Упр.1045 ГДЗ Колягин Ткачёва 10 класс (Алгебра)

1045. 1) sin48°; 2) cos 164°; 3) tg92°; 4) sin 4пи/3; 5) cos 5пи/3.

1. $$\sin 48^\circ = 2\sin \frac{48^\circ}{2}\cos \frac{48^\circ}{2} = 2\sin 24^\circ \cos 24^\circ.$$
2. $$\cos 164^\circ = \cos^2 \frac{164^\circ}{2} — \sin^2 \frac{164^\circ}{2} = \cos^2 82^\circ — \sin^2 82^\circ.$$
3. $$\tg 92^\circ = \frac{2\tg \frac{92^\circ}{2}}{1-\tg^2 \frac{92^\circ}{2}} = \frac{2\tg 46^\circ}{1-\tg^2 46^\circ}.$$
4. $$\sin \frac{4\pi}{3} = 2\sin \frac{1}{2}\cdot \frac{4\pi}{3}\cos \frac{1}{2}\cdot \frac{4\pi}{3} = 2\sin \frac{2\pi}{3}\cos \frac{2\pi}{3}.$$
5. $$\cos \frac{5\pi}{3} = \cos^2 \frac{1}{2}\cdot \frac{5\pi}{3} — \sin^2 \frac{1}{2}\cdot \frac{5\pi}{3} = \cos^2 \frac{5\pi}{6} — \sin^2 \frac{5\pi}{6}.$$

Ответ

$$\sin 48^\circ = 2\sin 24^\circ \cos 24^\circ;$$
$$\cos 164^\circ = \cos^2 82^\circ — \sin^2 82^\circ;$$
$$\tg 92^\circ = \frac{2\tg 46^\circ}{1-\tg^2 46^\circ};$$
$$\sin \frac{4\pi}{3} = 2\sin \frac{2\pi}{3}\cos \frac{2\pi}{3};$$
$$\cos \frac{5\pi}{3} = \cos^2 \frac{5\pi}{6} — \sin^2 \frac{5\pi}{6}.$$