Упр.793 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) 3(cos пи/3 + isin пи/3);
2) (1+i)(cos 3пи/8 + isin 3пи/8)(cosпи/8 + isin пи/8);
3) cos 7пи/10 + isin 7пи/10/cos пи/5 + isin пи/5;
4) i/cos пи/4 + isin пи/4.

1. $$3\left(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}\right)=3\left(\frac12+i\cdot \frac{\sqrt3}{2}\right)=\frac32+\frac{3\sqrt3}{2}i.$$
2. $$\begin{aligned} (1+i)\left(\cos \frac{3\pi}{8}+i\sin \frac{3\pi}{8}\right)\left(\cos \frac{\pi}{8}+i\sin \frac{\pi}{8}\right) &= (1+i)\left(\cos \left(\frac{3\pi}{8}+\frac{\pi}{8}\right)+i\sin \left(\frac{3\pi}{8}+\frac{\pi}{8}\right)\right) \\ &= (1+i)\left(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2}\right) \\ &= (1+i)(0+i\cdot 1) \\ &= (1+i)i=i+i^2=-1+i. \end{aligned}$$
3. $$\begin{aligned} \frac{\cos \frac{7\pi}{10}+i\sin \frac{7\pi}{10}}{\cos \frac{\pi}{5}+i\sin \frac{\pi}{5}} &= \cos \left(\frac{7\pi}{10}-\frac{\pi}{5}\right)+i\sin \left(\frac{7\pi}{10}-\frac{\pi}{5}\right) \\ &= \cos \frac{\pi}{2}+i\sin \frac{\pi}{2} \\ &= 0+i\cdot 1=i. \end{aligned}$$
4. $$\begin{aligned} \frac{i}{\cos \frac{\pi}{4}+i\sin \frac{\pi}{4}} &= \frac{i}{\frac{\sqrt2}{2}+i\frac{\sqrt2}{2}} = \frac{\sqrt2\, i}{1+i} = \frac{\sqrt2\, i(1-i)}{(1+i)(1-i)} \\ &= \frac{\sqrt2\, i-\sqrt2\, i^2}{1-i^2} = \frac{\sqrt2\, i+\sqrt2}{1+1} = \frac{\sqrt2+\sqrt2\, i}{2} = \frac{\sqrt2}{2}+\frac{\sqrt2}{2}i. \end{aligned}$$

Ответ

1) $$\frac32+\frac{3\sqrt3}{2}i$$; 2) $$-1+i$$; 3) $$i$$; 4) $$\frac{\sqrt2}{2}+\frac{\sqrt2}{2}i$$.