Question

Решите уравнение: 2 cos (п/4 - x/2) = 1

Answer 1

$2 cos (\frac{\pi}{4} - \frac{x}{2}) = 1\\cos(\frac{x}{2}-\frac{\pi}{4})=-\frac{1}{2}\\\frac{x}{2}-\frac{\pi}{4}=\pm(\pi-arccos\frac{1}{2})+2\pi n\\\frac{x}{2}-\frac{\pi}{4}=\pm(\pi-\frac{\pi}{3})+2\pi n\\\frac{x}{2}-\frac{\pi}{4}=\pm \frac{2\pi}{3}+2\pi n\\\frac{x}{2}=\pm \frac{2\pi}{3}+\frac{\pi}{4}+2\pi n\\x=\pm\frac{4\pi}{3}+\frac{\pi}{2}+4\pi n, \; n\in Z;\\\\x_1=\frac{11\pi}{6}+4\pi n, \; n\in Z;\\x_2=-\frac{5\pi}{6}+4\pi n, \; n\in Z.$