Страница 192 номер 626, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

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

2) \(\sin{5x} = \sin x\)
\(\sin{5x} - \sin x = 0,\)
\(2\sin{2x}\cos{3x} = 0,\)
\(\sin{2x} = 0\) или \(\cos{3x} = 0\)
\(2x = \pi n\) или \(3x = \frac{\pi}{2} + \pi n\)
\(x_{1} = \dfrac{\pi}{2}n,\)
\(x_{2} = \dfrac{\pi}{6} + \dfrac{\pi}{3}n, n \in Z;\)

3) \(\sin{2x} = \cos{3x}\)
\(\cos{3x} - \sin{2x} = 0,\)
\(\sin\left( \dfrac{\pi}{2} + 3x \right) - \sin{2x} = 0,\)
\(2\sin\left( \dfrac{\pi}{4} + \dfrac{x}{2} \right)\cos\left( \dfrac{\pi}{4} + \dfrac{5x}{2} \right) = 0,\)
\(\sin\left( \frac{\pi}{4} + \frac{x}{2} \right) = 0\) или \(\cos\left( \frac{\pi}{4} + \frac{5x}{2} \right) = 0\)
\(\frac{\pi}{4} + \frac{x}{2} = \pi n\) или \(\frac{\pi}{4} + \frac{5x}{2} = \frac{\pi}{2} + \pi n\)
\(\frac{x}{2} = - \frac{\pi}{4} + \pi n\) или \(\frac{5x}{2} = \frac{\pi}{4} + \pi n\)
\(x_{1} = - \dfrac{\pi}{2} + 2\pi n,\)
\(x_{2} = \dfrac{\pi}{10} + \dfrac{2\pi}{5}n, n \in Z;\)

4) \(\sin x + \cos{3x} = 0\)
\(\cos\left( \dfrac{\pi}{2} - x \right) + \cos{3x} = 0,\)
\(2\cos\left( \dfrac{\pi}{4} + x \right)\cos\left( \dfrac{\pi}{4} - 2x \right) = 0,\)
\(\cos\left( \frac{\pi}{4} + x \right) = 0\) или \(\cos\left( \frac{\pi}{4} - 2x \right) = 0\)
\(\frac{\pi}{4} + x = \frac{\pi}{2} + \pi n\) или \(\frac{\pi}{4} - 2x = \frac{\pi}{2} + \pi n\)
\(x = \frac{\pi}{4} + \pi n\) или \(2x = - \frac{\pi}{4} - \pi n\)
\(x_{1} = \dfrac{\pi}{4} + \pi n,\)
\(x_{2} = - \dfrac{\pi}{8} - \dfrac{\pi}{2}n, n \in Z.\)