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Question Number 145763 by gsk2684 last updated on 07/Jul/21
ifz=(3+isinθ4−icosθ)ispurelyrealand Π2<θ<Πthenfindarg(sinθ+icosθ)?
Answered by mathmax by abdo last updated on 08/Jul/21
zreal⇒z=z−⇒3+isinθ4−icosθ=3−isinθ4+icosθ⇒ (3+isinθ)(4+icosθ)=(3−isinθ)(4−icosθ)⇒ 12+3icosθ+4isinθ−sinθcosθ=12−3icosθ−4isinθ−sinθcosθ⇒ 6icosθ+8isinθ=0⇒2i(3cosθ+4sinθ)=0⇒3cosθ+4sinθ=0⇒ 5(35cosθ+45sinθ)=0⇒5cos(θ−α)=0(e)withcosα=35andsinα=45 ⇒tanα=43⇒α=arctan(43) (e)⇒θ−α=π2+kπ(k∈Z) sinθ+icosθ=sin(α+π2+kπ)+icos(α+π2+kπ) =cos(α+kπ)−isin(α+kπ) =(−1)kcosα−i(−1)ksinα =(−1)ke−iα=(−1)ke−iarctan(43) forπ2<θ<πwesolveπ2<α+π2+kπ<π....(findk)
Commented bygsk2684 last updated on 08/Jul/21
thankyou
Commented bymathmax by abdo last updated on 09/Jul/21
youarewelcome
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