Вычислите: 1) sin 225°; 2) cos 570°; 3) tg 390°; 4) ctg 765°; 5) sin(−21π/4); 6) cos(31π/6); 7) tg(17π/3); 8) ctg(−19π/4).

1. \(\sin 225^\circ=\sin(180^\circ+45^\circ)=-\sin45^\circ=-\frac{\sqrt2}{2}.\)
2. \(\cos570^\circ=\cos(360^\circ+210^\circ)=\cos210^\circ=-\frac{\sqrt3}{2}.\)
3. \(\operatorname{tg}390^\circ=\operatorname{tg}(360^\circ+30^\circ)=\frac{\sqrt3}{3}.\)
4. \(\operatorname{ctg}765^\circ=\operatorname{ctg}(720^\circ+45^\circ)=1.\)
5. \(\sin\left(-\frac{21\pi}{4}\right)=\sin\left(-\frac{5\pi}{4}\right)=\frac{\sqrt2}{2}.\)
6. \(\cos\frac{31\pi}{6}=\cos\frac{7\pi}{6}=-\frac{\sqrt3}{2}.\)
7. \(\operatorname{tg}\frac{17\pi}{3}=\operatorname{tg}\frac{2\pi}{3}=-\sqrt3.\)
8. \(\operatorname{ctg}\left(-\frac{19\pi}{4}\right)=\operatorname{ctg}\left(-\frac{3\pi}{4}\right)=1.\)

Ответ: 1) \(-\frac{\sqrt2}{2}\); 2) \(-\frac{\sqrt3}{2}\); 3) \(\frac{\sqrt3}{3}\); 4) 1; 5) \(\frac{\sqrt2}{2}\); 6) \(-\frac{\sqrt3}{2}\); 7) \(-\sqrt3\); 8) 1.