To solve [tex]\(\sin^{-1}\left(\cos \left(\frac{2 \pi}{3}\right)\right)\)[/tex], we need to approach the problem step-by-step.
### Step 1: Calculate [tex]\(\cos \left(\frac{2 \pi}{3}\right)\)[/tex]
The angle [tex]\(\frac{2 \pi}{3}\)[/tex] radians is in the second quadrant. In the second quadrant:
[tex]\[ \cos \left(\frac{2 \pi}{3}\right) = -\cos \left(\pi - \frac{2 \pi}{3}\right) \][/tex]
[tex]\[ \cos \left(\pi - \frac{2 \pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) \][/tex]
We know that:
[tex]\[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]
So:
[tex]\[ \cos \left(\frac{2 \pi}{3}\right) = -\frac{1}{2} \][/tex]
Therefore:
[tex]\[ \cos \left(\frac{2 \pi}{3}\right) = -0.4999999999999998 \][/tex]
### Step 2: Find [tex]\(\sin^{-1}(-0.5)\)[/tex]
Now, we need to calculate the inverse sine (arcsine) of [tex]\(-0.5\)[/tex]. The inverse sine function, [tex]\(\sin^{-1}(x)\)[/tex], returns values in the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].
We need to find an angle [tex]\(\theta\)[/tex] such that:
[tex]\[ \sin(\theta) = -0.5 \][/tex]
that also lies in the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].
We know that:
[tex]\[ \sin \left(- \frac{\pi}{6}\right) = -\frac{1}{2} \][/tex]
So:
[tex]\[ \sin^{-1}\left(-\frac{1}{2}\right) = - \frac{\pi}{6} \][/tex]
Therefore:
[tex]\[ \sin^{-1}\left(\cos \left(\frac{2 \pi}{3}\right)\right) = - \frac{\pi}{6} = -0.5235987755982986 \][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{-\pi}{6}}
\][/tex]
