To explain this, I will name two generic complex.
c_1 = a*cis(alpha)c1=a⋅cis(α) and c_2 = b*cis(beta)c2=b⋅cis(β)
The product between c_1c1 and c_2c2 is:
ab*cis(alpha)cis(beta) =ab⋅cis(α)cis(β)=
ab*(cos(alpha)+isin(alpha)) (cos(beta)+isin(beta)) =ab⋅(cos(α)+isin(α))(cos(β)+isin(β))=
ab*({cos(alpha)cos(beta)-sin(alpha)sin(beta)} +ab⋅({cos(α)cos(β)−sin(α)sin(β)}+
{i(sin(alpha)sin(beta)+cos(alpha)sin(beta)}) ={i(sin(α)sin(β)+cos(α)sin(β)})=
ab*{cos(a+b)+isin(a+b)}ab⋅{cos(a+b)+isin(a+b)}//
Therefore, we can assume that the product of the two complex numbers c_1c1 and c_2c2 can be generaly given by the form above.
Ex.:
(2*cis(pi)) * (3*cis(2pi)) = 6*cis(3pi) = 6*cis(pi)(2⋅cis(π))⋅(3⋅cis(2π))=6⋅cis(3π)=6⋅cis(π)
Hope it helps.
