Упр.1483 ГДЗ Макарычев Миндюк 9 класс (Углубленный) (Алгебра)
а) (sin(37°)+sin(23°))/(sin(37°)-sin(23°));
б) (cos(20°)-cos(140°))/(cos(20°)+cos(140°));
в) (sin(55°)-sin(35°))/(cos(55°)+cos(35°));
г) (cos(25°)-cos(85°))/(sin(25°)+sin(85°));
д) (cos(?)-cos(?))/(cos(?)+cos(?));
е) (sin(?)+sin(?))/(sin(?)-sin(?));
ж) (sin(45°+?)+sin(45°-?))/(sin(45°+?)-sin(45°-?));
з) (cos(45°-?)+cos(45°+?))/(sin(45°-?)+sin(45°+?)).
Используем формулы:
$$\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2},$$
$$\sin x-\sin y=2\cos\frac{x+y}{2}\sin\frac{x-y}{2},$$
$$\cos x-\cos y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2},$$
$$\cos x+\cos y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}.$$
$$ \frac{\sin 37^\circ+\sin 23^\circ}{\sin 37^\circ-\sin 23^\circ} = \frac{2\sin 30^\circ\cos 7^\circ}{2\cos 30^\circ\sin 7^\circ} = \frac{\sqrt{3}}{3}\ctg 7^\circ. $$
$$ \frac{\cos 20^\circ-\cos 140^\circ}{\cos 20^\circ+\cos 140^\circ} = \frac{2\sin 80^\circ\sin 60^\circ}{2\cos 80^\circ\cos 60^\circ} = \sqrt{3}\tg 80^\circ. $$
$$ \frac{\sin 55^\circ-\sin 35^\circ}{\cos 55^\circ+\cos 35^\circ} = \frac{2\cos 45^\circ\sin 10^\circ}{2\cos 45^\circ\cos 10^\circ} = \tg 10^\circ. $$
$$ \frac{\cos 25^\circ-\cos 85^\circ}{\sin 25^\circ+\sin 85^\circ} = \frac{2\sin 55^\circ\sin 30^\circ}{2\sin 55^\circ\cos 30^\circ} = \tg 30^\circ = \frac{\sqrt{3}}{3}. $$
$$ \frac{\cos \alpha-\cos \beta}{\cos \alpha+\cos \beta} = \frac{-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}}{2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}} = \tg\frac{\alpha+\beta}{2}\,\tg\frac{\beta-\alpha}{2}. $$
$$ \frac{\sin \alpha+\sin \beta}{\sin \alpha-\sin \beta} = \frac{2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}}{2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}} = \tg\frac{\alpha+\beta}{2}\,\ctg\frac{\alpha-\beta}{2}. $$
$$ \frac{\sin(45^\circ+a)+\sin(45^\circ-a)}{\sin(45^\circ+a)-\sin(45^\circ-a)} = \frac{2\sin 45^\circ\cos a}{2\cos 45^\circ\sin a} = \ctg a. $$
$$ \frac{\cos(45^\circ-a)+\cos(45^\circ+a)}{\sin(45^\circ-a)+\sin(45^\circ+a)} = \frac{2\cos 45^\circ\cos a}{2\sin 45^\circ\cos a} = 1. $$
Ответ
а) $$\frac{\sqrt{3}}{3}\ctg 7^\circ$$; б) $$\sqrt{3}\tg 80^\circ$$; в) $$\tg 10^\circ$$; г) $$\frac{\sqrt{3}}{3}$$; д) $$\tg\frac{\alpha+\beta}{2}\,\tg\frac{\beta-\alpha}{2}$$; е) $$\tg\frac{\alpha+\beta}{2}\,\ctg\frac{\alpha-\beta}{2}$$; ж) $$\ctg a$$; з) $$1$$.
