To find the exact value of [tex]\(\cos^{-1}\left(\cos\left(\frac{8\pi}{13}\right)\right)\)[/tex], we need to understand that the [tex]\(\cos^{-1}\)[/tex] function, or arccosine, returns values in the principal range [tex]\([0, \pi]\)[/tex].
### Step-by-Step Solution:
1. Initial Angle:
[tex]\[
x = \frac{8\pi}{13}
\][/tex]
2. Adjustment to Principal Value:
Since [tex]\(\arccos(\cos(x))\)[/tex] returns the principal value in the range [tex]\([0, \pi]\)[/tex], we need to determine where [tex]\(\frac{8\pi}{13}\)[/tex] falls in relation to [tex]\([0, \pi]\)[/tex].
3. Comparison with [tex]\(\pi\)[/tex]:
To do this, compare [tex]\(\frac{8\pi}{13}\)[/tex] with [tex]\(\pi\)[/tex]:
[tex]\[
\frac{8\pi}{13} \quad vs \quad \pi
\][/tex]
We know:
[tex]\[
\pi \approx 3.14159265359
\][/tex]
Calculate:
[tex]\[
\frac{8\pi}{13} = \frac{8 \times 3.14159265359}{13} \approx 1.933287786824488
\][/tex]
4. Adjusting the Angle:
Since [tex]\(\frac{8\pi}{13} \approx 1.933287786824488\)[/tex] is less than [tex]\(\pi\)[/tex], it already lies within the interval [tex]\([0, \pi]\)[/tex]. Hence, we do not need to adjust it further.
5. Conclusion:
The angle [tex]\(\frac{8\pi}{13}\)[/tex] is already within the principal range for the [tex]\(\cos^{-1}\)[/tex] function. Therefore:
[tex]\[
\cos^{-1}\left(\cos\left(\frac{8\pi}{13}\right)\right) = \frac{8\pi}{13}
\][/tex]
### Final Answer:
[tex]\(\cos^{-1}\left(\cos\left(\frac{8\pi}{13}\right)\right) = \frac{8\pi}{13}\)[/tex]
