Question

For $x \in[0,3 \pi]$, the number of integral values of $x$ satisfying $\sec ^{2} x-4 \tan x < 0$ is

Answer 1

$\sec ^{2} x-4 \tan x < 0$
$\Rightarrow 1+\tan ^{2} x-4 \tan x < 0$

$\Rightarrow 2-\sqrt{3} < \tan x < 2+\sqrt{3} $
$\Rightarrow n \pi+\frac{\pi}{12} < x < \frac{5 \pi}{12}+n \pi, n \in I $
$\Rightarrow \frac{\pi}{12} < x < \frac{5 \pi}{12}, \frac{13 \pi}{12} < x < \frac{17 \pi}{12}, \frac{25 \pi}{12} < x < \frac{29 \pi}{12}$
$\therefore $ Integral values of $x$ can be $1,4$ and $7$ .