Однородные тригонометрические уравнения. Задача 12

ОТВЕТ: \(\dfrac{\pi }{2} + \pi k;\quad \dfrac{\pi }{3} + \pi k;\quad \,k \in Z.\)

\(\sqrt 3 {\cos ^2}x = \cos x\sin x\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\sqrt 3 {\cos ^2}x-\cos x\sin x = 0\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,\cos x\left( {\sqrt 3 \cos x-\sin x} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos x = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sqrt 3 \cos x-\sin x = 0.}\end{array}} \right.\)

\(\cos x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \dfrac{\pi }{2} + \pi k,\,\,\,\,\,\,\,\,k\, \in \,Z.\)

Уравнение \(\sqrt 3 \cos x-\sin x = 0\) является однородным тригонометрическим уравнением первой степени:

\(\sqrt 3 \cos x-\sin x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\sqrt 3 -{\rm{tg}}\,x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{tg}}\,x = \sqrt 3 \,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \dfrac{\pi }{3} + \pi k,\,\,\,\,\,\,k\, \in \,Z.\)

Ответ:  \(\dfrac{\pi }{2} + \pi k;\quad \dfrac{\pi }{3} + \pi k;\quad \,k \in Z.\)