Упр.626 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) cos 5пи/3 + isin 5пи/3;
2) 3(cos 2пи + isin2пи);
3) корень 2 (cos 3пи/4 + isin 3пи/4);
4) 4(cos 9пи/2 + isin 9пи/2);
5) cos 7пи/6 + isin 7пи/6;
6) cos 13пи/3 + isin 13пи/3.

1. $$\cos \frac{5\pi}{3}+i\sin \frac{5\pi}{3}=\cos \frac{5\pi}{3}+i\left(-\frac{\sqrt{3}}{2}\right)=\frac{1}{2}-\frac{\sqrt{3}}{2}i.$$
2. $$3(\cos 2\pi+i\sin 2\pi)=3(1+i\cdot 0)=3.$$
3. $$\sqrt{2}\left(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4}\right)=\sqrt{2}\left(-\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}\right)=-1+i.$$
4. $$4\left(\cos \frac{9\pi}{2}+i\sin \frac{9\pi}{2}\right)=4\left(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2}\right)=4(0+i\cdot 1)=4i.$$
5. $$\cos \frac{7\pi}{6}+i\sin \frac{7\pi}{6}=\cos \left(\pi+\frac{\pi}{6}\right)+i\sin \left(\pi+\frac{\pi}{6}\right)=-\cos \frac{\pi}{6}-i\sin \frac{\pi}{6}=-\frac{\sqrt{3}}{2}-\frac{1}{2}i.$$
6. $$\cos \frac{13\pi}{3}+i\sin \frac{13\pi}{3}=\cos \left(4\pi+\frac{\pi}{3}\right)+i\sin \left(4\pi+\frac{\pi}{3}\right)=\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}=\frac{1}{2}+\frac{\sqrt{3}}{2}i.$$

Ответ

1) $$\frac{1}{2}-\frac{\sqrt{3}}{2}i$$; 2) $$3$$; 3) $$-1+i$$; 4) $$4i$$; 5) $$-\frac{\sqrt{3}}{2}-\frac{1}{2}i$$; 6) $$\frac{1}{2}+\frac{\sqrt{3}}{2}i$$.