Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 678

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\[1)\ \frac{\sin{2x}}{\sin x} = 0\]

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

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

\[x = \frac{\text{πn}}{2}.\]

\[\sin x \neq 0\]

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

\[Ответ:\ \ \frac{\pi}{2} + \pi n.\]

\[2)\ \frac{\sin{3x}}{\sin x} = 0\]

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

\[3x = \arcsin 0 + \pi n = \pi n\]

\[x = \frac{\text{πn}}{3}.\]

\[\sin x \neq 0\]

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

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

\[3)\ \frac{\cos{2x}}{\cos x} = 0\]

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

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

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

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

\[x = \frac{\pi}{4} + \frac{\text{πn}}{2}.\]

\[\cos x \neq 0\]

\[x \neq \arccos 0 + \pi n \neq \frac{\pi}{2} + \pi n.\]

\[Ответ:\ \ \frac{\pi}{4} + \frac{\text{πn}}{2}.\]

\[4)\ \frac{\cos{3x}}{\cos x} = 0\]

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

\[3x = \arccos 0 + \pi n\]

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

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

\[x = \frac{\pi}{6} + \frac{\text{πn}}{3}.\]

\[\cos x \neq 0\]

\[x \neq \arccos 0 + \pi n\]

\[x \neq \frac{\pi}{2} + \pi n.\]

\[Ответ:\ \pm \frac{\pi}{6} + \pi n.\]

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

\[\sin x = 0\]

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

\[\sin{5x} \neq 0\]

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

\[x \neq \frac{\text{πn}}{5}.\]

\[Ответ:\ \ решений\ нет.\]

\[6)\ \frac{\cos x}{\cos{7x}} = 0\]

\[\cos x = 0\]

\[x = \arccos 0 + \pi n\]

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

\[\cos{7x} \neq 0\]

\[7x \neq \arccos 0 + \pi n\]

\[7x \neq \frac{\pi}{2} + \pi n\]

\[x \neq \frac{1}{7} \bullet \left( \frac{\pi}{2} + \pi n \right)\]

\[x \neq \frac{\pi}{14} + \frac{\text{πn}}{7}.\]

\[Ответ:\ \ решений\ нет.\]