Let z1 and z2 be two complex numbers such that z1 + iz2 = 0 and arg (z1z2) = pi.

Find arg (z1).

let z1= cos@ + isin@ and z2= cosQ+icosQ

multiply z1.z2 and simplify further you will get cos(@+Q) + i sin(Q+@)

given @+Q = pi

.'. z1z2=-1

also simplify z1 +iz2 by subsitituting the polar form.

you get [cos@-sinQ] + i[sin@ +cosQ]=0

only possible when

cos@ = sinQ and sin@=-cosQ

solvee these trigonometric equations you get @-Q=pi/2

.'.@=3pi/4 and Q=pi/4

.'.arg(z1)= 3pi/4 and arg(z2)= pi/4