To determine the exact value of [tex]\(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\)[/tex] in radians, we can use our understanding of the tangent and inverse tangent functions.
1. Recognize the standard values and their inverses: We know that the tangent of certain key angles has specific values. One such value is [tex]\(\frac{\sqrt{3}}{3}\)[/tex].
2. Identity involving the tangent function: The tangent function for [tex]\(\frac{\pi}{6}\)[/tex] (30 degrees) gives us:
[tex]\[
\tan\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3}
\][/tex]
3. Inverse of the tangent function: Taking the inverse tangent of [tex]\(\frac{\sqrt{3}}{3}\)[/tex], we obtain:
[tex]\[
\tan^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{6}
\][/tex]
4. Odd function property: The tangent function is an odd function, meaning that [tex]\(\tan(-\theta) = -\tan(\theta)\)[/tex]. Therefore:
[tex]\[
\tan\left(-\frac{\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{3}
\][/tex]
5. Find the desired inverse tangent value: Given [tex]\(\tan\left(-\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{3}\)[/tex], the inverse tangent function gives us:
[tex]\[
\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right) = -\frac{\pi}{6}
\][/tex]
In conclusion, the exact value of [tex]\(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\)[/tex] in radians is:
[tex]\[
\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right) = -\frac{\pi}{6}
\][/tex]
