Question

The set of all $x\in (-\pi ,\pi )$ satisfying $|4\sin x-1|<\sqrt{5}$ is given by

• A. $(-\frac{\pi }{10},\frac{3\pi }{10})$
• B. $(\frac{\pi }{10},\frac{3\pi }{10})$
• C. $(\frac{\pi }{10},-\frac{3\pi }{10})$
• D. None of these

Answer 1

Answer: (A)

$(-\frac{\pi }{10},\frac{3\pi }{10})$

We are given the equation $|4\sin x-1|<\sqrt{5}$

$\Rightarrow -\sqrt{5}<4\sin x-1<\sqrt{5}$

$\Rightarrow -\left(\frac{\sqrt{5}-1}{4}\right)<\sin x<\left(\frac{\sqrt{5}+1}{4}\right)$

$\Rightarrow -\sin \frac{\pi }{10}<\sin x<\cos \frac{\pi }{5}$

$\Rightarrow \sin (-\frac{\pi }{10})<\sin x<\sin (\frac{\pi }{2}-\frac{\pi }{5})$

$\Rightarrow \sin (-\frac{\pi }{10})<\sin x<\sin \left(\frac{3\pi }{10}\right)$

$\Rightarrow x\in (-\frac{\pi }{10},\frac{3\pi }{10})$