Страница 193 номер 635, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\cos x\cos{2x} = \sin x\sin{2x}\)
\(\cos x\left( \cos^{2}x - \sin^{2}x \right) - 2\sin^{2}x\cos x = 0,\)
\(\cos x\left( \cos^{2}x - \sin^{2}x - 2\sin^{2}x \right) = 0,\)
\(\cos x = 0\) или \(1 - 4\sin^{2}x = 0\), \(\sin x = \pm \frac{1}{2}\)
\(x_{1} = \dfrac{\pi}{2} + \pi n,\)
\(x_{2} = \pm \dfrac{\pi}{6} + \pi n, n \in Z;\)

2) \(\sin{2x}\cos x = \cos{2x}\sin x\)
\(2\sin x\cos^{2}x - \left( \cos^{2}x - \sin^{2}x \right)\sin x = 0,\)
\(\sin x\left( \cos^{2}x + \sin^{2}x \right) = 0,\)
\(\sin x = 0,\)
\(x = \pi n, n \in Z;\)

3) \(\sin{3x} = \sin{2x}\cos x\)
\(\sin{2x}\cos x + \sin x\cos{2x} = \sin{2x}\cos x,\)
\(\sin x\cos{2x} = 0,\)
\(\sin x = 0\) или \(\cos{2x} = 0\)
\(x_{1} = \pi n,\)
\(x_{2} = \dfrac{\pi}{4} + \dfrac{\pi}{2}n, n \in Z;\)

4) \(\cos{5x}\cos x = \cos{4x}\)
\(\cos{5x}\cos x = \cos{5x}\cos x + \sin{5x}\sin x,\)
\(\sin{5x}\sin x = 0,\)
\(\sin{5x} = 0\) или \(\sin x = 0\)
\(x_{1} = \dfrac{\pi}{5}n,\)
\(x_{2} = \pi n\), \(n \in Z\) – входит в \(x_{1}\).

Ответ: \(x = \frac{\pi}{5}n\), \(n \in Z\).