Question Number 226779 by Spillover last updated on 14/Dec/25
$${By}\:{using}\:{De}\:{Moivres}\:{theorm} \\ $$$${simplify} \\ $$$$\left({a}\right)\frac{\left(\mathrm{cos}\:\frac{\pi}{\mathrm{2}}−{i}\mathrm{sin}\:\frac{\pi}{\mathrm{2}}\right)\left(\mathrm{cos}\:\frac{\pi}{\mathrm{3}}+{i}\mathrm{sin}\:\frac{\pi}{\mathrm{3}}\right)}{\mathrm{cos}\:\frac{\pi}{\mathrm{3}}−{i}\mathrm{sin}\:\frac{\pi}{\mathrm{3}}} \\ $$$$\left({b}\right)\frac{\mathrm{cos}\:\frac{\pi}{\mathrm{8}}+{i}\mathrm{sin}\:\frac{\pi}{\mathrm{8}}}{\mathrm{cos}\:\frac{\pi}{\mathrm{6}}+{i}\mathrm{sin}\:\frac{\pi}{\mathrm{6}}} \\ $$
Answered by Frix last updated on 14/Dec/25
$$\frac{\mathrm{e}^{−\mathrm{i}\frac{\pi}{\mathrm{2}}} \mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{3}}} }{\mathrm{e}^{−\mathrm{i}\frac{\pi}{\mathrm{3}}} }=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{6}}} =\mathrm{cos}\:\frac{\pi}{\mathrm{6}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\pi}{\mathrm{6}} \\ $$$$ \\ $$$$\frac{\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{8}}} }{\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{6}}} }=\mathrm{e}^{−\mathrm{i}\frac{\pi}{\mathrm{24}}} =\mathrm{cos}\:\frac{\pi}{\mathrm{24}}\:−\mathrm{i}\:\mathrm{sin}\:\frac{\pi}{\mathrm{24}} \\ $$