Упр.24.1 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) sin(?)cos(4?)+cos(?)sin(4?);
2) cos(3?/8)cos(?/8)-sin(3?/8)sin(?/8);
3) sin(?)sin(?+?)+cos(?)cos(?+?);
4) sin(53°)cos(7°)-cos(53°)sin(-7°);
5) cos(?+?)+2sin(?)sin(?).

1. $$\sin \alpha \cos 4\alpha+\cos \alpha \sin 4\alpha=\sin(\alpha+4\alpha)=\sin 5\alpha.$$

2. $$\cos \frac{3\pi}{8}\cos \frac{\pi}{8}-\sin \frac{3\pi}{8}\sin \frac{\pi}{8} =\cos\left(\frac{3\pi}{8}+\frac{\pi}{8}\right) =\cos \frac{\pi}{2}=0.$$

3. $$\sin \alpha \sin(\alpha+\beta)+\cos \alpha \cos(\alpha+\beta)=\cos \beta.$$

4. $$\sin 53^\circ \cos 7^\circ-\cos 53^\circ \sin(-7^\circ)$$
   $$=\sin 53^\circ \cos 7^\circ+\cos 53^\circ \sin 7^\circ$$
   $$=\sin(53^\circ+7^\circ)=\sin 60^\circ=\frac{\sqrt{3}}{2}.$$

5. $$\cos(\alpha+\beta)+2\sin \alpha \sin \beta$$
   $$=\cos \alpha \cos \beta-\sin \alpha \sin \beta+2\sin \alpha \sin \beta$$
   $$=\cos \alpha \cos \beta+\sin \alpha \sin \beta=\cos(\alpha-\beta).$$

Ответ

1) $$\sin 5\alpha$$; 2) $$0$$; 3) $$\cos \beta$$; 4) $$\frac{\sqrt{3}}{2}$$; 5) $$\cos(\alpha-\beta)$$.