A1. Решите уравнение: a) 2 sin x = 1; б) cos x = -√2/2; в) sin x = 3/5; г) √3tgx = 1.

Решение:

• a) 2 sin x = 1
  • \[ \sin x = \frac{1}{2} \]
  • \[ x = \frac{\pi}{6} + 2\pi k, \quad x = \frac{5\pi}{6} + 2\pi k, \quad k \in \mathbb{Z} \]
• б) cos x = -√2/2
  • \[ x = \frac{3\pi}{4} + 2\pi k, \quad x = \frac{5\pi}{4} + 2\pi k, \quad k \in \mathbb{Z} \]
• в) sin x = 3/5
  • \[ x = \arcsin\left(\frac{3}{5}\right) + 2\pi k, \quad x = \pi - \arcsin\left(\frac{3}{5}\right) + 2\pi k, \quad k \in \mathbb{Z} \]
• г) √3tgx = 1
  • \[ \text{tg} x = \frac{1}{\sqrt{3}} \]
  • \[ x = \frac{\pi}{6} + \pi k, \quad k \in \mathbb{Z} \]