Страница 415 номер 1431, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\begin{cases} \sin x\cos y = - \frac{1}{2} \\ \operatorname{tg} x\operatorname{ctg} y = 1 \end{cases}\)
\(\begin{cases} \sin x\cos y = - \frac{1}{2} \\ \frac{\sin x}{\cos x} \cdot \frac{\cos y}{\sin y} = 1 \end{cases},\)
\(\cos x\sin y = - \dfrac{1}{2},\)
\(\sin x\cos y - \cos x\sin y = - \dfrac{1}{2} + \dfrac{1}{2} = 0,\)
\(\sin(x - y) = 0,\)
\(x - y = \pi n, n \in Z;\)
\(x = \pi n + y, n \in Z;\)
\(\sin(y + \pi n)\cos y = - \dfrac{1}{2}:\)

а) \(n = 2m\): \(\sin y\cos y = - \frac{1}{2}\),
\(\sin{2y} = - 1,\)
\(2y = - \dfrac{\pi}{2} + 2\pi k, k \in Z\)
\(y = - \dfrac{\pi}{4} + \pi k, k \in Z;\)

б) \(n = 2m + 1\): \(- \sin y\cos y = - \frac{1}{2}\)
\(\sin{2y} = 1,\)
\(y = \dfrac{\pi}{4} + \pi k, k \in Z;\)

Ответ: \(\left( - \frac{\pi}{4} + \pi k + 2\pi m; - \frac{\pi}{4} + \pi k \right)\), \(\left( \frac{5\pi}{4} + \pi k + 2\pi m;\ \frac{\pi}{4} + \pi k \right)\), \(k,m \in Z\);

2) \(\begin{cases} \sin x\sin y = \frac{1}{4} \\ 3\operatorname{tg} x = \operatorname{ctg} y \end{cases}\)
\(\begin{cases} \sin x\sin y = \frac{1}{4} \\ \frac{\sin x}{\cos x} \cdot \frac{\sin y}{\cos y} = \frac{1}{3} \end{cases},\)
\(\cos x\cos y = \dfrac{3}{4},\)
\(\cos x\cos y - \sin x\sin y = \dfrac{3}{4} - \dfrac{1}{4} = \dfrac{1}{2},\)
\(\cos(x + y) = \dfrac{1}{2},\)
\(x + y = \pm \dfrac{\pi}{3} + 2\pi n, n \in Z\)
\(x = \pm \dfrac{\pi}{3} - y + 2\pi n, n \in Z;\)
\(\sin x\sin y = \dfrac{1}{2}\left( \cos(x - y) - \cos(x + y) \right),\)
\(\dfrac{1}{4} = \dfrac{1}{2}\left( \cos(x - y) - \dfrac{1}{2} \right),\)
\(\cos(x - y) = 1,\)
\(x - y = 2\pi m, m \in Z;\)
\(\begin{cases} x = \pm \frac{\pi}{3} - y + 2\pi n,\ n \in Z \\ y = x - 2\pi m,\ m \in Z \end{cases},\)
\(\begin{cases} 2x = \pm \frac{\pi}{3} + 2\pi n + 2\pi m \\ y = x - 2\pi m \end{cases}, n,m \in Z\)
\(\begin{cases} x = \frac{\pi}{6} + \pi(n + m) \\ y = \frac{\pi}{6} + \pi(n - m) \end{cases}\) или \(\begin{cases} x = - \frac{\pi}{6} + \pi(n + m) \\ y = - \frac{\pi}{6} + \pi(n - m) \end{cases}\), \(n,m \in Z\).