Упр.22.8 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)
1) 4cos(225°)-6cos(120°)+3ctg(300°)+tg(240°);
2) (6cos^2(-240°)ctg(210°))/(sin(-300°)cos^2(180°));
3) sin(7?/4)cos(2?/3)tg(4?/3)ctg(7?/6);
4) (cos(64°)cos(4°)-cos(86°)cos(26°))/(cos(71°)cos(41°)-cos(49°)cos(19°)).
$$4\cos 225^\circ-6\cos 120^\circ+3\ctg 300^\circ+\tg 240^\circ$$
$$=4\cos(180^\circ+45^\circ)-6\cos(90^\circ+30^\circ)+3\ctg(270^\circ+30^\circ)+\tg(270^\circ-30^\circ)$$
$$=4(-\cos 45^\circ)-6(-\sin 30^\circ)+3(-\tg 30^\circ)+\ctg 30^\circ$$
$$=4\left(-\frac{\sqrt2}{2}\right)-6\left(-\frac12\right)+3\left(-\frac{\sqrt3}{3}\right)+\sqrt3$$
$$=-2\sqrt2+3-\sqrt3+\sqrt3=3-2\sqrt2$$
$$\frac{6\cos^2(-240^\circ)\ctg 210^\circ}{\sin(-300^\circ)\cos^2 180^\circ}$$
$$=\frac{6\cos^2(180^\circ+60^\circ)\ctg(180^\circ+30^\circ)}{-\sin(270^\circ+30^\circ)\cos^2 0^\circ}$$
$$=\frac{6(-\cos 60^\circ)^2\cdot \ctg 30^\circ}{(-\cos 30^\circ)\cdot 1^2}$$
$$=\frac{6\left(\frac12\right)^2\cdot \sqrt3}{\frac{\sqrt3}{2}}=3$$
$$\sin\frac{7\pi}{4}\cdot \cos\frac{2\pi}{3}\cdot \tg\frac{4\pi}{3}\cdot \ctg\frac{7\pi}{6}$$
$$=\sin\left(2\pi-\frac{\pi}{4}\right)\cdot \cos\left(\pi-\frac{\pi}{3}\right)\cdot \tg\left(\pi+\frac{\pi}{3}\right)\cdot \ctg\left(\pi+\frac{\pi}{6}\right)$$
$$=-\sin\frac{\pi}{4}\cdot \left(-\cos\frac{\pi}{3}\right)\cdot \tg\frac{\pi}{3}\cdot \ctg\frac{\pi}{6}$$
$$=-\frac{\sqrt2}{2}\cdot \left(-\frac12\right)\cdot \sqrt3\cdot \sqrt3=\frac{3\sqrt2}{4}$$
$$\frac{\cos 64^\circ\cdot \cos 4^\circ-\cos 86^\circ\cdot \cos 26^\circ}{\cos 71^\circ\cdot \cos 41^\circ-\cos 49^\circ\cdot \cos 19^\circ}$$
$$=\frac{\cos 64^\circ\cdot \cos 4^\circ-\cos(90^\circ-4^\circ)\cdot \cos(90^\circ-64^\circ)}{\cos 71^\circ\cdot \cos 41^\circ-\cos(90^\circ-41^\circ)\cdot \cos(90^\circ-71^\circ)}$$
$$=\frac{\cos 64^\circ\cdot \cos 4^\circ-\sin 4^\circ\cdot \sin 64^\circ}{\cos 71^\circ\cdot \cos 41^\circ-\sin 41^\circ\cdot \sin 71^\circ}$$
$$=\frac{\cos(64^\circ+4^\circ)}{\cos(71^\circ+41^\circ)}=\frac{\cos 68^\circ}{\cos 112^\circ}$$
$$=\frac{\cos 68^\circ}{-\cos 68^\circ}=-1$$
Ответ
1) $$3-2\sqrt2$$; 2) $$3$$; 3) $$\frac{3\sqrt2}{4}$$; 4) $$-1$$.
