Упр.1083 ГДЗ Колягин Ткачёва 10 класс (Алгебра)

1) cos 630° — sin 1470°-ctg 1125°;
2) tg 1800° -sin495° + cos945°;
3) 3cos3660° + sin (-1560°) + cos (-450°);
4) cos4455° — cos(-945°) + tg 1035° — ctg(-1500°).

1. $$\cos 630^\circ-\sin 1470^\circ-\ctg 1125^\circ$$
   $$=\cos(360^\circ+270^\circ)-\sin(1440^\circ+30^\circ)-\ctg(1080^\circ+45^\circ)$$
   $$=\cos 270^\circ-\sin 30^\circ-\ctg 45^\circ$$
   $$=0-\frac12-1=-\frac32.$$

2. $$\tg 1800^\circ-\sin 495^\circ+\cos 945^\circ$$
   $$=\tg 0^\circ-\sin(360^\circ+135^\circ)+\cos(720^\circ+225^\circ)$$
   $$=0-\sin 135^\circ+\cos 225^\circ$$
   $$=-\frac{\sqrt2}{2}-\frac{\sqrt2}{2}=-\sqrt2.$$

3. $$3\cos 3660^\circ+\sin(-1560^\circ)+\cos(-450^\circ)$$
   $$=3\cos(3600^\circ+60^\circ)+\sin(-1800^\circ+240^\circ)+\cos(-360^\circ-90^\circ)$$
   $$=3\cos 60^\circ+\sin 240^\circ+\cos 90^\circ$$
   $$=3\cdot\frac12-\frac{\sqrt3}{2}+0=\frac{3-\sqrt3}{2}.$$

4. $$\cos 4455^\circ-\cos(-945^\circ)+\tg 1035^\circ-\ctg(-1500^\circ)$$
   $$=\cos(4320^\circ+135^\circ)-\cos(-1080^\circ+135^\circ)+\tg(1080^\circ-45^\circ)-\ctg(-1440^\circ-60^\circ)$$
   $$=\cos 135^\circ-\cos 135^\circ-\tg 45^\circ+\ctg 60^\circ$$
   $$=0-1+\frac{\sqrt3}{3}=\frac{\sqrt3-3}{3}.$$

Ответ

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