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

1. Вычислить: 1) $$\cos\frac{23\pi}{4}-\sin\frac{15\pi}{4}-\operatorname{ctg}\left(-\frac{11\pi}{2}\right)$$; 2) $$\sin\frac{25\pi}{3}-\cos\left(-\frac{17\pi}{2}\right)-\operatorname{tg}\frac{10\pi}{3}$$; 3) $$\sin(-7\pi)-2\cos\frac{31\pi}{3}-\operatorname{tg}\frac{7\pi}{4}$$; 4) $$\cos(-9\pi)+2\sin\left(-\frac{49\pi}{6}\right)-\operatorname{ctg}\left(-\frac{21\pi}{4}\right)$$.

1) $$\cos \frac{23\pi}{4}-\sin \frac{15\pi}{4}-\ctg\left(-\frac{11\pi}{2}\right)$$

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

Ответ: $$\sqrt2$$

2) $$\sin \frac{25\pi}{3}-\cos\left(-\frac{17\pi}{2}\right)-\tg \frac{10\pi}{3}$$

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

Ответ: $$-\frac{\sqrt3}{2}$$

3) $$\sin(-7\pi)-2\cos \frac{31\pi}{3}-\tg \frac{7\pi}{4}$$

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

Ответ: $$0$$

4) $$\cos(-9\pi)+2\sin\left(-\frac{49\pi}{6}\right)-\ctg\left(-\frac{21\pi}{4}\right)$$

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

Ответ: $$-1$$