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Question Number 73913 by arkanmath7@gmail.com last updated on 16/Nov/19

$$Ineedthesol.plz$$$$findtheimaginaryandrealpartsof$$$$logsin(a+ib)?$$

Answered by Tanmay chaudhury last updated on 16/Nov/19

$$Log(sinacosib+cosasinib)$$$$Log(sina.coshb+\mathrm{c}osa.isinhb)$$$$Rcos\alpha =sina.coshb$$$$Rsin\alpha =cosa.sinhb$$$$Log(Rcos\alpha +iRsin\alpha )$$$$Log\left[{Re}^{i\alpha }\right]\Rrightarrow Log\left[{Re}^{i(2n\pi +\alpha )}\right]$$$$Log\left[{Re}^{i(2n\pi +\alpha )}\right]=logR+i(2n\pi +\alpha )$$$$R={[{(sina.coshb)}^2+{(cosa.sinhb)}^2]}^{\frac{1}{2}}$$$$tan\alpha =\frac{cosa.sinhb}{sina.coshb}\Rrightarrow \alpha ={tan}^{-1}\left(\frac{cosa.sinhb}{sina.coshb}\right)$$

Commented by arkanmath7@gmail.com last updated on 16/Nov/19

$$thnx$$

Answered by Smail last updated on 16/Nov/19

$$ln\left(sin(a+ib)\right)=x+iy$$$$sin(a+ib)=e^x{e}^{iy}$$$$sin\left(a\right)cos\left(ib\right)+sin\left(ib\right)cos\left(a\right)=e^x{e}^{iy}$$$$sin\left(a\right)cosh\left(b\right)+isinh\left(b\right)cos\left(a\right)=e^x{e}^{iy}$$$$\frac{sinh\left(b\right)cos\left(a\right)}{sin\left(a\right)cosh\left(b\right)}=tan\left(y\right)$$$$y={tan}^{-1}\left(tanh\left(b\right)cot\left(a\right)\right))$$$$e^x=\sqrt{{sin}^2\left(a\right){cosh}^2\left(b\right)+{sinh}^2\left(b\right){cos}^2\left(a\right)}$$$$x=\frac{1}{2}ln({sin}^2\left(a\right){cosh}^2\left(b\right)+{sinh}^2\left(b\right){cos}^2\left(a\right))$$$$sin(a+ib)=\frac{1}{2}ln({sin}^2\left(a\right){cosh}^2\left(b\right)+{sinh}^2\left(b\right){cos}^2\left(a\right))+{itan}^{-1}\left(tanh\left(b\right)cot\left(a\right)\right)$$

Commented by Smail last updated on 16/Nov/19

$$ln\left(sin(a+ib)\right)=\frac{1}{2}ln({sin}^2\left(a\right){cosh}^2\left(b\right)+{sinh}^2\left(b\right){cos}^2\left(a\right))+{itan}^{-1}\left(tanh\left(b\right)cot\left(a\right)\right)$$$$$$

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