Question

The range of sin^-1x + cos^-1x + tan^-1x is _______________________.

Answer 1

Domain of the given function = $\left[-1,1\right]\cap \mathbf{R}$ = $\left[-1,1\right]$

Now,

For $-1\le x\le 1$,

${\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2}$ and $-\frac{\mathrm{\pi }}{4}\le {\tan }^{-1}x\le \frac{\mathrm{\pi }}{4}$

$\therefore \frac{\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{4}\le {\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x\le \frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{4}$

$\Rightarrow \frac{\mathrm{\pi }}{4}\le {\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x\le \frac{3\mathrm{\pi }}{4}$

Thus, the range of the given function is $\left[\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4}\right]$.

The range of sin⁻¹x + cos⁻¹x + tan⁻¹x is $\underline{\left[\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4}\right]}$.