To find the exact values of [tex]\( s \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] that satisfy the condition [tex]\(\cos^2(s) = \frac{3}{4}\)[/tex], follow these steps:
1. Solve for [tex]\(\cos(s)\)[/tex]:
Given [tex]\(\cos^2(s) = \frac{3}{4}\)[/tex], we take the square root of both sides to find:
[tex]\[
\cos(s) = \pm \sqrt{\frac{3}{4}}
\][/tex]
Simplifying the square root, we get:
[tex]\[
\cos(s) = \pm \frac{\sqrt{3}}{2}
\][/tex]
2. Identify the angles whose cosine value equals [tex]\(\frac{\sqrt{3}}{2}\)[/tex] or [tex]\(-\frac{\sqrt{3}}{2}\)[/tex]:
- For [tex]\(\cos(s) = \frac{\sqrt{3}}{2}\)[/tex], [tex]\( s \)[/tex] could be:
[tex]\[
s = \frac{\pi}{6} \quad \text{or} \quad s = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}
\][/tex]
- For [tex]\(\cos(s) = -\frac{\sqrt{3}}{2}\)[/tex], [tex]\( s \)[/tex] could be:
[tex]\[
s = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \quad \text{or} \quad s = \pi + \frac{\pi}{6} = \frac{7\pi}{6}
\][/tex]
3. List all the solutions within the interval [tex]\([0, 2\pi)\)[/tex]:
We have four values of [tex]\( s \)[/tex] that satisfy the condition:
[tex]\[
s = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}
\][/tex]
Thus, the exact values of [tex]\( s \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] that satisfy [tex]\(\cos^2(s) = \frac{3}{4}\)[/tex] are:
[tex]\[
\boxed{\frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}}
\][/tex]
