Question Number 145524 by physicstutes last updated on 05/Jul/21
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{argument}\:\mathrm{of}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{below} \\ $$$$\left(\mathrm{i}\right)\:{z}\:=\:\mathrm{1}+{e}^{\frac{\pi}{\mathrm{6}}{i}} \\ $$$$\left(\mathrm{ii}\right)\:{z}\:=\:\mathrm{1}\:−{e}^{\frac{\pi}{\mathrm{6}}{i}} \\ $$
Answered by Olaf_Thorendsen last updated on 05/Jul/21
$$\left({i}\right)\:{z}\:=\:\mathrm{1}+{e}^{{i}\frac{\pi}{\mathrm{6}}} \\ $$$${z}\:=\:{e}^{{i}\frac{\pi}{\mathrm{12}}} \left({e}^{−{i}\frac{\pi}{\mathrm{12}}} +{e}^{{i}\frac{\pi}{\mathrm{12}}} \right) \\ $$$${z}\:=\:\mathrm{2cos}\frac{\pi}{\mathrm{12}}{e}^{{i}\frac{\pi}{\mathrm{12}}} \\ $$$$\mathrm{arg}{z}\:=\:\frac{\pi}{\mathrm{12}}\:\left[\mathrm{2}\pi\right] \\ $$$$\left(\mathrm{ii}\right)\:{z}\:=\:\mathrm{1}\:−{e}^{\frac{\pi}{\mathrm{6}}{i}} \\ $$$${z}\:=\:{e}^{{i}\frac{\pi}{\mathrm{12}}} \left({e}^{−{i}\frac{\pi}{\mathrm{12}}} \:−{e}^{{i}\frac{\pi}{\mathrm{12}}} \right) \\ $$$${z}\:=\:−\mathrm{2}{i}\mathrm{sin}\frac{\pi}{\mathrm{12}}{e}^{{i}\frac{\pi}{\mathrm{12}}} \\ $$$${z}\:=\:{e}^{{i}\pi} \mathrm{2}{e}^{{i}\frac{\pi}{\mathrm{2}}} \mathrm{sin}\frac{\pi}{\mathrm{12}}{e}^{{i}\frac{\pi}{\mathrm{12}}} \\ $$$${z}\:=\:\mathrm{2sin}\frac{\pi}{\mathrm{12}}{e}^{{i}\left(\pi+\frac{\pi}{\mathrm{2}}+\frac{\pi}{\mathrm{12}}\right)} \\ $$$${z}\:=\:\mathrm{2sin}\frac{\pi}{\mathrm{12}}{e}^{{i}\frac{\mathrm{19}\pi}{\mathrm{12}}} \\ $$$$\mathrm{arg}{z}\:=\:\frac{\mathrm{19}\pi}{\mathrm{12}}\:\left[\mathrm{2}\pi\right] \\ $$
Commented by physicstutes last updated on 05/Jul/21
$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$$$ \\ $$