Question Number 137468 by ZiYangLee last updated on 03/Apr/21
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\left(\mathrm{cos}\:\mathrm{1}°+{i}\mathrm{sin}\:\mathrm{1}°\right)\left(\mathrm{cos}\:\mathrm{2}°+{i}\mathrm{sin}\:\mathrm{2}°\right)\ldots\left(\mathrm{cos}\:\mathrm{359}°+{i}\mathrm{sin}\:\mathrm{359}°\right). \\ $$
Answered by mr W last updated on 03/Apr/21
$$={e}^{\frac{\mathrm{1}}{\mathrm{180}}\pi{i}} {e}^{\frac{\mathrm{2}}{\mathrm{180}}\pi{i}} ...{e}^{\frac{\mathrm{359}}{\mathrm{180}}\pi{i}} \\ $$$$={e}^{\frac{\mathrm{1}+\mathrm{2}+...\mathrm{359}}{\mathrm{180}}\pi{i}} \\ $$$$={e}^{\mathrm{359}\pi{i}} \\ $$$$={e}^{\pi{i}} \\ $$$$=−\mathrm{1} \\ $$