Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 1272

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

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

\[\bullet \cos^{2}x - 2\sin^{2}x \bullet \cos^{2}x = a\]

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

\[\bullet \cos^{2}x = a\]

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

\[1 - a = \frac{1}{2} \bullet 4\sin^{2}x \bullet \cos^{2}x\]

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

\[\sin^{2}{2x} = 2 - 2a\]

\[\frac{1 - \cos{4x}}{2} = 2 - 2a\]

\[1 - \cos{4x} = 4 - 4a\]

\[- \cos{4x} = 3 - 4a\]

\[\cos{4x} = 4a - 3\]

\[4x = \pm \arccos(4a - 3) + 2\pi n\]

\[x = \pm \frac{1}{4}\arccos(4a - 3) + \frac{\text{πn}}{2}.\]

\[Допустимые\ значения\ числа\ a:\]

\[- 1 \leq \cos{4x} \leq 1\]

\[- 1 \leq 4a - 3 \leq 1\]

\[2 \leq 4a \leq 4\]

\[\frac{1}{2} \leq a \leq 1.\]

\[Ответ:\ \ \frac{1}{2} \leq a \leq 1;\]

\[\ \ x = \pm \frac{1}{4}\arccos(4a - 3) + \frac{\text{πn}}{2}.\]