Question

If sin⁻¹ x + sin⁻¹ y + sin⁻¹ z = $\frac{3\mathrm{\pi }}{2}$, then write the value of x + y + z.

Answer 1

$$\begin{gathered} {\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{3\mathrm{\pi }}{2} \\ \Rightarrow {\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2}\left[\mathrm{As}\mathrm{the}\mathrm{maximum}\mathrm{value}\mathrm{in}\mathrm{the}\mathrm{range}\mathrm{of}{\sin }^{-1}x\mathrm{is}\frac{\mathrm{\pi }}{2} \\ \mathrm{And}\mathrm{here}\mathrm{sum}\mathrm{of}\mathrm{three}\mathrm{inverse}\mathrm{of}\mathrm{sine}\mathrm{is}3\mathrm{times}\frac{\mathrm{\pi }}{2}. \\ \mathrm{i}.\mathrm{e}.,\mathrm{every}\sin \mathrm{inverse}\mathrm{function}\mathrm{is}\mathrm{equal}\mathrm{to}\frac{\mathrm{\pi }}{2}\mathrm{here}.\right] \\ \Rightarrow {\sin }^{-1}x=\frac{\mathrm{\pi }}{2},{\sin }^{-1}y=\frac{\mathrm{\pi }}{2}\mathrm{and}{\sin }^{-1}z=\frac{\mathrm{\pi }}{2} \\ \Rightarrow x=1,y=1\mathrm{and}z=1 \\ \therefore x+y+z=1+1+1=3 \end{gathered}$$