To find the exact values of [tex]\( s \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] that satisfy the equation [tex]\(\sin^2(s) = \frac{1}{4}\)[/tex], let's go through the steps methodically:
1. Given Equation:
[tex]\[
\sin^2(s) = \frac{1}{4}
\][/tex]
2. Solve for [tex]\(\sin(s)\)[/tex]:
[tex]\[
\sin(s) = \pm \frac{1}{2}
\][/tex]
This means [tex]\(\sin(s)\)[/tex] can be either [tex]\(\frac{1}{2}\)[/tex] or [tex]\(-\frac{1}{2}\)[/tex].
3. Identify the Angles in the Interval [tex]\([0, 2\pi)\)[/tex]:
- When [tex]\(\sin(s) = \frac{1}{2}\)[/tex], the angles [tex]\( s \)[/tex] within [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
s = \frac{\pi}{6}, \quad s = \frac{5\pi}{6}
\][/tex]
- When [tex]\(\sin(s) = -\frac{1}{2}\)[/tex], the angles [tex]\( s \)[/tex] within [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
s = \frac{7\pi}{6}, \quad s = \frac{11\pi}{6}
\][/tex]
4. Collect all Solutions:
Therefore, the solutions [tex]\( s \)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
s = \frac{\pi}{6}, \quad \frac{5\pi}{6}, \quad \frac{7\pi}{6}, \quad \frac{11\pi}{6}
\][/tex]
5. Write the Final Answer:
[tex]\[
s = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}
\][/tex]
So, the exact values of [tex]\( s \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] that satisfy [tex]\(\sin^2(s) = \frac{1}{4}\)[/tex] are:
[tex]\(\boxed{\frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}}\)[/tex].
