To evaluate the expression [tex]\(\sin^{-1}\left(\cos\left(\frac{5\pi}{6}\right)\right)\)[/tex], we need to follow several steps, considering the properties and relationships of trigonometric functions and their inverses.
### Step-by-Step Solution:
1. Determine [tex]\(\cos\left(\frac{5\pi}{6}\right)\)[/tex]:
We start by evaluating the cosine of the angle [tex]\(\frac{5\pi}{6}\)[/tex].
[tex]\[
\cos\left(\frac{5\pi}{6}\right)
\][/tex]
The angle [tex]\(\frac{5\pi}{6}\)[/tex] lies in the second quadrant, where the cosine is negative. The reference angle for this is:
[tex]\[
\pi - \frac{5\pi}{6} = \frac{\pi}{6}
\][/tex]
In the second quadrant, the cosine of [tex]\(\frac{5\pi}{6}\)[/tex] is the negative of the cosine of [tex]\(\frac{\pi}{6}\)[/tex]:
[tex]\[
\cos\left(\frac{5\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right)
\][/tex]
We know that:
[tex]\[
\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\][/tex]
Therefore:
[tex]\[
\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}
\][/tex]
2. Evaluate [tex]\(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)[/tex]:
Now we need to find the value of [tex]\(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)[/tex].
The inverse sine function, [tex]\(\sin^{-1}(x)\)[/tex], gives us an angle whose sine is [tex]\(x\)[/tex]. We need to find the angle [tex]\(\theta\)[/tex] such that:
[tex]\[
\sin(\theta) = -\frac{\sqrt{3}}{2}
\][/tex]
The sine value [tex]\(-\frac{\sqrt{3}}{2}\)[/tex] corresponds to an angle in the fourth quadrant, where sine is negative. The reference angle corresponding to [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is:
[tex]\[
\theta = -\frac{\pi}{3}
\][/tex]
Since [tex]\(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)[/tex], we have:
[tex]\[
\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}
\][/tex]
### Final Answer:
Hence, the value of the expression [tex]\(\sin^{-1}\left(\cos\left(\frac{5\pi}{6}\right)\right)\)[/tex] is:
[tex]\[
\boxed{-\frac{\pi}{3}}
\][/tex]
In terms of approximate numerical values:
[tex]\[
-\frac{\pi}{3} \approx -1.0471975511965979
\][/tex]
Thus, the entire computation confirms that:
[tex]\[
\sin^{-1}\left(\cos\left(\frac{5\pi}{6}\right)\right) = -\frac{\pi}{3} \approx -1.0471975511965979
\][/tex]
