Упр.531 ГДЗ Алимов 10-11 класс (Алгебра)

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

1. $$\cos \frac{23\pi}{4}-\sin \frac{15\pi}{4}-\ctg\left(-\frac{11\pi}{2}\right)$$
   $$=\cos\left(6\pi-\frac{\pi}{4}\right)-\sin\left(4\pi-\frac{\pi}{4}\right)-\ctg\left(-6\pi+\frac{\pi}{2}\right)$$
   $$=\cos\frac{\pi}{4}+\sin\frac{\pi}{4}-\ctg\frac{\pi}{2}$$
   $$=\frac{\sqrt2}{2}+\frac{\sqrt2}{2}-0=\sqrt2.$$

2. $$\sin \frac{25\pi}{3}-\cos\left(-\frac{17\pi}{2}\right)-\tg \frac{10\pi}{3}$$
   $$=\sin\left(8\pi+\frac{\pi}{3}\right)-\cos\left(-8\pi-\frac{\pi}{2}\right)-\tg\left(3\pi+\frac{\pi}{3}\right)$$
   $$=\sin\frac{\pi}{3}-\cos\frac{\pi}{2}-\tg\frac{\pi}{3}$$
   $$=\frac{\sqrt3}{2}-0-\sqrt3=-\frac{\sqrt3}{2}.$$

3. $$\sin(-7\pi)-2\cos \frac{31\pi}{3}-\tg \frac{7\pi}{4}$$
   $$=\sin(-8\pi+\pi)-2\cos\left(10\pi+\frac{\pi}{3}\right)-\tg\left(2\pi-\frac{\pi}{4}\right)$$
   $$=\sin\pi-2\cos\frac{\pi}{3}-\tg\left(-\frac{\pi}{4}\right)$$
   $$=0-2\cdot\frac12-(-1)=-1+1=0.$$

4. $$\cos(-9\pi)+2\sin\left(-\frac{49\pi}{6}\right)-\ctg\left(-\frac{21\pi}{4}\right)$$
   $$=\cos(-10\pi+\pi)+2\sin\left(-8\pi-\frac{\pi}{6}\right)-\ctg\left(-5\pi-\frac{\pi}{4}\right)$$
   $$=\cos\pi+2\sin\left(-\frac{\pi}{6}\right)-\ctg\left(-\frac{\pi}{4}\right)$$
   $$=-1+2\cdot\left(-\frac12\right)-(-1)=-1-1+1=-1.$$

Ответ

1) $$\sqrt2$$; 2) $$-\frac{\sqrt3}{2}$$; 3) $$0$$; 4) $$-1$$.