Решить уравнение (656-665). 656 1) cos (4 – 2x) = -1/2; 3) √2 cos (2x + π/4) + 1 = 0; 657 1) 2 sin (3x - π/4) + 1 = 0; 3) 3 + 4 sin (2x + 1) = 0; 658 1) (1 + √2 cos x) (1 - 4 sin x cos x) = 0; 2) (1 - √2 cos x) (1 + 2 sin 2x cos 2x) = 0. 659 1) tg (2x + π/4) = -1; 3) √3 - tg (x - π/5) = 0; 660 1) 2 sin² x + sin x = 0; 3) cos² x - 2 cos x = 0; 661 1) 6 sin² x - cos x + 6 = 0; 662 1) tg² x + 3 tg x = 0; 3) tg x - 12 ctg x + 1 = 0.

Решение:

656.

1. 1)

   \( \cos (4 - 2x) = -\frac{1}{2} \)
   \( 4 - 2x = \pm \frac{2\pi}{3} + 2\pi k, k \in \mathbb{Z} \)
   \( -2x = -4 \pm \frac{2\pi}{3} + 2\pi k \)
   \( x = 2 \mp \frac{\pi}{3} - \pi k \)

2. 3)

   \( \sqrt{2} \cos \left(2x + \frac{\pi}{4}\right) + 1 = 0 \)
   \( \cos \left(2x + \frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \)
   \( 2x + \frac{\pi}{4} = \pm \frac{3\pi}{4} + 2\pi k, k \in \mathbb{Z} \)
   \( 2x = -\frac{\pi}{4} \pm \frac{3\pi}{4} + 2\pi k \)
   \( 2x_1 = -\frac{\pi}{4} + \frac{3\pi}{4} + 2\pi k = \frac{2\pi}{4} + 2\pi k = \frac{\pi}{2} + 2\pi k \Rightarrow x_1 = \frac{\pi}{4} + \pi k \)
   \( 2x_2 = -\frac{\pi}{4} - \frac{3\pi}{4} + 2\pi k = -\pi + 2\pi k \Rightarrow x_2 = -\frac{\pi}{2} + \pi k \)

657.

1. 1)

   \( 2 \sin \left(3x - \frac{\pi}{4}\right) + 1 = 0 \)
   \( \sin \left(3x - \frac{\pi}{4}\right) = -\frac{1}{2} \)
   \( 3x - \frac{\pi}{4} = \left(-\frac{\pi}{6}\right) + 2\pi k \quad \text{или} \quad 3x - \frac{\pi}{4} = \pi - \left(-\frac{\pi}{6}\right) + 2\pi k \)
   \( 3x = \frac{\pi}{4} - \frac{\pi}{6} + 2\pi k = \frac{3\pi - 2\pi}{12} + 2\pi k = \frac{\pi}{12} + 2\pi k \Rightarrow x = \frac{\pi}{36} + \frac{2\pi k}{3} \)
   \( 3x = \frac{\pi}{4} + \frac{7\pi}{6} + 2\pi k = \frac{3\pi + 14\pi}{12} + 2\pi k = \frac{17\pi}{12} + 2\pi k \Rightarrow x = \frac{17\pi}{36} + \frac{2\pi k}{3} \)

2. 3)

   \( 3 + 4 \sin (2x + 1) = 0 \)
   \( \sin (2x + 1) = -\frac{3}{4} \)
   \( 2x + 1 = \arcsin \left(-\frac{3}{4}\right) + 2\pi k \quad \text{или} \quad 2x + 1 = \pi - \arcsin \left(-\frac{3}{4}\right) + 2\pi k \)
   \( 2x = -1 + \arcsin \left(-\frac{3}{4}\right) + 2\pi k \Rightarrow x = -\frac{1}{2} + \frac{1}{2} \arcsin \left(-\frac{3}{4}\right) + \pi k \)
   \( 2x = -1 + \pi - \arcsin \left(-\frac{3}{4}\right) + 2\pi k \Rightarrow x = -\frac{1}{2} + \frac{\pi}{2} - \frac{1}{2} \arcsin \left(-\frac{3}{4}\right) + \pi k \)

658.

1. 1)

   \( (1 + \sqrt{2} \cos x) (1 - 4 \sin x \cos x) = 0 \)
   \( 1 + \sqrt{2} \cos x = 0 \quad \text{или} \quad 1 - 4 \sin x \cos x = 0 \)
   \( \cos x = -\frac{1}{\sqrt{2}} \Rightarrow x = \pm \frac{3\pi}{4} + 2\pi k \)
   \( 1 - 2 \sin (2x) = 0 \Rightarrow \sin (2x) = \frac{1}{2} \Rightarrow 2x = \frac{\pi}{6} + 2\pi k \quad \text{или} \quad 2x = \frac{5\pi}{6} + 2\pi k \Rightarrow x = \frac{\pi}{12} + \pi k \quad \text{или} \quad x = \frac{5\pi}{12} + \pi k \)

2. 2)

   \( (1 - \sqrt{2} \cos x) (1 + 2 \sin 2x \cos 2x) = 0 \)
   \( 1 - \sqrt{2} \cos x = 0 \Rightarrow \cos x = \frac{1}{\sqrt{2}} \Rightarrow x = \pm \frac{\pi}{4} + 2\pi k \)
   \( 1 + \sin (4x) = 0 \Rightarrow \sin (4x) = -1 \Rightarrow 4x = -\frac{\pi}{2} + 2\pi k \Rightarrow x = -\frac{\pi}{8} + \frac{\pi k}{2} \)

659.

