Question: Find the principal value of \begin{align} cos^{-1}\left(-\frac{1}{2}\right)\end{align}

Answer:

\begin{align} Let\;\; cos^{-1}\left(-\frac{1}{2}\right)=y, \;\;Then,\;\; cos y = -\frac{1}{2} = - cos\left(\frac{\pi}{3}\right)=cos\left(\pi - \frac{\pi}{3}\right) = cos\left(\frac{2\pi}{3}\right)\end{align}

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

\begin{align} \left[0,\pi\right] and \;\;cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\end{align}

Therefore, the principal value of

\begin{align} cos^{-1}\left(-\frac{1}{2}\right) is \frac{2\pi}{3}\end{align}

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Question: Find the principal value of \begin{align} cos^{-1}\left(-\frac{1}{2}\right)\end{align}

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