Упр.8.52 ГДЗ Погорелов 7-9 класс (Геометрия)

1. Найдите синус, косинус, тангенс и котангенс углов: 1) $$120^\circ$$; 2) $$135^\circ$$; 3) $$150^\circ$$.

Используем формулы приведения:

$$\sin(180^\circ-a)=\sin a,\qquad \cos(180^\circ-a)=-\cos a,$$

$$\tg(180^\circ-a)=-\tg a,\qquad \ctg(180^\circ-a)=-\ctg a.$$

1. $$180^\circ-120^\circ=60^\circ.$$

   $$\sin120^\circ=\sin60^\circ=\frac{\sqrt3}{2},$$

   $$\cos120^\circ=-\cos60^\circ=-\frac12,$$

   $$\tg120^\circ=-\tg60^\circ=-\sqrt3,$$

   $$\ctg120^\circ=-\ctg60^\circ=-\frac{1}{\sqrt3}.$$

2. $$180^\circ-135^\circ=45^\circ.$$

   $$\sin135^\circ=\sin45^\circ=\frac{1}{\sqrt2},$$

   $$\cos135^\circ=-\cos45^\circ=-\frac{1}{\sqrt2},$$

   $$\tg135^\circ=-\tg45^\circ=-1,$$

   $$\ctg135^\circ=-\ctg45^\circ=-1.$$

3. $$180^\circ-150^\circ=30^\circ.$$

   $$\sin150^\circ=\sin30^\circ=\frac12,$$

   $$\cos150^\circ=-\cos30^\circ=-\frac{\sqrt3}{2},$$

   $$\tg150^\circ=-\tg30^\circ=-\frac{1}{\sqrt3},$$

   $$\ctg150^\circ=-\ctg30^\circ=-\sqrt3.$$

Ответ:

1) $$\sin120^\circ=\frac{\sqrt3}{2},\ \cos120^\circ=-\frac12,\ \tg120^\circ=-\sqrt3,\ \ctg120^\circ=-\frac{1}{\sqrt3}.$$

2) $$\sin135^\circ=\frac{1}{\sqrt2},\ \cos135^\circ=-\frac{1}{\sqrt2},\ \tg135^\circ=-1,\ \ctg135^\circ=-1.$$

3) $$\sin150^\circ=\frac12,\ \cos150^\circ=-\frac{\sqrt3}{2},\ \tg150^\circ=-\frac{1}{\sqrt3},\ \ctg150^\circ=-\sqrt3.$$