Задача 35. Решите уравнение: \({\sin ^3}x + {\cos ^3}x = \sin x-\cos x\)
ОТВЕТ: \(\dfrac{\pi }{2} + \pi k;\;\,\,\,\;k \in Z.\)
\({\sin ^3}x + {\cos ^3}x = \sin x-\cos x\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,{\sin ^3}x + {\cos ^3}x = \sin x \left( {{{\sin }^2}x + {{\cos }^2}x} \right)-\cos x\left( {{{\sin }^2}x + {{\cos }^2}x} \right)\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,{\sin ^3}x + {\cos ^3}x = {\sin ^3}x + \sin x{\cos ^2}x-\cos x{\sin ^2}x-{\cos ^3}x\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\cos x{\sin ^2}x-\sin x{\cos ^2}x + 2{\cos ^3}x = 0\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\cos x\left( {{{\sin }^2}x-\sin x\cos x + 2{{\cos }^2}x} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos x = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{{\sin }^2}x-\sin x\cos x + 2{{\cos }^2}x = 0.}\end{array}} \right.\) \(\cos x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \dfrac{\pi }{2} + \pi k,\,\,\,\,\,\,\,\,k\, \in \,Z.\) Уравнение \({\sin ^2}x-\sin x\cos x + 2{\cos ^2}x = 0\) является однородным тригонометрическим уравнением второй степени: \({\sin ^2}x-\sin x\cos x + 2{\cos ^2}x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{t}}{{\rm{g}}^2}x-{\rm{tg}}\,x + 2 = 0.\) Пусть \({\rm{tg}}\,x = t,\,\,\,\,\,\,\,t\, \in \,R\). Тогда: \({t^2}-t + 2 = 0;\,\,\,\,\,\,\,\,\,\,\,\,D = 1-8 = -7 < 0.\) Следовательно, последнее уравнение не имеет решений. Ответ: \(\dfrac{\pi }{2} + \pi k;\;\,\,\,\;k \in Z.\)