    Question

# The principal value of ${\mathrm{cos}}^{-1}\left(-\frac{1}{2}\right)$ is equal to

Open in App
Solution

## Finding the principal value of ${\mathrm{cos}}^{-1}\left(-\frac{1}{2}\right)$:As we know that, ${\mathrm{cos}}^{-1}\left(-x\right)=\mathrm{\pi }-{\mathrm{cos}}^{-1}\left(x\right)$$\therefore {\mathrm{cos}}^{-1}\left(-\frac{1}{2}\right)=\mathrm{\pi }-{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right).....\left(i\right)$Now, Let, $\begin{array}{rcl}{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)& =& y\end{array}$$\begin{array}{rcl}& ⇒& \mathrm{cos}\left(y\right)=\frac{1}{2}\\ \mathrm{cos}\left(y\right)& =& \mathrm{cos}\left(\frac{\mathrm{\pi }}{3}\right)\\ & ⇒& y=\frac{\mathrm{\pi }}{3}\end{array}$So, $\begin{array}{rcl}{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)& =& \frac{\mathrm{\pi }}{3}\end{array}$$\begin{array}{rcl}\therefore {\mathrm{cos}}^{-1}\left(-\frac{1}{2}\right)& =& \mathrm{\pi }-{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)\left(fromeq.\left(i\right)\right)\\ & =& \mathrm{\pi }-\frac{\mathrm{\pi }}{3}\\ & =& \frac{2\mathrm{\pi }}{3}\end{array}$Hence, the principal value of ${\mathrm{cos}}^{-1}\left(-\frac{1}{2}\right)$ is $\frac{2\mathrm{\pi }}{3}$.  Suggest Corrections  0      Similar questions  Related Videos   Implicit Differentiation
MATHEMATICS
Watch in App  Explore more