Question

The set of all $x$ in $(-\pi ,\pi )$ satisying $|4\cos x-1|<\sqrt{5}$ is

• A. $(-\pi ,\frac{-4\pi }{5})\cup (\frac{4\pi }{5},\pi )$
• B. $(-\frac{4\pi }{5},-\frac{\pi }{5})\cup (\frac{\pi }{5},\frac{4\pi }{5})$
• C. $(-\frac{3\pi }{5},-\frac{\pi }{5})\cup (\frac{\pi }{5},\frac{3\pi }{5})$
• D. $(-\pi ,-\frac{\pi }{5})\cup (\frac{\pi }{5},\frac{3\pi }{5})$

Answer 1

The correct option is C $(-\frac{3\pi }{5},-\frac{\pi }{5})\cup (\frac{\pi }{5},\frac{3\pi }{5})$

$-\sqrt{5}<4\cos x-1<\sqrt{5},\frac{(1-\sqrt{5})}{4}<\cos x<\frac{(1+\sqrt{5})}{4}$
Here in $(-\pi ,\pi )$
$\cos (\pm \frac{\pi }{5})=\frac{(1+\sqrt{5})}{4}\cos (\pm \frac{3\pi }{5})=\frac{(1-\sqrt{5})}{4}$

From the graph $x\in (-\frac{3\pi }{5},-\frac{\pi }{5})\cup (\frac{\pi }{5},\frac{3\pi }{5})$