Question: Find the principal value of \begin{align} tan^{-1}\left(-\sqrt3\right)\end{align}

Answer:

\begin{align} Let \;\; tan^{-1}\left(-\sqrt3\right)=y \;\;Then\;\; tan y = -\sqrt3 = -tan\left(\frac{\pi}{3}\right)= tan\left(-\frac{\pi}{3}\right)\end{align}

We know that the range of the principal value branch of tan⁻¹ is

\begin{align} \left(-\frac{\pi}{2},\frac{\pi}{2}\right) and \;\;tan\left(-\frac{\pi}{3}\right) = -\sqrt3\end{align}

Therefore, the principal value of

\begin{align} tan^{-1}\left(-\sqrt 3\right) is -\frac{\pi}{3}\end{align}

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Question: Find the principal value of \begin{align} tan^{-1}\left(-\sqrt3\right)\end{align}

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