To solve the equation [tex]\(\cos^2(x) = \frac{1}{2}\)[/tex] for [tex]\(x \in [0, 2\pi]\)[/tex], follow these steps:
1. Simplify the Equation:
[tex]\[
\cos^2(x) = \frac{1}{2}
\][/tex]
By taking the square root of both sides, we obtain:
[tex]\[
\cos(x) = \pm \frac{1}{\sqrt{2}}
\][/tex]
Simplifying [tex]\(\pm \frac{1}{\sqrt{2}}\)[/tex]:
[tex]\[
\cos(x) = \pm \frac{\sqrt{2}}{2}
\][/tex]
2. Solve for [tex]\(x\)[/tex] with [tex]\(\cos(x) = \frac{\sqrt{2}}{2}\)[/tex]:
We need to find the angles where the cosine value is [tex]\(\frac{\sqrt{2}}{2}\)[/tex]. The values of [tex]\(x\)[/tex] in the interval [tex]\([0, 2\pi]\)[/tex] that satisfy [tex]\(\cos(x) = \frac{\sqrt{2}}{2}\)[/tex] are:
[tex]\[
x = \frac{\pi}{4} \quad \text{and} \quad x = \frac{7\pi}{4}
\][/tex]
3. Solve for [tex]\(x\)[/tex] with [tex]\(\cos(x) = -\frac{\sqrt{2}}{2}\)[/tex]:
Similarly, we need to find the angles where the cosine value is [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]. The values of [tex]\(x\)[/tex] in the interval [tex]\([0, 2\pi]\)[/tex] that satisfy [tex]\(\cos(x) = -\frac{\sqrt{2}}{2}\)[/tex] are:
[tex]\[
x = \frac{3\pi}{4} \quad \text{and} \quad x = \frac{5\pi}{4}
\][/tex]
4. Combine All Solutions:
Collecting all solutions, the values of [tex]\(x\)[/tex] that satisfy [tex]\(\cos^2(x) = \frac{1}{2}\)[/tex] in the interval [tex]\([0, 2\pi]\)[/tex] are:
[tex]\[
x = \frac{\pi}{4}, \frac{7\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}
\][/tex]
Therefore, the solutions to [tex]\(\cos^2(x) = \frac{1}{2}\)[/tex] for [tex]\(x \in [0, 2\pi]\)[/tex] are:
[tex]\[
x = \frac{\pi}{4}, \frac{7\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}
\][/tex]
