Given:
#Z_1=2(cos(18^@)+isin(18^@))" [1]"#
#Z_2=alpha(cos(40^@)+isin(40^@))" [2]"#
#Z_3=32(cos(theta)+isin(theta))" [3]"#
#(Z_1^10)/(Z_2-Z_3)=4i" [4]"#
Substitute equation [1] into equation [4]:
#((2(cos(18^@)+isin(18^@)))^10)/(Z_2-Z_3)=4i#
Use De Moivre's formula:
#(2^10(cos(18^@xx10)+isin(18^@xx10)))/(Z_2-Z_3)=4i#
#(1024(cos(180^@)+isin(180^@)))/(Z_2-Z_3)=4i#
#(1024(-1+i0))/(Z_2-Z_3)=4i#
#(-1024+i0)/(Z_2-Z_3)=4i" [4.1]"#
Multiply both sides of equation [4.1] by #Z_3-Z_2#:
#1024+i0=4i(Z_3-Z_2)" [4.2]"#
Substitute equations [2] and [3] into equation [4.2]:
#1024+i0=4i(32(cos(theta)+isin(theta))-alpha(cos(40^@)+isin(40^@)))#
Perform the multiplication:
#1024+i0=4i(32cos(theta)+i32sin(theta))-alphacos(40^@)-ialphasin(40^@))#
#1024+i0=i128cos(theta)-128sin(theta)-i4alphacos(40^@)+4alphasin(40^@))#
Separating the real and imaginary parts into two equations:
#1024 = 4alphasin(40^@)-128sin(theta)" [5]"#
#i0 = i128cos(theta)-i4alphacos(40^@)" [6]"#
Solve equation [6] for #alpha# in terms of #theta#:
#0 = 32cos(theta)-alphacos(40^@)#
#alpha = 32cos(theta)/cos(40^@)" [6.1]"#
Substitute equation [6.1] into equation [5]:
#1024 = 4(32cos(theta)/cos(40^@))sin(40^@)-128sin(theta)#
Multiply both sides by #cos(40^@)/128#:
#8cos(40^@) = cos(theta)sin(40^@)-sin(theta)cos(40^@)#
Multiply both sides by -1:
#-8cos(40^@) = sin(theta)cos(40^@)-cos(theta)sin(40^@)#
The right side is the identity for #sin(A-B)# where #A=theta# and #B = 40^@#:
#sin(theta-40^@) = -8cos(40^@)#
The above equation does not have a solution, because the right side is outside of the the range of the sine function, #-1 <= sin(x) <= 1#. This means that there are not real values for #theta# and #alpha#
