Задача 72186 ...
Условие
Решение
[m]3(\cos \frac{\pi}{5}+i \cdot \sin \frac{\pi}{5}) \cdot 5(\cos \frac{\pi}{7}+i \cdot \sin \frac{\pi}{7}) =[/m]
[m] =3 \cdot 5(\cos \frac{\pi}{5}\cos \frac{\pi}{7}+i \cdot \sin \frac{\pi}{5}\cos \frac{\pi}{7}+[/m]
[m]+\cos \frac{\pi}{5} \cdot i \cdot \sin \frac{\pi}{7}+i \cdot \sin \frac{\pi}{5} \cdot i \cdot \sin \frac{\pi}{7}) =[/m]
[m]=15 \cdot ((\cos \frac{\pi}{5}\cos \frac{\pi}{7} - \sin \frac{\pi}{5} \sin \frac{\pi}{7}) +[/m]
[m]+i(\sin \frac{\pi}{5}\cos \frac{\pi}{7}+\cos \frac{\pi}{5} \sin \frac{\pi}{7})) = [/m]
Воспользуемся формулами синуса суммы и косинуса суммы
[m]=15(cos(\frac{\pi}{5} +\frac{\pi}{7}) + i \cdot \sin (\frac{\pi}{5} +\frac{\pi}{7})) = [/m]
[m]=15(cos(\frac{7\pi}{35} +\frac{5\pi}{35}) + i \cdot \sin (\frac{7\pi}{35} +\frac{5\pi}{35})) = 15(cos \frac{12\pi}{35} + i \cdot \sin \frac{12\pi}{35})[/m]
Все решения
[m]2(cos\frac{π}{5}+i sin\frac{π}{5})\cdot 4 (cos\frac{π}{7}+i sin\frac{π}{7}) =2e^{\frac{π}{5}\cdot i}\cdot 4e^{\frac{π}{7}\cdot i}=2\cdot 4\cdot e^{\frac{π}{7}\cdot i+\frac{π}{7}\cdot i}=8\cdot e^{\frac{7π+5π}{35}\cdot i}=8\cdot (cos\frac{12π}{35} +isin\frac{12π}{35})[/m]
