Упр.530 ГДЗ Алимов 10-11 класс (Алгебра)

1) cos 630° — sin 1470° — ctg 1125°;
2) tg 1800° — sin 495° + cos 945°;
3) 3 cos 3660° + sin (-1560°) + cos (-450°);
4) cos 4455° — cos (-945°) + tg 1035° — ctg (-1500°).

1. $$\cos 630^\circ-\sin 1470^\circ-\ctg 1125^\circ$$

   $$\cos(720^\circ-90^\circ)-\sin(1440^\circ+30^\circ)-\ctg(1080^\circ+45^\circ)$$

   $$=\cos 90^\circ-\sin 30^\circ-\ctg 45^\circ=0-\frac12-1=-\frac32$$

2. $$\tg 1800^\circ-\sin 495^\circ+\cos 945^\circ$$

   $$\tg(10\cdot 180^\circ+0^\circ)-\sin(360^\circ+180^\circ-45^\circ)+\cos(2\cdot 360^\circ+180^\circ+45^\circ)$$

   $$=\tg 0^\circ-\sin 45^\circ-\cos 45^\circ=0-\frac{\sqrt2}{2}-\frac{\sqrt2}{2}=-\sqrt2$$

3. $$3\cos 3660^\circ+\sin(-1560^\circ)+\cos(-450^\circ)$$

   $$=3\cos(10\cdot 360^\circ+60^\circ)+\sin(-5\cdot 360^\circ+180^\circ+60^\circ)+\cos(-360^\circ-90^\circ)$$

   $$=3\cos 60^\circ+\sin 240^\circ+\cos(-90^\circ)$$

   $$=3\cdot \frac12-\sin 60^\circ-0=\frac32-\frac{\sqrt3}{2}=\frac{3-\sqrt3}{2}$$

4. $$\cos 4455^\circ-\cos(-945^\circ)+\tg 1035^\circ-\ctg(-1500^\circ)$$

   $$=\cos(12\cdot 360^\circ+90^\circ+45^\circ)-\cos(-3\cdot 360^\circ+90^\circ+45^\circ)+\tg(6\cdot 180^\circ-45^\circ)-\ctg(-8\cdot 180^\circ-60^\circ)$$

   $$=\cos 135^\circ-\cos 135^\circ+\tg(-45^\circ)-\ctg(-60^\circ)$$

   $$=0-1+\frac{\sqrt3}{3}=-1+\frac{\sqrt3}{3}=\frac{\sqrt3-3}{3}$$

Ответ

1) $$-\frac32$$; 2) $$-\sqrt2$$; 3) $$\frac{3-\sqrt3}{2}$$; 4) $$\frac{\sqrt3-3}{3}$$.