SOLUTION OF PROBLEM: 7

Let z and w are two non-zero compex numbers such that |z| =|w| and arg(z) + arg(w)=180(degree), then z is

SOLUTION: we can have two different ways to approach the solutions

CONVENTIONAL METHOD: 

Let z = r e(ip) , where r is |z| and p is arg(z)

also, w = r e(iq), where r is |w| and q is ar(w)

as given |z|=|w| and p + q =180(degree), we can write w = r e(i(180-p))= r(cos(180-p)+i sin(180-p))

w = r(-cos(p) +i sin(p)) = -r( cos(p) - isin(p)) = -1/z =-z( bar).

Hence, w = -z(bar)

NOTE: e(ip) is e to the power ip.

GRAPHICAL METHOD: