Question

Решить уравнение 2 cos⁡(π/4+x)-√2=0.

Answer 1

$2\cos\left(\dfrac{\pi}{4} +x\right)-\sqrt{2} =0$

$2\cos\left(\dfrac{\pi}{4} +x\right)=\sqrt{2}$

$\cos\left(\dfrac{\pi}{4} +x\right)=\dfrac{\sqrt{2} }{2}$

$\dfrac{\pi}{4} +x=\pm\arccos\dfrac{\sqrt{2} }{2} +2\pi n$

$\dfrac{\pi}{4} +x=\pm\dfrac{\pi }{4} +2\pi n$

$x=-\dfrac{\pi}{4}\pm\dfrac{\pi }{4} +2\pi n$

$x_1=-\dfrac{\pi}{4}-\dfrac{\pi }{4} +2\pi n=-\dfrac{\pi }{2} +2\pi n,\ n\in\mathbb{Z}$

$x_2=-\dfrac{\pi}{4}+\dfrac{\pi }{4} +2\pi n=2\pi n,\ n\in\mathbb{Z}$

ответ: $-\dfrac{\pi }{2} +2\pi n;\ 2\pi n,\ n\in\mathbb{Z}$