To find the exact value of [tex]\(\tan^{-1}(-\sqrt{3})\)[/tex], let's proceed step by step.
1. Understanding [tex]\(\tan^{-1}(x)\)[/tex]: The function [tex]\(\tan^{-1}(x)\)[/tex], also known as the arctangent function, returns the angle whose tangent is [tex]\(x\)[/tex]. We need to find the angle [tex]\(\theta\)[/tex] such that [tex]\(\tan(\theta) = -\sqrt{3}\)[/tex].
2. Identifying common tangents: Values of [tex]\(\theta\)[/tex] that have a tangent involving [tex]\(\sqrt{3}\)[/tex] are typically [tex]\(\frac{\pi}{3}\)[/tex] and [tex]\(\frac{2\pi}{3}\)[/tex], but these correspond to [tex]\(\sqrt{3}\)[/tex] and [tex]\(-\sqrt{3}/3\)[/tex], respectively.
3. Finding [tex]\(\theta\)[/tex] with [tex]\(\tan(\theta) = -\sqrt{3}\)[/tex]: We need to locate an angle where the tangent function gives [tex]\(-\sqrt{3}\)[/tex]. This happens at an angle where the sine and cosine functions yield a ratio of [tex]\(-\sqrt{3}\)[/tex]. From trigonometric identities and the unit circle:
- [tex]\(\tan\left(-\frac{\pi}{3}\right) = \tan\left(2\pi - \frac{\pi}{3}\right)= \tan\left( \frac{5\pi}{3}\right) = -\sqrt{3}\)[/tex]
- This suggests that one solution lies at [tex]\(\theta = -\frac{\pi}{3}\)[/tex] within the principal range of the arctangent function [tex]\((- \frac{\pi}{2}, \frac{\pi}{2})\)[/tex].
4. Conclusion: Since the arctangent function returns values in the range of [tex]\((-\frac{\pi}{2}, \frac{\pi}{2})\)[/tex] and the appropriate angle corresponding to [tex]\(\tan(\theta) = -\sqrt{3}\)[/tex] in that range is [tex]\(-\frac{\pi}{3}\)[/tex], we conclude:
[tex]\[
\tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3}
\][/tex]
Hence, the exact value of [tex]\(\tan^{-1}(-\sqrt{3})\)[/tex] is [tex]\(-\frac{\pi}{3}\)[/tex].
