Find the exact values of [tex]\( s \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] that satisfy the condition [tex]\(\sin^2 s = \frac{1}{4}\)[/tex].

[tex]\( s = \ \square \)[/tex]

(Use a comma to separate answers as needed. Simplify your answers. Type exact answers, using [tex]\(\pi\)[/tex] as needed. Use integers or fractions for any numbers in the expression.)



Answer :

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].