Упр.29.3 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) ctg(arcsin(v2/2)); 4) ctg(2arcctg(-v3));
2) cos(2arctg(1)); 5) sin(3arcsin(-v3/2));
3) cos((1/2)arcsin(v3/2)); 6) sin(arcsin(v2/2)+2arctg(1)).

1. $$\ctg\left(\arcsin\frac{\sqrt2}{2}\right)=\ctg\frac{\pi}{4}=1.$$
2. $$\cos\left(2\arctg 1\right)=\cos\left(2\cdot\frac{\pi}{4}\right)=\cos\frac{\pi}{2}=0.$$
3. $$\cos\left(\frac12\arcsin\frac{\sqrt3}{2}\right)=\cos\left(\frac12\cdot\frac{\pi}{3}\right)=\cos\frac{\pi}{6}=\frac{\sqrt3}{2}.$$
4. $$\ctg\left(2\arcctg(-\sqrt3)\right)=\ctg\left(2\left(\pi-\arcctg\sqrt3\right)\right)=\ctg\left(2\pi-2\cdot\frac{\pi}{6}\right)$$
   $$=\ctg\left(2\pi-\frac{\pi}{3}\right)=\ctg\left(-\frac{\pi}{3}\right)=-\ctg\frac{\pi}{3}=-\frac{\sqrt3}{3}.$$
5. $$\sin\left(3\arcsin\left(-\frac{\sqrt3}{2}\right)\right)=\sin\left(-3\arcsin\frac{\sqrt3}{2}\right)=-\sin\left(3\cdot\frac{\pi}{3}\right)=-\sin\pi=0.$$
6. $$\sin\left(\arcsin\frac{\sqrt2}{2}+2\arctg 1\right)=\sin\left(\frac{\pi}{4}+2\cdot\frac{\pi}{4}\right)=\sin\left(\frac{\pi}{4}+\frac{\pi}{2}\right)$$
   $$=\sin\frac{3\pi}{4}=\frac{\sqrt2}{2}.$$

Ответ

1) $$1$$; 2) $$0$$; 3) $$\frac{\sqrt3}{2}$$; 4) $$-\frac{\sqrt3}{3}$$; 5) $$0$$; 6) $$\frac{\sqrt2}{2}$$.