Упр.634 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) (2(cos 4пи/3 + isin 4пи/3))3;
2) (cos пи/6 + isin пи/6)6;
3) (1/2(cos пи/8 + isin пи/8))4;
4) (cos(-пи/6) + isin (-пи/6))9.

Используем формулу Муавра:

$$\bigl(r(\cos \varphi+i\sin \varphi)\bigr)^n=r^n(\cos n\varphi+i\sin n\varphi).$$

1. $$\bigl(2(\cos \frac{4\pi}{3}+i\sin \frac{4\pi}{3})\bigr)^3$$

   $$=2^3\left(\cos 3\cdot \frac{4\pi}{3}+i\sin 3\cdot \frac{4\pi}{3}\right)$$

   $$=8(\cos 4\pi+i\sin 4\pi)=8(1+i\cdot 0)=8.$$

2. $$\left(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}\right)^6$$

   $$=\cos \left(6\cdot \frac{\pi}{6}\right)+i\sin \left(6\cdot \frac{\pi}{6}\right)$$

   $$=\cos \pi+i\sin \pi=-1+i\cdot 0=-1.$$

3. $$\left(\frac12\left(\cos \frac{\pi}{8}+i\sin \frac{\pi}{8}\right)\right)^4$$

   $$=\left(\frac12\right)^4\left(\cos 4\cdot \frac{\pi}{8}+i\sin 4\cdot \frac{\pi}{8}\right)$$

   $$=\frac{1}{16}\left(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2}\right)=\frac{1}{16}(0+i\cdot 1)=\frac{i}{16}.$$

4. $$\left(\cos \left(-\frac{\pi}{6}\right)+i\sin \left(-\frac{\pi}{6}\right)\right)^9$$

   $$=\cos \left(9\cdot \left(-\frac{\pi}{6}\right)\right)+i\sin \left(9\cdot \left(-\frac{\pi}{6}\right)\right)$$

   $$=\cos \left(-\frac{3\pi}{2}\right)+i\sin \left(-\frac{3\pi}{2}\right)=0+i\cdot 1=i.$$

Ответ

1) $$8$$; 2) $$-1$$; 3) $$\frac{i}{16}$$; 4) $$i$$.