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]\[ \left[ \frac{3\pi}{2}, 2\pi \right]; \cos s = \frac{\sqrt{2}}{2} \][/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 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]