Страница 192 номер 629, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\sqrt{3}\sin x\cos x = \sin^{2}x\)
\(\sin x\left( \sqrt{3}\cos x - \sin x \right) = 0,\)
\(\sin x = 0\) или \(\sqrt{3} - \operatorname{tg} x = 0\)
\(x_{1} = \pi n,\)
\(x_{2} = \dfrac{\pi}{3} + \pi n, n \in Z;\)

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

3) \(\sin{4x} + \sin^{2}{2x} = 0\)
\(2\sin{2x}\cos{2x} + \sin^{2}{2x} = 0,\)
\(\sin{2x}\left( 2\cos{2x} + \sin{2x} \right) = 0,\)
\(\sin{2x} = 0\) или \(2 + \operatorname{tg} {2x} = 0\)
\(x_{1} = \dfrac{\pi}{2}n,\)
\(x_{2} = - \dfrac{1}{2}\operatorname{arctg} 2 + \dfrac{\pi}{2}n, n \in Z;\)

4) \(\sin{2x} + 2\cos^{2}x = 0\)
\(2\sin x\cos x + 2\cos^{2}x = 0,\)
\(2\cos x\left( \sin x + \cos x \right) = 0,\)
\(2\cos x = 0\) или \(\operatorname{tg} x + 1 = 0\)
\(x_{1} = \dfrac{\pi}{2} + \pi n,\)
\(x_{2} = - \dfrac{\pi}{4} + \pi n, n \in Z.\)