To determine the exact value of [tex]\(\arcsin \left(\sin \left(\frac{5 \pi}{4}\right)\right)\)[/tex], let's go through the problem in a systematic, step-by-step manner.
1. Calculate [tex]\(\sin \left(\frac{5\pi}{4}\right)\)[/tex]:
- The angle [tex]\(\frac{5\pi}{4}\)[/tex] lies in the third quadrant of the unit circle.
- The reference angle for [tex]\(\frac{5\pi}{4}\)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex].
- In the third quadrant, sine values are negative.
- Therefore, [tex]\(\sin \left(\frac{5\pi}{4}\right) = -\sin \left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex].
Hence,
[tex]\[
\sin \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
2. Determine [tex]\(\arcsin \left( -\frac{\sqrt{2}}{2} \right)\)[/tex]:
- The function [tex]\(\arcsin(x)\)[/tex] is defined as the inverse of the sine function in the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].
- We need to find the angle [tex]\(\theta\)[/tex] within the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex] for which [tex]\(\sin(\theta) = -\frac{\sqrt{2}}{2}\)[/tex].
- The angle that satisfies [tex]\(\sin(\theta) = -\frac{\sqrt{2}}{2}\)[/tex] within the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex] is [tex]\(-\frac{\pi}{4}\)[/tex].
Hence,
[tex]\[
\arcsin\left(\sin\left(\frac{5\pi}{4}\right)\right) = \arcsin\left(-\frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4}
\][/tex]
Therefore, the exact value of [tex]\(\arcsin \left(\sin \left(\frac{5 \pi}{4}\right)\right)\)[/tex] is
[tex]\[
\boxed{-\frac{\pi}{4}}
\][/tex]
