To find the exact value of [tex]\(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)\)[/tex], we need to determine the angle whose sine is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
First, recall that the sine function returns the y-coordinate of a point on the unit circle. We need to find the angle where the y-coordinate is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
The unit circle shows that [tex]\(\sin(\theta) = \frac{\sqrt{2}}{2}\)[/tex] for specific angles. Specifically, we know that:
[tex]\[
\sin(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
45 degrees is in the first quadrant, which is within the range of the inverse sine function [tex]\(\sin^{-1}\)[/tex], where the output is restricted to angles between [tex]\(-90^\circ\)[/tex] and [tex]\(90^\circ\)[/tex]. This confirms that 45 degrees is a valid answer.
We should also consider the answer choices:
a. [tex]\(45^\circ\)[/tex]
b. [tex]\(150^\circ\)[/tex]
c. [tex]\(30^\circ\)[/tex]
d. [tex]\(315^\circ\)[/tex]
Given that the inverse sine function only returns values in the first and fourth quadrants (angles between [tex]\(-90^\circ\)[/tex] and [tex]\(90^\circ\)[/tex]), the angles 150 degrees and 315 degrees are not in this range:
- 150 degrees is in the second quadrant.
- 315 degrees is in the fourth quadrant, but it corresponds to a negative angle of [tex]\(-45^\circ\)[/tex], which is not a valid answer given the sine value.
Thus, the correct and exact value of [tex]\(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)\)[/tex] is:
\[
\boxed{45^\circ}
