Question

If sin^-1x = $\frac{\mathrm{\pi }}{5}$ for some x ∊ (-1, 1), then the value of cos^-1 x is ____________________.

Answer 1

Given: ${\sin }^{-1}x=\frac{\mathrm{\pi }}{5},\forall x\in \left(-1,1\right)$

We know

$$\begin{gathered} {\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2} \\ \Rightarrow \frac{\mathrm{\pi }}{5}+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2} \\ \Rightarrow {\cos }^{-1}x=\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{5}=\frac{3\mathrm{\pi }}{10} \end{gathered}$$

Thus, the value of cos⁻¹x is $\frac{3\mathrm{\pi }}{10}$.

If sin⁻¹x = $\frac{\mathrm{\pi }}{5}$ for some x ∈ (−1, 1), then the value of cos⁻¹ x is $\underline{\frac{3\mathrm{\pi }}{10}}$.