Question

How do you find the exact value of ${\cos }^{-1}\left(-\frac{1}{2}\right)$?

Answer 1

Compute the value of the given expression.

Given: ${\cos }^{-1}\left(-\frac{1}{2}\right)$

Using identity ${\cos }^{-1}(-x)=\mathrm{\pi }-{\cos }^{-1}\left(\mathrm{x}\right)$.

$\Rightarrow {\cos }^{-1}\left(-\frac{1}{2}\right)=\mathrm{\pi }-{\cos }^{-1}\left(\frac{1}{2}\right)$

Us we know that ${\cos }^{-1}\left(\frac{1}{2}\right)=\frac{\mathrm{\pi }}{3}$.

$$\begin{gathered} {\cos }^{-1}\left(-\frac{1}{2}\right)=\mathrm{\pi }-\frac{\mathrm{\pi }}{3} \\ \Rightarrow {\cos }^{-1}\left(-\frac{1}{2}\right)=\frac{2\mathrm{\pi }}{3} \end{gathered}$$

Hence, the exact value of the expression ${\cos }^{-1}\left(-\frac{1}{2}\right)$ is $\frac{2\mathrm{\pi }}{3}$ .