To evaluate [tex]\(\text{arccsc}(\sqrt{2})\)[/tex], we need to first recall what the arccsc function represents. The [tex]\(\text{arccsc}(x)\)[/tex] function gives the angle [tex]\(\theta\)[/tex] such that [tex]\(\csc(\theta) = x\)[/tex].
The cosecant function is defined as the reciprocal of the sine function, i.e., [tex]\(\csc(\theta) = \frac{1}{\sin(\theta)}\)[/tex]. Therefore, finding [tex]\(\text{arccsc}(\sqrt{2})\)[/tex] means finding the angle [tex]\(\theta\)[/tex] for which:
[tex]\[
\csc(\theta) = \sqrt{2}
\][/tex]
This implies:
[tex]\[
\sin(\theta) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
We now need to determine all angles [tex]\(\theta\)[/tex] within the principal range of the cosecant function, specifically for [tex]\(\theta \in [0, \pi]\)[/tex], where [tex]\(\sin(\theta) = \frac{\sqrt{2}}{2}\)[/tex].
We know that the sine function achieves the value [tex]\(\frac{\sqrt{2}}{2}\)[/tex] at the angles:
[tex]\[
\theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{3\pi}{4}
\][/tex]
Therefore, the angles [tex]\(\theta\)[/tex] that satisfy [tex]\(\csc(\theta) = \sqrt{2}\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{3\pi}{4}
\][/tex]
Thus, the solution to [tex]\(\text{arccsc}(\sqrt{2})\)[/tex] is:
[tex]\[
\left\{ \frac{\pi}{4}, \frac{3\pi}{4} \right\}
\][/tex]
The given options are:
A. [tex]\(\left\{\frac{\pi}{6}, \frac{5 \pi}{6}\right\}\)[/tex]
B. [tex]\(\left\{\frac{\pi}{3} \pm 2 \pi n, \frac{5 \pi}{3} \pm 2 \pi n\right\}\)[/tex]
C. [tex]\(\frac{\pi}{3}\)[/tex]
Since none of these options directly match the set of angles [tex]\(\left\{\frac{\pi}{4}, \frac{3\pi}{4}\right\}\)[/tex], it seems there might be a misunderstanding or an error in the provided answer choices. Therefore, the correct answer based on our evaluation is:
[tex]\[
\left\{\frac{\pi}{4}, \frac{3\pi}{4}\right\}
\][/tex]
