Question

The set of all $x$ in $(-\pi ,\pi )$ satisfying $|4sinx-1|<\sqrt{5}$ is $xϵ(-\pi ,-\frac{k\pi }{10})\cup (-\frac{\pi }{10},\frac{\pi }{10})\cup (\frac{k\pi }{10},\pi )$. Find the value of $k$.

Answer 1

$|4sin\left(x\right)-1|\le \sqrt{5}$

$-\sqrt{5}\le 4sin\left(x\right)-1\le \sqrt{5}$

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

Now

$sin\left({18}^0\right)=\frac{\sqrt{5}-1}{4}$ and $sin{54}^0=\frac{\sqrt{5}+1}{4}$.

Hence

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

Hence

$xϵ(-\pi ,-(\pi -\frac{\pi }{10}),\cup (\frac{-\pi }{10},\frac{\pi }{10})\cup ((\pi -\frac{\pi }{10}),\pi )$
$xϵ(-\pi ,-\frac{9\pi }{10}),\cup (\frac{-\pi }{10},\frac{\pi }{10})\cup (\frac{9\pi }{10},\pi )$

Hence $K=9$.