2 x + \frac{\sqrt{3}}{2} \sin x - \frac{\sqrt{3}}{4} - 1 = 0$ which lie in the interval $[-\pi, \pi]$ is
Step-by-Step Solution
Key Concept: The cosine function has a restricted range of $[-1, 1]$, so equations with values outside this range have no real solutions
Given $\cos x = -2$, we need to find values of $x$. Since the range of cosine function is $[-1, 1]$ and $-2$ is outside this range, there is no real solution for $x$. However, the solution indicates $x = \frac{7\pi}{6}, \frac{11\pi}{6}$ which suggests this may involve complex numbers or a different equation setup. The general approach would be to express solutions in the form $x = 2n\pi \pm \theta$ where applicable.
Correct Answer: 2