Practicing Success
$sin^{-1}\begin{Bmatrix}sin \left(\frac{2x^2+4}{x^2+1}\right)\end{Bmatrix} < \pi - 3,$ if |
x ∈ [-1, 0] x ∈ [0, 1] x ∈ (-1, 1) x ∈ [1, ∞] |
x ∈ (-1, 1) |
We have, $\frac{2x^2+4}{x^2+1}=\frac{2(x^2+1)+2}{x^2+1}= 2+\frac{2}{x^2+1}$ $∴ 2 < \frac{2x^2+4}{x^2+1}$ ≤ 4 for all x ∈ R $⇒ \pi - 4 < \pi -\frac{2x^2+4}{x^2+1} < \pi - 2 $ for all x ∈ R $ -\frac{\pi}{2} < \pi - \frac{2x^2+4}{x^2+1}<\frac{\pi}{2}$ for all x ∈ R $∴ sin^{-1}\begin{Bmatrix}sin\left(\frac{2x^2+4}{x^2+1}\right)\end{Bmatrix}= sin^{-1} \begin{Bmatrix}sin\left(\pi - \frac{2x^2+4}{x^2+1}\right)\end{Bmatrix}$ $= \pi - \frac{2x^2+4}{x^2-1}$ Hence, $ sin^{-1}\begin{Bmatrix}sin\left(\frac{2x^2+4}{x^2+1}\right)\end{Bmatrix} , \pi - 3 $ $ ⇒ \pi - \frac{2x^2+4}{x^2+1} < \pi - 3 $ $ ⇒ \frac{2x^2+4}{x^2+1} > 3 $ $ ⇒ 2+\frac{2}{x^2+1} ⇒ 2 > 3 > x^2 + 1 ⇒ x^2 -1 < 0 ⇒ x ∈ (-1, 1)$ |