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

1. $$\cos 4x=1$$
2. $$\cos 2x=-1$$
3. $$\cos\frac{x}{4}=-1$$
4. $$2\cos\frac{x}{3}=\sqrt{3}$$
5. $$\cos\left(x+\frac{\pi}{3}\right)=0$$
6. $$\cos\left(2x-\frac{\pi}{4}\right)=0$$

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