Упр.635 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) i(cos 5пи/3 + isin 5пи/3)/ cos пи/6 + isin пи/6;
2) 1/ cos 4пи/3 — isin 4пи/3;
3) (cos пи/3 — isin пи/3)(1+ корень 3i)7/i;
4) (1/2 + корень 3/2i)(cos пи/3 — isin пи/3);
5) (t8 + корень 3t5/4)5;
6) (2i)7/(-корень 2 + i корень 2)6.

1. $$i\frac{\cos \frac{5\pi}{3}+i\sin \frac{5\pi}{3}}{\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}}=i\left(\cos \left(\frac{5\pi}{3}-\frac{\pi}{6}\right)+i\sin \left(\frac{5\pi}{3}-\frac{\pi}{6}\right)\right)$$
   $$=i\left(\cos \frac{3\pi}{2}+i\sin \frac{3\pi}{2}\right)=i(0-i)=-i^2=1.$$
2. $$\frac{1}{\cos \frac{4\pi}{3}-i\sin \frac{4\pi}{3}}=\frac{1}{\cos \left(-\frac{4\pi}{3}\right)+i\sin \left(-\frac{4\pi}{3}\right)}$$
   $$=\cos \frac{4\pi}{3}+i\sin \frac{4\pi}{3}=-\frac12-\frac{\sqrt3}{2}i.$$
3. $$\frac{\left(\cos \frac{\pi}{3}-i\sin \frac{\pi}{3}\right)\left(1+\sqrt3\,i\right)^7}{i}$$
   $$=\frac{1}{i}\left(\cos \left(-\frac{\pi}{3}\right)+i\sin \left(-\frac{\pi}{3}\right)\right)\left(2\left(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}\right)\right)^7$$
   $$=\frac{1}{i}\cdot 2^7\left(\cos \left(-\frac{\pi}{3}+\frac{7\pi}{3}\right)+i\sin \left(-\frac{\pi}{3}+\frac{7\pi}{3}\right)\right)$$
   $$=\frac{128}{i}\left(\cos 2\pi+i\sin 2\pi\right)=\frac{128}{i}=-128i.$$
4. $$\left(\frac12+\frac{\sqrt3}{2}i\right)\left(\cos \frac{\pi}{3}-i\sin \frac{\pi}{3}\right)$$
   $$=\left(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}\right)\left(\cos \left(-\frac{\pi}{3}\right)+i\sin \left(-\frac{\pi}{3}\right)\right)$$
   $$=\cos 0+i\sin 0=1.$$
5. $$\left(\frac{i^8+\sqrt3\,i^5}{4}\right)^5=\left(\frac{1+\sqrt3\,i}{4}\right)^5$$
   $$=\left(\frac{2\left(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}\right)}{4}\right)^5 =\left(\frac12\right)^5\left(\cos \frac{5\pi}{3}+i\sin \frac{5\pi}{3}\right)$$
   $$=\frac{1}{32}\left(\frac12-\frac{\sqrt3}{2}i\right)=\frac{1}{64}-\frac{\sqrt3}{64}i.$$
6. $$\frac{(2i)^7}{(-\sqrt2+i\sqrt2)^6}=\frac{\left(2\left(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2}\right)\right)^7}{\left(2\left(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4}\right)\right)^6}$$
   $$=\frac{2^7\left(\cos \frac{7\pi}{2}+i\sin \frac{7\pi}{2}\right)}{2^6\left(\cos \frac{9\pi}{2}+i\sin \frac{9\pi}{2}\right)}$$
   $$=2\left(\cos \left(\frac{7\pi}{2}-\frac{9\pi}{2}\right)+i\sin \left(\frac{7\pi}{2}-\frac{9\pi}{2}\right)\right)$$
   $$=2(\cos(-\pi)+i\sin(-\pi))=2(-1+0i)=-2.$$

Ответ

1) $$1$$; 2) $$-\frac12-\frac{\sqrt3}{2}i$$; 3) $$-128i$$; 4) $$1$$; 5) $$\frac{1}{64}-\frac{\sqrt3}{64}i$$; 6) $$-2$$.