Find the exact values of s in the interval left bracket 0 comma 2 pi right parenthesis that satisfy the given condition cosine squared s equals one half
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We are asked to find the exact values of \( s \) in the interval \( [0, 2\pi) \) that satisfy the equation:

\[
\cos^2(s) = \frac{1}{2}
\]

### Step 1: Take the square root of both sides
We know that \( \cos^2(s) = \frac{1}{2} \) implies:

\[
\cos(s) = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}
\]

So we need to find the values of \( s \) where \( \cos(s) = \frac{\sqrt{2}}{2} \) and \( \cos(s) = -\frac{\sqrt{2}}{2} \).

### Step 2: Identify the angles
The cosine of \( s \) is positive in the first and fourth quadrants, and negative in the second and third quadrants.

1. For \( \cos(s) = \frac{\sqrt{2}}{2} \), the reference angle is \( \frac{\pi}{4} \). Thus, the solutions are:
   - \( s = \frac{\pi}{4} \) (in the first quadrant)
   - \( s = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \) (in the fourth quadrant)

2. For \( \cos(s) = -\frac{\sqrt{2}}{2} \), the reference angle is still \( \frac{\pi}{4} \). Thus, the solutions are:
   - \( s = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \) (in the second quadrant)
   - \( s = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \) (in the third quadrant)

### Step 3: List the solutions
The exact values of \( s \) that satisfy the equation \( \cos^2(s) = \frac{1}{2} \) in the interval \( [0, 2\pi) \) are:

\[
s = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}
\]

Would you like further clarification on any part of the solution? Here are 5 related questions to explore:

1. What is the unit circle and how does it relate to solving trigonometric equations?
2. How do you find reference angles in different quadrants?
3. What are some general methods for solving trigonometric equations?
4. What is the significance of the interval \( [0, 2\pi) \) in trigonometry?
5. How can you solve other trigonometric equations like \( \sin^2(s) = \frac{1}{2} \)?

**Tip:** Memorizing the key angles and their trigonometric values on the unit circle makes solving such problems much easier.

Find the exact values of s in the interval [0, 2π) that satisfy the given condition: cos^2(s) = 1/2.

Find the exact values of s that satisfy cos^2(s) = 1/2 in the interval [0, 2π). This solution explores trigonometric concepts like cosine function, reference angles, and quadrants. Suitable for high school students in Grades 10-12.

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