Упр.661 ГДЗ Колягин Ткачёва 11 класс (Алгебра)
- Выполнить действия над комплексными числами в тригонометрической форме:
- $$3(\cos 130^\circ+i\sin 130^\circ)(\cos 140^\circ+i\sin 140^\circ)$$;
- $$(\cos \frac{5\pi}{6}+i\sin \frac{5\pi}{6})\sqrt{3}(\cos(-\frac{\pi}{12})+i\sin(-\frac{\pi}{12}))$$;
- $$\frac{\cos 50^\circ+i\sin 50^\circ}{2(\cos 20^\circ+i\sin 20^\circ)}$$;
- $$\frac{2(\cos(-\frac{5\pi}{12})+i\sin(-\frac{5\pi}{12}))}{\sqrt{2}(\cos(-\frac{\pi}{12})+i\sin(-\frac{\pi}{12}))}$$.
$$3(\cos 130^\circ+i\sin 130^\circ)(\cos 140^\circ+i\sin 140^\circ)$$
$$=3\bigl(\cos(130^\circ+140^\circ)+i\sin(130^\circ+140^\circ)\bigr)$$
$$=3(\cos 270^\circ+i\sin 270^\circ).$$$$\left(\cos \frac{5\pi}{6}+i\sin \frac{5\pi}{6}\right)\sqrt{3}\left(\cos\left(-\frac{\pi}{12}\right)+i\sin\left(-\frac{\pi}{12}\right)\right)$$
$$=\sqrt{3}\left(\cos\left(\frac{5\pi}{6}-\frac{\pi}{12}\right)+i\sin\left(\frac{5\pi}{6}-\frac{\pi}{12}\right)\right)$$
$$=\sqrt{3}\left(\cos \frac{9\pi}{12}+i\sin \frac{9\pi}{12}\right)$$
$$=\sqrt{3}\left(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4}\right).$$$$\frac{\cos 50^\circ+i\sin 50^\circ}{2(\cos 20^\circ+i\sin 20^\circ)}$$
$$=\frac12\left(\cos(50^\circ-20^\circ)+i\sin(50^\circ-20^\circ)\right)$$
$$=\frac12(\cos 30^\circ+i\sin 30^\circ).$$$$\frac{2\left(\cos\left(-\frac{5\pi}{12}\right)+i\sin\left(-\frac{5\pi}{12}\right)\right)}{\sqrt{2}\left(\cos\left(-\frac{\pi}{12}\right)+i\sin\left(-\frac{\pi}{12}\right)\right)}$$
$$=\sqrt{2}\left(\cos\left(-\frac{5\pi}{12}+\frac{\pi}{12}\right)+i\sin\left(-\frac{5\pi}{12}+\frac{\pi}{12}\right)\right)$$
$$=\sqrt{2}\left(\cos\left(-\frac{4\pi}{12}\right)+i\sin\left(-\frac{4\pi}{12}\right)\right)$$
$$=\sqrt{2}\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right).$$
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) $$\sqrt{2}\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right).$$
