tan x로 치환하는 삼각부등식

$\tan x = t$라 하면 $\displaystyle t ^{2} - ( 1 + \sqrt{3} ) t + \sqrt{3} < 0$ $\displaystyle ( t - 1 ) ( t - \sqrt{3} ) < 0$이므로 $\displaystyle 1 < t < \sqrt{3}$ 즉 $\displaystyle 1 < \tan x < \sqrt{3}$ $\displaystyle \frac{\pi}{2} < x < \pi$에서는 $\tan x < 0$이므로 해가 없고, $\displaystyle 0 < x < \frac{\pi}{2}$에서 $\displaystyle \frac{\pi}{4} < x < \frac{\pi}{3}$ 따라서 $\displaystyle \alpha = \frac{\pi}{4}$, $\displaystyle \beta = \frac{\pi}{3}$이므로 $\displaystyle \beta - \alpha = \frac{\pi}{12}$