Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 868

\[1)\ 4\sin^{4}x + \sin^{2}{2x} = 2\]

\[4\sin^{4}x + 4\sin^{2}x \bullet \cos^{2}x = 2\]

\[4\sin^{2}x \bullet \left( \sin^{2}x + \cos^{2}x \right) = 2\]

\[2\sin^{2}x \bullet 1 = 1\]

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

\[\sin x = \pm \frac{\sqrt{2}}{2}\]

\[x = \pm \arcsin\frac{\sqrt{2}}{2} + \pi n = \pm \frac{\pi}{4} + \pi n.\]

\[Ответ:\ \pm \frac{\pi}{4} + \pi n.\ \]

\[2)\ \sin^{4}\frac{x}{3} + \cos^{4}\frac{x}{3} = \frac{5}{8}\]

\[1^{2} - \frac{1}{2}\sin^{2}\frac{2x}{3} = \frac{5}{8}\]

\[\frac{1}{2}\sin^{2}\frac{2x}{3} = \frac{3}{8}\]

\[\sin^{2}\frac{2x}{3} = \frac{3}{4}\]

\[\sin\frac{2x}{3} = \pm \frac{\sqrt{3}}{2}\]

\[\frac{2x}{3} = \pm \arcsin\frac{\sqrt{3}}{2} + \pi n = \pm \frac{\pi}{3} + \pi n\]

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

\[Ответ:\ \pm \frac{\pi}{2} + \frac{3\pi n}{2}.\]