Задача 11. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {{9^{2\,{\text{tg}}\,x + \cos y}} = 3\,\,\,\,\,\,\,} \\ {{9^{\cos y}}-{{81}^{{\text{tg}}\,x}} = 2} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\pi \,n,\; \pm \dfrac{\pi }{3} + 2\pi \,k} \right),\,\,\,\,n,k \in Z.\)
Решение
\({9^{2{\rm{tg}}\,x + \cos y}} = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,2\left( {2{\rm{tg}}\,x + \cos y} \right) = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\cos y = \dfrac{1}{2}-2{\rm{tg}}\,x.\)
Подставим во второе уравнение:
\({9^{\frac{1}{2}-2{\rm{tg}}\,x}}-{81^{{\rm{tg}}\,x}} = 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\dfrac{3}{{{{81}^{{\rm{tg}}\,x}}}}-{81^{{\rm{tg}}\,x}} = 2\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{81^{2{\rm{tg}}\,x}} + 2 \cdot {81^{{\rm{tg}}\,x}}-3 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{{81}^{{\rm{tg}}\,x}} = -3,}\\{{{81}^{{\rm{tg}}\,x}} = 1\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{81^{{\rm{tg}}\,x}} = {81^0}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{tg}}\,x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \pi n,\,\,\,n\, \in \,Z.\)
Так как \({\rm{tg}}\,x = 0\), то \(\cos y = \dfrac{1}{2}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,y = \pm \dfrac{\pi }{3} + 2\pi k,\,\,\,k\, \in \,Z.\)
Ответ: \(\left( {{\rm{\pi }}\,n,\; \pm \dfrac{{\rm{\pi }}}{3} + 2{\rm{\pi }}\,k} \right),\,\,\,\,n,k \in Z.\)