Question

Используя метод вс аргумента покажите , что уравнение
sin4x-cos4x=корень 2

Answer 1

$\displaystyle sin4x-cos4x=\sqrt2\ \Big|:\sqrt2\\\\\frac{1}{\sqrt2}\, sin4x-\frac{1}{\sqrt2}\, cos4x=\frac{\sqrt2}{\sqrt2}\\\\\\cos\frac{\pi}{4}\cdot sin4x-sin\frac{\pi}{4}\, cos4x=1\\\\\\sin\Big(4x-\frac{\pi}{4}\Big)=1\\\\\\4x-\frac{\pi} {4}=\frac{\pi}{2}+2\pi n\ ,\ n\in Z\\\\\\4x=\frac{\pi} {4}+\frac{\pi}{2}+\pi n\ ,\ n\in Z\\\\\\4x=\frac{3\pi }{4}+2\pi n\ ,\ n\in Z\\\\\\x=\frac{3\pi }{16}+\frac{\pi n}{2}\ ,\ n\in Z \ \ -\ \ otvet$