if-z-z-1-3-i-find-

Question Number 149142 by mathdanisur last updated on 03/Aug/21

$${if}\:\:\:{z}\:+\:\mid{z}\mid\:=\:\mathrm{1}\:+\:\sqrt{\mathrm{3}}\:\boldsymbol{{i}} \\ $$$${find}\:\:\:\boldsymbol{\varphi}\:=\:? \\ $$

Answered by liberty last updated on 03/Aug/21

z=a+bi ∣z∣=(√(a^2 +b^2 )) ⇒a+bi+(√(a^2 +b^2 )) = 1+(√3) i { ((b=(√3))),((a+(√(a^2 +3)) =1 ⇒(√(a^2 +3)) =1−a)) :} ⇒a^2 +3=a^2 −2a+1 ⇒a=−1 then z=−1+(√3) i tan ϕ=((√3)/(−1))=−(√3) ϕ=−(π/3)

$$\mathrm{z}=\mathrm{a}+\mathrm{b}{i} \\ $$$$\mid\mathrm{z}\mid=\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} } \\ $$$$\Rightarrow\mathrm{a}+\mathrm{b}{i}+\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:=\:\mathrm{1}+\sqrt{\mathrm{3}}\:{i} \\ $$$$\begin{cases}{{b}=\sqrt{\mathrm{3}}}\\{{a}+\sqrt{{a}^{\mathrm{2}} +\mathrm{3}}\:=\mathrm{1}\:\Rightarrow\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{3}}\:=\mathrm{1}−\mathrm{a}}\end{cases} \\ $$$$\Rightarrow\mathrm{a}^{\mathrm{2}} +\mathrm{3}=\mathrm{a}^{\mathrm{2}} −\mathrm{2a}+\mathrm{1} \\ $$$$\Rightarrow\mathrm{a}=−\mathrm{1}\:\mathrm{then}\:\mathrm{z}=−\mathrm{1}+\sqrt{\mathrm{3}}\:{i} \\ $$$$\mathrm{tan}\:\varphi=\frac{\sqrt{\mathrm{3}}}{−\mathrm{1}}=−\sqrt{\mathrm{3}}\: \\ $$$$\varphi=−\frac{\pi}{\mathrm{3}} \\ $$

Commented by mathdanisur last updated on 03/Aug/21

$${Cool},\:{Thank}\:{You}\:{Ser} \\ $$

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