Question

Write the value of $2{\sin }^{-1}\frac{1}{2}+{\cos }^{-1}\left(-\frac{1}{2}\right)$.

Answer 1

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