To find the exact value of [tex]\( s \)[/tex] in the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex] such that [tex]\(\sin s = \frac{1}{2}\)[/tex], we will use our knowledge of the unit circle and trigonometric functions.
1. Identify known angles with [tex]\(\sin\)[/tex] values of [tex]\(\frac{1}{2}\)[/tex]:
The sine function has a value of [tex]\(\frac{1}{2}\)[/tex] at [tex]\(\theta = \frac{\pi}{6}\)[/tex] (or 30 degrees) and [tex]\(\theta = \pi - \frac{\pi}{6}\)[/tex] because [tex]\(\sin (\pi - x) = \sin x\)[/tex].
2. Locate these angles on the unit circle:
- [tex]\(\theta = \frac{\pi}{6}\)[/tex]: This angle is in the first quadrant.
- [tex]\(\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\)[/tex]: This angle is in the second quadrant.
3. Check the intervals:
We are given the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex].
- [tex]\(\frac{\pi}{6}\)[/tex] is not within this interval as it lies in the first quadrant.
- [tex]\(\frac{5\pi}{6}\)[/tex] is within the interval from [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex].
4. Confirm the sine value in the second quadrant:
- At [tex]\(\theta = \frac{5\pi}{6}\)[/tex], [tex]\(\sin \left(\frac{5\pi}{6}\right) = \sin \left(\pi - \frac{\pi}{6}\right) = \sin \left(\frac{\pi}{6}\right) = \frac{1}{2}\)[/tex].
Thus, the exact value of [tex]\( s \)[/tex] in the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex] such that [tex]\(\sin s = \frac{1}{2}\)[/tex] is:
[tex]\[
s = \frac{5\pi}{6}
\][/tex]
This is the simplified exact form using [tex]\(\pi\)[/tex].
