Question

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

Answer 1

${\cos }^{-1}\left(-x\right)=\mathrm{\pi }-{\cos }^{-1}x,x\in \left[-1,1\right]$
Now, we have

$$\begin{gathered} {\sin }^{-1}\left(\frac{1}{2}\right)+{\cos }^{-1}\left(-\frac{1}{2}\right)={\sin }^{-1}\left(\frac{1}{2}\right)+\mathrm{\pi }-{\cos }^{-1}\left(\frac{1}{2}\right) \\ ={\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{6}\right)+\mathrm{\pi }-{\cos }^{-1}\left(\cos \frac{\mathrm{\pi }}{3}\right) \\ =\frac{\mathrm{\pi }}{6}+\mathrm{\pi }-\frac{\mathrm{\pi }}{3} \\ =\frac{5\mathrm{\pi }}{6} \end{gathered}$$
∴ ${\sin }^{-1}\left(\frac{1}{2}\right)+{\cos }^{-1}\left(-\frac{1}{2}\right)=\frac{5\mathrm{\pi }}{6}$