Find the real part the imaginary part and the absolute value

Find the real part, the imaginary part, and the absolute value of sin (x - iy).

Solution

sin(x-iy)=sinx.cosiy-cosx.siniy

Since siniy=i sinhy ,cosiy=coshy

Sin(x-iy)=sinx.coshy- cosx.i sinhy

= sinx.1/2 (e^y +e^-y) -cosx.i.1/2(e^x-e^-x)

Real part=1/2 sinx(e^y+e^-y)

Imaginary part=-1/2 cosx(e^x-e^-x)   

Absolute value=1/2{sinx(e^y+e^-y)+cosx(e^x-e^-x)}