655 Вычислить:

Решение:

1) \( 2 \arcsin \frac{\sqrt{3}}{2} + 3 \arcsin \left(-\frac{1}{2}\right) \)

1. \( \arcsin \frac{\sqrt{3}}{2} = \frac{\pi}{3} \)
2. \( \arcsin \left(-\frac{1}{2}\right) = -\frac{\pi}{6} \)
3. \( 2 \cdot \frac{\pi}{3} + 3 \cdot \left(-\frac{\pi}{6}\right) = \frac{2\pi}{3} - \frac{\pi}{2} = \frac{4\pi - 3\pi}{6} = \frac{\pi}{6} \)

2) \( \arcsin \frac{1}{\sqrt{2}} - 4 \arcsin 1 \)

1. \( \arcsin \frac{1}{\sqrt{2}} = \frac{\pi}{4} \)
2. \( \arcsin 1 = \frac{\pi}{2} \)
3. \( \frac{\pi}{4} - 4 \cdot \frac{\pi}{2} = \frac{\pi}{4} - 2\pi = \frac{\pi - 8\pi}{4} = -\frac{7\pi}{4} \)

3) \( \arccos \left(-\frac{1}{2}\right) - \arcsin \frac{\sqrt{3}}{2} \)

1. \( \arccos \left(-\frac{1}{2}\right) = \frac{2\pi}{3} \)
2. \( \arcsin \frac{\sqrt{3}}{2} = \frac{\pi}{3} \)
3. \( \frac{2\pi}{3} - \frac{\pi}{3} = \frac{\pi}{3} \)

4) \( \arccos (-1) - \arcsin (-1) \)

1. \( \arccos (-1) = \pi \)
2. \( \arcsin (-1) = -\frac{\pi}{2} \)
3. \( \pi - \left(-\frac{\pi}{2}\right) = \pi + \frac{\pi}{2} = \frac{3\pi}{2} \)

5) \( 2 \operatorname{arctg} 1 + 3 \operatorname{arctg} \left(-\frac{1}{\sqrt{3}}\right) \)

1. \( \operatorname{arctg} 1 = \frac{\pi}{4} \)
2. \( \operatorname{arctg} \left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} \)
3. \( 2 \cdot \frac{\pi}{4} + 3 \cdot \left(-\frac{\pi}{6}\right) = \frac{\pi}{2} - \frac{\pi}{2} = 0 \)

6) \( 4 \operatorname{arctg} (-1) + 3 \operatorname{arctg} \sqrt{3} \)

1. \( \operatorname{arctg} (-1) = -\frac{\pi}{4} \)
2. \( \operatorname{arctg} \sqrt{3} = \frac{\pi}{3} \)
3. \( 4 \cdot \left(-\frac{\pi}{4}\right) + 3 \cdot \frac{\pi}{3} = -\pi + \pi = 0 \)

Ответ: 1) \(\frac{\pi}{6}\); 2) -\(\frac{7\pi}{4}\); 3) \(\frac{\pi}{3}\); 4) \(\frac{3\pi}{2}\); 5) 0; 6) 0.