Упр.674 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

Задача

1) (3(cos 7пи/8 + isin 7пи/8))4;
2) (cos2- + isin20)12;
3) (2(cos(-20) + isin (-20)))3;
4) 1/(cos пи/20 + isin пи/20)5.

Подробный ответ

Применим формулу Муавра:
$$\left(3\left(\cos \frac{7\pi}{8}+i\sin \frac{7\pi}{8}\right)\right)^4=3^4\left(\cos \frac{28\pi}{8}+i\sin \frac{28\pi}{8}\right).$$
$$=81\left(\cos \frac{7\pi}{2}+i\sin \frac{7\pi}{2}\right)=81\left(\cos \frac{3\pi}{2}+i\sin \frac{3\pi}{2}\right)=-81i.$$
$$\left(\cos 20^\circ+i\sin 20^\circ\right)^{12}=\cos 240^\circ+i\sin 240^\circ.$$
$$\cos 240^\circ=\cos \frac{4\pi}{3}=-\frac12,\qquad \sin 240^\circ=\sin \frac{4\pi}{3}=-\frac{\sqrt3}{2}.$$
Следовательно,
$$\left(\cos 20^\circ+i\sin 20^\circ\right)^{12}=-\frac12-\frac{\sqrt3}{2}i.$$
$$\left(2\left(\cos(-20^\circ)+i\sin(-20^\circ)\right)\right)^3=2^3\left(\cos(-60^\circ)+i\sin(-60^\circ)\right).$$
$$=8\left(\frac12-\frac{\sqrt3}{2}i\right)=4-4\sqrt3\,i.$$
$$\frac{1}{\left(\cos \frac{\pi}{20}+i\sin \frac{\pi}{20}\right)^5}=\left(\cos \frac{\pi}{20}+i\sin \frac{\pi}{20}\right)^{-5}.$$
По формуле Муавра:
$$=\cos\left(-5\cdot \frac{\pi}{20}\right)+i\sin\left(-5\cdot \frac{\pi}{20}\right)=\cos\left(-\frac{\pi}{4}\right)+i\sin\left(-\frac{\pi}{4}\right).$$
$$=\frac{\sqrt2}{2}-\frac{\sqrt2}{2}i.$$