Упр.633 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) cos пи/2 + isin пи/2/cos пи/3 + isin пи/3;
2) 8(cos 5пи/4 + isin 5пи/4)/2(cos 3пи/4 + isin 3пи/4);
3) корень 3(cos(-пи/3) + isin(-пи/3))/ cos(-пи/6) + isin(-пи/6);
4) cos30 + isin 30/2(cos(-15) + isin(-15));
5) корень 12(cos 20 + isin20)/корень 3(cos50 + isin 50);
6) cos7 + isin7/cos2+isin2.

1. $$\frac{\cos \frac{\pi}{2}+i\sin \frac{\pi}{2}}{\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}}=\cos\left(\frac{\pi}{2}-\frac{\pi}{3}\right)+i\sin\left(\frac{\pi}{2}-\frac{\pi}{3}\right)=\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}.$$
2. $$\frac{8\left(\cos \frac{5\pi}{4}+i\sin \frac{5\pi}{4}\right)}{2\left(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4}\right)}=4\left(\cos\left(\frac{5\pi}{4}-\frac{3\pi}{4}\right)+i\sin\left(\frac{5\pi}{4}-\frac{3\pi}{4}\right)\right)=4\left(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2}\right).$$
3. $$\frac{\sqrt{3}\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right)}{\cos\left(-\frac{\pi}{6}\right)+i\sin\left(-\frac{\pi}{6}\right)}=\sqrt{3}\left(\cos\left(-\frac{\pi}{3}+\frac{\pi}{6}\right)+i\sin\left(-\frac{\pi}{3}+\frac{\pi}{6}\right)\right)=\sqrt{3}\left(\cos\left(-\frac{\pi}{6}\right)+i\sin\left(-\frac{\pi}{6}\right)\right).$$
4. $$\frac{\cos 30^\circ+i\sin 30^\circ}{2\left(\cos(-15^\circ)+i\sin(-15^\circ)\right)}=\frac12\left(\cos(30^\circ-(-15^\circ))+i\sin(30^\circ-(-15^\circ))\right)=\frac12\left(\cos 45^\circ+i\sin 45^\circ\right).$$
5. $$\frac{\sqrt{12}\left(\cos 20^\circ+i\sin 20^\circ\right)}{\sqrt{3}\left(\cos 50^\circ+i\sin 50^\circ\right)}=\frac{\sqrt{12}}{\sqrt{3}}\left(\cos(20^\circ-50^\circ)+i\sin(20^\circ-50^\circ)\right)=2\left(\cos(-30^\circ)+i\sin(-30^\circ)\right).$$
6. $$\frac{\cos 7+i\sin 7}{\cos 2+i\sin 2}=\cos(7-2)+i\sin(7-2)=\cos 5+i\sin 5.$$

Ответ

1) $$\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}$$;
2) $$4\left(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2}\right)$$;
3) $$\sqrt{3}\left(\cos\left(-\frac{\pi}{6}\right)+i\sin\left(-\frac{\pi}{6}\right)\right)$$;
4) $$\frac12\left(\cos 45^\circ+i\sin 45^\circ\right)$$;
5) $$2\left(\cos(-30^\circ)+i\sin(-30^\circ)\right)$$;
6) $$\cos 5+i\sin 5$$.