Упр.1095 ГДЗ Колягин Ткачёва 10 класс (Алгебра)

1) cos 105° +cos75°;
2) sin 105°-sin75°;
3) cos 11пи/12 + cos 5пи/12;
4) cos 11пи/12-cos5 пи/12;
5) sin 7пи/12-sin пи/12;
6) sin 105° + sin 165°.

1. $$\cos 105^\circ+\cos 75^\circ=2\cos\frac{105^\circ+75^\circ}{2}\cos\frac{105^\circ-75^\circ}{2}$$
   $$=2\cos 90^\circ\cos 15^\circ=2\cdot 0\cdot \cos 15^\circ=0.$$

2. $$\sin 105^\circ-\sin 75^\circ=2\cos\frac{105^\circ+75^\circ}{2}\sin\frac{105^\circ-75^\circ}{2}$$
   $$=2\cos 90^\circ\sin 15^\circ=2\cdot 0\cdot \sin 15^\circ=0.$$

3. $$\cos\frac{11\pi}{12}+\cos\frac{5\pi}{12}=2\cos\frac{\frac{11\pi}{12}+\frac{5\pi}{12}}{2}\cos\frac{\frac{11\pi}{12}-\frac{5\pi}{12}}{2}$$
   $$=2\cos\frac{16\pi}{24}\cos\frac{6\pi}{24}=2\cos\frac{2\pi}{3}\cos\frac{\pi}{4}$$
   $$=2\cdot\left(-\frac12\right)\cdot\frac{\sqrt2}{2}=-\frac{\sqrt2}{2}.$$

4. $$\cos\frac{11\pi}{12}-\cos\frac{5\pi}{12}=-2\sin\frac{\frac{11\pi}{12}+\frac{5\pi}{12}}{2}\sin\frac{\frac{11\pi}{12}-\frac{5\pi}{12}}{2}$$
   $$=-2\sin\frac{16\pi}{24}\sin\frac{6\pi}{24}=-2\sin\frac{2\pi}{3}\sin\frac{\pi}{4}$$
   $$=-2\cdot\frac{\sqrt3}{2}\cdot\frac{\sqrt2}{2}=-\frac{\sqrt6}{2}.$$

5. $$\sin\frac{7\pi}{12}-\sin\frac{\pi}{12}=2\cos\frac{\frac{7\pi}{12}+\frac{\pi}{12}}{2}\sin\frac{\frac{7\pi}{12}-\frac{\pi}{12}}{2}$$
   $$=2\cos\frac{8\pi}{24}\sin\frac{6\pi}{24}=2\cos\frac{\pi}{3}\sin\frac{\pi}{4}$$
   $$=2\cdot\frac12\cdot\frac{\sqrt2}{2}=\frac{\sqrt2}{2}.$$

6. $$\sin 105^\circ+\sin 165^\circ=2\sin\frac{105^\circ+165^\circ}{2}\cos\frac{105^\circ-165^\circ}{2}$$
   $$=2\sin 135^\circ\cos(-30^\circ)=2\cdot\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}=\frac{\sqrt6}{2}.$$

Ответ

$$1)\ 0;\quad 2)\ 0;\quad 3)\ -\frac{\sqrt2}{2};\quad 4)\ -\frac{\sqrt6}{2};\quad 5)\ \frac{\sqrt2}{2};\quad 6)\ \frac{\sqrt6}{2}.$$