To determine the exact value of [tex]\(\arctan \left(-\frac{\sqrt{3}}{3}\right)\)[/tex], we should first recall the meaning of the arctan (inverse tangent) function. The arctan function returns the angle whose tangent is the given number, and it typically outputs values in the interval [tex]\(-\frac{\pi}{2}\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex].
Given that we need to find [tex]\(\arctan (\text{value})\)[/tex] for [tex]\(\text{value} = -\frac{\sqrt{3}}{3}\)[/tex], we should look for an angle in [tex]\(-\frac{\pi}{2}\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] whose tangent equals [tex]\(-\frac{\sqrt{3}}{3}\)[/tex].
Now, let's consider the standard angles and their tangent values:
- [tex]\(\tan \left(-\frac{\pi}{6}\right) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}\)[/tex]
Given this information:
[tex]\[
\tan \left(-\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{3}
\][/tex]
Thus:
[tex]\[
\arctan \left(-\frac{\sqrt{3}}{3}\right) = -\frac{\pi}{6}
\][/tex]
Now we compare this result with the provided choices:
1) [tex]\(-\frac{2 \pi}{3}\)[/tex]
2) [tex]\(-\frac{\pi}{6}\)[/tex]
3) [tex]\(\frac{5 \pi}{6}\)[/tex]
4) [tex]\(-\frac{\pi}{4}\)[/tex]
5) [tex]\(\frac{\pi}{3}\)[/tex]
6) [tex]\(\frac{2 \pi}{3}\)[/tex]
Indeed, the correct value from the choices given that matches [tex]\(\arctan \left(-\frac{\sqrt{3}}{3}\right)\)[/tex] is:
[tex]\[
\boxed{-\frac{\pi}{6}}
\][/tex]
