Use #z = ( x, y ) = r ( cos theta, sin theta ), r = sqrt ( x^2 + y^2 )>= 0,#
#0 <= theta = arctan ( y/x),in Q_1# or #Q-4#
#= arccos ( y/r), theta in Q_2#
#= pi + arctan(y/x), theta in Q_3#
The complex #z = ( x + i y ) = r ( cos theta + i sin theta ) = r e^(i
theta )#.
Here, z = u/v,
#Q_2 u = - 2 + i = sqrt 5 e^(i cos^(-1)((-2)/sqrt 5)# and
#Q_3 v = - 8 - 5i = sqrt89e^(iarctan( 5/8)#. Now,
z = qsqrt(5/89)( e^(i cos^(-1)((-2)/sqrt 5))/(e^(i(pi +arctan( 5/8)))
#= sqrt( 5/89) e^i ( cos^(-1)((-2)/sqrt 5) - (pi +arctan( 5/8)))#
#= sqrt( 5/89) e^i(153.3435^o - 180^o - 32.0054^o)#
#= sqrt( 5/89) e^i(-58.662^o)#
#= sqrt( 5/89)( cos 58.662^o - i sin 58.662^p )#
The answer is 1/89( 11- 18 i ).
My answer appears to be very very close.
