Упр.9.2 ГДЗ Никольский 10 класс (Алгебра)

1. $$\cos 15^\circ=\cos(45^\circ-30^\circ)=\cos45^\circ\cos30^\circ+\sin45^\circ\sin30^\circ$$

   $$=\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\cdot\frac12=\frac{\sqrt6+\sqrt2}{4}$$

2. $$\cos 75^\circ=\cos(45^\circ+30^\circ)=\cos45^\circ\cos30^\circ-\sin45^\circ\sin30^\circ$$

   $$=\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}-\frac{\sqrt2}{2}\cdot\frac12=\frac{\sqrt6-\sqrt2}{4}$$

3. $$\cos 105^\circ=\cos(60^\circ+45^\circ)=\cos60^\circ\cos45^\circ-\sin60^\circ\sin45^\circ$$

   $$=\frac12\cdot\frac{\sqrt2}{2}-\frac{\sqrt3}{2}\cdot\frac{\sqrt2}{2}=\frac{\sqrt2-\sqrt6}{4}$$

Ответ

$$\cos 15^\circ=\frac{\sqrt6+\sqrt2}{4},\quad \cos 75^\circ=\frac{\sqrt6-\sqrt2}{4},\quad \cos 105^\circ=\frac{\sqrt2-\sqrt6}{4}$$