Упр.661 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) 3(cos 130° + isin 130°)(cos 140° + isin 140°);
2) (cos 5пи/6 + isin 5пи/6) * корень 3 (cos(-пи/12) + isin (-пи/12));
3) cos50 + isin50/2(cos20 + isin20);
4) 2(cos(-5пи/12) + isin (-5пи/12))/ корень 2(cos(-пи/12) + isin(-пи/12).

1. $$3(\cos 130^\circ+i\sin 130^\circ)(\cos 140^\circ+i\sin 140^\circ)$$
   $$=3\bigl(\cos(130^\circ+140^\circ)+i\sin(130^\circ+140^\circ)\bigr)$$
   $$=3(\cos 270^\circ+i\sin 270^\circ).$$

2. $$\left(\cos \frac{5\pi}{6}+i\sin \frac{5\pi}{6}\right)\sqrt{3}\left(\cos\left(-\frac{\pi}{12}\right)+i\sin\left(-\frac{\pi}{12}\right)\right)$$
   $$=\sqrt{3}\left(\cos\left(\frac{5\pi}{6}-\frac{\pi}{12}\right)+i\sin\left(\frac{5\pi}{6}-\frac{\pi}{12}\right)\right)$$
   $$=\sqrt{3}\left(\cos \frac{9\pi}{12}+i\sin \frac{9\pi}{12}\right)$$
   $$=\sqrt{3}\left(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4}\right).$$

3. $$\frac{\cos 50^\circ+i\sin 50^\circ}{2(\cos 20^\circ+i\sin 20^\circ)}$$
   $$=\frac12\left(\cos(50^\circ-20^\circ)+i\sin(50^\circ-20^\circ)\right)$$
   $$=\frac12(\cos 30^\circ+i\sin 30^\circ).$$

4. $$\frac{2\left(\cos\left(-\frac{5\pi}{12}\right)+i\sin\left(-\frac{5\pi}{12}\right)\right)}{\sqrt{2}\left(\cos\left(-\frac{\pi}{12}\right)+i\sin\left(-\frac{\pi}{12}\right)\right)}$$
   $$=\sqrt{2}\left(\cos\left(-\frac{5\pi}{12}+\frac{\pi}{12}\right)+i\sin\left(-\frac{5\pi}{12}+\frac{\pi}{12}\right)\right)$$
   $$=\sqrt{2}\left(\cos\left(-\frac{4\pi}{12}\right)+i\sin\left(-\frac{4\pi}{12}\right)\right)$$
   $$=\sqrt{2}\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right).$$

Ответ

1) $$3(\cos 270^\circ+i\sin 270^\circ)$$;
2) $$\sqrt{3}\left(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4}\right)$$;
3) $$\frac12(\cos 30^\circ+i\sin 30^\circ)$$;
4) $$\sqrt{2}\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right).$$