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If z = x + i y is a complex number, then the value of expression [sinh x.cosh (iy) + cosh x.sinh (iy)] will be: 

If z = x + i y is a complex number, then the value of expression [sinh x.cosh (iy) + cosh x.sinh (iy)] will be:  Correct Answer sinh z

Formula used:

If complex number z = x + iy then hyperbolic sine can be expressed as:

  • sinh(z) = sinh(x+iy) = sinh(x).cos(y) + i cosh(x).sin(y)
  • cosh (z) = cos (iz)
  • sinh (z) = - i cos(iz)
  • cos(z) = cosh (iz)
  • sin (z) = - i sinh (iz)
  • sinh (iz) = i sin (z)
  • sin(iz) = i sinh (z)

 

Calculation:

Let Z =     ----(1)

Form the above property, we can say that,

cosh (iy) = cos y and

sinh (iy) = i sinh y   

Therefore, from equation (1)

Z = Additional Information

Some of the important identities involving the hyperbolic functions are:

  • cosh(z)2 - sinh(z)2 = 1
  • sinh(z1+ z2) = sinh(z1)cosh(z2) + cosh(z1)sinh(z2)
  • cosh(z1+ z2) = cosh(z1)cosh (z2) + sinh(z1)sinh(z2)
  • sinh(z + 2π  = sinh(z)
  • cosh(z + 2π) = cosh(z)
  • cosh(-z) = cosh(z)
  • sinh(-z) = - sinh(z)

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