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If-z-cos-isin-0-lt-lt-pi-6-then-prove-that-argument-of-1-z-4-2-pi-2-




Question Number 43363 by rahul 19 last updated on 10/Sep/18
If z= cos θ + isin θ , 0<θ<(π/6) , then prove  that argument of  1−z^4  = 2θ − (π/2) .
Ifz=cosθ+isinθ,0<θ<π6,thenprovethatargumentof1z4=2θπ2.
Commented by maxmathsup by imad last updated on 10/Sep/18
we have z =e^(iθ)  ⇒1−z^4  =1−e^(4iθ)    =1−cos(4θ)−isin(4θ)  =2 sin^2 (2θ) −2isin(2θ)cos(2θ) =−2isin(2θ){cos(2θ) +isin(2θ)}  =2 sin(2θ)(−i) e^(i(2θ))   we have  0<2θ<(π/3) ⇒sin(2θ)>0 ⇒∣1−z^4 ∣ =2 sin(2θ)  and  arg(1−z^4 ) ≡arg(−i) +2θ  [2π] ⇒arg(1−z^4 )≡ −(π/2) +2θ[2π] .
wehavez=eiθ1z4=1e4iθ=1cos(4θ)isin(4θ)=2sin2(2θ)2isin(2θ)cos(2θ)=2isin(2θ){cos(2θ)+isin(2θ)}=2sin(2θ)(i)ei(2θ)wehave0<2θ<π3sin(2θ)>0⇒∣1z4=2sin(2θ)andarg(1z4)arg(i)+2θ[2π]arg(1z4)π2+2θ[2π].
Answered by ajfour last updated on 10/Sep/18
1−z^4 =1−(x+iy)^4     = (1−x^4 −y^4 +6x^2 y^2 )+4i(xy^3 −x^3 y)    tan φ = ((4xy(y^2 −x^2 ))/(1−(x^2 −y^2 )^2 +4x^2 y^2 ))      = ((4m(m^2 −1))/((1+m^2 )^2 −(1−m^2 )^2 +4m^2 ))      (if  m=tan θ )  ⇒   tan φ = ((m^2 −1)/(2m)) = −cot 2θ                  = tan (2θ−(π/2))  ⇒ φ (argument of 1−z^4 )= 2θ−(π/2) .
1z4=1(x+iy)4=(1x4y4+6x2y2)+4i(xy3x3y)tanϕ=4xy(y2x2)1(x2y2)2+4x2y2=4m(m21)(1+m2)2(1m2)2+4m2(ifm=tanθ)tanϕ=m212m=cot2θ=tan(2θπ2)ϕ(argumentof1z4)=2θπ2.
Commented by rahul 19 last updated on 10/Sep/18
thanks sir.
thankssir.
Answered by tanmay.chaudhury50@gmail.com last updated on 10/Sep/18
z=cosθ+isinθ=e^(iθ)   1−z^4   =1−e^(i4θ)   =1−(cos4θ+isin4θ)  =1−cos4θ−isin4θ  =2sin^2 2θ−i×2sin2θ.cos2θ  so tanα=((−2sin2θcos2θ)/(2sin^2 2θ))=−cot2θ  tan∝=−tan((Π/2)−2θ)  tanα=tan(2θ−(Π/2))  α=2θ−(Π/2)
z=cosθ+isinθ=eiθ1z4=1ei4θ=1(cos4θ+isin4θ)=1cos4θisin4θ=2sin22θi×2sin2θ.cos2θsotanα=2sin2θcos2θ2sin22θ=cot2θtan∝=tan(Π22θ)tanα=tan(2θΠ2)α=2θΠ2
Commented by rahul 19 last updated on 10/Sep/18
thanks sir.
thankssir.
Answered by tanmay.chaudhury50@gmail.com last updated on 10/Sep/18
z=cosθ+isinθ=e^(iθ)   1−z^4   =1−e^(i4θ)   =1−(cos4θ+isin4θ)  =1−cos4θ−isin4θ  =2sin^2 2θ−i×2sin2θ.cos2θ  so tanα=((−2sin2θcos2θ)/(2sin^2 2θ))=−cot2θ  tan∝=−tan((Π/2)−2θ)  tanα=tan(2θ−(Π/2))  α=2θ−(Π/2)
z=cosθ+isinθ=eiθ1z4=1ei4θ=1(cos4θ+isin4θ)=1cos4θisin4θ=2sin22θi×2sin2θ.cos2θsotanα=2sin2θcos2θ2sin22θ=cot2θtan∝=tan(Π22θ)tanα=tan(2θΠ2)α=2θΠ2

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