Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 1263

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

\[\left( \sin x + \cos y \right)^{2} = \sin^{2}x +\]

\[+ 2\sin x\cos y + \cos^{2}y = \frac{1}{2} +\]

\[+ 2\sin x\cos y\]

\[\sin x\cos y = \frac{1}{4}\]

\[Пусть\ a = \sin x;b = \cos y:\]

\[\left\{ \begin{matrix} a + b = 1 \\ ab = \frac{1}{4}\text{\ \ \ \ \ } \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} b = 1 - a\ \ \ \ \ \ \\ a(1 - a) = \frac{1}{4} \\ \end{matrix} \right.\ \]

\[a^{2} - a + \frac{1}{4} = 0\]

\[\left( a - \frac{1}{2} \right)^{2} = 0\]

\[a = \frac{1}{2} \rightarrow b = 1 - a = \frac{1}{2}.\]

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

\[\left\{ \begin{matrix} x = ( - 1)^{n} \cdot \frac{\pi}{6} + \pi k \\ y = \pm \frac{\pi}{3} + 2\pi n\ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[2)\ \left\{ \begin{matrix} 4\sin x - 2\sin y = 3 \\ 2\cos x - \cos y = 0\ \ \\ \end{matrix} \right.\ \]

\[\left( 4\sin x - 2\sin y \right)^{2} +\]

\[+ \left( 4\cos x - 2\cos y \right)^{2} = 9\]

\[20 - 16\sin x\sin y -\]

\[- 16\cos x\cos y = 9\]

\[\cos(x - y) = \frac{11}{16}\]

\[x - y = \pm \arccos\frac{11}{16} + 2\pi k\]

\[1)\ x = y + \arccos\frac{11}{16} + 2\pi k\]

\[\cos\left( y - \arccos\frac{11}{16} \right) - \cos y = 0\]

\[- \frac{5}{16} - \frac{3\sqrt{11}}{16}tg\ y = 0\]

\[tg\ y = - \frac{\sqrt{15}}{9}\]

\[y = - arctg\frac{\sqrt{15}}{9} + \pi n.\]

\[2)\ x = y - \arccos\frac{11}{16} + 2\pi k\]

\[\cos\left( y - \arccos\frac{11}{16} \right) - \cos y = 0\]

\[- \frac{5}{16} + \frac{3\sqrt{11}}{16}\text{tg\ }\sqrt{15} = 0\]

\[tg\ y = \frac{\sqrt{15}}{9}\]

\[y = arctg\frac{\sqrt{15}}{9} + \pi n\]

\[\left\{ \begin{matrix} y = - arctg\frac{\sqrt{15}}{9} + \pi n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x = - arctg\frac{\sqrt{15}}{9} + \arccos\frac{11}{16} + \pi n + 2\pi k \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} y = arctg\frac{\sqrt{15}}{9} + \pi n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x = \text{arctg}\frac{\sqrt{15}}{9} - \arccos\frac{11}{16} - \pi n + 2\pi k \\ \end{matrix} \right.\ \]