To solve for \(\theta\) when \(\cos \theta = -0.5\), follow these steps:

1. The cosine of an angle is negative in the second and third quadrants. This means that \(\theta\) lies in one of these quadrants.
2. Find the reference angle: 
   \[
   \theta_{\text{ref}} = \cos^{-1}(0.5)
   \]
   \[
   \theta_{\text{ref}} = 60^\circ \text{ or } \frac{\pi}{3} \text{ radians}
   \]
3. Use this reference angle to find \(\theta\) in the second and third quadrants:
   - In the second quadrant: \(\theta = 180^\circ - 60^\circ = 120^\circ\) or \(\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\)
   - In the third quadrant: \(\theta = 180^\circ + 60^\circ = 240^\circ\) or \(\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}\)

Thus, the solutions are:
\[
\theta = 120^\circ \text{ or } 240^\circ \quad \text{or} \quad \theta = \frac{2\pi}{3} \text{ or } \frac{4\pi}{3}
\]

Would you like further details on these steps or more information?

Here are 5 related questions:
1. How do we determine which quadrant an angle lies in?
2. What is the cosine function's range and periodicity?
3. Can we solve \(\cos \theta = -0.5\) for more angles beyond \(0^\circ\) to \(360^\circ\)?
4. How does \(\cos^{-1}\) (inverse cosine) function work on a calculator?
5. What are the applications of cosine in real-world problems?

**Tip:** Always make sure your calculator is in the correct mode (degrees or radians) when solving trigonometric equations!

Discover how to solve cos(θ) = -0.5 step by step by finding the reference angle and determining the solution in both the second and third quadrants. This problem covers trigonometric concepts including the cosine function and inverse cosine, suitable for high school students in Grades 9-12.

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