Однородные тригонометрические уравнения. Задача 11

Задача 11. Решите уравнение: \({\sin ^2}x-\sin x\cos x = 0\)

Ответ

ОТВЕТ: \(\pi k;\quad \dfrac{\pi }{4} + \pi k;\quad \,k \in Z.\)

Решение

\({\sin ^2}x-\sin x\cos x = 0\).

\({\sin ^2}x-\sin x\cos x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\sin x\left( {\sin x-\cos x} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\sin x = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sin x-\cos x = 0.}\end{array}} \right.\)

\(\sin x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \pi k,\,\,\,\,\,\,\,\,k\, \in \,Z.\)

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

\(\sin x-\cos x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{tg}}\,x-1 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{tg}}\,x = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \dfrac{\pi }{4} + \pi k,\,\,\,\,\,\,\,k\, \in \,Z.\)

Ответ: \(\pi k;\quad \dfrac{\pi }{4} + \pi k;\quad \,k \in Z.\)