Упр.573 ГДЗ Алимов 10-11 класс (Алгебра)

2) cos2x=-1;
3) корень cosx/4=-1;
4) 2cosx/3=корень 3;
5) cos(x+пи/3)=0;
6) cos(2x-пи/4)=0.

1. $$\cos 4x=1$$

   $$4x=2\pi n,\quad n\in\mathbb Z$$

   $$x=\frac{\pi n}{2},\quad n\in\mathbb Z$$

2. $$\cos 2x=-1$$

   $$2x=\pi+2\pi n,\quad n\in\mathbb Z$$

   $$x=\frac{\pi}{2}+\pi n,\quad n\in\mathbb Z$$

3. $$\sqrt2\cos\frac{x}{4}=-1$$

   $$\cos\frac{x}{4}=-\frac{1}{\sqrt2}=-\frac{\sqrt2}{2}$$

   $$\frac{x}{4}=\pm\frac{3\pi}{4}+2\pi n,\quad n\in\mathbb Z$$

   $$x=\pm 3\pi+8\pi n,\quad n\in\mathbb Z$$

4. $$2\cos\frac{x}{3}=\sqrt3$$

   $$\cos\frac{x}{3}=\frac{\sqrt3}{2}$$

   $$\frac{x}{3}=\pm\frac{\pi}{6}+2\pi n,\quad n\in\mathbb Z$$

   $$x=\pm\frac{\pi}{2}+6\pi n,\quad n\in\mathbb Z$$

5. $$\cos\left(x+\frac{\pi}{3}\right)=0$$

   $$x+\frac{\pi}{3}=\frac{\pi}{2}+\pi n,\quad n\in\mathbb Z$$

   $$x=\frac{\pi}{6}+\pi n,\quad n\in\mathbb Z$$

6. $$\cos\left(2x-\frac{\pi}{4}\right)=0$$

   $$2x-\frac{\pi}{4}=\frac{\pi}{2}+\pi n,\quad n\in\mathbb Z$$

   $$2x=\frac{3\pi}{4}+\pi n$$

   $$x=\frac{3\pi}{8}+\frac{\pi n}{2},\quad n\in\mathbb Z$$

Ответ

1) $$x=\frac{\pi n}{2},\ n\in\mathbb Z$$; 2) $$x=\frac{\pi}{2}+\pi n,\ n\in\mathbb Z$$; 3) $$x=\pm 3\pi+8\pi n,\ n\in\mathbb Z$$; 4) $$x=\pm\frac{\pi}{2}+6\pi n,\ n\in\mathbb Z$$; 5) $$x=\frac{\pi}{6}+\pi n,\ n\in\mathbb Z$$; 6) $$x=\frac{3\pi}{8}+\frac{\pi n}{2},\ n\in\mathbb Z$$.