Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 665

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\[1)\sin{3x} = \sin{5x}\]

\[\sin{5x} - \sin{3x} = 0\]

\[2 \bullet \sin\frac{5x - 3x}{2} \bullet \cos\frac{5x + 3x}{2} = 0\]

\[\sin\frac{2x}{2} \bullet \cos\frac{8x}{2} = 0\]

\[\sin x \bullet \cos{4x} = 0\]

\[1)\ \sin x = 0\]

\[x = \arcsin 0 + \pi n = \pi n.\]

\[2)\ \cos{4x} = 0\]

\[4x = \arccos 0 + \pi n = \frac{\pi}{2} + \pi n\]

\[x = \frac{1}{4} \bullet \left( \frac{\pi}{2} + \pi n \right) = \frac{\pi}{8} + \frac{\text{πn}}{4}.\]

\[Ответ:\ \ \pi n;\ \ \frac{\pi}{8} + \frac{\text{πn}}{4}\text{.\ }\]

\[2)\cos^{2}{3x} - \cos{3x} \bullet \cos{5x} = 0\]

\[\cos{3x} \bullet \left( \cos{3x} - \cos{5x} \right) = 0\]

\[\cos{3x} \bullet ( - 2) \bullet \sin\frac{3x + 5x}{2} \bullet \sin\frac{3x - 5x}{2} = 0\]

\[- 2 \bullet \cos{3x} \bullet \sin\frac{8x}{2} \bullet \sin\left( - \frac{2x}{2} \right) = 0\]

\[2 \bullet \cos{3x} \bullet \sin{4x} \bullet \sin x = 0\]

\[1)\ \cos{3x} = 0\]

\[3x = \arccos 0 + \pi n = \frac{\pi}{2} + \pi n\]

\[x = \frac{1}{3} \bullet \left( \frac{\pi}{2} + \pi n \right) = \frac{\pi}{6} + \frac{\text{πn}}{3}.\]

\[2)\ \sin{4x} = 0\]

\[4x = \arcsin 0 + \pi n = \pi n\]

\[x = \frac{1}{4} \bullet \pi n = \frac{\text{πn}}{4}.\]

\[3)\ \sin x = 0\]

\[x = \arcsin 0 + \pi n = \pi n.\]

\[Ответ:\ \ \frac{\pi}{6} + \frac{\text{πn}}{3};\ \ \frac{\text{πn}}{4}\text{.\ }\]

\[3)\cos x = \cos{3x}\]

\[\cos x - \cos{3x} = 0\]

\[- 2 \bullet \sin\frac{x + 3x}{2} \bullet \sin\frac{x - 3x}{2} = 0\]

\[- 2 \bullet \sin\frac{4x}{2} \bullet \sin\left( - \frac{2x}{2} \right) = 0\]

\[2 \bullet \sin{2x} \bullet \sin x = 0\]

\[1)\ \sin{2x} = \arcsin 0 + \pi n = \pi n\]

\[x = \frac{1}{2} \bullet \pi n = \frac{\text{πn}}{2}.\]

\[\sin x = \arcsin 0 + \pi n = \pi n.\]

\[Ответ:\ \ \frac{\text{πn}}{2}\text{.\ }\]

\[4)\sin x \bullet \sin{5x} - \sin^{2}{5x} = 0\]

\[\sin{5x} \bullet \left( \sin x - \sin{5x} \right) = 0\]

\[\sin{5x} \bullet 2 \bullet \sin\frac{x - 5x}{2} \bullet \cos\frac{x + 5x}{2} = 0\]

\[2 \bullet \sin{5x} \bullet \sin\left( - \frac{4x}{2} \right) \bullet \cos\frac{6x}{2} = 0\]

\[- 2 \bullet \sin{5x} \bullet \sin{2x} \bullet \cos{3x} = 0\]

\[1)\ \sin{5x} = 0\]

\[5x = \arcsin 0 + \pi n = \pi n\]

\[x = \frac{1}{5} \bullet \pi n = \frac{\text{πn}}{5}.\]

\[2)\ \sin{2x} = 0\]

\[2x = \arcsin 0 + \pi n = \pi n\]

\[x = \frac{1}{2} \bullet \pi n = \frac{\text{πn}}{2}.\]

\[3)\ \cos{3x} = 0\]

\[3x = \arccos 0 + \pi n = \frac{\pi}{2} + \pi n\]

\[x = \frac{1}{3} \bullet \left( \frac{\pi}{2} + \pi n \right) = \frac{\pi}{6} + \frac{\text{πn}}{3}.\]

\[Ответ:\ \ \frac{\text{πn}}{5};\ \ \frac{\text{πn}}{2};\ \ \frac{\pi}{6} + \frac{\text{πn}}{3}\text{.\ }\]