To solve the equation \( \cos(x) = -0.5 \), we need to find the values of \( x \) where the cosine of \( x \) equals \(-0.5\).

1. **General solution for cosine:**

The cosine function is periodic with a period of \( 2\pi \), meaning that solutions repeat every \( 2\pi \). Cosine equals \(-0.5\) at specific angles.

We know that:
\[
\cos(\pi - \frac{\pi}{3}) = \cos\left(\frac{2\pi}{3}\right) = -0.5
\]
and cosine is also negative in the third quadrant:
\[
\cos(\pi + \frac{\pi}{3}) = \cos\left(\frac{4\pi}{3}\right) = -0.5
\]

Thus, the general solutions for \( x \) are:
\[
x = \frac{2\pi}{3} + 2n\pi \quad \text{or} \quad x = \frac{4\pi}{3} + 2n\pi
\]
where \( n \) is any integer.

2. **Principal solutions:**

For specific values of \( x \) within one period, the solutions are:
\[
x = \frac{2\pi}{3}, \quad x = \frac{4\pi}{3}
\]

Would you like more details on trigonometric solutions or any clarification?

Here are five related questions to expand on this:

1. How does the periodic nature of cosine influence its solutions?
2. What is the importance of reference angles in trigonometry?
3. How would you solve \( \sin(x) = -0.5 \)?
4. Can you explain why cosine is negative in the second and third quadrants?
5. How can the unit circle be used to verify the solutions?

**Tip:** Always check the quadrant when solving trigonometric equations to determine the correct sign of the function!

This guide explains how to solve cos(x) = -0.5 using trigonometric identities and periodic properties of the cosine function. Learn about the unit circle, quadrant rules, and periodicity for solving trigonometric equations. Suitable for students in Grades 10-12.

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