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Question Number 146782 by mathdanisur last updated on 15/Jul/21
Find the modulus of a complex  number:  Z = cos 40 + i sin 40 +1 = ?
$${Find}\:{the}\:{modulus}\:{of}\:{a}\:{complex} \\ $$$${number}: \\ $$$${Z}\:=\:{cos}\:\mathrm{40}\:+\:{i}\:{sin}\:\mathrm{40}\:+\mathrm{1}\:=\:? \\ $$
Answered by Ar Brandon last updated on 15/Jul/21
z=e^(40i) +1=e^(20i) (e^(20i) +e^(−20i) )     =e^(20i) (2cos20)⇒∣z∣=2cos(20)
$$\mathrm{z}=\mathrm{e}^{\mathrm{40i}} +\mathrm{1}=\mathrm{e}^{\mathrm{20i}} \left(\mathrm{e}^{\mathrm{20i}} +\mathrm{e}^{−\mathrm{20i}} \right) \\ $$$$\:\:\:=\mathrm{e}^{\mathrm{20i}} \left(\mathrm{2cos20}\right)\Rightarrow\mid\mathrm{z}\mid=\mathrm{2cos}\left(\mathrm{20}\right) \\ $$
Commented by mathdanisur last updated on 15/Jul/21
cool Ser thankyou  how did you change it and write it  ase e^(40i) .?
$${cool}\:{Ser}\:{thankyou} \\ $$$${how}\:{did}\:{you}\:{change}\:{it}\:{and}\:{write}\:{it} \\ $$$${ase}\:\boldsymbol{{e}}^{\mathrm{40}\boldsymbol{{i}}} .? \\ $$
Commented by Ar Brandon last updated on 16/Jul/21
e^(iθ) =cosθ+isinθ
$$\mathrm{e}^{\mathrm{i}\theta} =\mathrm{cos}\theta+\mathrm{isin}\theta \\ $$
Commented by mathdanisur last updated on 16/Jul/21
thanks Ser
$${thanks}\:{Ser} \\ $$

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