12В. а) Решите уравнение \({\rm{t}}{{\rm{g}}^2}x + \left( {1 + \sqrt 3 } \right){\rm{tg}}\,x{\rm{ + }}\sqrt 3 = 0\);
б) Найдите все корни принадлежащие промежутку \(\left[ {\dfrac{{5\pi }}{2};4\pi } \right].\)
a) \({\rm{t}}{{\rm{g}}^2}x + \left( {1 + \sqrt 3 } \right){\rm{tg}}\,x + \sqrt 3 = 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,{\rm{t}}{{\rm{g}}^2}x + {\rm{tg}}\,x + \sqrt 3 {\rm{tg}}\,x + \sqrt 3 = 0\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,{\rm{tg}}\,x\left( {{\rm{tg}}\,x + 1} \right) + \sqrt 3 \left( {{\rm{tg}}\,x + 1} \right) = 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left( {{\rm{tg}}\,x + 1} \right)\left( {{\rm{tg}}\,x + \sqrt 3 } \right) = 0\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{\rm{tg }}x = — 1,\;\;}\\{{\rm{tg }}x = — \sqrt 3 }\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = — \dfrac{\pi }{4} + \pi k,}\\{x = — \dfrac{\pi }{3} + \pi k,}\end{array}} \right.\,\,\,\,\,\,k \in Z.\) \(x = — \dfrac{\pi }{3} + 3\pi = \dfrac{{8\pi }}{3};\,\,\,x = — \dfrac{\pi }{3} + 4\pi = \dfrac{{11\pi }}{3};\) \(x = — \dfrac{\pi }{4} + 3\pi = \dfrac{{11\pi }}{4};\,\,\,\,x = — \dfrac{\pi }{4} + 4\pi = \dfrac{{15\pi }}{4}.\) Ответ: а) \( — \dfrac{\pi }{4} + \pi k,\,\,\,\,\,\, — \dfrac{\pi }{3} + \pi k,\,\,\,\,\,\,k\, \in \,Z;\) б) \(\dfrac{{8\pi }}{3};\,\,\,\dfrac{{11\pi }}{4};\;\;\,\dfrac{{11\pi }}{3};\;\;\dfrac{{15\pi }}{4}.\)
б) Отберём корни, принадлежащие отрезку \(\left[ {\dfrac{{5\pi }}{2};4\pi } \right],\) с помощью тригонометрической окружности. Получим значения: