Упр.1084 ГДЗ Колягин Ткачёва 10 класс (Алгебра)

1) cos 23pi/4 — sin 15pi/4-ctg(-11pi/2)
2) cos 25pi/3 — cos(-17pi/2)-tg 10pi/3
3) sin(-7pi)-2cos 31pi/3 -tg 7pi/4
4) cos(-9pi)+2sin (-49pi/6)-ctg(-21pi/4)

1. $$\cos \frac{23\pi}{4}-\sin \frac{15\pi}{4}-\ctg\left(-\frac{11\pi}{2}\right)$$
   $$=\cos\left(6\pi-\frac{\pi}{4}\right)-\sin\left(4\pi-\frac{\pi}{4}\right)-\ctg\left(-6\pi+\frac{\pi}{2}\right)$$
   $$=\cos \frac{\pi}{4}+\sin \frac{\pi}{4}-\ctg \frac{\pi}{2}$$
   $$=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}-0=\sqrt{2}.$$
2. $$\cos \frac{25\pi}{3}-\cos\left(-\frac{17\pi}{2}\right)-\tg \frac{10\pi}{3}$$
   $$=\cos\left(8\pi+\frac{\pi}{3}\right)-\cos\left(-8\pi-\frac{\pi}{2}\right)-\tg\left(3\pi+\frac{\pi}{3}\right)$$
   $$=\cos \frac{\pi}{3}-\cos \frac{\pi}{2}-\tg \frac{\pi}{3}$$
   $$=\frac{1}{2}-0-\sqrt{3}=\frac{1-2\sqrt{3}}{2}.$$
3. $$\sin(-7\pi)-2\cos \frac{31\pi}{3}-\tg \frac{7\pi}{4}$$
   $$=\sin(-8\pi+\pi)-2\cos\left(10\pi+\frac{\pi}{3}\right)-\tg\left(2\pi-\frac{\pi}{4}\right)$$
   $$=\sin \pi-2\cos \frac{\pi}{3}-\tg \frac{7\pi}{4}$$
   $$=0-2\cdot \frac{1}{2}-(-1)=0.$$
4. $$\cos(-9\pi)+2\sin\left(-\frac{49\pi}{6}\right)-\ctg\left(-\frac{21\pi}{4}\right)$$
   $$=\cos(-10\pi+\pi)+2\sin\left(-8\pi-\frac{\pi}{6}\right)-\ctg\left(-5\pi-\frac{\pi}{4}\right)$$
   $$=\cos \pi-2\sin \frac{\pi}{6}+\ctg \frac{\pi}{4}$$
   $$=-1-2\cdot \frac{1}{2}+1=-1.$$

Ответ

1) $$\sqrt{2}$$; 2) $$\frac{1-2\sqrt{3}}{2}$$; 3) $$0$$; 4) $$-1$$.