Question

1) решите уравнение cos 4x - √2 cos 2x-1=0 (нужно само решение! ) 2) найдите все корни, принадлежащие промежутку [-pi; pi]

Answer 1

$Cos4x-\sqrt{2}Cos2x-1=0\\\\2Cos^{2}2x-1-\sqrt{2}Cos2x-1=0\\\\2Cos^{2}2x-\sqrt{2}Cos2x-2=0|:\sqrt{2}\\\\\sqrt{2}Cos^{2}2x-Cos2x-\sqrt{2}=0\\\\Cos2x=m;-1\leq m\leq 1\\\\\sqrt{2}m^{2} -m-\sqrt{2} =0\\\\D=(-1)^{2}-4*\sqrt{2}*(-\sqrt{2})=1+8=9=3^{2}\\\\m_{1} =\frac{1-3}{2\sqrt{2} }=-\frac{2}{2\sqrt{2} }=-\frac{\sqrt{2} }{2}\\\\m_{2}=\frac{1+3}{2\sqrt{2} }=\frac{4}{2\sqrt{2}}=\sqrt{2}1\\\\Cos2x=-\frac{\sqrt{2} }{2}\\\\2x=\pm arcCos(-\frac{\sqrt{2} }{2})+2\pi n,n\in z$

$2x=\pm\frac{3\pi }{4}+2\pi n,n\in z\\\\x=\pm\frac{3\pi }{8}+\pi n,n\in z\\\\1)x=\frac{3\pi }{8}+\pi n,n\in z\\\\n=-1\\\\x_{1} =-\frac{5\pi }{8}\\\\n=0\\\\x_{2}=\frac{3\pi }{8} \\\\2)x=-\frac{3\pi }{8}+\pi n,n\in z\\\\n=0\\\\x_{3}=-\frac{3\pi }{8}\\\\n=1\\\\x_{4}=\frac{5\pi }{8}$