To find the exact value of [tex]\(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)[/tex] in radians, let's go through the following steps:
1. Understand the function: The expression [tex]\(\sin^{-1}(x)\)[/tex], also known as [tex]\(\arcsin(x)\)[/tex], denotes the angle [tex]\(\theta\)[/tex] such that [tex]\(\sin(\theta) = x\)[/tex]. This angle [tex]\(\theta\)[/tex] lies in the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].
2. Recognize the sine value: We need to find the angle [tex]\(\theta\)[/tex] for which [tex]\(\sin(\theta) = -\frac{\sqrt{3}}{2}\)[/tex].
3. Identify the reference angle: The reference angle for [tex]\(-\frac{\sqrt{3}}{2}\)[/tex] is the positive angle that would give us [tex]\(\frac{\sqrt{3}}{2}\)[/tex]. We know that:
[tex]\[
\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
4. Determine the correct quadrant: Since [tex]\(\sin(\theta) = -\frac{\sqrt{3}}{2}\)[/tex], [tex]\(\theta\)[/tex] must be in the fourth quadrant and the angle should be negative because the arcsine function's range is [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].
5. Find the exact angle in radians: Reflecting the angle [tex]\(\frac{\pi}{3}\)[/tex] in the fourth quadrant gives us [tex]\(-\frac{\pi}{3}\)[/tex].
Thus, the exact value of [tex]\(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)[/tex] in radians in terms of [tex]\(\pi\)[/tex] is:
[tex]\[
\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}
\][/tex]
Therefore, we have:
[tex]\[
\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}
\][/tex]
