Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 685

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\[1)\ \left\{ \begin{matrix} \sin y \bullet \cos y = \frac{1}{2}\text{\ \ \ \ \ \ \ } \\ \sin{2x} + \sin{2y} = 0 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} \frac{1}{2}\sin{2y} = \frac{1}{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \sin{2x} + \sin{2y} = 0 \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} \sin{2y} = 1\ \ \ \ \ \ \ \ \\ \sin{2x} + 1 = 0 \\ \end{matrix} \right.\ \text{\ \ }\]

\[\left\{ \begin{matrix} \sin{2y = 1}\text{\ \ \ } \\ \sin{2x} = - 1 \\ \end{matrix} \right.\ \]

\[\sin{2x} = - 1\]

\[2x = - \arcsin 1 + 2\pi n\]

\[2x = - \frac{\pi}{2} + 2\pi n\]

\[x = \frac{1}{2} \bullet \left( - \frac{\pi}{2} + 2\pi n \right)\]

\[x = - \frac{\pi}{4} + \pi n.\]

\[\sin{2y} = 1\]

\[2y = \arcsin 1 + 2\pi n\]

\[2y = \frac{\pi}{2} + 2\pi n\]

\[y = \frac{1}{2} \bullet \left( \frac{\pi}{2} + 2\pi n \right)\]

\[y = \frac{\pi}{4} + \pi n.\]

\[Ответ:\ \ x = - \frac{\pi}{4} + \pi n;\ \ \]

\[\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }y = \frac{\pi}{4} + \pi n.\]

\[2)\ \left\{ \begin{matrix} \sin x + \sin y = 1\ \ \ \ \\ \cos x - \cos y = \sqrt{3} \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 2 \bullet \sin\frac{x + y}{2} \bullet \cos\frac{x - y}{2} = 1\ \ \ \ \ \\ - 2 \bullet \sin\frac{x + y}{2} \bullet \sin\frac{x - y}{2} = \sqrt{3} \\ \end{matrix} \right.\ \ \]

\[Разделим\ второе\ уравнение\ \]

\[на\ первое:\]

\[- tg\frac{x - y}{2} = \sqrt{3}\]

\[\text{tg}\frac{x - y}{2} = - \sqrt{3}\]

\[\frac{x - y}{2} = - arctg\ \sqrt{3} + \pi n =\]

\[= - \frac{\pi}{3} + \pi n\]

\[x - y = 2 \bullet \left( - \frac{\pi}{3} + \pi n \right)\]

\[x = - \frac{2\pi}{3} + 2\pi n\]

\[x = y - \frac{2\pi}{3} + 2\pi n.\]

\[\sin\left( y - \frac{2\pi}{3} + 2\pi n \right) + \sin y = 1\]

\[\sin\left( y - \frac{2\pi}{3} \right) + \sin y = 1\]

\[- \frac{1}{2}\sin y - \frac{\sqrt{3}}{2}\cos y + \sin y = 1\]

\[\frac{1}{2}\sin y - \frac{\sqrt{3}}{2}\cos y = 1\]

\[\cos\frac{\pi}{3} \bullet \sin y - \sin\frac{\pi}{3} \bullet \cos y = 1\]

\[\sin\left( y - \frac{\pi}{3} \right) = 1\]

\[y - \frac{\pi}{3} = \arcsin 1 + 2\pi k\]

\[y = \frac{\pi}{2} + 2\pi k\]

\[y = \frac{\pi}{2} + \frac{\pi}{3} + 2\pi k\]

\[y = \frac{3\pi}{6} + \frac{2\pi}{6} + 2\pi k\]

\[y = \frac{5\pi}{6} + 2\pi k.\]

\[x = y - \frac{2\pi}{3} + 2\pi n =\]

\[= \frac{5\pi}{6} - \frac{4\pi}{6} + 2\pi n + 2\pi k =\]

\[= \frac{\pi}{6} + 2\pi(n + k).\]

\[Ответ:\ \ x = \frac{\pi}{6} + 2\pi(n + k);\ \ \]

\[\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }y = \frac{5\pi}{6} + 2\pi k.\]