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The problem you shared asks to find the exact values of \( s \) in the interval \( [0, 2\pi] \) that satisfy the condition:

\[
\cos s = -\frac{1}{2}
\]

Let's solve it step by step:

1. **Step 1: Reference the unit circle**
   We know that \( \cos s = -\frac{1}{2} \) corresponds to angles where the cosine value is \( -\frac{1}{2} \). In the unit circle, cosine is negative in the second and third quadrants.

2. **Step 2: Identify the reference angle**
   The reference angle for \( \cos^{-1} \left( \frac{1}{2} \right) \) is \( \frac{\pi}{3} \). This is because \( \cos \frac{\pi}{3} = \frac{1}{2} \), so the reference angle will help us find the angles where cosine is negative.

3. **Step 3: Determine the angles**
   In the second quadrant, the angle that satisfies \( \cos s = -\frac{1}{2} \) is:
   \[
   s = \pi - \frac{\pi}{3} = \frac{2\pi}{3}
   \]
   In the third quadrant, the angle is:
   \[
   s = \pi + \frac{\pi}{3} = \frac{4\pi}{3}
   \]

4. **Step 4: Final Answer**
   The exact values of \( s \) in the interval \( [0, 2\pi] \) that satisfy \( \cos s = -\frac{1}{2} \) are:
   \[
   s = \frac{2\pi}{3}, \frac{4\pi}{3}
   \]

Would you like a more detailed explanation of any step or have any questions?

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Here are 5 related questions that can expand your understanding:
1. What is the cosine value of angles in the first and fourth quadrants?
2. How would the solution change if the equation were \( \cos s = \frac{1}{2} \)?
3. Can you explain how to derive reference angles for other trigonometric values?
4. What other methods could be used to solve \( \cos s = -\frac{1}{2} \)?
5. How does the unit circle help in solving trigonometric equations?

**Tip**: Remember, cosine represents the x-coordinate of a point on the unit circle, and this helps in determining whether the value is positive or negative depending on the quadrant.

Find the exact values of s in the interval [0, 2π] that satisfy the condition: cos s = -1/2

Learn how to find the exact values of s in the interval [0, 2π] that satisfy the equation cos s = -1/2. This trigonometry problem utilizes the unit circle and reference angles to find solutions, making it ideal for students in Grades 9-12.

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