To solve the problem of finding the exact value of the inverse sine function where [tex]\(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)\)[/tex], we need to utilize the properties of the unit circle and the specific range for the inverse sine function.
1. Understanding the Inverse Sine Function:
The inverse sine function, [tex]\(\sin^{-1}\)[/tex] (also known as arcsin), returns the angle whose sine value is the given input. The range of [tex]\(\sin^{-1}\)[/tex] is limited to [tex]\([-90^{\circ}, 90^{\circ}]\)[/tex] or [tex]\([-\frac{\pi}{2}, \frac{\pi}{2}]\)[/tex] in radians. This means that we only consider angles in quadrants I and IV.
2. Exact Value on the Unit Circle:
We need to find an angle [tex]\(\theta\)[/tex] in the range [tex]\([-90^{\circ}, 90^{\circ}]\)[/tex] such that [tex]\(\sin(\theta) = \frac{\sqrt{2}}{2}\)[/tex].
3. Identifying the Angle:
On the unit circle, the sine of the angle represents the y-coordinate of the point where the terminal side of the angle intersects the unit circle. We are given [tex]\(\sin(\theta) = \frac{\sqrt{2}}{2}\)[/tex].
4. Common Angles:
We recall from trigonometry that [tex]\(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\)[/tex] and that [tex]\(45^{\circ}\)[/tex] (or [tex]\(\frac{\pi}{4}\)[/tex] radians) is within the first quadrant, which is within the range of the inverse sine function.
5. Checking Provided Options:
- Option a: [tex]\(45^{\circ}\)[/tex]. We've established that [tex]\(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\)[/tex], which is correct.
- Option b: [tex]\(150^{\circ}\)[/tex]. While [tex]\(\sin(150^{\circ}) = \frac{\sqrt{2}}{2}\)[/tex] is true, this angle is in the second quadrant and outside the range for [tex]\(\sin^{-1}\)[/tex].
Thus, the correct answer is:
a. [tex]\(45^{\circ}\)[/tex].
