Question

решите уравнение 3arctg^2x-2piactgx-p^2=0

Answer 1

Ответ:

$3arctg^2x-2\pi \cdot arctgx-\pi ^2=0\\\\t=arctgx\ \ ,\ \ -\dfrac{\pi}{2}<t<\dfrac{\pi}{2}\ \ , \ \ \ 3t^2-2\pi \, t-\pi ^2=0\ ,\\\\D\4=\pi ^2+3\pi ^2=4\pi ^2\ \ \Rightarrow \ \ t_1=\dfrac{\pi +2\pi }{3}=\pi \ \ ,\ \ t_2=\dfrac{\pi -2\pi }{3}=-\dfrac{\pi }{3}\\\\\\t_1=\pi \notin \Big[-\dfrac{\pi}{2}\, ;\, \dfrac{\pi}{2}\, \Big]\\\\\\arctgx=-\dfrac{\pi }{3}\ \ \to \ \ \ tg\Big(arctx\Big)=tg\Big(-\dfrac{\pi}{3}\Big)\ \ ,\ \ \ x=-\sqrt3$