Упр.17.23 ГДЗ Никольский Потапов 11 класс (Алгебра)
а) 4(cos 4пи/3 + isin 4пи/3);
б) cos 2пи/3 + isin 2пи/3;
в) 25(cos 3пи/4 + isin 3пи/4);
г) 36(cos 4пи/5 + isin 4пи/5).
Для числа вида $$z=r(\cos \varphi+i\sin \varphi)$$ его квадратные корни находятся по формуле
$$\alpha_k=\sqrt r\left(\cos \frac{\varphi+2\pi k}{2}+i\sin \frac{\varphi+2\pi k}{2}\right), \quad k=0,1.$$
$$z=4\left(\cos \frac{4\pi}{3}+i\sin \frac{4\pi}{3}\right), \quad r=4,\ \varphi=\frac{4\pi}{3}.$$
$$\alpha_k=2\left(\cos \frac{\frac{4\pi}{3}+2\pi k}{2}+i\sin \frac{\frac{4\pi}{3}+2\pi k}{2}\right), \quad k=0,1.$$
При $$k=0$$:
$$\alpha_0=2\left(\cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3}\right)=2\left(-\frac12+\frac{\sqrt3}{2}i\right)=-1+\sqrt3\,i.$$
При $$k=1$$:
$$\alpha_1=2\left(\cos \frac{5\pi}{3}+i\sin \frac{5\pi}{3}\right)=2\left(\frac12-\frac{\sqrt3}{2}i\right)=1-\sqrt3\,i.$$
$$z=\cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3}, \quad r=1,\ \varphi=\frac{2\pi}{3}.$$
$$\alpha_k=\cos \frac{\frac{2\pi}{3}+2\pi k}{2}+i\sin \frac{\frac{2\pi}{3}+2\pi k}{2}, \quad k=0,1.$$
При $$k=0$$:
$$\alpha_0=\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}=\frac12+\frac{\sqrt3}{2}i.$$
При $$k=1$$:
$$\alpha_1=\cos \frac{4\pi}{3}+i\sin \frac{4\pi}{3}=-\frac12-\frac{\sqrt3}{2}i.$$
$$z=25\left(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4}\right), \quad r=25,\ \varphi=\frac{3\pi}{4}.$$
$$\alpha_k=5\left(\cos \frac{\frac{3\pi}{4}+2\pi k}{2}+i\sin \frac{\frac{3\pi}{4}+2\pi k}{2}\right), \quad k=0,1.$$
При $$k=0$$:
$$\alpha_0=5\left(\cos \frac{3\pi}{8}+i\sin \frac{3\pi}{8}\right).$$
При $$k=1$$:
$$\alpha_1=5\left(\cos \frac{11\pi}{8}+i\sin \frac{11\pi}{8}\right).$$
$$z=36\left(\cos \frac{4\pi}{5}+i\sin \frac{4\pi}{5}\right), \quad r=36,\ \varphi=\frac{4\pi}{5}.$$
$$\alpha_k=6\left(\cos \frac{\frac{4\pi}{5}+2\pi k}{2}+i\sin \frac{\frac{4\pi}{5}+2\pi k}{2}\right), \quad k=0,1.$$
При $$k=0$$:
$$\alpha_0=6\left(\cos \frac{2\pi}{5}+i\sin \frac{2\pi}{5}\right).$$
При $$k=1$$:
$$\alpha_1=6\left(\cos \frac{7\pi}{5}+i\sin \frac{7\pi}{5}\right).$$
Ответ
а) $$-1+\sqrt3\,i,\ 1-\sqrt3\,i$$;
б) $$\frac12+\frac{\sqrt3}{2}i,\ -\frac12-\frac{\sqrt3}{2}i$$;
в) $$5\left(\cos \frac{3\pi}{8}+i\sin \frac{3\pi}{8}\right),\ 5\left(\cos \frac{11\pi}{8}+i\sin \frac{11\pi}{8}\right)$$;
г) $$6\left(\cos \frac{2\pi}{5}+i\sin \frac{2\pi}{5}\right),\ 6\left(\cos \frac{7\pi}{5}+i\sin \frac{7\pi}{5}\right)$$.
