Question

If Z1 and Z2 are two complex numbers , prove that amp(Z1×Z2)= amp (Z1) + amp(Z2)​

Answer 1

Therefore $amp(z_1\times z_2)=amp(z_1)+amp(z_2)$

Step-by-step explanation:

Let z₁= a+ib      and    z₂=c+id

$amp(z_1)=tan^{-1}(\frac{b}{a})$       and      $amp(z_2)=tan^{-1}(\frac{d}{c})$

$amp(z_1)+amp(z_2)=tan^{-1}(\frac{b}{a})+tan^{-1}(\frac{d}{c})$

                              $=tan^{-1}\frac{\frac{b}{a}+\frac{d}{c} }{1-\frac{b}{a}\frac{d}{c}}$

                             $=tan^{-1} (\frac{bc+ad}{ac-bd} )$

z₁×z₂ =(a+ib) (c+id)

        =ac+ibc+iad+i²bd

        =ac-bd+i(bc+ad)

$amp(z_1\times z_2)=tan^{-1} (\frac{bc+ad}{ac-bd} )$

Therefore $amp(z_1\times z_2)=amp(z_1)+amp(z_2)$