Question

If 3 sin^-1x = π-cos^-1x, then x = __________________.

Answer 1

$$\begin{gathered} 3{\sin }^{-1}x=\mathrm{\pi }-{\cos }^{-1}x \\ \Rightarrow 3\left(\frac{\mathrm{\pi }}{2}-{\cos }^{-1}x\right)=\mathrm{\pi }-{\cos }^{-1}x\left({\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow \frac{3\mathrm{\pi }}{2}-3{\cos }^{-1}x=\mathrm{\pi }-{\cos }^{-1}x \end{gathered}$$
$$\begin{gathered} \Rightarrow 2{\cos }^{-1}x=\frac{3\mathrm{\pi }}{2}-\mathrm{\pi }=\frac{\mathrm{\pi }}{2} \\ \Rightarrow {\cos }^{-1}x=\frac{\mathrm{\pi }}{4} \\ \Rightarrow x=\cos \frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}} \end{gathered}$$

If 3sin⁻¹x = $\mathrm{\pi }$ − cos⁻¹x, then x = $\underline{\frac{1}{\sqrt{2}}}$.