Question

if cos(x+iy)=r(cos alpha+i sin alpha) prove that y=(1)/(2)log(sin(x-alpha))/(sin(x+alpha))​

Answer 1

Answer

y=(1)/(2)log(sin(x-α))/(sin(x+α))​

Step-by-step explanation

 given,

      cos(x+iy)=r(cosα+i sin α)

To prove ,

        y=(1)/(2)log(sin(x-α))/(sin(x+α))​

solution,

    cos(x+iy)=r(cosα+i sin α)  

    cosx . cosiy - sinx . siniy = rcosα +irsinα -------------- (1)

we know that,

cosiy = е^y + е^-y/2

siniy = е^y - е^-y/2

then (1) becomes,

cosx (е^y + е^-y/2) - sinx(е^y - е^-y/2) = rcosα +irsinα

comparing coeffients of cosx and sinx,

е^y + е^-y/2 =  rcosα  & е^y - е^-y/2 =   rsinα ---------(2)

we know that ,

sin(x+α) = sinx cosα + cosx sinα

sin(x-α) = sinx cosα - cosx sinα

removing the exponential from (2) , we get,

   

 y=(1)/(2)log(sin(x-α))/(sin(x+α))​

          hence proved