To find the exact value of [tex]\( s \)[/tex] in the interval [tex]\(\left[0, \frac{\pi}{2}\right]\)[/tex] where [tex]\(\sin s = \frac{\sqrt{2}}{2}\)[/tex], follow these steps:
1. Recall Known Sine Values:
We need to identify the angle [tex]\( s \)[/tex] whose sine value is [tex]\(\frac{\sqrt{2}}{2}\)[/tex]. From the unit circle and trigonometric values, we know that:
[tex]\[
\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
2. Identify the Interval:
The interval given is [tex]\(\left[0, \frac{\pi}{2}\right]\)[/tex]. Within this interval, [tex]\(\frac{\pi}{4}\)[/tex] (or 45 degrees) is the angle whose sine value is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
3. Express in Radians:
Since [tex]\(\frac{\pi}{4}\)[/tex] lies within the interval [tex]\(\left[0, \frac{\pi}{2}\right]\)[/tex], we have:
[tex]\[
s = \frac{\pi}{4}
\][/tex]
Thus, the exact value of [tex]\( s \)[/tex] that satisfies [tex]\(\sin s = \frac{\sqrt{2}}{2}\)[/tex] within the interval [tex]\(\left[0, \frac{\pi}{2}\right]\)[/tex] is:
[tex]\[
s = \frac{\pi}{4} \text{ radians}
\][/tex]
Alternatively, the numerical value of [tex]\(\frac{\pi}{4}\)[/tex] in decimal form is approximately [tex]\(0.7853981633974483\)[/tex] radians. So the angle [tex]\( s \)[/tex] can also be expressed as:
[tex]\[
s \approx 0.7853981633974483 \text{ radians}
\][/tex]
Therefore, [tex]\( s = \frac{\pi}{4} \text{ radians}\)[/tex].
