Упр.632 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1. Используем формулу умножения комплексных чисел в тригонометрической форме:
   $$\left(\cos \alpha+i\sin \alpha\right)\left(\cos \beta+i\sin \beta\right)=\cos(\alpha+\beta)+i\sin(\alpha+\beta).$$

   $$\left(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}\right)\left(\cos \frac{\pi}{12}+i\sin \frac{\pi}{12}\right)=\cos \left(\frac{\pi}{6}+\frac{\pi}{12}\right)+i\sin \left(\frac{\pi}{6}+\frac{\pi}{12}\right)$$
   $$=\cos \frac{\pi}{4}+i\sin \frac{\pi}{4}.$$

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

3. $$\sqrt{3}\left(\cos \frac{5\pi}{24}+i\sin \frac{5\pi}{24}\right)\cdot 2\left(\cos \frac{\pi}{8}+i\sin \frac{\pi}{8}\right)$$
   $$=2\sqrt{3}\left(\cos \left(\frac{5\pi}{24}+\frac{\pi}{8}\right)+i\sin \left(\frac{5\pi}{24}+\frac{\pi}{8}\right)\right)$$
   $$=2\sqrt{3}\left(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}\right).$$

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

5. $$\sqrt{2}\left(\cos 55^\circ+i\sin 55^\circ\right)\cdot \sqrt{2}\left(\cos 35^\circ+i\sin 35^\circ\right)$$
   $$=2\left(\cos(55^\circ+35^\circ)+i\sin(55^\circ+35^\circ)\right)$$
   $$=2\left(\cos 90^\circ+i\sin 90^\circ\right).$$

6. $$\left(\cos 7+i\sin 7\right)\left(\cos 3+i\sin 3\right)$$
   $$=\cos(7+3)+i\sin(7+3)$$
   $$=\cos 10+i\sin 10.$$

Ответ

1) $$\cos \frac{\pi}{4}+i\sin \frac{\pi}{4}$$;
2) $$2\left(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}\right)$$;
3) $$2\sqrt{3}\left(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3}\right)$$;
4) $$12\left(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}\right)$$;
5) $$2\left(\cos 90^\circ+i\sin 90^\circ\right)$$;
6) $$\cos 10+i\sin 10$$.