| Задача 1. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {\sin x\,\sin y = \frac{1}{4}} \\ {\cos x\,\cos y = \frac{3}{4}} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( { — \frac{\pi }{6} + \pi \left( {n + k} \right), — \frac{\pi }{6} + \pi \left( {n — k} \right)} \right),\;\;\left( {\frac{\pi }{6} + \pi \left( {n + k} \right),\frac{\pi }{6} + \pi \left( {n — k} \right)} \right),\,\,n,k \in Z.\)
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| Задача 2. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {\sin x\,\cos y = \frac{1}{4}} \\ {\cos \,x\,\sin \,y = \frac{3}{4}} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\frac{\pi }{6} + \pi \left( {n + k} \right),\frac{\pi }{3} + \pi \left( {n — k} \right)} \right)\,,\;\,\left( { — \frac{\pi }{6} + \pi \left( {n + k} \right),\frac{{2\pi }}{3} + \pi \left( {n — k} \right)} \right),\,\,n,k \in Z.\)
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| Задача 3. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {y — x = \frac{1}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {\cos \,\,\left( {\,\pi \,x} \right)\,\,\cos \,\left( {\,\pi \,y} \right) = \frac{{\sqrt 2 }}{2}} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {n,\;\frac{1}{4} + n} \right),\,\,\,\left( { — \frac{1}{4} + n,\;n} \right),\,\,\,n \in Z.\)
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| Задача 4. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {\cos \,\,\left( {x — y} \right) = \frac{1}{2}\,\,\,} \\ {\cos \,\,\left( {x + y} \right) = — \frac{1}{2}} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\frac{{\rm{\pi }}}{6} + {\rm{\pi }}\left( {n + k} \right),\;\frac{{\rm{\pi }}}{2} + {\rm{\pi }}\left( {n-k} \right)} \right),\,\,\,\left( {\frac{{\rm{\pi }}}{2} + {\rm{\pi }}\left( {n + k} \right),\,\frac{{\rm{\pi }}}{6} + {\rm{\pi }}\left( {n-k} \right)} \right),\,\)
\(\left( {-\frac{{\rm{\pi }}}{6} + {\rm{\pi }}\left( {n + k} \right),\;-\frac{{\rm{\pi }}}{2} + {\rm{\pi }}\left( {n-k} \right)} \right),\,\,\,\left( {-\frac{{\rm{\pi }}}{2} + {\rm{\pi }}\left( {n + k} \right),\,-\frac{{\rm{\pi }}}{6} + {\rm{\pi }}\left( {n-k} \right)} \right),\,\,\,\,n,k \in Z.\)
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| Задача 5. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {x + y = \frac{{3\pi }}{4}\,\,\,\,\,\,\,} \\ {{\text{tg}}\,\,x — {\text{tg}}\,\,y = 2} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\frac{{5\pi }}{{12}} — \pi \,n,\;\frac{\pi }{3} + \pi \,n} \right),\;\,\left( {\frac{\pi }{{12}} — \pi \,n,\;\frac{{2\pi }}{3} + \pi \,n} \right),\,\,\,n \in Z.\)
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| Задача 6. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {\sin x + \cos y = 1} \\ {x + y = \frac{\pi }{3}\,\,\,\,\,\,\,\,\,\,\,\,\,} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {{{\left( { — 1} \right)}^n}\arcsin \sqrt {2 — \sqrt 3 } — \frac{\pi }{{12}} + \pi \,n,\;\frac{{5\pi }}{{12}} — {{\left( { — 1} \right)}^n}\arcsin \sqrt {2 — \sqrt 3 } — \pi \,n} \right),\,\,\,n \in Z.\)
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| Задача 7. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {x + y = \frac{{2\pi }}{3}} \\ {\frac{{\sin x}}{{\sin y}} = 2\,\,\,\,\,} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\frac{\pi }{{\text{2}}} + \pi \,n,\;\,\frac{\pi }{6} — \pi \,n} \right),\,\,\,n \in Z.\)
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| Задача 8. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {{\text{tg}}\,x + {\text{tg}}\,y = 2\,\,} \\ {\cos x\,\cos y = \frac{1}{2}} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\frac{\pi }{4} + \pi \,n,\;\frac{\pi }{4} + \pi \left( {2k — n} \right)} \right),\,\,\,n,k \in Z.\)
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| Задача 9. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {\sin x + \cos y = 0\,\,\,\,\,\,\,\,\,\,\,} \\ {{{\sin }^2}x + {{\cos }^2}y = \frac{1}{2}\,\,\,\,\,\,} \\ {0 < x < \pi ,\quad 0 < y < \pi } \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\frac{\pi }{6},\;\frac{{2\pi }}{3}} \right),\;\,\left( {\frac{{5\pi }}{6},\;\frac{{2\pi }}{3}} \right).\)
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| Задача 10. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {{{\sin }^2}x + {{\cos }^2}y = \frac{{11}}{{16}}\,\,\,\,\,\,\,} \\ {\sin \frac{{x + y}}{2}\,\cos \frac{{x — y}}{2} = \frac{5}{8}} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {{{\left( { — 1} \right)}^n}\frac{\pi }{6} + \pi \,n,\;{{\left( { — 1} \right)}^k}\arcsin \frac{3}{4} + \pi \,k} \right),\,\,\,\,n,k \in Z.\)
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| 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: \(\left( {\pi \,n,\; \pm \frac{\pi }{3} + 2\pi \,k} \right),\,\,\,\,n,k \in Z.\)
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| Задача 12. Решите систему уравнений \(\left\{ {\begin{array}{*{20}{c}} {\sin x + \sin y = \sin \,\left( {x + y} \right)} \\ {\left| {\,x\,} \right| + \left| {\,y\,} \right| = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\frac{1}{2},\; — \frac{1}{2}} \right);\;\,\left( { — \frac{1}{2},\;\frac{1}{2}} \right);\;\,\left( {1,\;0} \right);\;\,\left( { — 1,\;0} \right);\,\,\,\left( {0,\;1} \right);\;\,\left( {0,\; — 1} \right).\)
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