Question

Решите систему

sinxsiny=√2/4
cosxcosy=√2/4

Answer 1

$\begin{cases}\sin x\sin y=\cfrac{\sqrt{2}}{4}\\ \cos x\cos y=\cfrac{\sqrt{2}}{4}\end{cases}\Leftrightarrow \begin{cases}\sin x\sin y=\cos x\cos y\\\sin x\sin y+\cos x\cos y=\cfrac{\sqrt{2}}{2}\end{cases}\Leftrightarrow \begin{cases}\cos(x+y)=0\\\cos (x-y)=\cfrac{\sqrt{2}}{2}\end{cases}$

$\begin{cases}x+y=\pm\cfrac{\pi}{2}+2\pi k\\x-y=\pm \cfrac{\pi}{4}+2\pi k\end{cases}\Rightarrow 2x=\pm\cfrac{\pi}{2}+2\pi k\pm \cfrac{\pi}{4}+2\pi k\Rightarrow x=\left \{ \pm\cfrac{\pi}{8}+2\pi k,\pm\cfrac{3\pi}{8}+2\pi k \right \}$

$x=\pm \cfrac{\pi}{8}+2\pi k\Rightarrow y=\pm \cfrac{\pi}{2}+2\pi k\mp \cfrac{\pi}{8}+2\pi k=\left \{ \pm\cfrac{3\pi}{8}+2\pi k,\pm\cfrac{5\pi}{8}+2\pi k \right \}\\x=\pm \cfrac{3\pi}{8}+2\pi k\Rightarrow y=\pm \cfrac{\pi}{2}+2\pi k\mp \cfrac{3\pi}{8}+2\pi k=\left \{ \pm\cfrac{\pi}{8}+2\pi k,\pm\cfrac{7\pi}{8}+2\pi k \right \}$

Везде $k\in \mathbb{Z}$