№4. Вычислите: 1) sin (-3π/4) + cos (-π/4) + sin π/4 · cos π/2 + cos 0 · sin π/2 2) cos 5π/3 + cos 4π/3 + sin 3π/2 · sin 5π/8 · cos 3π/2 3) sin π/4 + cos (-3π/4) – 2 · sin (-π/6) + 2 cos 5π/6 4) 3 cos π/3 – 2 sin 2π/3 + 7 cos (-2π/3) – sin (-5π/4) 5) 3 cos 7π/4 + 2 sin 3π/4 – sin (-9π/4) + 7 cos 13π/2 6) 2 sin (-5π/6) + 11 cos (-π/3) + sin 7π/6 – 8 cos 2π/3

Выполним вычисления.

1. $$\sin(-\frac{3\pi}{4}) + \cos(-\frac{\pi}{4}) + \sin(\frac{\pi}{4}) \cdot \cos(\frac{\pi}{2}) + \cos(0) \cdot \sin(\frac{\pi}{2})$$ $$\sin(-\frac{3\pi}{4}) = -\sin(\frac{3\pi}{4}) = -\sin(\pi - \frac{\pi}{4}) = -\sin(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$$ $$\cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$ $$\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$ $$\cos(\frac{\pi}{2}) = 0$$ $$\cos(0) = 1$$ $$\sin(\frac{\pi}{2}) = 1$$ $$-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot 0 + 1 \cdot 1 = 0 + 0 + 1 = 1$$

Ответ: 1

1. $$\cos(\frac{5\pi}{3}) + \cos(\frac{4\pi}{3}) + \sin(\frac{3\pi}{2}) \cdot \sin(\frac{5\pi}{8}) \cdot \cos(\frac{3\pi}{2})$$ $$\cos(\frac{5\pi}{3}) = \cos(2\pi - \frac{\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2}$$ $$\cos(\frac{4\pi}{3}) = \cos(\pi + \frac{\pi}{3}) = -\cos(\frac{\pi}{3}) = -\frac{1}{2}$$ $$\sin(\frac{3\pi}{2}) = -1$$ $$\cos(\frac{3\pi}{2}) = 0$$ $$\frac{1}{2} - \frac{1}{2} + (-1) \cdot \sin(\frac{5\pi}{8}) \cdot 0 = 0 + 0 = 0$$

Ответ: 0

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

Ответ: $$1 - \sqrt{3}$$

1. $$3 \cdot \cos(\frac{\pi}{3}) - 2 \cdot \sin(\frac{2\pi}{3}) + 7 \cdot \cos(-\frac{2\pi}{3}) - \sin(-\frac{5\pi}{4})$$ $$\cos(\frac{\pi}{3}) = \frac{1}{2}$$ $$\sin(\frac{2\pi}{3}) = \sin(\pi - \frac{\pi}{3}) = \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$$ $$\cos(-\frac{2\pi}{3}) = \cos(\frac{2\pi}{3}) = \cos(\pi - \frac{\pi}{3}) = -\cos(\frac{\pi}{3}) = -\frac{1}{2}$$ $$\sin(-\frac{5\pi}{4}) = -\sin(\frac{5\pi}{4}) = -\sin(\pi + \frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$ $$3 \cdot \frac{1}{2} - 2 \cdot \frac{\sqrt{3}}{2} + 7 \cdot (-\frac{1}{2}) - \frac{\sqrt{2}}{2} = \frac{3}{2} - \sqrt{3} - \frac{7}{2} - \frac{\sqrt{2}}{2} = -2 - \sqrt{3} - \frac{\sqrt{2}}{2}$$

Ответ: $$-2 - \sqrt{3} - \frac{\sqrt{2}}{2}$$

1. $$3 \cdot \cos(\frac{7\pi}{4}) + 2 \cdot \sin(\frac{3\pi}{4}) - \sin(-\frac{9\pi}{4}) + 7 \cdot \cos(\frac{13\pi}{2})$$ $$\cos(\frac{7\pi}{4}) = \cos(2\pi - \frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$ $$\sin(\frac{3\pi}{4}) = \sin(\pi - \frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$$ $$\sin(-\frac{9\pi}{4}) = -\sin(\frac{9\pi}{4}) = -\sin(2\pi + \frac{\pi}{4}) = -\sin(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$$ $$\cos(\frac{13\pi}{2}) = \cos(6\pi + \frac{\pi}{2}) = \cos(\frac{\pi}{2}) = 0$$ $$3 \cdot \frac{\sqrt{2}}{2} + 2 \cdot \frac{\sqrt{2}}{2} - (-\frac{\sqrt{2}}{2}) + 7 \cdot 0 = \frac{3\sqrt{2}}{2} + \sqrt{2} + \frac{\sqrt{2}}{2} + 0 = 3\sqrt{2}$$

Ответ: $$3\sqrt{2}$$

1. $$2 \cdot \sin(-\frac{5\pi}{6}) + 11 \cdot \cos(-\frac{\pi}{3}) + \sin(\frac{7\pi}{6}) - 8 \cdot \cos(\frac{2\pi}{3})$$ $$\sin(-\frac{5\pi}{6}) = -\sin(\frac{5\pi}{6}) = -\sin(\pi - \frac{\pi}{6}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2}$$ $$\cos(-\frac{\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2}$$ $$\sin(\frac{7\pi}{6}) = \sin(\pi + \frac{\pi}{6}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2}$$ $$\cos(\frac{2\pi}{3}) = \cos(\pi - \frac{\pi}{3}) = -\cos(\frac{\pi}{3}) = -\frac{1}{2}$$ $$2 \cdot (-\frac{1}{2}) + 11 \cdot \frac{1}{2} + (-\frac{1}{2}) - 8 \cdot (-\frac{1}{2}) = -1 + \frac{11}{2} - \frac{1}{2} + 4 = 3 + \frac{10}{2} = 3 + 5 = 8$$

Ответ: 8