Find the exact value of [tex]\( s \)[/tex] in the given interval that has the given circular function value. Do not use a calculator.

[tex]\[
\left[0, \frac{\pi}{2}\right] ; \sin s=\frac{\sqrt{2}}{2}
\][/tex]

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

(Simplify your answer. Type an exact answer, using [tex]\(\pi\)[/tex] as needed. Use integers or fractions for any numbers.)



Answer :

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].