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Question Number 145763 by gsk2684 last updated on 07/Jul/21

if z=(((3+i sin θ)/(4−i cos θ)))is purely real and   (Π/2)<θ<Π then find arg(sin θ +i cos θ)?

ifz=(3+isinθ4icosθ)ispurelyrealand Π2<θ<Πthenfindarg(sinθ+icosθ)?

Answered by mathmax by abdo last updated on 08/Jul/21

z real ⇒z=z^−  ⇒((3+isinθ)/(4−icosθ))=((3−isinθ)/(4+icosθ)) ⇒  (3+isinθ)(4+icosθ)=(3−isinθ)(4−icosθ) ⇒  12+3icosθ+4isinθ−sinθ cosθ =12−3icosθ−4isinθ−sinθ cosθ ⇒  6icosθ+8isinθ =0 ⇒2i(3cosθ +4sinθ)=0 ⇒3cosθ +4sinθ =0 ⇒  5((3/5)cosθ +(4/5)sinθ)=0 ⇒5cos(θ−α)=0 (e) with cosα=(3/5)and sinα=(4/5)  ⇒tanα=(4/3) ⇒α=arctan((4/3))  (e)⇒θ−α=(π/2)+kπ   (k∈Z)  sinθ +icosθ=sin(α+(π/2)+kπ)+icos(α+(π/2)+kπ)  =cos(α+kπ)−isin(α+kπ)  =(−1)^k  cosα −i(−1)^k  sinα  =(−1)^k  e^(−iα)  =(−1)^k  e^(−iarctan((4/3)))   for (π/2)<θ<π we solve (π/2)<α+(π/2)+kπ<π....(findk)

zrealz=z3+isinθ4icosθ=3isinθ4+icosθ (3+isinθ)(4+icosθ)=(3isinθ)(4icosθ) 12+3icosθ+4isinθsinθcosθ=123icosθ4isinθsinθcosθ 6icosθ+8isinθ=02i(3cosθ+4sinθ)=03cosθ+4sinθ=0 5(35cosθ+45sinθ)=05cos(θα)=0(e)withcosα=35andsinα=45 tanα=43α=arctan(43) (e)θα=π2+kπ(kZ) sinθ+icosθ=sin(α+π2+kπ)+icos(α+π2+kπ) =cos(α+kπ)isin(α+kπ) =(1)kcosαi(1)ksinα =(1)keiα=(1)keiarctan(43) forπ2<θ<πwesolveπ2<α+π2+kπ<π....(findk)

Commented bygsk2684 last updated on 08/Jul/21

thank you

thankyou

Commented bymathmax by abdo last updated on 09/Jul/21

you are welcome

youarewelcome

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