To find the value of [tex]\( s \)[/tex] in the interval [tex]\(\left[\frac{3\pi}{2}, 2\pi\right]\)[/tex] where [tex]\(\cos(s) = \frac{\sqrt{2}}{2}\)[/tex], we can follow these detailed steps:
1. Identify the general angles where [tex]\(\cos(s) = \frac{\sqrt{2}}{2}\)[/tex]:
The cosine function equals [tex]\(\frac{\sqrt{2}}{2}\)[/tex] at specific known reference angles. These angles are:
[tex]\[
s = \frac{\pi}{4} \quad \text{and} \quad s = \frac{7\pi}{4}
\][/tex]
2. Determine which angle lies within the specified interval:
We need to check each of these angles to see if they fall within the interval [tex]\(\left[\frac{3\pi}{2}, 2\pi\right]\)[/tex]:
- The angle [tex]\(\frac{\pi}{4}\)[/tex] is in the first quadrant, and [tex]\(\frac{\pi}{4}\)[/tex] is less than [tex]\(\frac{3\pi}{2}\)[/tex], so it does not lie in the desired interval.
- The angle [tex]\(\frac{7\pi}{4}\)[/tex] can be expressed in radians. Since:
[tex]\[
2\pi = \frac{8\pi}{4}
\][/tex]
and since:
[tex]\[
\frac{3\pi}{2} = \frac{6\pi}{4}
\][/tex]
we can see that:
[tex]\[
\frac{7\pi}{4}
\][/tex]
lies between [tex]\(\frac{6\pi}{4}\)[/tex] and [tex]\(\frac{8\pi}{4}\)[/tex], so it falls within the interval [tex]\(\left[\frac{3\pi}{2}, 2\pi\right]\)[/tex].
3. Conclusion:
Therefore, the exact value of [tex]\( s \)[/tex] in the interval [tex]\(\left[\frac{3\pi}{2}, 2\pi\right]\)[/tex] that satisfies [tex]\(\cos(s) = \frac{\sqrt{2}}{2}\)[/tex] is:
[tex]\[
s = \frac{7\pi}{4}
\][/tex]
So, the solution is:
[tex]\[
s = \frac{7\pi}{4} \text{ radians}
\][/tex]
