If |Z1|=|Z2|\ \text{and}\ arg z1 \sim arg z2=\pi, how do

elvishwitchxyp 2022-01-17 Answered
If |Z1|=|Z2| and argz1argz2=π, how do you show that Z1+Z2=0?
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Expert Answer

temzej9
Answered 2022-01-18 Author has 30 answers
The proof uses Euler’s Identity, actually its reciprocal: eiπ=1.
Let r=|z1|=|z2| and θ=arg(z1) so arg(z2)=arg(z1)π=θπ.
We get z1=reiθ
z2=rei(θπ)=reiθeiπ=reiθ
z1+z2=reiθ±reiθ=0
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godsrvnt0706
Answered 2022-01-19 Author has 31 answers
Let |z1|=|z2|=r
arg(z1)=θ
Arg(z2)=ϕ
z1=reiθ……(1)
z2=reiϕ………(2)
And θϕ=π
eix=cosx+isinx
ei(π+x)=cos(π+x)+isin(π+x)
ei(π+x)=cosxisinx=(cosx+isinx)=eix
Here θ=π+ϕ
So this reduces to
eiθ=eiϕ
Multiplying by r on both sides
reiθ=reiϕ
From equations (1) and (2)
The above expression becomes
z1=z2
That is
z1+z2=0.
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