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Separatic into real and imaginary parts of sin-1 (eio).
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  • Given:sin1(eiθ)
  • Letsin1(eiθ)=x+iy

eiθ=sin(x+iy)

cosθ+isinθ=sinx cos hy+icosx sin hy

  • Comparing real and imaginary parts,

cosθ=sinx cos hyEquation 1

sinx=cosθcoshy

sinθ=cosx sinhyEquation 2

cosx=sinθsinhy

  • But sin2x+cos2x=1

cos2θcosh2y+sin2θsinh2y=1

cos2θ sin h2y +sin2θ cosh2y=cosh2y×sinh2y

(1sin2θ) sinh2y +sin2θ (1+sinh2y)=(1+sinh2y)×sinh2y$

sinh2y sin2θ sinh2y +sin2θ +sin2θ sinh2y=sinh2y+sinh4y

sin2θ=sinh4y

sinθ=sinh2yEquation 3

sinθ=sinhy

y=sinh1sinθ

  • Squaring equation 2,sin2θ=cos2x sinh2y

sin2θ=cos2x sinθFrom Equation 3

cos2x=sinθ

cosx=sinθ

x=cos1(sinθ)

  • Thus,sin1(eiθ)=x+iy=cos1(sinθ)+isinh1sinθ
  • Hence,Real part is x=cos1(sinθ)and Imaginary part is y=sinh1sinθ
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