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

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