Math, asked by avyangdvn18, 10 months ago

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

Answers

Answered by jitendra420156
3

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)

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