若z1的模=3,z2的模=5,(z1-z2)的模=7求z1的共轭*z2

Question

若z1的模=3,z2的模=5,(z1-z2)的模=7
求z1的共轭*z2

Answer 1

设Z1=3cosx+3isinx Z2=5cosy+5isiny
(│Z1-Z2│)^2=(3cosx-5cosy)^2+(3sinx-5siny)^2=7*7=49
9cosx*cosx+9sinx*sinx+25cosy*cosy+25siny*siny-30cosx*cosy=49
解得 cos(x-y)=-1/2
所以 sin(y-x)=√3/2 或者 -√3/2
z1的共轭*z2=（3cosx-3isinx）（5cosy+5isiny ）
=15[(cosx*cosy+sinx*siny）+i(cosx*siny-sinx*cosy)]
=7.5[cos(x-y)+isin(y-x)]
=7.5[-1/2±i√3/2]
=3.75(-1/2±i√3/2)