Упр.7.47 ГДЗ Никольский 10 класс (Алгебра)

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

   $$=\frac{\sqrt2}{2}+\left(-\frac{\sqrt2}{2}\right)-2\left(-\frac12\right)+2\left(-\frac{\sqrt3}{2}\right)$$

   $$=1-\sqrt3$$

2. $$3\cos \frac{\pi}{3}-2\sin \frac{2\pi}{3}+7\cos\left(-\frac{2\pi}{3}\right)-\sin\left(-\frac{5\pi}{4}\right)$$

   $$=3\cdot \frac12-2\cdot \frac{\sqrt3}{2}+7\cdot \left(-\frac12\right)-\frac{\sqrt2}{2}$$

   $$=\frac32-\sqrt3-\frac72-\frac{\sqrt2}{2}=-2-\sqrt3-\frac{\sqrt2}{2}$$

3. $$3\cos \frac{7\pi}{4}+2\sin \frac{3\pi}{4}-\sin\left(-\frac{9\pi}{4}\right)+7\cos \frac{13\pi}{2}$$

   $$=3\cdot \frac{\sqrt2}{2}+2\cdot \frac{\sqrt2}{2}+\frac{\sqrt2}{2}+7$$

   $$=7+\frac{7\sqrt2}{2}$$

4. $$2\sin\left(-\frac{5\pi}{6}\right)+11\cos\left(-\frac{7\pi}{3}\right)+\sin \frac{7\pi}{6}-8\cos \frac{11\pi}{3}$$

   $$=2\cdot \left(-\frac12\right)+11\cdot \frac12+\left(-\frac12\right)-8\cdot \frac12$$

   $$=-1+\frac{11}{2}-\frac12-4=0$$

Ответ

$$1-\sqrt3;\ -2-\sqrt3-\frac{\sqrt2}{2};\ 7+\frac{7\sqrt2}{2};\ 0$$