To solve the equation $\cos(2\theta) = -\frac{\sqrt{3}}{2}$, let's break it down step by step:

Step 1: Identify the reference angle

We know that $\cos(\theta) = -\frac{\sqrt{3}}{2}$ corresponds to angles where the cosine function is negative. In the unit circle, the cosine value of $-\frac{\sqrt{3}}{2}$ happens at specific reference angles.

$\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6} \quad \text{and} \quad \frac{7\pi}{6}$

This means that for $\cos(2\theta) = -\frac{\sqrt{3}}{2}$, we have two possible reference angles for $2\theta$: $2\theta = \frac{5\pi}{6} + 2k\pi \quad \text{or} \quad 2\theta = \frac{7\pi}{6} + 2k\pi$ where $k$ is any integer, accounting for the periodic nature of the cosine function.

Step 2: Solve for $\theta$

Now, we solve for $\theta$ by dividing both sides of each equation by 2: $\theta = \frac{5\pi}{12} + k\pi \quad \text{or} \quad \theta = \frac{7\pi}{12} + k\pi$ where $k$ is any integer.

Step 3: General formula

Thus, the general solution to $\cos(2\theta) = -\frac{\sqrt{3}}{2}$ is: $\theta = \frac{5\pi}{12} + k\pi \quad \text{or} \quad \theta = \frac{7\pi}{12} + k\pi$ where $k$ is any integer.

Summary

The general solution for all values of $\theta$ that satisfy $\cos(2\theta) = -\frac{\sqrt{3}}{2}$ is: $\theta = \frac{5\pi}{12} + k\pi \quad \text{or} \quad \theta = \frac{7\pi}{12} + k\pi$ where $k \in \mathbb{Z}$.

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Would you like more details on any part of this process or have additional questions? Here are a few related questions to explore:

1. How does the unit circle relate to the cosine function?
2. What is the impact of different values of $k$ on the solutions?
3. Can you derive similar general solutions for other trigonometric equations like $\sin(2\theta) = \frac{\sqrt{2}}{2}$?
4. How do you determine the general solution for sine functions in a similar way?
5. What are the periodic properties of trigonometric functions?

Tip: Always remember that the general solution to a trigonometric equation involves considering both the reference angles and the periodic nature of the trigonometric functions.