1. 1)

   \( \tan \left(2x + \frac{\pi}{4}\right) = -1 \)
   \( 2x + \frac{\pi}{4} = -\frac{\pi}{4} + \pi k \)
   \( 2x = -\frac{\pi}{2} + \pi k \)
   \( x = -\frac{\pi}{4} + \frac{\pi k}{2} \)

2. 3)

   \( \sqrt{3} - \tan \left(x - \frac{\pi}{5}\right) = 0 \)
   \( \tan \left(x - \frac{\pi}{5}\right) = \sqrt{3} \)
   \( x - \frac{\pi}{5} = \frac{\pi}{3} + \pi k \)
   \( x = \frac{\pi}{5} + \frac{\pi}{3} + \pi k = \frac{3\pi + 5\pi}{15} + \pi k = \frac{8\pi}{15} + \pi k \)

660.

1. 1)

   \( 2 \sin^2 x + \sin x = 0 \)
   \( \sin x (2 \sin x + 1) = 0 \)
   \( \sin x = 0 \Rightarrow x = \pi k \)
   \( 2 \sin x + 1 = 0 \Rightarrow \sin x = -\frac{1}{2} \Rightarrow x = -\frac{\pi}{6} + 2\pi k \quad \text{или} \quad x = \frac{7\pi}{6} + 2\pi k \)

2. 3)

   \( \cos^2 x - 2 \cos x = 0 \)
   \( \cos x (\cos x - 2) = 0 \)
   \( \cos x = 0 \Rightarrow x = \frac{\pi}{2} + \pi k \)
   \( \cos x - 2 = 0 \Rightarrow \cos x = 2 \) — решений нет.

661.

1. 1)

   \( 6 \sin^2 x - \cos x + 6 = 0 \)
   \( 6 (1 - \cos^2 x) - \cos x + 6 = 0 \)
   \( 6 - 6 \cos^2 x - \cos x + 6 = 0 \)
   \( -6 \cos^2 x - \cos x + 12 = 0 \)
   \( 6 \cos^2 x + \cos x - 12 = 0 \)
   \( \cos x = \frac{-1 \pm \sqrt{1^2 - 4(6)(-12)}}{2(6)} = \frac{-1 \pm \sqrt{1 + 288}}{12} = \frac{-1 \pm \sqrt{289}}{12} = \frac{-1 \pm 17}{12} \)
   \( \cos x = \frac{16}{12} = \frac{4}{3} \) — решений нет.
   \( \cos x = \frac{-18}{12} = -\frac{3}{2} \) — решений нет.

662.

1. 1)

   \( \tan^2 x + 3 \tan x = 0 \)
   \( \tan x (\tan x + 3) = 0 \)
   \( \tan x = 0 \Rightarrow x = \pi k \)
   \( \tan x = -3 \Rightarrow x = \arctan (-3) + \pi k \)

2. 3)

   \( \tan x - 12 \cot x + 1 = 0 \)
   \( \tan x - \frac{12}{\tan x} + 1 = 0 \)
   \( \tan^2 x + \tan x - 12 = 0 \)
   Пусть \( y = \tan x \). \( y^2 + y - 12 = 0 \)
   \( (y+4)(y-3) = 0 \)
   \( y = -4 \) или \( y = 3 \)
   \( \tan x = -4 \Rightarrow x = \arctan (-4) + \pi k \)
   \( \tan x = 3 \Rightarrow x = \arctan 3 + \pi k \)

Ответ: 656. 1) \( x = 2 \mp \frac{\pi}{3} - \pi k \); 3) \( x = \frac{\pi}{4} + \pi k \), \( x = -\frac{\pi}{2} + \pi k \). 657. 1) \( x = \frac{\pi}{36} + \frac{2\pi k}{3} \), \( x = \frac{17\pi}{36} + \frac{2\pi k}{3} \); 3) \( x = -\frac{1}{2} + \frac{1}{2} \arcsin \left(-\frac{3}{4}\right) + \pi k \), \( x = -\frac{1}{2} + \frac{\pi}{2} - \frac{1}{2} \arcsin \left(-\frac{3}{4}\right) + \pi k \). 658. 1) \( x = \pm \frac{3\pi}{4} + 2\pi k \), \( x = \frac{\pi}{12} + \pi k \), \( x = \frac{5\pi}{12} + \pi k \); 2) \( x = \pm \frac{\pi}{4} + 2\pi k \), \( x = -\frac{\pi}{8} + \frac{\pi k}{2} \). 659. 1) \( x = -\frac{\pi}{4} + \frac{\pi k}{2} \); 3) \( x = \frac{8\pi}{15} + \pi k \). 660. 1) \( x = \pi k \), \( x = -\frac{\pi}{6} + 2\pi k \), \( x = \frac{7\pi}{6} + 2\pi k \); 3) \( x = \frac{\pi}{2} + \pi k \). 661. 1) Решений нет. 662. 1) \( x = \pi k \), \( x = \arctan (-3) + \pi k \); 3) \( x = \arctan (-4) + \pi k \), \( x = \arctan 3 + \pi k \).