Question Number 146778 by mathdanisur last updated on 15/Jul/21
$${Find}\:{the}\:{modulus}\:{of}\:{a}\:{complex} \\ $$$${number}: \\ $$$${Z}\:=\:{cos}\:\mathrm{40}\:+\:{i}\:{sin}\:\mathrm{20}\:+\:\mathrm{1}\:=\:? \\ $$
Answered by mathmax by abdo last updated on 15/Jul/21
$$\mathrm{40}\rightarrow\frac{\mathrm{40}\pi}{\mathrm{180}}=\frac{\mathrm{2}\pi}{\mathrm{9}}\:\mathrm{and}\:\mathrm{20}\rightarrow\frac{\mathrm{20}\pi}{\mathrm{180}}=\frac{\pi}{\mathrm{9}}\:\Rightarrow\mathrm{Z}=\mathrm{cos}\left(\mathrm{2}\frac{\pi}{\mathrm{9}}\right)+\mathrm{1}+\mathrm{isin}\left(\frac{\pi}{\mathrm{9}}\right) \\ $$$$=\mathrm{1}+\mathrm{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{9}}\right)+\mathrm{isin}\left(\frac{\pi}{\mathrm{9}}\right)\:\Rightarrow\mid\mathrm{Z}\mid=\sqrt{\left(\mathrm{1}+\mathrm{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{9}}\right)^{\mathrm{2}} \:+\mathrm{sin}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{9}}\right)\right.} \\ $$
Commented by mathdanisur last updated on 15/Jul/21
$${thankyou}\:{Ser} \\ $$