Find the principal value of ${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$.

Step-by-step solution

1. Let $\theta ={\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$. By definition $\theta \in [0,\pi ]$ and $\cos \theta =-\frac{\sqrt{3}}{2}$.
2. The reference angle whose cosine is $\frac{\sqrt{3}}{2}$ is $\pi /6$.
3. Since the cosine is negative and the principal value must lie in $[0,\pi ]$, $\theta$ is in the second quadrant.
4. Therefore $\theta =\pi -\frac{\pi }{6}=\frac{5\pi }{6}$.

Concept tested: principal value of inverse cosine

Published 6 September 2026 · How this solution was prepared and checked